Representation Theory

Section 0 Introduction

Representation theory began with an observation by Dedekind. For a finite abelian group $𝐺$ and formal variables $𝑥_{𝑔}$ for $𝑔 \in 𝐺$ , define a $| 𝐺 | \times | 𝐺 |$ matrix $𝑋_{𝐺}$ by defining the entry at row $𝑔$ column $ℎ$ to be $𝑥_{𝑔 \cdot ℎ}$ . Dedekind computed its determinant $det 𝑋_{𝐺}$ for various abelian groups and observed that it can always be factored into linear factors.

He then wrote a letter to Frobenius to ask for a proof. Frobenius then studied this behaviour for abelian and non-abelian groups, and gave a proof as well as a generalization to non-abelian groups:

THM 0.1

Suppose $𝐺$ is a finite abelian group. Then
$det 𝑋_{𝐺} = \prod_{𝜒 : 𝐺 \to ℂ^{\times}} (\sum_{𝑔 \in 𝐺} 𝜒 (𝑔) 𝑥_{𝑔}),$
where the product is over all group homomorphisms $𝜒 : 𝐺 \to ℂ^{\times}$ . Such a homomorphism is called a group character.
Suppose $𝐺$ is a finite group, not necessarily abelian. Then
$det 𝑋_{𝐺} = \prod_{𝑗 = 1}^{𝑟} 𝑝_{𝑗} {(𝑥)}^{deg 𝑝_{𝑗}},$
where $𝑝_{𝑗}$ are pairwise non-proportional irreducible polynomials and $𝑟$ is the number of conjugacy classes of $𝐺$ .

This marks the beginning of representation theory.

Section 1 Basic Notions of Representation Theory

Throughout this section $𝑘$ will denote a general field as the base field.

1.1 Representation of Algebras

1.1.1 Associative Algebras

An (associative) algebra over $𝑘$ is a vector space over $𝑘$ equipped with a bilinear associative multiplication with unit. An associative algebra is called commutative if the multiplication is commutative. An (algebra) homomorphism is a linear map between algebras that preserves multiplication and unit.

EX 1.1

If $𝑉$ is a vector space, then $𝐴 = GL (𝑉)$ is an algebra, where multiplication is composition.
$𝐴 = 𝑘 ⟨ 𝑥_{1}, \dots, 𝑥_{𝑛} ⟩$ , the free algebra over $𝑘$ , is generated by $𝑥_{1}, \dots, 𝑥_{𝑛}$ with no relations.
For a group $𝐺$ , the group algebra $𝐴 = 𝑘 [𝐺]$ has basis ${𝑎_{𝑔} : 𝑔 \in 𝐺}$ and multiplication defined by $𝑎_{𝑔} \cdot 𝑎_{ℎ} = 𝑎_{𝑔 \cdot ℎ}$ .

A left / right ideal of an algebra is a vector subspace $𝐼 \subseteq 𝐴$ such that for every $𝑎 \in 𝐴$ , $𝑎 𝐼 \subseteq 𝐼$ / $𝐼 𝑎 \subseteq 𝐼$ . A (two-sided) ideal is a subspace that is both a left ideal and a right ideal. If $𝐼$ is an ideal, we can form the quotient algebra $𝐴 / 𝐼 = {𝑎 + 𝐼 : 𝑎 \in 𝐴}$ , with the quotient (algebra) homomorphism $𝜋 : 𝐴 \to 𝐴 / 𝐼, 𝑎 \mapsto 𝑎 + 𝐼$ . The ideal generated by a subset $𝑆$ is denoted $⟨ 𝑆 ⟩$ .

An algebra can be generated by generators and relations. Let $𝑘 ⟨ 𝑥_{1}, \dots, 𝑥_{𝑛} ⟩$ be the free algebra, and $𝑓_{1}, \dots, 𝑓_{𝑚} \in 𝑘 ⟨ 𝑥_{1}, \dots, 𝑥_{𝑛} ⟩$ be the relations; then $𝐴 = 𝑘 ⟨ 𝑥_{1}, \dots, 𝑥_{𝑛} ⟩ / ⟨ 𝑓_{1}, \dots, 𝑓_{𝑚} ⟩$ is called the algebra generated by $𝑥_{1}, \dots, 𝑥_{𝑛}$ subject to the relations $𝑓_{1}, \dots, 𝑓_{𝑚}$ .

1.1.2 Representations

DEF 1.2

A representation of an associative algebra

𝐴

, also called a left $𝐴$ -module, is a vector space

𝑉

with an algebra homomorphism

𝜌_{𝑉} : 𝐴 \to End (𝑉)

. We denote

𝜌_{𝑉} (𝑎) (𝑣)

𝑎 \cdot 𝑣

EX 1.3

$𝑉 = 𝐴$ itself can be made into a representation, by defining $𝜌_{𝑉} (𝑎)$ to be the left multiplication by $𝑎$ . This is called the regular representation.
If $𝐴 = 𝑘 ⟨ 𝑥_{1}, \dots, 𝑥_{𝑛} ⟩$ , then a representation of $𝐴$ is equivalent to a vector space $𝑉$ with $𝑛$ linear operators (in correspondence to the variables $𝑥_{𝑖}$ ).

A representation is called faithful if $𝜌_{𝑉}$ is injective.

A subrepresentation of $𝑉$ is a subspace $𝑊 \subseteq 𝑉$ , such that $𝜌_{𝑉} (𝑊) \subseteq 𝑊$ . A representation is called irreducible or simple if its only subrepresentations is $0$ and itself.

If $𝑉_{1}$ and $𝑉_{2}$ are two representations, a (representation) homomorphism $𝑉_{1} \to 𝑉_{2}$ is a linear map $𝜑 : 𝑉_{1} \to 𝑉_{2}$ such that $𝜑 (𝑎 \cdot 𝑣) = 𝑎 \cdot 𝜑 (𝑣)$ for all $𝑎 \in 𝐴$ and $𝑣 \in 𝑉_{1}$ . $𝜑$ is called a isomorphism if it’s an isomorphism of vector spaces. In this case, $𝑉_{1}$ is isomorphic to $𝑉_{2}$ .

If $𝑉_{1}$ and $𝑉_{2}$ are two representations, we can make their direct sum $𝑉_{1} \oplus 𝑉_{2}$ into a representation by $𝑎 \cdot (𝑣_{1} \oplus 𝑣_{2}) = (𝑎 \cdot 𝑣_{1}) \oplus (𝑎 \cdot 𝑣_{2})$ . This is called the direct sum of representations. A representation $𝑉$ is called indecomposable if it’s not isomorphic to a direct sum.

By definition irreducible representations are indecomposable; the converse is in general not true. The basic questions of representation theory is then the classification of irreducible and indecomposable representations.

PROP 1.4

Suppose $𝑉_{1}$ and $𝑉_{2}$ are two representations of $𝐴$ , $𝜑$ is a non-zero representation homomorphism $𝑉_{1} \to 𝑉_{2}$ .

If $𝑉_{1}$ is irreducible, then $𝜑$ is injective.
If $𝑉_{2}$ is irreducible, then $𝜑$ is surjective.
If $𝑉_{1}$ and $𝑉_{2}$ are both irreducible, then $𝜑$ is an isomorphism.

COR 1.5

If $𝑘$ is additionally algebraically closed and $𝑉$ is a finite dimensional irreducible representation of $𝐴$ , then all representation homomorphisms $𝑉 \to 𝑉$ are scalar multiplications.

If $𝑘$ is algebraically closed and $𝐴$ is additionally commutative, then every irreducible representation of $𝐴$ is $1$ -dimensional.

EX 1.6

$𝐴 = 𝑘 ⟨ 𝑥 ⟩$ where $𝑘$ is algebraically closed. Since $𝐴$ is commutative, irreducible representations of $𝐴$ are $𝑉 = 𝑘$ and $𝜌_{𝑉} (𝑥) = 𝜆$ for some $𝜆 \in 𝑘$ .
A representation of $𝐴$ is a vector space with a linear operator; therefore, indecomposable representations of $𝐴$ are linear operators that has only one Jordan block.
$𝐴 = 𝑘 [𝐺]$ for some group $𝐺$ . A representation of $𝐴$ is a vector space $𝑉$ with a homomorphism $𝐺 \to GL (𝑉)$ ; such a homomorphism is called a representation of the group $𝐺$ .

1.2 Representation of Quivers

A quiver is a directed graph, possibly with self-loops and parallel edges. For an edge $ℎ$ pointing from $ℎ^{'}$ to $ℎ^{″}$ , $ℎ^{'}$ is called the source and $ℎ^{″}$ is called the target. The source of $ℎ$ will always be denoted $ℎ^{'}$ , and the target will always be denoted $ℎ^{″}$ .

A representation of a quiver $𝑄$ is an assignment to every vertex $𝑔$ a vector space $𝑉_{𝑘}$ and to every edge $ℎ^{'} \to ℎ^{″}$ a linear map $𝑉_{ℎ^{'}} \to 𝑉_{ℎ^{″}}$ .

For a quiver $𝑄$ with finitely many vertices, we can define its path algebra $𝑃_{𝑄}$ , generated by the set of paths¹ in $𝑄$ , and multiplication given by path concatenation, zero if two paths cannot be concatenated.

At each vertex $𝑖$ , there is a trivial path $𝑝_{𝑖}$ , consisting of only one vertex, $𝑖$ itself. If $𝑄$ has finitely many vertices, then the unit of the path algebra $𝑃_{𝑄}$ is $1 = \sum_{vertex 𝑖} 𝑝_{𝑖}$ .

PROP 1.7

A representation of a quiver

𝑄

is the same as a representation of its path algebra

𝑃_{𝑄}

1.3 Representation of Lie Algebras

For introductory materials on Lie algebras and their representations, we refer to the Lie Groups and Lie Algebras course notes.

A representation of a Lie algebra $𝔤$ is a Lie algebra homomorphism $𝜌 : 𝔤 \to 𝔤𝔩 (𝑉)$ .

EX 1.8 adjoint representation

𝑉 = 𝔤

, then

𝜌 : 𝔤 \to 𝔤𝔩 (𝑉)

𝑥 \mapsto [𝑥, -]

is a Lie algebra representation, called the adjoint representation. This

𝜌

is usually denoted

ad

, and

{ad}_{𝑥} (𝑦) = [𝑥, 𝑦]

Let $𝔤$ be a Lie algebra, with basis ${𝑥_{1}, \dots, 𝑥_{𝑛}}$ . Suppose $[𝑥_{𝑖}, 𝑥_{𝑗}] = \sum 𝑐_{𝑖 𝑗}^{𝑘} 𝑥_{𝑘}$ , where $𝑐_{𝑖 𝑗}^{𝑘} \in 𝑘$ are called the structure constants. The universal enveloping algebra, $𝒰︀ (𝑔)$ , is the associative algebra generated by $𝑥_{1}, \dots, 𝑥_{𝑛}$ subject to relations $𝑥_{𝑖} 𝑥_{𝑗} - 𝑥_{𝑗} 𝑥_{𝑖} = \sum 𝑐_{𝑖 𝑗}^{𝑘} 𝑥_{𝑘}$ . Representations of a Lie algebra are the same as representations of its universal enveloping algebra.

THM 1.9

The ordered monomials

𝑥_{1}^{𝑑_{1}} \dots 𝑥_{𝑛}^{𝑑_{𝑛}}

with

𝑑_{𝑖} \in ℤ_{⩾ 0}

form a basis of

𝒰︀ (𝑔)

If $𝑉$ and $𝑊$ are representations of $𝔤$ , then we can form a Lie algebra representation on $𝑉 \otimes 𝑊$ , the tensor representation, by

\begin{aligned} 𝜌_{𝑉 \otimes 𝑊} : 𝔤 & \to 𝔤𝔩 (𝑉 \otimes 𝑊), \\ 𝑥 & \mapsto 𝜌_{𝑉} (𝑥) \otimes id - id \otimes 𝜌_{𝑊} (𝑥), \end{aligned}

and a representation on the dual space $𝑉^{*}$ , the dual representation, by

\begin{aligned} 𝜌_{𝑉^{*}} (𝑥) : 𝔤 & \to 𝔤𝔩 (𝑉^{*}), \\ 𝑥 & \mapsto - 𝜌_{𝑉} {(𝑥)}^{⊤}, \end{aligned}

$⊤$ denoting adjoint. The form of these representations are deduced from the tensor and dual representation of its corresponding Lie group representation.

1.4 Example: Representations of ${𝔰𝔩}_{2}$

Suppose ${𝔰𝔩}_{2}$ is the special linear Lie algebra over $ℂ$ . ${𝔰𝔩}_{2}$ has a basis

𝑒 = (\begin{matrix} 0 & 1 \\ 0 \end{matrix}), ℎ = (\begin{matrix} 1 \\ - 1 \end{matrix}), 𝑓 = (\begin{matrix} 0 \\ 1 & 0 \end{matrix}),

with relations

[ℎ, 𝑒] = 2 𝑒, [ℎ, 𝑓] = - 2 𝑓, [𝑒, 𝑓] = ℎ .

Therefore, a representation of ${𝔰𝔩}_{2}$ is the same as a vector space $𝑉$ over $ℂ$ with three linear operators $𝐸$ , $𝐹$ , $𝐻$ , satisfying $𝐻 𝐸 - 𝐸 𝐻 = 2 𝐸$ , $𝐻 𝐹 - 𝐹 𝐻 = - 2 𝐹$ , $𝐸 𝐹 - 𝐹 𝐸 = 𝐻$ . We now discuss its finite dimensional irreducible representations.

Step 1. Pick an eigenvalue of $𝐻$ with largest real part, and denote $𝑉 (𝜆)$ its generalized eigenspace, $𝑉 (𝜆) = {𝑣 \in 𝑉 : {(𝐻 - 𝜆 𝐼)}^{𝑛} 𝑣 = 0 for some 𝑛 ⩾ 0}$ . Since $(𝐻 - (𝜆 + 2) 𝐼) 𝐸 = 𝐸 (𝐻 - 𝜆 𝐼)$ , we have ${(𝐻 - (𝜆 + 2) 𝐼)}^{𝑛} 𝐸 = 𝐸 {(𝐻 - 𝜆 𝐼)}^{𝑛}$ ; therefore, for every $𝑣 \in 𝑉 (𝜆)$ , $𝐸 𝑣 \in 𝑉 (𝜆 + 2)$ . But $𝜆$ has the largest real part, so ${𝐸 |}_{𝑉 (𝜆)} = 0$ .

Step 2. Let $𝑃_{𝑛} (𝑥) = 𝑛! 𝑥 (𝑥 - 1) \dots (𝑥 - 𝑛 + 1)$ . By induction, if $𝑤 \in 𝑉$ such that $𝐸 𝑤 = 0$ , then for any $𝑛 > 0$ , $𝐸^{𝑛} 𝐹^{𝑛} 𝑤 = 𝑃_{𝑛} (𝐻) 𝑤$ .

Step 3. If $𝑣 \in 𝑉 (𝜆)$ , then by a similar argument as Step 1, $𝐹 𝑣 = 𝑉 (𝜆 - 2)$ , so $𝐹^{𝑛} 𝑣 \in 𝑉 (𝜆 - 2 𝑛)$ . Since $dim 𝑉 < \infty$ , $𝐻$ has only finitely many eigenvalues; therefore, for some $𝑁$ , $𝑉 (𝜆 - 2 𝑁) = 0$ , so $𝐹^{𝑁} 𝑣 = 0$ for $𝑣 \in 𝑉 (𝜆)$ .

Step 4. By Step 2, ${𝐸^{𝑁} 𝐹^{𝑁} |}_{𝑉 (𝜆)} = 0 = {𝑃_{𝑁} (𝐻) |}_{𝑉 (𝜆)}$ . As $𝑃_{𝑁}$ has no multiple roots, this tells us that $𝐻$ is diagonalizable on $𝑉 (𝜆)$ ; therefore, $𝑉 (𝜆)$ is the eigenspace of $𝜆$ , $𝑉 (𝜆) = {𝑣 \in 𝑉 : 𝐻 𝑣 = 𝜆 𝑣}$ .

Step 5. Suppose $𝑣 \in 𝑉 (𝜆)$ . Let $𝑁$ be the smallest integer such that $𝐹^{𝑁} 𝑣 = 0$ . By induction $𝐸 𝐹^{𝑘} 𝑣 = 𝑘 𝐹^{𝑘 - 1} 𝐻 𝑣 - (𝑘 - 1) 𝑘 𝐹^{𝑘 - 1} 𝑣 = 𝑘 (𝜆 - 𝑘 + 1) 𝐹^{𝑘 - 1} 𝑣$ ; let $𝑘 = 𝑁$ , then $𝐸 𝐹^{𝑁} 𝑣 = 0$ but $𝐹^{𝑁 - 1} 𝑣 \neq 0$ . Therefore, $𝑁 (𝜆 - 𝑁 + 1) = 0$ , so $𝜆 = 𝑁 - 1$ .

Step 6. Therefore, for any positive integer $𝑁$ , there is an irreducible representation of ${𝔰𝔩}_{2}$ with basis

𝑣_{0} = 𝑣, 𝑣_{1} = 𝐹 𝑣, 𝑣_{2} = \frac{𝐹^{2} 𝑣}{2!}, \dots, 𝑣_{𝑁 - 1} = \frac{𝐹^{𝑁 - 1} 𝑣}{(𝑁 - 1)!},

with actions

and this representation is denoted $𝑉_{𝜆}$ where $𝜆 = 𝑁 - 1$ ; moreover they are all irreducible representations of ${𝔰𝔩}_{2}$ .

Section 2 General Results on Representation Theory

From here on we will assume that the base field $𝑘$ is an algebraically closed field.

2.1 Semisimple Representations

DEF 2.1

A representation

𝑉

𝐴

is called semisimple if it is the direct sum of some irreducible representations.

EX 2.2

𝑉

is an irreducible representation of

𝐴

and

𝜌_{𝑉} : 𝐴 \to End (𝑉)

, then the vector space

End (𝑉)

can be made into a representation of

𝐴

𝑎 \cdot 𝐴 = 𝜌_{𝑣} (𝑎) 𝐴

for

𝐴 \in End (𝑉)

End (𝑉)

is (representation) isomorphic to

𝑉^{\oplus 𝑛}

where

𝑛 = dim 𝑉

, so

End (𝑉)

is a semisimple representation of

𝐴

Let us denote $𝑉^{\oplus 𝑛}$ by $𝑛 𝑉$ . Every semisimple representation of $𝐴$ has the form $𝑉 = ⨁_{𝑖 = 1}^{𝑛} 𝑛_{𝑖} 𝑉_{𝑖}$ , where $𝑉_{𝑖}$ are pairwise non-isomorphic finite dimensional irreducible representations of $𝐴$ . If $𝑊_{𝑖}$ is a vector space of dimension $𝑛_{𝑖}$ , then $𝑛_{𝑖} 𝑉_{𝑖} ≅ 𝑉_{𝑖} \otimes 𝑊_{𝑖}$ , where the action of $𝐴$ on $𝑊_{𝑖}$ is trivial ( $𝑊_{𝑖}$ only records the multiplicity of $𝑉_{𝑖}$ ). By Schur’s lemma, $𝑊_{𝑖}$ is isomorphic as a vector space to ${Hom}_{𝐴} (𝑉_{𝑖}, 𝑉)$ ; therefore,

𝑉 = ⨁_{𝑖 = 1}^{𝑛} 𝑉_{𝑖} \otimes {Hom}_{𝐴} (𝑉_{𝑖}, 𝑉) .

This is called the isotypical decomposition.

PROP 2.3

Suppose

𝑉_{𝑖}

(

1 ⩽ 𝑖 ⩽ 𝑚

) are pairwise non-isomorphic finite dimensional irreducible representations of

𝐴

, and

𝑉 = ⨁_{𝑖 = 1}^{𝑚} 𝑛_{𝑖} 𝑉_{𝑖}

. Suppose

𝑊

is a subrepresentation of

𝑉

. Then

𝑊 ≅ ⨁_{𝑖 = 1}^{𝑚} 𝑟_{𝑖} 𝑉_{𝑖}

where

𝑟_{𝑖} ⩽ 𝑛_{𝑖}

, and the inclusion

𝜑 : 𝑊 ↪︎ 𝑉

is of the form

𝜑 = ⨁_{𝑖 = 1}^{𝑚} 𝜑_{𝑖}

, where

𝜑_{𝑖} : 𝑟_{𝑖} 𝑉_{𝑖} \to 𝑛_{𝑖} 𝑉_{𝑖}

(𝑣_{1}, \dots, 𝑣_{𝑟_{𝑖}}) \mapsto (𝑣_{1}, \dots, 𝑣_{𝑟_{𝑖}}) \cdot 𝑋_{𝑖}

and

𝑋_{𝑖}

is a

𝑟_{𝑖} \times 𝑛_{𝑖}

matrix of full rank.

COR 2.4

𝑉

is a finite dimensional representation of

𝐴

𝑣_{1}, \dots, 𝑣_{𝑛}

are linearly independent vectors in

𝑉

. Then for any

𝑤_{1}, \dots, 𝑤_{𝑛} \in 𝑉

, there is some

𝑎 \in 𝐴

such that

𝑎 𝑣_{𝑖} = 𝑤_{𝑖}

for all

𝑖

COR 2.5

Suppose $𝑉 = ⨁_{𝑖 = 1}^{𝑚} 𝑉_{𝑖}$ where $(𝑉_{𝑖}, 𝜌_{𝑖})$ are pairwise non-isomorphic finite dimensional irreducible representations of $𝐴$ .
Then $⨁_{𝑖 = 1}^{𝑚} 𝜌_{𝑖} : 𝐴 \to ⨁_{𝑖 = 1}^{𝑚} End (𝑉_{𝑖})$ is surjective.
Suppose $𝑉$ is a finite dimensional representation of $𝐴$ .
Then $𝑉$ is irreducible if and only if $𝜌 : 𝐴 \to End (𝑉)$ is surjective.

2.2 Semisimple Algebras

DEF 2.6

An associative algebra

𝐴

is called semisimple if every finite dimensional representation of it is semisimple.

We will classify all semisimple algebras in this section. We first look at an important class of semisimple algebras, the matrix algebras.

2.2.1 The Matrix Algebra

Let $𝑑_{𝑖}$ ( $1 ⩽ 𝑖 ⩽ 𝑟$ ) be distinct integers, $𝑉_{𝑖} = 𝑘^{𝑑_{𝑖}}$ and $𝑀_{𝑖} = End (𝑉_{𝑖}) = {Mat}_{𝑑_{𝑖} \times 𝑑_{𝑖}} (𝑘)$ . Let $𝐴 = ⨁_{𝑖 = 1}^{𝑟} 𝑀_{𝑖}$ . We will discuss the representations of $𝐴$ .

Obviously, each $𝑉_{𝑖}$ itself is a representation of $𝐴$ by the projection ${pr}_{𝑖} : 𝐴 \to 𝑀_{𝑖} = End (𝑉_{𝑖})$ , and this representation is irreducible since $𝐴$ contains all operators on $𝑉_{𝑖}$ .

Suppose now that $𝑋$ is an $𝑛$ -dimensional representation of $𝐴$ . We can form the dual representation on the dual space $𝑋^{*}$ ; however this representation is for the opposite algebra $𝐴^{op}$ , the same algebra but with the order of multiplication reversed. This representation is given by $(𝑎 \cdot 𝑓) (𝑣) = 𝑓 (𝑎 \cdot 𝑣)$ , for $𝑎 \in 𝐴$ , $𝑓 \in 𝑋^{*}$ and $𝑣 \in 𝑋$ . In the particular case of a matrix algebra, we actually have $𝐴^{op} ≅ 𝐴$ , by the isomorphism $𝐵 \mapsto 𝐵^{⊤}$ ; therefore, $𝑋^{*}$ is a representation of $𝐴$ .

Pick a basis ${𝑦_{1}, \dots, 𝑦_{𝑛}}$ of $𝑋^{*}$ and let $𝜑 : 𝐴^{\oplus 𝑛} \to 𝑋^{*}$ , $(𝑎_{1}, \dots, 𝑎_{𝑛}) \mapsto \sum_{𝑖 = 1}^{𝑛} 𝑎_{𝑖} 𝑦_{𝑖}$ . Since $𝐴$ contains all scalar multiplications, $𝜑$ is surjective. Therefore, $𝜑^{*}$ gives a injection $𝑋 \to {(𝐴^{\oplus 𝑛})}^{*}$ .

Through the map ${Mat}_{𝑑 \times 𝑑} (𝑘) \to {Mat}_{𝑑 \times 𝑑} {(𝑘)}^{*}$ , $𝐵 \mapsto tr (𝐵^{⊤} \cdot -)$ , ${(𝐴^{\oplus 𝑛})}^{*}$ is isomorphic to $𝐴^{\oplus 𝑛}$ as representations of $𝐴$ . Therefore,

𝜑^{*} : 𝑋 \to {(𝐴^{\oplus 𝑛})}^{*} ≅ 𝐴^{\oplus 𝑛} = {(⨁_{𝑖 = 1}^{𝑟} End (𝑉_{𝑖}))}^{\oplus 𝑛} ≅ {(⨁_{𝑖 = 1}^{𝑟} 𝑑_{𝑖} 𝑉_{𝑖})}^{\oplus 𝑛} .

Therefore, there is an injection $𝑋 \to ⨁_{𝑖 = 1}^{𝑟} 𝑛 𝑑_{𝑖} 𝑉_{𝑖}$ ; by Proposition 2.3, $𝑋$ is the direct sum of some $𝑉_{𝑖}$ ’s.

In corollary, all irreducible representations of $𝐴$ are $𝑉_{𝑖}$ , and all representations of $𝐴$ are direct sums of $𝑉_{𝑖}$ .

2.2.2 Classification of Semisimple Algebras

In this section, we will see that every semisimple algebra is isomorphic to a matrix algebra. Also, we will justify the name “semisimple”: a semisimple algebra is equivalent to a direct sum of simple algebras (ones with no non-trivial ideals). We will use the concept of filtrations; a more detailed examination of filtrations will be postponed.

Suppose $𝑉$ is a finite dimensional representation of $𝐴$ . A finite filtration of $𝑉$ is a sequence of subrepresentations

0 = 𝑉_{0} \subseteq 𝑉_{1} \subseteq \dots \subseteq 𝑉_{𝑛} = 𝑉 .

Any finite dimensional representation $𝑉$ admits a finite filtration $0 = 𝑉_{0} \subseteq 𝑉_{1} \subseteq \dots \subseteq 𝑉_{𝑛} = 𝑉$ , such that each quotient $𝑉_{𝑖} / 𝑉_{𝑖 - 1}$ is irreducible.

DEF 2.7

The radical of an algebra

𝐴

, denoted

rad 𝐴

, is the set of elements

𝑎 \in 𝐴

that, for all irreducible representation

(𝑉, 𝜌)

𝐴

𝜌 (𝑎) = 0

PROP 2.8

The radical of

𝐴

is a two-sided ideal. It is the largest nilpotent ideal in

𝐴

; every nilpotent ideal is contained in

rad 𝐴

THM 2.9

Any finite dimensional algebra $𝐴$ has only a finite number o finite dimensional irreducible representations up to isomorphism. If we list them as $𝑉_{1}, \dots, 𝑉_{𝑟}$ , then

𝐴 / rad 𝐴 ≅ ⨁_{𝑖 = 1}^{𝑟} End (𝑉_{𝑖}) .

COR 2.10

For any finite dimensional algebra $𝐴$ ,

\sum_{\begin{matrix} 𝑉 finite dimensional \\ irreducible representation \end{matrix}} {(dim 𝑉)}^{2} ⩽ dim 𝐴 .

EX 2.11

Let

𝐴

be the algebra of all upper-triangular matrices. For

1 ⩽ 𝑖 ⩽ 𝑛

let

(𝑉_{𝑖} = 𝑘, 𝜌_{𝑖})

be the representation given by

𝜌_{𝑖} (𝑋) = 𝑋_{𝑖 𝑖}

. They are clearly irreducible and pairwise non-isomorphic. Let

𝐵

be the subalgebra of strictly upper-triangular matrices. Then

𝐵

is nilpotent and

rad 𝐴 \supseteq 𝐵

. But

𝐴 / rad 𝐴

𝑛

dimensional and

\sum_{𝑖 = 1}^{𝑛} {(dim 𝑉_{𝑖})}^{2} = 𝑛

, it turns out that

rad 𝐴 = 𝐵

and

𝑉_{𝑖}

are all irreducible representations.

PROP 2.12

Finite dimensional simple algebras are exactly those isomorphic to ${Mat}_{𝑑 \times 𝑑} (𝑘)$ for some $𝑑$ .
For finite dimensional algebras, the following are equivalent:
1. $rad 𝐴 = 0$ .
2. $\sum_{𝑉 finite dimensional irreducible} {(dim 𝑉)}^{2} = dim 𝐴$ .
3. Any finite dimensional representation of $𝐴$ is semisimple, i.e. $𝐴$ semisimple.
4. The regular representation of $𝐴$ is semisimple.
5. $𝐴 ≅ ⨁_{𝑖 = 1}^{𝑟} {Mat}_{𝑑_{𝑖} \times 𝑑_{𝑖}} (𝑘)$ for some $𝑑_{𝑖}$ ; i.e., $𝐴$ is the direct sum of some simple algebras.

2.3 Characters

Throughout this section $𝐴$ will be an arbitrary associative algebra.

DEF 2.13

Suppose $(𝑉, 𝜌)$ is a representation of $𝐴$ . The character of it is

\begin{aligned} 𝜒_{𝑉} : 𝐴 & \to 𝑘, \\ 𝑎 & \mapsto tr 𝜌 (𝑎) . \end{aligned}

If $𝑉$ is irreducible, call $𝜒$ irreducible.

If we let $[𝐴, 𝐴]$ be the commutator subalgebra, the subalgebra generated by elements of the form $𝑥 𝑦 - 𝑦 𝑥$ , then characters actually factors through $𝐴 / [𝐴, 𝐴]$ .

THM 2.14

Characters of distinct irreducible representations of $𝐴$ are linearly independent.
If $𝐴$ is finite dimensional and semisimple, then all irreducible characters of $𝐴$ are distinct, and they form a basis for ${(𝐴 / [𝐴, 𝐴])}^{*}$ .

2.4 Jordan-Hölder Theorem and Krull-Schmidt Theorem

THM 2.15

Suppose that $𝑉$ is a finite dimensional representation of $𝐴$ . Then $𝑉$ admits a finite filtration $0 = 𝑉_{0} \subseteq 𝑉_{1} \subseteq \dots \subseteq 𝑉_{𝑛} = 𝑉$ such that each $𝑉_{𝑖} / 𝑉_{𝑖 - 1}$ is irreducible.

Moreover, every two such filtrations are equal in length, and the quotients $𝑉_{𝑖} / 𝑉_{𝑖 - 1}$ are isomorphic up to a permutation. The number $𝑛$ is called the length of $𝑉$ , and the quotients are called the Jordan-Hölder series of $𝑉$ .

LEM 2.16

Suppose $𝑊$ is indecomposable and $𝜃, 𝜃^{'} \in {End}_{𝐴} (𝑊)$ .

$𝜃$ is either an isomorphism or nilpotent.
If $𝜃$ and $𝜃^{'}$ are both nilpotent, then so is $𝜃 + 𝜃^{'}$ .

THM 2.17

Any finite dimensional representation of an algebra

𝐴

can be uniquely (up to isomorphisms and a permutation) decomposed into a direct sum of indecomposable representations.

2.5 Representations of Tensor Products

If $𝐴$ and $𝐵$ are algebras, then their tensor product $𝐴 \otimes 𝐵$ is also an algebra, with multiplication $(𝑎 \otimes 𝑏) \cdot (𝑎^{'} \otimes 𝑏^{'}) = (𝑎 𝑎^{'}) \otimes (𝑏 𝑏^{'})$ . If $𝑉$ and $𝑊$ are representations of $𝐴$ and $𝐵$ respectively, then $𝑉 \otimes 𝑊$ is also a representation with componentwise action. Recall that $End (𝑉 \otimes 𝑊) = End (𝑉) \otimes End (𝑊)$ .

THM 2.18

If $𝑉$ and $𝑊$ are irreducible, then $𝑉 \otimes 𝑊$ is an irreducible representation of $𝐴 \otimes 𝐵$ .
Any finite dimensional representation of $𝐴 \otimes 𝐵$ can be uniquely expressed in the form $𝑉 \otimes 𝑊$ .

The density theorem says that $𝐴 \to End (𝑉)$ and $𝐵 \to End (𝑊)$ are surjective. Therefore, $𝐴 \otimes 𝐵 \to End (𝑉 \otimes 𝑊)$ is also surjective, hence the irreducibility.
By taking $𝐴^{'} = im 𝜌_{𝑉}$ and $𝐵^{'} = im 𝜌_{𝑊}$ , we may assume that $𝐴 = 𝐴^{'}$ and $𝐵 = 𝐵^{'}$ are finite dimensional.
Claim. $rad (𝐴 \otimes 𝐵) = rad 𝐴 \otimes 𝐵 + 𝐴 \otimes rad 𝐵$ .
If the Claim is true, then
$\begin{aligned} 𝐴 \otimes 𝐵 / rad (𝐴 \otimes 𝐵) = & 𝐴 / rad 𝐴 \otimes 𝐵 / rad 𝐵 \\ = & (⨁_{𝑉 fin. diml. irr.} End (𝑉)) \otimes (⨁_{𝑊 fin. diml. irr} End (𝑊)) \\ = & ⨁_{𝑉, 𝑊} End (𝑉 \otimes 𝑊), \end{aligned}$
therefore proving the result.
Prove of the claim: let $𝐼 = rad 𝐴 \otimes 𝐵 + 𝐴 \otimes rad 𝐵$ . $𝐼$ is nilpotent, and $𝐴 \otimes 𝐵 / 𝐼 = ⨁_{𝑉, 𝑊} End (𝑉 \otimes 𝑊)$ by the above calculation is semisimple. Therefore, $𝐼$ is the largest nilpotent ideal, hence the radical.

Section 3 Representations of Finite Groups

Let $𝐺$ be a finite group, $𝑘$ algebraically closed and $𝑘 [𝐺]$ the group algebra.

THM 3.1

𝑘 [𝐺]

is semisimple if and only if

| 𝐺 | \in 𝑘

is not zero (or equivalently

char 𝑘 > 0

and

char 𝑘 ∣ | 𝐺 |

Therefore, our general results of semisimple algebras apply. In particular, $𝑘 [𝐺] = ⨁_{𝑉 irr} End (𝑉)$ , so

| 𝐺 | = \sum_{𝑉 irr} {(dim 𝑉)}^{2} .

This is called the sum of squares formula.

3.1 Characters

Define the vector space of class functions

𝐹_{c} (𝐺, 𝑘) = {𝑓 : 𝐺 \to 𝑘 : \forall 𝑔, ℎ, 𝑓 (𝑔 ℎ 𝑔^{- 1}) = 𝑓 (ℎ)} .

Suppose $𝐺$ is a finite group and $| 𝐺 | \neq 0$ in $𝑘$ . Recall that the character $𝜒_{𝑉}$ of a representation $(𝑉, 𝜌_{𝑉})$ is defined as $𝜒_{𝑉} (𝑔) = tr 𝜌_{𝑉} (𝑔)$ .

THM 3.2

Characters of irreducible representations form a basis of $𝐹_{c} (𝐺, 𝑘)$ .

In particular, the number of irreducible representations of $𝐺$ is equal to the conjugacy classes of $𝐺$ .

COR 3.3

char 𝑘 = 0

, then two representations of

𝐺

are isomorphic if and only if their character are equal.

We are particularly interested in the case $𝑘 = ℂ$ . We can define a $𝐺$ -invariant Hermitian inner product on $𝐹_{c} (𝐺, ℂ)$ by $(𝑓_{1}, 𝑓_{2}) = \frac{1}{| 𝐺 |} \sum_{𝑔 \in 𝐺} 𝑓_{1} (𝑔) 𝑓_{2} (𝑔)$ .

THM 3.4

If $𝑉$ and $𝑊$ are representations, then $(𝜒_{𝑉}, 𝜒_{𝑊}) = dim {Hom}_{𝐺} (𝑉, 𝑊)$ .

In particular, the characters of irreducible representations are orthogonal.

Let $𝑝 = \frac{1}{| 𝐺 |} \sum_{𝑔 \in 𝐺} 𝑔 \in 𝑘 [𝐺]$ be the symmetrizer. $𝜌_{𝑉} (𝑝)$ is a projection from $𝑉$ to the stabilizer subspace $𝑉^{𝐺}$ , so $𝜒_{𝑈} (𝑝) = dim 𝑉^{𝐺}$ .

Therefore,

\begin{aligned} (𝜒_{𝑉}, 𝜒_{𝑊}) & = \frac{1}{| 𝐺 |} \sum_{𝑔 \in 𝐺} 𝜒_{𝑉} (𝑔) 𝜒_{𝑊} (𝑔) \\ = \frac{1}{| 𝐺 |} \sum_{𝑔 \in 𝐺} 𝜒_{𝑉 \otimes 𝑊^{*}} (𝑔) \\ = tr 𝜌_{𝑉 \otimes 𝑊^{*}} (𝑝) \\ = {dim (𝑉 \otimes 𝑊^{*})}^{𝐺} \\ = dim {Hom}_{𝐺} (𝑊, 𝑉) . \end{aligned}

Therefore, by the isotypical decomposition, the regular representation of $𝐺$ can be decomposed as $⨁_{𝑉 irr} 𝑉^{\oplus (𝜒_{regular}, 𝜒_{𝑉})}$ .

3.2 Complex Representations

A finite dimensional representation $𝑉$ over $ℂ$ of a group $𝐺$ is called unitary if there exists a $𝐺$ -invariant positive definition Hermitian form on $𝑉$ .

By taking the symmetrizer, any finite dimensional representation of a finite group $𝐺$ has a unitary structure. If $𝑉$ is moreover irreducible, then this structure is unique up to scaling by a positive real.

As a corollary, by taking orthogonal complements, every finite dimensional representation of a finite group $𝐺$ is completely reducible.

Let $𝑉$ over $ℂ$ be a irreducible representation of $𝐺$ and ${𝑣_{1}, \dots, 𝑣_{𝑛}}$ an orthonormal basis of $𝑉$ with respect to the unitary structure. Let $𝑡_{𝑖 𝑗}^{𝑉} (𝑔) = (𝜌_{𝑉} (𝑔) 𝑣_{𝑖}, 𝑣_{𝑗})$ .

PROP 3.5

If $𝑉$ and $𝑊$ are both irreducible representations of $𝐺$ , then

(𝑡_{𝑖 𝑗}^{𝑉}, 𝑡_{𝑘 𝑙}^{𝑊}) = {\begin{cases} \frac{1}{dim 𝑉} if 𝑉 ≅ 𝑊 and 𝑖 = 𝑘, 𝑗 = 𝑙 \\ 0 otherwise. \end{cases}

In particular ${𝑡_{𝑖 𝑗}^{𝑉}}$ form an orthogonal basis for $𝐹 (𝐺, ℂ) = {𝑓 : 𝐺 \to ℂ}$ .

3.2.1 The Character Table

Since the number of irreducible representations of a finite group is equal to the number of conjugacy classes, we can draw a table with columns the conjugacy classes, and rows the irreducible representations. On column $𝐶$ row $𝑉$ , we fill in the character $𝜒_{𝑉} (𝐶)$ (conjugate elements share the same character). This forms a square matrix on $ℂ$ , called the character table of $𝐺$ .

There are several properties of this square matrix.

The conjugacy class of $𝑒$ is $𝑒$ itself, and $𝜒_{𝑉} (𝑒) = dim 𝑉$ . Therefore, column $𝑒$ are the dimensions of irreducible representations. By the sum of squares formula, the sum of squares of the first column is equal to $| 𝐺 |$ .
Irreducible characters are orthogonal. Therefore, for any two different rows $𝑉_{1}$ , $𝑉_{2}$ , $\sum_{𝐶} | 𝐶 | \cdot 𝜒_{𝑉_{1}} (𝐶) 𝜒_{𝑉_{2}} (𝐶) = 0$ . For any row $𝑉$ , $\sum_{𝐶} | 𝐶 | \cdot {| 𝜒_{𝑉} (𝐶) |}^{2} = | 𝐺 |$ .
Therefore, by properties of a orthogonal square matrix, the columns of a character table are also orthogonal.

EX 3.6

Character tables of $𝔖_{3}$ , $𝔖_{4}$ and $𝔄_{4}$ .

The character table of $𝔖_{3}$ can be constructed from the knowledge of

a trivial representation,
a 1-dimensional sign representation,
sum of squares formula, and
orthogonality of columns.

$𝜒$	$𝑒$ $# = 1$	$(12)$ $# = 3$	$(123)$ $# = 2$
trivial	$1$	$1$	$1$
sign	$1$	$- 1$	$1$
( $ℂ^{2}$ )	$2$	$0$	$- 1$

The 2-dimensional representation can be generalized constructed as follows. $𝔖_{𝑛}$ acts by permutation of basis on $ℂ^{𝑛}$ , and there is an obvious invariant subspace $𝑣_{1} + \dots + 𝑣_{𝑛} = 0$ . $ℂ^{𝑛} / (𝑣_{1} + \dots + 𝑣 + 𝑛 = 0)$ is an irreducible representation of $𝔖_{𝑛}$ .

The character table of $𝔖_{4}$ can be constructed from the knowledge of

a trivial representation and a sign representation,
a 3-dimensional representation, constructed as above,
the tensor of two representations,
orthogonality and sum of squares formula.

$𝜒$	$𝑒$ $# = 1$	$(12)$ $# = 3$	$(12) (34)$ $# = 3$	$(123)$ $# = 8$	$(1234)$ $# = 6$
trivial	$1$	$1$	$1$	$1$	$1$
sign	$1$	$- 1$	$1$	$1$	$- 1$
$ℂ^{3}$	$3$	$1$	$- 1$	$0$	$- 1$
$ℂ^{3} \otimes sign$	$3$	$- 1$	$- 1$	$0$	$- 1$
( $ℂ^{2}$ )	$2$	$0$	$2$	$- 1$	$0$

Now what is this last 2-dimensional representation? Recall the last row of the character table of $𝔖_{3}$ : they coincide. Actually, this representation of $𝔖_{4}$ is a pullback of that representation of $𝔖_{3}$ .

Consider finite groups $𝐺$ , $𝐻$ and a surjective homomorphism $𝜑 : 𝐺 \to 𝐻$ . Then for any irreducible representation $𝑉$ of $𝐻$ , the composition $𝐺 \to 𝐻 \to GL (𝑉)$ is a representation of $𝐺$ , and is irreducible by the density theorem. This representation is called the pullback representation.

Now consider $𝔄_{4}$ . There is a surjective homomorphism $𝔄_{4} \to ℤ_{3}$ by quotienting out $(12) (34)$ . Therefore we get the trivial representation and two pullback representations:

$𝜒$	$𝑒$ $# = 1$	$(12) (34)$ $# = 3$	$(123)$ $# = 4$	$(132)$ $# = 4$
trivial	$1$	$1$	$1$	$1$
pullback	$1$	$1$	$𝜁$	$𝜁^{2}$
pullback	$1$	$1$	$𝜁^{2}$	$𝜁$
( $ℂ^{3}$ )	$3$	$0$	$0$	$- 1$

The last row is by restriction: if $𝐻$ is a subgroup of $𝐺$ , then $𝐻 \to 𝐺 \to GL (𝑉)$ is a representation of $𝐻$ .

3.2.2 Frobenius Determinant

For a finite group $𝐺$ and formal variables $𝑥_{𝑔}$ for $𝑔 \in 𝐺$ , define a $| 𝐺 | \times | 𝐺 |$ matrix $𝑋_{𝐺}$ by defining the entry at row $𝑔$ column $ℎ$ to be $𝑥_{𝑔^{- 1} \cdot ℎ}$ .

THM 3.7

Suppose $𝐺$ is a finite group, not necessarily abelian. Then

det 𝑋_{𝐺} = \prod_{𝑗 = 1}^{𝑟} 𝑝_{𝑗} {(𝑥)}^{deg 𝑝_{𝑗}},

where $𝑝_{𝑗}$ are pairwise non-proportional irreducible polynomials and $𝑟$ is the number of conjugacy classes of $𝐺$ .

3.2.3 Frobenius-Schur Indicator

Let $𝐺$ be a finite group and $𝑉$ a finite dimensional complex representation.

Say $𝑉$ is of

complex type or $ℂ$ -type if $𝑉$ is not representation isomorphic to its dual representation $𝑉^{*}$ .
real type or $ℝ$ -type if $𝑉$ admits a non-degenerate symmetric $𝐺$ -invariant bilinear form.
quaternionic type or $ℍ$ -type if $𝑉$ admits a non-degenerate anti-symmetric $𝐺$ -invarient bilinear form.

Every finite dimensional complex representation $𝑉$ is either of complex type, or of real type (in which case $𝑉 = 𝑈 \otimes_{ℝ} ℂ$ for some real representation $𝑅$ ), or of quaternionic type.

THM 3.8

For a representation $𝑉$ , let the Frobenius-Schur inducator be

FS (𝑉) = \frac{1}{| 𝐺 |} \sum_{𝑔 \in 𝐺} 𝜒_{𝑉} (𝑔^{2}) .

Then

FS (𝑉) = {\begin{cases} 1 & if 𝑉 is of ℝ -type, \\ - 1 & if 𝑉 is of ℍ -type, \\ 1 & if 𝑉 is of ℂ -type. \end{cases}

For a matrix $𝐴$ , $tr 𝐴^{2} = {tr}_{𝑆^{2} 𝑉} (𝐴 \otimes 𝐴) - {tr}_{Λ^{2} 𝑉} (𝐴 \otimes 𝐴)$ . Therefore, letting $𝑝$ be the symmetrizer,

FS (𝑉) = 𝜒_{𝑆^{2} 𝑉} (𝑝) - 𝜒_{Λ^{2} 𝑉} (𝑝) = dim {(𝑆^{2} 𝑉)}^{𝐺} - dim {(Λ^{2} 𝑉)}^{𝐺} .

Now notice that $Hom (𝑉^{*}, 𝑉) ≅ {(𝑉 \otimes 𝑉)}^{𝐺} ≅ {(𝑆^{2} 𝑉)}^{𝐺} \oplus {(Λ^{2} 𝑉)}^{𝐺}$ . The dimensions of the direct summand could be $0, 0$ , $1, 0$ or $0, 1$ , depending on whether $𝑉$ is of $ℂ$ -type, $ℝ$ -type or $ℍ$ -type.

3.2.4 Frobenius Divisibility

Denote by $𝔸$ the ring of algebraic integers.

THM 3.9 Frobenius divisibility theorem

𝐺

is a finite group,

𝑉

is a complex irreducible representation, then

dim 𝑉 ∣ | 𝐺 |

LEM 3.10

Suppose

𝐶

is a conjugacy class in

𝐺

. Then

𝜆 ≔ 𝜒_{𝑉} (𝐶) \cdot \frac{| 𝐶 |}{dim 𝑉}

is an algebraic integer.

Let $𝑝_{𝐶} = \sum_{𝑔 \in 𝐶} 𝑔$ . Schur’s lemma implies that $𝑝_{𝐶}$ acts on $𝑉$ as a scalar $𝜇$ , so $tr 𝑝_{𝐶} = 𝜇 dim 𝑉$ . Therefore, $𝜇 = 𝜒_{𝑉} (𝐶) \cdot \frac{| 𝐶 |}{dim 𝑉} = 𝜆$ .

$ℤ [𝐺]$ is a finitely generated $ℤ$ -module, and $ℤ$ is Noetherian; so any submodule of $ℤ [𝐺]$ is Noetherian. In particular, for any $𝑥 \in ℤ [𝐺]$ , $ℤ [𝑥]$ is finitely generated. Therefore, there is some $𝑛$ such that $ℤ [𝑥]$ is generated by $1, 𝑥, \dots, 𝑥^{𝑛}$ . Therefore $𝑥^{𝑛 + 1} = 𝑎_{𝑛} 𝑥^{𝑛} + \dots + 𝑎_{0}$ . Taking $𝑥 = 𝑝_{𝐶}$ and evaluate on $𝑉$ , we obtain $𝜆^{𝑛 + 1} = 𝑎_{𝑛} 𝜆^{𝑛} + \dots + 𝑎_{0}$ , so $𝜆$ is an algebraic integer.

$ℚ ∋ \frac{| 𝐺 |}{dim 𝑉} = \frac{| 𝐺 |}{dim 𝑉} ⟨ 𝜒_{𝑉}, 𝜒_{𝑉} ⟩ = \frac{1}{dim 𝑉} \sum_{𝐶} {| 𝜒_{𝑉} (𝐶) |}^{2} | 𝐶 | = \sum_{𝐶} 𝜆_{𝐶} 𝜒_{𝑉} (𝐶)$ , where $𝜆_{𝐶}$ defined as in the Lemma. The right hand side is an algebraic integer, so $\frac{| 𝐺 |}{dim 𝑉} \in ℚ \cap 𝔸 = ℤ$ .

COR 3.11

Actually,

dim 𝑉

divides

\frac{| 𝐺 |}{| 𝑍 (𝐺) |}

3.2.5 Burnside’s Theorem

THM 3.12

Let

𝐺

be a finite group. If a conjugacy class

𝐶

𝐺

has

𝑝^{𝑎}

elements (where

𝑝

is a prime and

𝑎 > 0

), then

𝐺

is not simple.

The irreducible representations of $𝐺$ can be classified into three types: the trivial representation; non-trivial irreducible representations with dimension divisible by $𝑝$ , denoted by $𝐷$ ; non-trivial irreducible representations with dimension not divisible by $𝑝$ , denoted by $𝑁$ . Then $\sum_{𝑉 irr.} 𝜒_{𝑉} (𝐶) dim 𝑉 = 0$ , so $𝑏 = \sum_{𝑉 \in 𝐷} \frac{1}{𝑝} \cdot 𝜒_{𝑉} (𝐶) dim 𝑉 \in 𝔸$ . So

0 = 1 + 𝑝 𝑏 + \sum_{𝑉 \in 𝑁} 𝜒_{𝑉} (𝐶) dim 𝑉 .

However, $𝔸 \cap ℚ = ℤ$ and $𝑏 \in 𝔸$ , so $𝜒_{𝑉} (𝐶)$ is a non-zero algebraic integer for some $𝑉 \subseteq 𝑁$ . $𝜒_{𝑉} (𝐶)$ is the sum of some roots of unity; if the average of some roots of unity is an algebraic integer, then either they sum to $0$ , or they are all equal. Therefore, $𝜌_{𝑉} (𝐶) = 𝜆 \cdot id$ , and $ker 𝜌_{𝑉}$ is a non-trivial normal subgroup of $𝐺$ .

COR 3.13 Burnside

| 𝐺 | = 𝑝^{𝑎} 𝑞^{𝑏}

for prime

𝑝

and

𝑞

, then

𝐺

is solvable.

3.3 Virtual Representations

A virtual representation of a group $𝐺$ is a formal linear combination of representations $\sum_{𝑖} 𝑛_{𝑖} 𝑉_{𝑖}$ where $𝑛_{𝑖} \in ℤ$ . The character of a virtual representation is the linear combination of the corresponding characters. If the inner product of a virtual character is $1$ and the total dimension is non-zero, then this virtual character is a single irreducible representation.

3.4 Restriction and Induction

If $𝐺$ is a finite group, $𝑉$ a representation of $𝐺$ and $𝐻$ is a subgroup of $𝐺$ , then the representation $𝜌_{𝑉} : 𝐺 \to GL (𝑉)$ restricts to a representation of $𝐻$ : $𝐻 ↪︎ 𝐺 \to GL (𝑉)$ . This representation is called the restriction, denoted ${Res}_{𝐻}^{𝐺} (𝑉)$ . Equivalently, ${Res}_{𝐻}^{𝐺} (𝑉) ≅ {Hom}_{𝑘 [𝐺]} (𝑘 [𝐺], 𝑉)$ where $𝑘 [𝐺]$ is seen as a $(𝑘 [𝐺], 𝑘 [𝐻])$ -bimodule; or $𝑘 [𝐺] \otimes_{𝑘 [𝐺]} 𝑉$ , where $𝑘 [𝐺]$ is seen as a $(𝑘 [𝐻], 𝑘 [𝐺])$ -bimodule.

The induced representation is the adjoint of the restricted representation. By the tensor-Hom adjoint pair, we have, for a left $𝐵$ -module $𝑁$ and a left $𝐴$ -module $𝑀$ ,

\begin{aligned} {Hom}_{𝐴} (𝐴 \otimes_{𝐵} 𝑁) & ≅ {Hom}_{𝐵} (𝑁, 𝑀), \\ {Hom}_{𝐴} (𝑁, 𝑀) & ≅ {Hom}_{𝐵} (𝑁, {Hom}_{𝐴} (𝐵, 𝑀)) . \end{aligned}

The module $𝐴 \otimes_{𝐵} 𝑁$ is called the induced $𝐴$ -module from $𝑁$ , and ${Hom}_{𝐴} (𝐵, 𝑀)$ is called the coinduced $𝐵$ -module from $𝑀$ .

Now suppose $𝑈$ is a representation of $𝐻$ . Then $𝑈$ is a left $𝑘 [𝐻]$ -module; we let the induced representation of $𝑈$ be the induced $𝑘 [𝐺]$ -module ${Ind}_{𝐻}^{𝐺} (𝑈) = 𝑘 [𝐺] \otimes_{𝑘 [𝐻]} 𝑈$ , and the coinduced representation of be the coinduced $𝑘 [𝐺]$ -module ${coInd}_{𝐻}^{𝐺} (𝑈) = {Hom}_{𝑘 [𝐻]} (𝑘 [𝐺], 𝑈)$ . For finite groups, the induced representation is isomorphic to the coinduced representation.

PROP 3.14

If $𝐾 \subseteq 𝐻 \subseteq 𝐾$ , then ${Ind}_{𝐾}^{𝐻} {Ind}_{𝐻}^{𝐺} = {Ind}_{𝐾}^{𝐺}$ .
If $𝑈$ is a representation of $𝐻 \subseteq 𝐺$ , then
$\begin{matrix} {Ind}_{𝐻}^{𝐺} (𝑈) ≅ {𝑓 : 𝐺 \to 𝑈 : 𝑓 (ℎ 𝑔) = 𝜌_{𝑈} (ℎ) 𝑓 (𝑔), \forall 𝑔 \in 𝐺, ℎ \in 𝐻} . \end{matrix}$
The action is given by $(𝑔 \cdot 𝑓) (𝑢) = 𝑓 (𝑥 𝑔)$ for $𝑥$ and $𝑔 \in 𝐺$ .
$dim {Ind}_{𝐻}^{𝐺} (𝑈) = dim 𝑈 \cdot \frac{| 𝐺 |}{| 𝐻 |}$ .

As a vector space, ${Ind}_{𝐻}^{𝐺} (𝑉)$ can be written as $⨁_{𝑖} 𝑔_{𝑖} \otimes 𝑈$ , where $𝑔_{𝑖}$ are coset representatives of $𝐻$ .

THM 3.15 Frobenius reciprocity

Suppose $𝐻$ is a subgroup of a finite group $𝐺$ , $𝑊$ and representation of $𝐻$ and $𝑉$ a representation of $𝐺$ . Then

\begin{matrix} {Hom}_{𝐺} ({Ind}_{𝐻}^{𝐺} (𝑈), 𝑉) ≅ {Hom}_{𝐻} (𝑈, {Res}_{𝐻}^{𝐺} (𝑉)); \\ {Hom}_{𝐺} (𝑉, {Ind}_{𝐻}^{𝐺} (𝑈)) ≅ {Hom}_{𝐻} ({Res}_{𝐻}^{𝐺} (𝑉), 𝑈) . \end{matrix}

In other words, $Res$ and $Ind$ are adjoint functors.

For complex representations, if $𝜒$ and $𝜓$ are two characters (or in general class functions) of $𝐻$ and $𝐺$ respectively,

{(𝜒, {Res}_{𝐻}^{𝐺} 𝜓)}_{𝐻} = {({Ind}_{𝐻}^{𝐺} 𝜒, 𝜓)}_{𝐺} .

We also have the formula

{Ind}_{𝐻}^{𝐺} (𝜑 \cdot {Res}_{𝐻}^{𝐺} (𝜓)) = {Ind}_{𝐻}^{𝐺} (𝜑) \cdot 𝜓 .

THM 3.16 Frobenius character formula

Suppose $𝐻$ is a subgroup of $𝐺$ , $char 𝑘 ∤ | 𝐻 |$ , $𝑈$ is a representation of $𝐻$ and $𝑉 = {Ind}_{𝐻}^{𝐺} (𝑈)$ . Then

𝜒_{𝑉} (𝑥) = \sum_{𝑔_{𝑖} 𝑥 𝑔_{𝑖}^{- 1} \in 𝐻} 𝜒_{𝑈} (𝑔_{𝑖} 𝑥 𝑔_{𝑖}^{- 1}) = \frac{1}{| 𝐻 |} \sum_{\begin{matrix} 𝑔 \in 𝐺 \\ 𝑔 𝑥 𝑔^{- 1} \in 𝐻 \end{matrix}} 𝜒_{𝑈} (𝑔 𝑥 𝑔^{- 1}),

where $𝑔_{𝑖}$ are coset representatives of $𝐻$ .

3.4.1 Mackey’s Irreducibility Criterion

Introduce the double cosets: if $𝐻$ and $𝐾$ are two subgroups of $𝐺$ , then $𝐺$ is the disjoint union of some $𝐾 𝑠 𝐻$ for some double coset representatives $𝑠$ . The set of double cosets (or a system of their representatives) may be denoted as $𝐾 \ 𝐺 / 𝐻$ .

Suppose $𝐻$ and $𝐾$ are subgroups of $𝐺$ and $(𝑊, 𝜌)$ is a representation of $𝐻$ . We wish to study ${Res}_{𝐾}^{𝐺} ({Ind}_{𝐺}^{𝐻} (𝑊))$ .

$𝑉 = {Ind}_{𝐺}^{𝐻} (𝑊)$ is the direct sum of images $𝑥 𝑊$ for $𝑥 \in 𝐺 / 𝐻$ . Let $𝑠 \in 𝐾 \ 𝐺 / 𝐻$ and let $𝑉 (𝑠)$ be the subspace of $𝑉$ generated by the images $𝑦 𝑊$ for $𝑦 \in 𝐾 𝑠 𝐻$ . Then ${Res}_{𝐾}^{𝐺} 𝑉 ≅ 𝑉$ as a vector space is also a direct sum of the $𝑉 (𝑠)$ ‘s and $𝑉 (𝑠)$ is stable under $𝐾$ , so they are also isomorphic as representations of $𝐾$ .

We can further decompose $𝑉 (𝑠)$ : the subgroup of $𝐾$ consisting of elements $𝑥$ such that $𝑥 (𝑠 𝑊) = 𝑠 𝑊$ is equal to $𝐻_{𝑠} = 𝐾 \cap 𝑠 𝐻 𝑠^{- 1}$ , so $𝑉 (𝑠)$ is a direct sum of the images $𝑥 (𝑠 𝑊)$ for $𝑥 \in 𝐾 / 𝐻_{𝑠}$ . This is to say $𝑉 (𝑠) = {Ind}_{𝐻_{𝑠}}^{𝐾} (𝑠 𝑊)$ . The representation $𝑠 𝑊$ is in turn isomorphic to $(𝑊_{𝑠}, 𝜌^{𝑠})$ where $𝜌^{𝑠} (𝑥) = 𝜌 (𝑠^{- 1} 𝑥 𝑠)$ . Therefore,

THM 3.17 Mackey’s formula

{Res}_{𝐾}^{𝐺} ({Ind}_{𝐻}^{𝐺} (𝑊)) ≅ ⨁_{𝑠 \in 𝑆} {Ind}_{𝐻_{𝑠}}^{𝐾} (𝑊_{𝑠})

as representations of

𝐾

Taking $𝐾 = 𝐻$ , we derive an irreducibility criterion for induced representations. Two representations $𝑊_{1}$ , $𝑊_{2}$ of $𝐺$ are called disjoint if ${Hom}_{𝐺} (𝑊_{1}, 𝑊_{2}) = 0$ .

COR 3.18 Mackey’s irreducibility criterion

The induced representation

𝑉 = {Ind}_{𝐻}^{𝐺} 𝑊

is irreducible, if and only if

𝑊

is irreducible, and the two representations

𝜌^{𝑠}

and

{Res}_{𝐻_{𝑠}}^{𝐻} (𝜌)

are disjoint. Here

𝐻_{𝑠} = 𝐻 \cap 𝑠 𝐻 𝑠^{- 1}

and

𝜌^{𝑠} (𝑥) = 𝜌 (𝑠^{- 1} 𝑥 𝑠)

COR 3.19

𝐻 ⊲ 𝐺

, then

{Ind}_{𝐻}^{𝐺} (𝜌)

is irreducible if and only if

𝜌

is irreducible and not isomorphic to any of its conjugates

𝜌^{𝑠}

for

𝑠 \notin 𝐻

EX 3.20 normal subgroups

Let

𝐴

be a normal subgroup of a group

𝐺

and let

(𝑉, 𝜌)

be an irreducible representation of

𝐺

. Let

𝑉 = ⨁ 𝑉_{𝑖}

be the isotypical decomposition of

{Res}_{𝐴}^{𝐺} (𝑉)

. For

𝑠 \in 𝐺

𝜌 (𝑠)

permutes the

𝑉_{𝑖}

; since

𝑉

is irreducible it permutes transitively. Let

𝑉_{𝑖_{0}}

be one of these

𝑉_{𝑖}

’s; if

𝑉_{𝑖_{0}} = 𝑉

then

{Res}_{𝐴}^{𝐺} (𝑉)

is isotypic, i.e. a direct sum of isomorphic irreducible representations. Otherwise let

𝐻

be the subgroup of

𝐺

consisting of

𝑠 \in 𝐺

such that

𝜌 (𝑠) 𝑉_{𝑖_{0}} = 𝑉_{𝑖_{0}}

. Then

𝐴 ⩽ 𝐻 < 𝐺

and

{Res}_{𝐴}^{𝐺} (𝑉)

is induced by the natural representation of

𝐻

𝑉_{𝑖_{0}}

. Therefore,

PROP 3.21

If $𝐴 ⊲ 𝐺$ , then

either there is a subgroup $𝐴 ⩽ 𝐻 < 𝐺$ and an irreducible representation $𝜎$ of $𝐻$ such that $𝜌$ is induced by $𝜎$ ; or
${Res}_{𝐴}^{𝐺} (𝜌)$ is isotypic.

EX 3.22 semidirect products by an abelian group

Suppose $𝐺 = 𝐴 ⋊ 𝐻$ where $𝐴$ is abelian. We wish to construct irreducible representations of $𝐺$ from certain subgroups of $𝐻$ . (This is the method of little groups of Wigner and Mackey).

Since $𝐴$ is abelian, its irreducible representations are of dimension $1$ and they form a group $\hat{𝐴} = Hom (𝐴, ℂ^{\times})$ . $𝐺$ acts on $\hat{𝐴}$ as $(𝑠 𝜒) (𝑎) = 𝜒 (𝑠^{- 1} 𝑎 𝑠)$ .

Let $𝜒_{𝑖}$ ( $𝑖 \in \hat{𝐴} / 𝐻$ ) be a system of representatives for the orbits of $𝐻$ in $\hat{𝐴}$ . For each $𝑖 \in \hat{𝐴} / 𝐻$ let $𝐻_{𝑖}$ be the stabilizer subgroup of $𝜒_{𝑖}$ , and let $𝐺_{𝑖} = 𝐴 ⋊ 𝐻_{𝑖}$ be the corresponding subgroup of $𝐺$ . Extend the functions $𝜒_{𝑖}$ to $𝐺_{𝑖}$ by $𝜒_{𝑖} (𝑎 ℎ) = 𝜒_{𝑖} (𝑎)$ for $𝑎 \in 𝐴$ and $ℎ \in 𝐻_{𝑖}$ . Since $ℎ$ fixes $𝜒_{𝑖}$ , $𝜒_{𝑖}$ is a $1$ -dimensional character of $𝐺$ .

Now let $𝜌$ be an irreducible representation of $𝐻_{𝑖}$ . By composing $𝜌$ with the projection $𝐺_{𝑖} \to 𝐻_{𝑖}$ we obtain as irreducible representation $\tilde{𝜌}$ of $𝐺_{𝑖}$ . Take the tensor product of $𝜒_{𝑖}$ and $\tilde{𝜌}$ ; we thus obtain an irreducible representation $𝜒_{𝑖} \otimes \tilde{𝜌}$ of $𝐺_{𝑖}$ . Let

𝜃_{𝑖, 𝜌} = {Ind}_{𝐺_{𝑖}}^{𝐺} (𝜒_{𝑖} \otimes \tilde{𝜌}) .

PROP 3.23

$𝜃_{𝑖, 𝜌}$ is irreducible.
$𝜃_{𝑖, 𝜌}$ and $𝜃_{𝑖^{'}, 𝜌^{'}}$ are isomorphic, if and only if $𝑖 = 𝑖^{'}$ and $𝜌 = 𝜌^{'}$ .
Every irreducible representation of $𝐺$ is isomorphic to some $𝜃_{𝑖, 𝜌}$ .

3.4.2 Artin’s Theorem

Recall the set of class functions $𝐹_{c} (𝐺, ℂ)$ and the inner product on it. Induction gives a map

\begin{aligned} {Ind}_{𝐻}^{𝐺} : 𝐹_{c} (𝐻, ℂ) & \to 𝐹_{c} (𝐺, ℂ), \\ 𝑓 & \mapsto (𝑥 \mapsto \frac{1}{| 𝐻 |} \sum_{\begin{matrix} 𝑔 \in 𝐺, \\ 𝑔 𝑥 𝑔^{- 1} \in 𝐻 \end{matrix}} 𝑓 (𝑔 𝑥 𝑔^{- 1})) . \end{aligned}

Let $𝜒_{1}, \dots, 𝜒_{𝑛}$ be the set of irreducible characters of $𝐺$ and let $𝑅 (𝐺)$ be the ring of virtual characters,

𝑅 (𝐺) = ℤ 𝜒_{1} \oplus \dots \oplus ℤ 𝜒_{𝑛} .

${Ind}_{𝐻}^{𝐺}$ and ${Res}_{𝐻}^{𝐺}$ induce ring homomorphisms $Ind : 𝑅 (𝐻) \to 𝑅 (𝐺)$ and $Res : 𝑅 (𝐺) \to 𝑅 (𝐻)$ , respectively. They are adjoints with respect to the bilinear forms ${(\cdot, \cdot)}_{𝐻}$ and ${(\cdot, \cdot)}_{𝐺}$ ,; moreover because $Ind (𝜑 \cdot Res (𝜓)) = Ind (𝜑) \cdot 𝜓$ , the image of $Ind$ is an ideal of $𝑅 (𝐺)$ .

THM 3.24 Artin’s theorem

Let $𝒳︀$ be a family of subgroups of $𝐺$ and $Ind : ⨁_{𝐻 \in 𝒳︀} 𝑅 (𝐻) \to 𝑅 (𝐺)$ . Then $𝐺$ is the union of all conjugates of the subgroups of $𝒳︀$ , if and only if the cokernel of $Ind$ is finite.

Since $𝑅 (𝐺)$ is finitely generated as an abelian group, the latter condition may be restated as, for every character $𝜒$ of $𝐺$ , there are virtual characters $𝜒_{𝐻} \in 𝑅 (𝐻)$ for $𝐻 \in 𝒳︀$ and $𝑑 \in ℤ_{+}$ such that $𝑑 \cdot 𝜒 = \sum_{𝐻 \in 𝒳︀} {Ind}_{𝐻}^{𝐺} (𝜒_{𝐻})$ .

Notice that the family of cyclic subgroups of $𝐺$ satisfies the first condition; therefore, as a corollary,

COR 3.25

Every character of

𝐺

is a linear combination with rational coefficients of characters induced by characters of cyclic subgroups of

𝐺

The first condition is equivalently stated as $Ind \otimes_{ℤ} ℂ : ⨁_{𝐻 \in 𝒳︀} 𝑅 (𝐻) \otimes_{ℤ} ℂ \to 𝑅 (𝐺) \otimes_{ℤ} ℂ$ is surjective, or by adjointness, $Res \otimes_{ℤ} ℂ$ is injective. If the first condition holds, then any class function on $𝐺$ which restricts to zero on each subgroup in $𝒳︀$ is itself zero because $𝐺$ is covered by conjugates of subgroups in $𝒳︀$ , so injectivity holds.

Conversely if the second condition holds let $𝑆$ be the union of conjugates of the subgroups in $𝒳︀$ . Every class function on $𝐺$ is of the form $\sum_{𝐻 \in 𝒳︀} {Ind}_{𝐻}^{𝐺} (𝑓_{𝐻})$ , so it vanishes outside of $𝑆$ . Therefore the complement of $𝑆$ must be empty, so $𝑆 = 𝐺$ is the union of all conjugates of subgroups in $𝒳︀$ .

If $𝐶$ is a cyclic group, let $𝜃_{𝐴}$ be a class function on $𝐴$ defined by

𝜃_{𝐴} (𝑥) = {\begin{cases} 𝑎 if 𝑥 genereates 𝐴 \\ 0 otherwise \end{cases}

Claim. If $𝐺$ is a finite group, then $| 𝐺 | = \sum_{𝐶 ⩽ 𝐺 cyclic} {Ind}_{𝐶}^{𝐺} (𝜃_{𝐶})$ .

The claim can be proved by Theorem 3.16. Also, this proves $𝜃_{𝐶} = | 𝐶 | - \sum_{𝐵 < 𝐶 cyclic} {Ind}_{𝐵}^{𝐴} (𝜃_{𝐵})$ , so by induction on $| 𝐶 |$ , $𝜃_{𝐶} \in 𝑅 (𝐶)$ . Therefore, the constant function $| 𝐺 |$ is in the image of $Ind$ . Since the image of $Ind$ is an ideal, $im Ind$ contains every element of the form $𝑔 𝜒$ , so $coker Ind$ is finite (actually its order is a factor of $| 𝐺 |$ ).

Section 4 Representations of Symmetric Groups

4.1 Young Tableux and Symmetric Functions

For materials on Young tableux and symmetric functions we refer to the book Young Tableux with Applications to Representation Theory and Geometry, especially section 6 Symmetric Functions, and also the corresponding notes. We fix some notation here.

The graded ring of symmetric functions on $𝑛$ variables is denoted by $Λ_{𝑛} = ⨁_{𝑟 ⩾ 0} Λ_{𝑛}^{𝑟}$ , and $Λ = {lim_{\leftarrow}}_{𝑛} Λ_{𝑛}$ .

The monomial symmetric function is $𝑚_{𝜆} (𝑥_{1}, \dots, 𝑥_{𝑛}) = \sum_{𝛼} 𝑥^{𝛼} \in Λ_{𝑛}^{𝑟}$ where $𝜆 ⊢ 𝑟$ and $𝛼$ varies over all permutations of $𝜆$ . $𝑚_{(1^{𝑟})}$ is called the elementary symmetric function $𝑒_{𝑟}$ , and $𝑒_{𝜆} = 𝑒_{𝜆_{1}} \cdot 𝑒_{𝜆_{2}} \cdot \dots$ . The complete symmetric function is $ℎ_{𝑟} (𝑥_{1}, \dots, 𝑥_{𝑛}) = \sum_{𝜆 ⊢ 𝑟} 𝑚_{𝜆}$ and $ℎ_{𝜆} = ℎ_{𝜆_{1}} \cdot ℎ_{𝜆_{2}} \cdot \dots$ . The power sum is $𝑝_{𝑟} = 𝑚_{(𝑟)}$ , and $𝑝_{𝜆} = 𝑝_{𝜆_{1}} \cdot 𝑝_{𝜆_{2}} \cdot \dots$ . Let $𝑠_{𝜆}$ be the Schur polynomial. These are all $ℤ$ -bases for $Λ$ .

An inner product on $Λ$ is defined by $⟨ 𝑠_{𝜆}, 𝑠_{𝜇} ⟩ = {\begin{cases} 1 if 𝜆 = 𝜇 \\ 0 otherwise \end{cases}$ . In this inner product, $ℎ_{𝜆}$ is dual to $𝑚_{𝜆}$ , and $𝑝_{𝜆}$ is dual to $\frac{𝑝_{𝜆}}{𝑧 (𝜆)}$ , where $𝑧 (𝜆) = \prod_{𝑟} 𝑟^{𝑚_{𝑟}} \cdot 𝑚_{𝑟!}$ , and $𝑚_{𝑟}$ is the number of times $𝑟$ appears in $𝜆$ . We obtain the following three identities:

\begin{matrix} \prod_{𝑖 = 1}^{𝑚} \prod_{𝑗 = 1}^{𝑙} \frac{1}{1 - 𝑥_{𝑖} 𝑦_{𝑗}} = \sum_{𝜆} ℎ_{𝜆} (𝑥) 𝑚_{𝜆 (𝑦)} = \sum_{𝜆} \frac{1}{𝑧 (𝜆)} 𝑝_{𝜆} (𝑥) 𝑝_{𝜆} (𝑦) = \sum_{𝜆} 𝑠_{𝜆} (𝑥) 𝑠_{𝜆} (𝑦) . \end{matrix}

$𝜔 : 𝑠_{𝜆} \mapsto 𝑠_{𝜆^{⊤}}$ is a ring automorphism, and also an involution and an isometry (i.e. preserving the inner product). Under this involution, $𝜔 (ℎ_{𝜆}) = 𝑒_{𝜆}$ , $𝜔 (𝑝_{𝜆}) = {(- 1)}^{\sum (𝑝_{𝑖} - 1)} 𝑝_{𝜆}$ .

The Jacobi-Trudy identity is

𝑠_{𝜆} = {det (ℎ_{𝜆_{𝑖} - 𝑖 + 𝑗})}_{1 ⩽ 𝑖, 𝑗 ⩽ 𝑛},

where $ℎ_{𝑟} = 0$ for $𝑟 < 0$ . The dual identity (by applying $𝜔$ ) is

𝑠_{𝜆} = {det (𝑒_{𝜆_{𝑖}^{⊤} - 𝑖 + 𝑗})}_{1 ⩽ 𝑖, 𝑗 ⩽ 𝑛} .

4.2 Specht Modules

Let $𝑇$ denote a Young diagram of shape $𝜆 ⊢ 𝑛$ numbered by integers $1$ to $𝑛$ with no repeats (not necessarily a Young tableau). The action of $𝔖_{𝑛}$ on $𝑇$ , $𝜎 \cdot 𝑇$ , puts number $𝜎 (𝑖)$ in the box in which $𝑇$ puts $𝑖$ . The column / row group of $𝑇$ is the subgroup of $𝔖_{𝑛}$ that permutes each column / row within themselves, denoted $𝐶_{𝑇}$ and $𝑅_{𝑇}$ .

Define

𝑏_{𝑇} = \sum_{𝑔 \in 𝐶_{𝑇}} {(- 1)}^{𝑔} 𝑔, 𝑎_{𝑇} = \sum_{𝑔 \in 𝑅_{𝑇}} 𝑔, 𝑐_{𝑇} = 𝑏_{𝑇} 𝑎_{𝑇} \in ℂ [𝔖_{𝑛}] .

Then for any $𝑟 \in 𝑅_{𝑇}$ and $𝑐 \in 𝐶_{𝑇}$ , $𝑏_{𝑇} 𝑟 = 𝑟 𝑏_{𝑇} = {(- 1)}^{𝑟} 𝑏_{𝑇}$ and $𝑐 𝑎_{𝑇} = 𝑎_{𝑇} 𝑐 = 𝑎_{𝑇}$ . In particular, $𝑎_{𝑇}^{2} = | 𝑅_{𝑇} | 𝑎_{𝑇}$ and $𝑏_{𝑇}^{2} = | 𝐶_{𝑇} | 𝑏_{𝑇}$ .

THM 4.1

𝑉_{𝜆} ≔ ℂ [𝔖_{𝑛}] \cdot 𝑐_{𝑈}

(where

𝑈

is any numbered Young diagram of shape

𝜆

) is an irreducible representation of

𝔖_{𝑛}

, and every irreducible representation of

𝔖_{𝑛}

is isomorphic to some

𝑉_{𝜆}

. These representations are called Specht modules.

As a corollary all (complex) representations of $𝔖_{𝑛}$ can be realized over $ℚ$ .

Define a tabloid ${𝑇}$ as the orbit of a numbered Young diagram $𝑇$ under the action of the row group, and let $𝑀_{𝜆}$ be the space with basis all tabloids. We can make the identification that $𝜎 \in 𝔖_{𝑛}$ corresponds to $𝜎 \cdot 𝑈$ ; then $𝑀_{𝜆}$ is a left $ℂ [𝔖_{𝑛}]$ -module, and $𝑉_{𝜆}$ is spanned by all $𝜎 \cdot 𝑏_{𝑈} \cdot {𝑈}$ where $𝜎 \in 𝔖_{𝑛}$ . Denote $𝑣_{𝑇} = 𝑏_{𝑇} \cdot {𝑇}$ , then $𝜎 \cdot 𝑣_{𝑇} = 𝑣_{𝜎 \cdot 𝑇}$ ; so $𝑉_{𝜆}$ is stable under $ℂ [𝔖_{𝑛}]$ , and in fact $𝑉_{𝜆}$ is essentially the subspace of $𝑀_{𝜆}$ spanned by $𝑏_{𝑇} \cdot {𝑇}$ for all numbered Young diagrams $𝑇$ of shape $𝜆$ .

LEM 4.2

Suppose $𝑇_{𝜆}$ and $𝑇_{𝜆^{'}}^{'}$ are two numbered Young diagrams and $𝜆$ does not strictly dominate $𝜆^{'}$ . Then either there are two distinct integers that occur in the same row of $𝑇^{'}$ and the same column of $𝑇$ , or $𝜆 = 𝜆^{'}$ and there is some $𝑝^{'} \in 𝑄_{𝜆^{'}}$ and $𝑞 \in 𝑃_{𝜆}$ such that $𝑝^{'} \cdot 𝑇^{'} = 𝑞 \cdot 𝑇$ . (The two cannot happen at the same time.)
If there is a pair of integers in the same row of $𝑇^{'}$ and in the same column of $𝑇$ , then $𝑏_{𝑇} \cdot {𝑇^{'}} = 0$ . Otherwise, $𝑏_{𝑇} \cdot {𝑇^{'}} = \pm 𝑣_{𝑇}$ .

As a corollary,

\begin{matrix} 𝑏_{𝑇} \cdot 𝑀_{𝜆} = 𝑏_{𝑇} \cdot 𝑉_{𝜆} = ℂ 𝑣_{𝑇} \neq 0, \\ 𝑏_{𝑇} \cdot 𝑀_{𝜆^{'}} = 𝑏_{𝑇} \cdot 𝑉_{𝜆^{'}} = 0 if 𝜆^{'} > 𝜆 . \end{matrix}

The $𝜆^{'} > 𝜆$ (lexicographic order) condition can be relaxed to $𝜆 ⋭ 𝜆^{'}$ (dominance order).

This proves irreducibility: any subrepresentation of $𝑉_{𝜆}$ must contain $𝑣_{𝑇}$ under the action of $𝑏_{𝑇}$ , and any subrepresentation containing $𝑣_{𝑇}$ generates $𝑉_{𝜆}$ under the action of $𝔖_{𝑛}$ . Moreover they are pairwise non-isomorphic by the second equation. Since the number of irreducible representations of $𝔖_{𝑛}$ is equal to the number of conjugacy classes, which is in turn equal to the number of partitions of $𝑛$ , we have thus produced all irreducible representations of $𝔖_{𝑛}$ .

Since the exact $𝑈$ defining $𝑉_{𝜆}$ does not matter, we choose any numbered diagram $𝑈$ and denote $𝑐_{𝑈}$ by $𝑐_{𝜆}$ .

Equation 38 has an important consequence that

PROP 4.3

For any

𝑥 \in ℂ [𝔖_{𝑛}]

, there is some scalar

𝜇 \in ℂ

such that

𝑏_{𝜆} 𝑥 𝑎_{𝜆} = 𝜇 𝑏_{𝜆} 𝑎_{𝜆}

. If

𝜆^{'} > 𝜆

(or, actually,

𝜆 ⋭ 𝜆^{'}

), then

𝑏_{𝜆} 𝑥 𝑎_{𝜆^{'}} = 0

The proof is just by Equation 38 and apply the identification $𝜎 ⟷ 𝜎 \cdot 𝑈$ .

PROP 4.4

𝑐_{𝜆}^{2} = \frac{𝑛!}{dim 𝑉_{𝜆}} 𝑐_{𝜆}

By the previous proposition

𝑐_{𝜆}^{2} = 𝑛_{𝜆} 𝑐_{𝜆}

. Let

𝜑

be the right multiplication by

\frac{𝑐_{𝜆}}{𝑛_{𝜆}}

. Then

tr 𝜑 = \frac{𝑛!}{𝑛_{𝜆}}

. However,

\frac{𝑐_{𝜆}}{𝑛_{𝜆}}

is an idempotent, so

𝜑

is a projection onto

im 𝜑

. So

tr 𝜑 = dim ℂ [𝔖_{𝑛}] \frac{𝑐_{𝜆}}{𝑛_{𝜆}} = dim 𝑉_{𝜆}

. Therefore,

𝑛_{𝜆} = \frac{𝑛!}{dim 𝑉_{𝜆}}

Recall that $𝑀_{𝜆}$ is the space with basis all tabloids of shape $𝜆$ . Again by Equation 38,

PROP 4.5

${Hom}_{ℂ [𝔖_{𝑛}]} (𝑀_{𝜆}, 𝑉_{𝜇}) = 0$ if $𝜇 ⋭ 𝜆$ . ${Hom}_{ℂ [𝔖_{𝑛}]} (𝑀_{𝜆}, 𝑉_{𝜆}) = 1$ .

In particular $𝑀_{𝜆} = 𝑉_{𝜆} \oplus ⨁_{𝜇 ⊳ 𝜆} 𝑘_{𝜇 𝜆} 𝑉_{𝜇}$ .

We need an algebraic lemma:

LEM 4.6

𝐴

is an algebra with unit and an idempotent

𝑒

𝑀

a left

𝐴

-module, then

{Hom}_{𝐴} (𝐴 𝑒, 𝑀) ≅ 𝑒 𝑀

Using this lemma, let $𝐴 = ℂ [𝔖_{𝑛}]$ , $𝑒 = 𝑎_{𝜆}$ (not quite an idempotent but up to a scalar) and $𝑀 = 𝑉_{𝜇}$ , we have ${Hom}_{𝔖_{𝑛}} (𝑀_{𝜆}, 𝑉_{𝜇}) = {Hom}_{𝐴} (𝐴 𝑎_{𝜆}, 𝐴 𝑐_{𝜇}) ≅ 𝑎_{𝜆} 𝐴 𝑏_{𝜇} 𝑎_{𝜇}$ . If $𝜇 ⋭ 𝜆$ , then $𝑎_{𝜆} 𝐴 𝑏_{𝜇} = 0$ (Proposition 4.3, inverse); therefore ${Hom}_{𝔖_{𝑛}} (𝑀_{𝜆}, 𝑉_{𝜇}) = 0$ .

We will prove that the numbers $𝑘_{𝜇 𝜆}$ are actually the Kostka numbers.

Denote $𝜒_{𝑀_{𝜆}}$ the character of the representation of $𝔖_{𝑛}$ on $𝑀_{𝜆}$ , and $𝐶_{𝜇}$ the conjugacy class of $𝔖_{𝑛}$ defined by the partition $𝜇$ . Then $𝜒_{𝑀_{𝜆}} (𝐶_{𝜇})$ is the number of tabloids fixed by $𝜎$ . By a binomial expansion, we can show that

PROP 4.7

The character $𝜒_{𝑀_{𝜆}}$ on the conjugacy class $𝐶_{𝜇}$ is equal to the coefficient before $𝑚_{𝜆}$ in $𝑝_{𝜇}$ .

That is,

𝑝_{𝜇} (𝑥) = \sum_{𝜆 ⊢ 𝑛} 𝜒_{𝑀_{𝜆}} (𝐶_{𝜇}) \cdot 𝑚_{𝜆} (𝑥),

or by dualizing,

ℎ_{𝜆} (𝑥) = \sum_{𝜇 ⊢ 𝑛} \frac{1}{𝑧_{𝜇}} 𝜒_{𝑀_{𝜆}} (𝐶_{𝜇}) 𝑝_{𝜇} (𝑥) .

REM 4.8

The convention in the class and Etingof’s book is different: they used

𝑐_{𝜆} = 𝑎_{𝜆} 𝑏_{𝜆}

instead of

𝑏_{𝜆} 𝑎_{𝜆}

here. The resulting representations are isomorphic.

4.3 Ring of Representations

Let $𝑅_{𝑛}$ be the Grothendieck group of $𝔖_{𝑛}$ , the free abelian group with basis given by isomorphism classes of finite dimensional representations of $𝔖_{𝑛}$ , such that $[𝑉] + [𝑊] = [𝑈]$ if $𝑈 ≅ 𝑉 \oplus 𝑊$ . Let $𝑅 = ⨁_{𝑛 ⩾ 0} 𝑅_{𝑛}$ .

$𝑉_{𝜆}$ forms a basis of $𝑅$ . Since the relation between $𝑀_{𝜆}$ and $𝑉_{𝜈}$ is upper-triangular (Proposition 4.5), $[𝑀_{𝜆}]$ is also a basis of $𝑅$ .

We can define a multiplication on $𝑅$ as follows. For $[𝑉] \in 𝑅_{𝑚}$ and $[𝑊] \in 𝑅_{𝑛}$ , consider $𝔖_{𝑚} \times 𝔖_{𝑛}$ as a subgroup of $𝔖_{𝑚 + 𝑛}$ , then $[𝑉] \circ [𝑊] ≔ [{Ind}_{𝔖_{𝑚} \times 𝔖_{𝑛}}^{𝔖_{𝑚 + 𝑛}} 𝑉 \otimes 𝑊]$ . This $𝑅$ is called the representation ring of $𝔖_{𝑛}$ .

This ring $𝑅$ is a commutative, associative, graded ring with unit, with a bilinear form $([𝑉], [𝑊]) = \frac{1}{𝑛!} \sum_{𝜎 \in 𝔖_{𝑛}} 𝜒_{𝑉} (𝜎) 𝜒_{𝑊} (𝜎)$ , and an involution $[𝑉] \mapsto [𝑉 \otimes sign]$ .

We show that there is a nice correspondence $𝜑$ between $𝑅$ and the ring of symmetric functions $Λ$ ; “nice” in terms that $𝜑$ is an isomorphism, an isometry (preserving the bilinear forms) and $𝜑$ preserves the involution.

This $𝜑$ is defined by $𝜑 : Λ \to 𝑅$ , $ℎ_{𝜆} \mapsto [𝑀_{𝜆}]$ .

PROP 4.9

This $𝜑$ is a ring isomorphism and an isometry.
$𝜑 (𝑠_{𝜆}) = [𝑉_{𝜆}]$ .
If $𝜔 : Λ \to Λ$ is the involution $ℎ_{𝜆} \mapsto 𝑒_{𝜆}$ and $𝜔 : 𝑅 \to 𝑅$ is the involution $[𝑉] \mapsto [𝑉 \otimes sign]$ , then $𝜑 \circ 𝜔 = 𝜔 \circ 𝜑$ .

$Λ = ℤ [ℎ_{1}, ℎ_{2}, \dots]$ . Therefore, to verify that $𝜑$ is a ring homomorphism we only need to prove, for $𝜆 = (𝜆_{1} ⩾ \dots ⩾ 𝜆_{𝑘})$ , that $[𝑀_{𝜆_{1}}] \circ \dots \circ [𝑀_{𝜆_{𝑘}}] = [𝑀_{𝜆}]$ . This follows from the fact that $[𝑀_{𝜆}] = {Ind}_{𝑅_{𝜆}}^{𝔖_{𝑛}} 1$ . Since $[𝑀_{𝜆}]$ forms a basis of $𝑅$ , $𝜑$ is a ring isomorphism.

Recall that $ℎ_{𝜆} = \sum_{𝜇} \frac{1}{𝑧_{𝜇}} 𝜒_{𝑀_{𝜆}} (𝐶 (𝜇)) 𝑝_{𝜇}$ . Construct an inverse $𝜓 : 𝑅 \to Λ \otimes ℚ$ by $[𝑉] \mapsto \sum_{𝜇 ⊢ 𝑛} \frac{1}{𝑧_{𝜇}} 𝜒_{𝑉} (𝐶 (𝜇)) 𝑝_{𝜇}$ . Then the composition $Λ \overset{𝜑}{\to} 𝑅 \overset{𝜓}{\to} Λ \otimes ℚ$ is $ℎ_{𝜆} \mapsto [𝑀_{𝜆}] \mapsto \sum_{𝜇} \frac{1}{𝑧_{𝜇}} 𝜒_{𝑀_{𝜆}} (𝐶 (𝜇)) 𝑝_{𝜇} = ℎ_{𝜆}$ is the identity map. Therefore $𝜓$ is the inverse of $𝜑$ and in particular the image of $𝜓$ lies in $Λ$ itself.

We next show $𝜑$ is an isometry. To do this it’s enough to show that $𝜓$ is an isometry, or $([𝑉], [𝑊]) = ⟨ 𝜓 ([𝑉]), 𝜓 ([𝑊]) ⟩$ . Recall that $⟨ 𝑝_{𝜆}, 𝑝_{𝜇} ⟩ = {\begin{cases} 𝑧_{𝜆} = 𝑧_{𝜇} & if 𝜆 = 𝜇 \\ 0 & if 𝜆 \neq 𝜇 \end{cases}$ . So

\begin{aligned} ⟨ 𝜓 ([𝑉]), 𝜓 ([𝑊]) ⟩ = & \sum_{𝜆 ⊢ 𝑛} \sum_{𝜇 ⊢ 𝑛} \frac{1}{𝑧_{𝜆} 𝑧_{𝜇}} 𝜒_{𝑉} (𝐶 (𝜆)) 𝜒_{𝑊} (𝐶 (𝜆)) ⟨ 𝑝_{𝜆}, 𝑝_{𝜇} ⟩ \\ \overset{𝜆 = 𝜇}{=} & \sum_{𝜇 ⊢ 𝑛} \frac{1}{𝑧_{𝜇}} 𝜒_{𝑉} (𝐶 (𝜇)) 𝜒_{𝑊} (𝐶 (𝜇)) \\ \overset{𝑧_{𝜆} = \frac{𝑛!}{| 𝐶 (𝜆) |}}{=} & \frac{1}{𝑛!} \sum_{𝑔 \in 𝔖_{𝑛}} 𝜒_{𝑉} (𝑔) 𝜒_{𝑊} (𝑔) \\ \overset{𝑉 and 𝑊 are real}{=} & ([𝑉], [𝑊]) . \end{aligned}

Recall that $ℎ_{𝜆} = 𝑠_{𝜆} + \sum_{𝜈 > 𝜆} 𝐾_{𝜈 𝜆} 𝑆_{𝜈}$ , where $𝐾_{𝜈 𝜆}$ are the Kostka numbers. Since this is an upper-triangular relation, we can solve $𝑠_{𝜆}$ in terms of $ℎ_{𝜆}$ : $𝑠_{𝜆} = ℎ_{𝜆} + \sum_{𝜈 > 𝜆} 𝑎_{𝜈 𝜆} ℎ_{𝜈}$ . Apply $𝜑$ : $𝜑 (𝑠_{𝜆}) = [𝑀_{𝜆}] + \sum_{𝜈 > 𝜆} 𝑎_{𝜈 𝜆} [𝑀_{𝜈}] = [𝑉_{𝜆}] + \sum_{𝜈 > 𝜆} 𝑏_{𝜈 𝜆} [𝑉_{𝜈}]$ . Since $𝜑$ is an isometry, it follows that $1 = ⟨ 𝑠_{𝜆}, 𝑠_{𝜆} ⟩ = (𝜑 (𝑠_{𝜆}), 𝜑 (𝑠_{𝜆})) = 1 + \sum 𝑏_{𝜈 𝜆}^{2}$ . Therefore, $𝑏_{𝜈} 𝜆 = 0$ and the right hand side must be $[𝑉_{𝜆}]$ alone.

To prove the third part, we prove that $𝜓 \circ 𝜔 = 𝜔 \circ 𝜓$ . For any representation $𝑉$ of $𝔖_{𝑛}$ ,

\begin{aligned} 𝜓 (𝑉 \otimes sign) = & \sum_{𝜇 ⊢ 𝑛} \frac{1}{𝑧_{𝜇}} 𝜒_{𝑉} (𝐶 (𝜇)) {(- 1)}^{\sum_{𝑘} (𝜇_{𝑘} - 1)} 𝑝_{𝜇} \\ = & \sum_{𝜇 ⊢ 𝑛} \frac{1}{𝑧_{𝜇}} 𝜒_{𝑉} (𝐶 (𝜇)) {(- 1)}^{| 𝜇 | - 𝑙 (𝜇)} 𝑝_{𝜇} \\ = & \sum_{𝜇 ⊢ 𝑛} \frac{1}{𝑧_{𝜇}} 𝜒_{𝑉} (𝐶 (𝜇)) 𝜔 (𝑝_{𝜇}) \\ = & 𝜔 (𝜓 (𝑉)) . \end{aligned}

In particular, $𝑀_{𝜆} = 𝑉_{𝜆} \oplus ⨁_{𝜇 ⊳ 𝜆} 𝐾_{𝜇 𝜆} 𝑉_{𝜇}$ .

With this result, we can transfer our knowledge about symmetric functions to representations of $𝔖_{𝑛}$ , and vice versa. For example, $𝑝_{𝜇} (𝑥) = \sum_{𝜆 ⊢ 𝑛} 𝜒_{𝑉_{𝜆}} (𝐶 (𝜇)) 𝑠_{𝜆} (𝑥)$ . For another example, recall the Littlewood-Richardson coefficients $𝑠_{𝜆} 𝑠_{𝜇} = \sum_{𝜈 ⊢ | 𝜆 | + | 𝜈 |} 𝑐_{𝜆 𝜇}^{𝜈} 𝑠_{𝜈}$ . By interpreting $𝑐_{𝜆 𝜇}^{𝜈}$ as a multiplicity of irreducible representation, we can proof they are non-negative integers.

If $𝑉$ is a representation of $𝔖_{𝑛}$ , define its Frobenius character by $𝐹_{𝑉} (𝑥) ≔ \sum_{𝜇 ⊢ 𝑛} \frac{1}{𝑧_{𝜇}} 𝜒_{𝑉} (𝐶 (𝜇)) 𝑝_{𝜇} (𝑥) \in Λ$ . Then

COR 4.10

𝐹_{𝑀_{𝜆}} (𝑥) = ℎ_{𝜆} (𝑥)

𝐹_{𝑉_{𝜆}} (𝑥) = 𝑠_{𝜆} (𝑥)

PROP 4.11 Frobenius character formula

If $𝜆 ⊢ 𝑛$ , $𝑁 ⩾ 𝑙 (𝜆)$ ( $𝑙 (𝜆)$ denoting the length of $𝜆$ ), then

𝜒_{𝑉_{𝜆}} (𝐶 (𝜇)) = [\prod_{𝑖 = 1}^{𝑁} 𝑥_{𝑖}^{𝜆_{𝑖} + 𝑁 - 𝑖}] (\prod_{1 ⩽ 𝑖 < 𝑗 ⩽ 𝑁} (𝑥_{𝑖} - 𝑥_{𝑗}) 𝑝_{𝜇} (𝑥_{1}, \dots, 𝑥_{𝑁})) .

$[𝑥^{𝜆}] (𝑝 (𝑥))$ denotes the coefficient in $𝑝 (𝑥)$ before $𝑥^{𝜆}$ .

We first assume that $𝑁 ⩾ 𝑛$ and let $𝑥_{𝑁 + 1} = 𝑥_{𝑁 + 2} = \dots = 0$ . Then

𝑝_{𝜇} (𝑥_{1}, \dots, 𝑥_{𝑁}) = \sum_{𝜆 ⊢ 𝑛} 𝜒_{𝑉_{𝜆}} (𝐶 (𝜇)) 𝑠_{𝜆} (𝑥_{1}, \dots, 𝑥_{𝑁}) .

For $𝛼 \in ℤ_{⩾ 0}^{𝑁}$ , define $𝑎_{𝛼} (𝑥) = {det (𝑥_{𝑖}^{𝛼_{𝑗}})}_{1 ⩽ 𝑖, 𝑗 ⩽ 𝑁}$ . Let $𝛿 = (𝑁 - 1, 𝑁 - 2, \dots, 0)$ . Then $𝑠_{𝜆} (𝑥_{1}, \dots, 𝑥_{𝑁}) = \frac{𝑎_{𝜆 + 𝛿} (𝑥)}{𝑎_{𝛿} (𝑥)}$ .

Since $𝑎_{𝛿} (𝑥) = \prod_{1 ⩽ 𝑖 < 𝑗 ⩽ 𝑁} (𝑥_{𝑖} - 𝑥_{𝑗})$ ,

\prod_{1 ⩽ 𝑖 < 𝑗 ⩽ 𝑁} (𝑥_{𝑖} - 𝑥_{𝑗}) \cdot 𝑝_{𝜇} (𝑥_{1}, \dots, 𝑥_{𝑁}) = \sum_{𝜆 ⊢ 𝑛} 𝜒_{𝑉_{𝜆}} (𝐶 (𝜇)) 𝑎_{𝜆 + 𝛿} (𝑥) .

To proof the proposition, we only need to observe that for $𝜆, 𝜈 ⊢ 𝑛$ ,

[𝑥^{𝜆 + 𝛿}] (𝑎_{𝜈 + 𝛿} (𝑥)) = {\begin{cases} 1 if 𝜈 = 𝜆 \\ 0 if 𝜈 \neq 𝜆 \end{cases},

which directly follows from the definition of $𝑎_{𝛼}$ , and the fact that the numbers in $𝜆 + 𝛿$ (and $𝜈 + 𝛿$ ) are strictly decreasing.

To prove the result for $𝑁 ⩾ 𝑙 (𝜆)$ , notice that the right hand side of the equation remains the same for all $𝑁 ⩾ 𝑙 (𝜆)$ , and we already proved the case $𝑁 ⩾ 𝑛$ .

For a box $(𝑖, 𝑗)$ in $𝜆$ , define the hook length $ℎ (𝑖, 𝑗)$ by the number of boxes in its row and to its right, or in its column and below it, including itself. Define $𝑓^{𝜆} = \prod_{(𝑖, 𝑗) \in 𝜆} ℎ (𝑖, 𝑗)$ .

THM 4.12 hook length formula

dim 𝑉_{𝜆} = \frac{𝑛!}{𝑓^{𝜆}} .

Let $𝜇 = (1^{𝑛})$ . By Proposition 4.11, denoting $𝑁 = 𝑙 (𝜆)$ ,

\begin{aligned} dim 𝑉_{𝜆} = & 𝜒_{𝑉_{𝜆}} 𝐶 (𝜇) \\ = & [\prod_{𝑖 = 1}^{𝑁} 𝑥_{𝑖}^{𝜆_{𝑖} + 𝑁 - 𝑖}] (\prod_{1 ⩽ 𝑖 < 𝑗 ⩽ 𝑁} (𝑥_{𝑖} - 𝑥_{𝑗}) {(𝑥_{1} + \dots + 𝑥_{𝑁})}^{𝑛}) \\ = & [\prod_{𝑖 = 1}^{𝑁} 𝑥_{𝑖}^{𝜆_{𝑖} + 𝑁 - 𝑖}] (\sum_{𝜎 \in 𝔖_{𝑁}} {(- 1)}^{𝜎} \prod_{𝑖 = 1}^{𝑁} 𝑥_{𝑖}^{𝑁 - 𝜎 (𝑖)} {(𝑥_{1} + \dots + 𝑥_{𝑁})}^{𝑛}) \\ = & \sum_{\begin{matrix} 𝜎 \in 𝔖_{𝑁} \\ 𝜎 (𝑖) ⩾ 𝑁 - 𝑙_{𝑖} \end{matrix}} {(- 1)}^{𝜎} \frac{𝑛!}{\prod_{𝑖 = 1}^{𝑁} (𝑙_{1} - 𝑁 + 𝜎 (𝑖))!} where 𝑙_{𝑖} = 𝜆_{𝑖} + 𝑁 - 𝑖 \\ = & \frac{𝑛!}{\prod_{𝑖 = 1}^{𝑁} 𝑙_{𝑖}!} \sum_{𝜎 \in 𝔖_{𝑁}} {(- 1)}^{𝜎} 𝑙_{𝑖} (𝑙_{𝑖} - 1) \dots (𝑙_{𝑖} - 𝑁 + 𝜎 (𝑖) + 1) \\ = & \frac{𝑛!}{\prod 𝑙_{𝑖}!} det (𝑙_{𝑖} \dots (𝑙_{𝑖} - 𝑁 + 𝑗 + 1)) \\ = & \frac{𝑛!}{\prod 𝑙_{𝑖}!} det (𝑙_{𝑖}^{𝑁 - 𝑗}) = \frac{𝑛!}{\prod 𝑙_{𝑖}!} \prod_{1 ⩽ 𝑖 < 𝑗 < 𝑁} (𝑙_{𝑖} - 𝑙_{𝑗}) . \end{aligned}

On the other hand, by a combinatorial argument,

\prod_{𝑗 = 1}^{𝜆_{1}} ℎ (1, 𝑗) = \prod_{\begin{matrix} 1 ⩽ 𝑘 ⩽ 𝜆_{𝑖} + 𝑁 - 𝑖 \\ 𝑘 \neq 𝜆_{𝑖} - 𝜆_{𝑗} + 𝑗 - 1, \forall 𝑗 \end{matrix}} 𝑘 .

Therefore the two sides are equal.

We also give a result for representations of $𝔄_{𝑛}$ .

THM 4.13

Suppose $𝜆 ⊢ 𝑛$ . Then

If $𝜆 \neq 𝜆^{⊤}$ , then ${Res}_{𝔄_{𝑛}}^{𝔖_{𝑛}} 𝑉_{𝜆}$ is irreducible.
If $𝜆 = 𝜆^{⊤}$ , then ${Res}_{𝔄_{𝑛}}^{𝔖_{𝑛}} 𝑉_{𝜆} = 𝑊_{𝜆} \oplus 𝑊_{𝜆}^{'}$ , where $𝑊_{𝜆} ≇ 𝑊_{𝜆}^{'}$ and both are irreducible representations of $𝔄_{𝑛}$ .
Any irreducible complex representation of $𝔄_{𝑛}$ appears inside some $𝑉_{𝜆}$ and this $𝜆$ is unique up to a transposition.

LEM 4.14

{Ind}_{𝔄_{𝑛}}^{𝔖_{𝑛}} 1 ≅ 1 \oplus sign

By Frobenius reciprocity, ${Hom}_{𝔄_{𝑛}} ({Res}_{𝔄_{𝑛}}^{𝔖_{𝑛}} 𝑉_{𝜆}, {Res}_{𝔄_{𝑛}}^{𝔖_{𝑛}} 𝑉_{𝜆}) = {Hom}_{𝔖_{𝜆}} (𝑉_{𝜆}, {Ind}_{𝔄_{𝑛}}^{𝔖_{𝑛}} {Res}_{𝔄_{𝑛}}^{𝔖_{𝑛}} 𝑉_{𝜆})$ .

{Ind}_{𝔄_{𝑛}}^{𝔖_{𝑛}} {Res}_{𝔄_{𝑛}}^{𝔖_{𝑛}} 𝑉_{𝜆} ≅ 𝑉_{𝜆} \otimes {Ind}_{𝔄_{𝑛}}^{𝔖_{𝑛}} 1 ≅ 𝑉_{𝜆} \oplus (𝑉_{𝜆} \otimes sign) ≅ 𝑉_{𝜆} \oplus 𝑉_{𝜆^{⊤}} .

Therefore, if $𝜆 \neq 𝜆^{⊤}$ , then ${Hom}_{𝔖_{𝑛}} (𝑉_{𝜆}, {Ind}_{𝔄_{𝑛}}^{𝔖_{𝑛}} {Res}_{𝔄_{𝑛}}^{𝔖_{𝑛}} 𝑉_{𝜆}) = ℂ$ , so ${Res}_{𝔄_{𝑛}}^{𝔖_{𝑛}} 𝑉_{𝜆}$ is irreducible. If $𝜆 = 𝜆^{⊤}$ , then ${Hom}_{𝔖_{𝑛}} (𝑉_{𝜆}, {Ind}_{𝔄_{𝑛}}^{𝔖_{𝑛}} {Res}_{𝔄_{𝑛}}^{𝔖_{𝑛}} 𝑉_{𝜆}) = ℂ^{2}$ , so ${Res}_{𝔄_{𝑛}}^{𝔖_{𝑛}} 𝑉_{𝜆}$ is the direct sum of two irreducible representations $𝑊_{𝜆} \oplus 𝑊_{𝜆}^{'}$ , and $𝑊_{𝜆}^{'} ≇ 𝑊_{𝜆}$ .

4.4 Representations of $GL (𝑉)$

4.4.1 Schur-Weyl Duality

THM 4.15 double centralizer theorem

Let $𝐸$ be a finite dimensional vector space over $𝑘$ , $𝐴, 𝐵 \subseteq End (𝐸)$ are two subalgebras, where $𝐴$ is semisimple and $𝐵 = {End}_{𝐴} (𝐸)$ . Then

$𝐴 = {End}_{𝐵} (𝐸)$ ;
$𝐵$ is semisimple; and
as a representation of $𝐴 \otimes 𝐵$ , $𝐸 ≅ ⨁_{𝑖} 𝑉_{𝑖} \otimes 𝑊_{𝑖}$ , where $𝑉_{𝑖}$ and $𝑊_{𝑖}$ are enumerations of irreducible representations of $𝐴$ and $𝐵$ respectively. In particular, there is a bijection between irreducible representations of $𝐴$ and irreducible representations of $𝐵$ .

Since

𝐴

is semisimple,

𝐴 = ⨁_{𝑖} End (𝑉_{𝑖})

, and

𝐸 = ⨁_{𝑖} 𝑉_{𝑖} \otimes 𝑊_{𝑖}

where

𝑊_{𝑖} = {Hom}_{𝐴} (𝑉_{𝑖}, 𝐸)

by the isotypical decomposition. Then

𝐵 = {End}_{𝐴} (𝐸) = {Hom}_{𝐴} (⨁_{𝑖} 𝑉_{𝑖} \otimes 𝑊_{𝑖}, ⨁_{𝑗} 𝑉_{𝑗} \otimes 𝑊_{𝑗}) = ⨁_{𝑖} End (𝑊_{𝑖})

, so

𝐵

is semisimple, so similarly

𝐴 = {End}_{𝐵} (𝐸)

Consider the case where $𝑉$ is a finite dimensional vector space over $ℂ$ and $𝐸 = 𝑉^{\otimes 𝑛}$ . Let $𝔖_{𝑛}$ act on $𝐸$ by permutation of tensor components, and $𝐴$ be the image of $ℂ [𝔖_{𝑛}] \to End (𝐸)$ under this action. Let $𝔤𝔩 (𝑉)$ be the Lie algebra $(End (𝑉), [\cdot, \cdot])$ .

THM 4.16 Schur-Weyl duality for

𝔤𝔩 (𝑉)

The algebra $𝐵 = {End}_{𝐴} (𝐸) = 𝑆^{𝑛} End (𝑉)$ is the image of the universal enveloping algebra $𝒰︀ (𝔤𝔩 (𝑉))$ under the natural action on $𝐸$ . That is, $𝐵$ is generated by

Δ_{𝑛} (𝑏) = 𝑏 \otimes 1 \otimes \dots \otimes 1 + 1 \otimes 𝑏 \otimes \dots \otimes 1 + \dots + 1 \otimes 1 \otimes \dots \otimes 𝑏

for $𝑏 \in 𝔤𝔩 (𝑉)$ as an algebra.

THM 4.17 Schur-Weyl duality for

GL (𝑉)

The image of

GL (𝑉)

End (𝐸)

spans

𝐵 = 𝑆^{𝑛} End (𝑉)

as a vector space.

COR 4.18 Schur-Weyl duality

As a representation of $𝔖_{𝑛} \times GL (𝑉)$ ,

𝑉^{\otimes 𝑛} ≅ ⨁_{𝜆 ⊢ 𝑛} 𝑉_{𝜆} \otimes 𝐿_{𝜆},

where $𝐿_{𝜆} = {Hom}_{𝔖_{𝑛}} (𝑉_{𝜆}, 𝑉^{\otimes 𝑛})$ is either an irreducible representation of $GL (𝑉)$ or zero, and the non-zero $𝐿_{𝜆}$ ’s are distinct.

For example, if $𝜆 = (𝑛)$ , then $𝑉_{𝜆} = 1$ , and $𝐿_{𝜆} = {(𝑉^{\otimes 𝑛})}^{𝔖_{𝑛}} = 𝑆^{𝑛} 𝑉$ ; if $𝜆 = (1^{𝑛})$ , then $𝐿_{𝜆} = Λ^{𝑛} 𝑉$ .

We now want to calculate the character of $𝐿_{𝜆}$ . Suppose $𝑁 = dim 𝑉$ , $𝑒_{1}, \dots, 𝑒_{𝑁}$ is a basis of $𝑉$ , $𝑔 \in GL (𝑉)$ and $𝜉_{1}, \dots, 𝜉_{𝑁}$ are eigenvalues of $𝑔$ . Suppose $𝑠 \in 𝐶 (𝜇) \subseteq 𝔖_{𝑛}$ , and $𝜇$ has $𝑚_{1}$ $1$ ′s, $𝑚_{2}$ $2$ ′s, $. .$ , $𝑚_{𝑘}$ $𝑘$ ’s. Then we can decompose $𝑠$ into cycles of length $𝑚_{1}, \dots, 𝑚_{𝑘}$ . For a cycle $𝑠_{0} = (12 \dots 𝑚)$ , $(𝑔^{\otimes 𝑚} 𝑠_{0}) (𝑒_{𝑖_{1}} \otimes \dots \otimes 𝑒_{𝑖_{𝑚}}) = 𝑔 𝑒_{𝑖_{𝑚}} \otimes 𝑔 𝑒_{𝑖_{1}} \otimes \dots \otimes 𝑔 𝑒_{𝑖_{𝑚 - 1}}$ ; therefore, the trace of $𝑔^{\otimes 𝑚} 𝑠_{0}$ on $𝑉^{\otimes 𝑚}$ is

\begin{aligned} {tr}_{𝑉^{\otimes 𝑚}} = & \sum_{1 ⩽ 𝑖_{1}, \dots, 𝑖_{𝑚} ⩽ 𝑁} 𝑔_{𝑖_{𝑚} 𝑖_{1}} 𝑔_{𝑖_{1} 𝑖_{2}} \dots 𝑔_{𝑖_{𝑚 - 1} 𝑖_{𝑚}} \\ = & {tr}_{𝑉} 𝑔^{𝑚} \\ = & 𝑝_{𝑚} (𝜉_{1}, \dots, 𝜉_{𝑁}) . \end{aligned}

Therefore, for $𝑠 \in 𝐶 (𝜇)$ , ${tr}_{𝑉^{\otimes 𝑛}} (𝑔^{\otimes 𝑛} 𝑠) = 𝑝_{𝜇} (𝜉_{1}, \dots, 𝜉_{𝑁})$ . Applying the decomposition is Corollary 4.18,

𝑝_{𝜇} (𝜉_{1}, \dots 𝜉_{𝑁}) = \sum_{𝜆 ⊢ 𝑛} 𝜒_{𝑉_{𝜆}} (𝐶 (𝜇)) 𝜒_{𝐿_{𝜆}} (𝑔) .

On the other hand, because the Frobenius character of $𝑉_{𝜆}$ is $\sum_{𝜇 ⊢ 𝑛} \frac{1}{𝑧_{𝜇}} 𝜒_{𝑉_{𝜆}} (𝐶 (𝜇)) 𝑝_{𝜇} = 𝑠_{𝜆}$ , we also have

𝑝_{𝜇} = \sum_{𝜆 ⊢ 𝑛} 𝜒_{𝑉_{𝜆}} (𝐶 (𝜇)) 𝑠_{𝜆} .

Therefore, $𝜒_{𝐿_{𝜆}} (𝑔) = 𝑠_{𝜆} (𝜉_{1}, \dots, 𝜉_{𝑁})$ . We thus obtained

THM 4.19 Weyl character formula

If $𝑙 (𝜆) > 𝑁$ , then $𝐿_{𝜆} = 0$ .

If $𝑙 (𝜆) < 𝑁$ , then $𝜒_{𝐿_{𝜆}} (𝑔) = 𝑠_{𝜆} (𝜉_{1}, \dots, 𝜉_{𝑁})$ .

Using the formula

𝑠_{𝜆} (𝑥_{1}, \dots, 𝑥_{𝑁}) = \frac{det {(𝑥_{𝑖}^{𝜆_{𝑗} + 𝑁 - 𝑗})}_{1 ⩽ 𝑖, 𝑗 ⩽ 𝑁}}{det {(𝑥_{𝑖}^{𝑁 - 𝑗})}_{1 ⩽ 𝑖, 𝑗 ⩽ 𝑁}},

taking $𝑔 = id$ , we obtain²

COR 4.20 Weyl dimension formula

dim 𝐿_{𝜆} = \prod_{1 ⩽ 𝑖 < 𝑗 ⩽ 𝑁} \frac{𝜆_{𝑖} - 𝜆_{𝑗} + 𝑗 - 𝑖}{𝑗 - 𝑖} .

COR 4.21

Let $𝜆 + (1^{𝑁})$ denote the partition of $𝑛 + 𝑁$ that adds to $1$ to every $𝜆_{𝑖}$ .

Then $𝐿_{𝜆 + (1^{𝑁})} ≅ 𝐿_{𝜆} \otimes Λ^{𝑁} 𝑉$ .

This corollary allows us to generalize $𝐿_{𝜆}$ to any $𝜆 \in ℤ^{𝑁}$ with $𝜆_{1} ⩾ \dots ⩾ 𝜆_{𝑁}$ , not necessarily non-negative. Take $𝑟 ⩾ 0$ such that $𝜆 + (𝑟^{𝑁})$ is indeed a partition (of $𝑛 + 𝑟 𝑁$ ). We then let

𝐿_{𝜆} ≔ 𝐿_{𝜆 + (𝑟^{𝑁})} \otimes {(Λ^{𝑁} 𝑉^{*})}^{𝑟} .

The representation $𝐿_{1^{𝑁}} ≅ Λ^{𝑁} 𝑉$ is called the determinant representation, denoted $det$ . Its dual representation is $Λ^{𝑁} 𝑉^{*} = {det}^{- 1}$ , so $Λ^{𝑁} 𝑉 \otimes Λ^{𝑁} 𝑉^{*} ≅ ℂ$ . Therefore, together with the corollary, this definition is independent on the choice of $𝑟$ .

4.4.2 Algebraic Representations of $GL (𝑉)$

Denote $𝐺 = GL (𝑉)$ . Define $𝑅$ to be the ring of polynomial functions on $𝐺$ , i.e. $𝑅 = ℂ [𝑔_{𝑖 𝑗}] [\frac{1}{det}]$ . Here $𝑔_{𝑖 𝑗}$ are viewed as formal variables, and $det$ is viewed as a polynomial in $𝑔_{𝑖 𝑗}$ . For example, if $𝑉 ≅ ℂ$ , then $𝐺 = ℂ^{\times}$ and $𝑅 = ℂ [𝑥, 𝑥^{- 1}]$ .

If we view $𝐺$ as the subvariety of $𝑈 = {Mat}_{𝑁 \times 𝑁} \times ℂ$ defined by the equation $𝑦 \cdot det 𝑔 = 1$ (where $(𝑔, 𝑦) \in {Mat}_{𝑁 \times 𝑁} \times ℂ$ ), then the function ring on $𝐺$ is exactly $ℂ [{Mat}_{𝑁 \times 𝑁}] [𝑦] / (𝑦 det - 1) = ℂ [𝑔_{𝑖 𝑗}] [\frac{1}{det}]$ .

A finite dimensional representation $𝑌$ of $GL (𝑉)$ is called algebraic (or rational or polynomial), if the matrix coefficients of the action on $𝑌$ belongs to $𝑅$ ; i.e., for all $𝑔 \in GL (𝑉)$ , $𝑣 \in 𝑌$ and $𝑢 \in 𝑌^{*}$ , $⟨ 𝑢, 𝑔 𝑣 ⟩ \in 𝑅$ .

By definition, any subrepresentation or quotient representation of an algebraic representation is also algebraic. $𝑉$ itself is algebraic, and so are $𝑉^{*}$ and $𝑉^{\otimes 𝑛}$ . Therefore, $𝐿_{𝜆}$ are algebraic.

Denote $Λ^{+} = {(𝜆_{1} ⩾ \dots ⩾ 𝜆_{𝑁}) \in ℤ^{𝑁}}$ .³

THM 4.22 Peter-Weyl

Let $𝐺 \times 𝐺$ acts on $𝑅$ as $((𝑔, ℎ) (𝜑)) (𝑥) = 𝜑 (𝑔^{- 1} 𝑥 ℎ)$ . Then as a $𝐺 \times 𝐺$ -representation, $𝑅 ≅ ⨁_{𝜆 \in Λ^{+}} 𝐿_{𝜆}^{*} \otimes 𝐿_{𝜆}$ .
Any algebraic representation of $𝐺$ is completely reducible.

$𝐺 \times 𝐺$ acts on ${Mat}_{𝑁 \times 𝑁}$ by $(𝑔, ℎ) 𝜑 = 𝑔 𝜑 ℎ^{- 1}$ . Then as a $𝐺 \times 𝐺$ -representation, ${Mat}_{𝑁 \times 𝑁} ≅ 𝑉 \otimes 𝑉^{*}$ , where the first $𝐺$ acts on $𝑉$ and the second $𝐺$ acts on $𝑉^{*}$ . $ℂ [{Mat}_{𝑁 \times 𝑁}] = Sym (𝑉^{*} \otimes 𝑉) = ⨁_{𝑛 ⩾ 0} 𝑆^{𝑛} (𝑉^{*} \otimes 𝑉)$ , and

{(𝑉^{*} \otimes 𝑉)}^{\otimes 𝑛} = (⨁_{\begin{matrix} 𝜆 ⊢ 𝑛 \\ 𝑙 (𝜆) ⩽ 𝑁 \end{matrix}} 𝐿_{𝜆}^{*} \otimes 𝑉_{𝜆}^{*}) \otimes (⨁_{\begin{matrix} 𝜇 ⊢ 𝑛 \\ 𝑙 (𝜇) ⩽ 𝑁 \end{matrix}} 𝐿_{𝜇} \otimes 𝑉_{𝜇}),

𝑆^{𝑛} (𝑉^{*} \otimes 𝑉) = {Hom}_{𝔖_{𝑛}} (ℂ, {(𝑉^{*} \otimes 𝑉)}^{\otimes 𝑛}) = ⨁_{\begin{matrix} 𝜆 ⊢ 𝑛 \\ 𝑙 (𝜆) ⩽ 𝑁 \end{matrix}} 𝐿_{𝜆}^{*} \otimes 𝐿_{𝜆}

by Schur’s lemma.

Let $𝜌$ be the representation of $𝐺 \times 𝐺$ on $ℂ^{\times}$ by $(𝑔, ℎ) \mapsto {(det 𝑔)}^{- 1} (det ℎ)$ , or as a $𝐺 \times 𝐺$ -representation, $𝜌 ≅ Λ^{𝑁} 𝑉^{*} \otimes Λ^{𝑁} 𝑉$ . Extend the $𝐺 \times 𝐺$ -action on $ℂ [{Mat}_{𝑁 \times 𝑁}]$ to $ℂ [{Mat}_{𝑁 \times 𝑁}] [𝑦]$ by requiring that $𝐺 \times 𝐺$ acts on $𝑦$ via $𝜌^{- 1}$ . Then

(𝑔, ℎ) \cdot (𝑦 det - 1) = ((𝑔, ℎ) \cdot 𝑦) ({(det 𝑔)}^{- 1} (det ℎ) \cdot det) - 1 = 𝑦 det - 1,

\begin{aligned} ℂ [{Mat}_{𝑁 \times 𝑁}] [𝑦] = & ⨁_{𝑑 ⩾ 0} ⨁_{\begin{matrix} 𝜆 ⊢ 𝑛 \\ 𝑙 (𝜆) ⩽ 𝑁 \end{matrix}} (𝐿_{𝜆}^{*} \otimes 𝐿_{𝜆}) 𝑦^{𝑑} \\ = & ⨁_{𝑑 ⩾ 0} ⨁_{\begin{matrix} 𝜆 ⊢ 𝑛 \\ 𝑙 (𝜆) ⩽ 𝑁 \end{matrix}} (𝐿_{𝜆}^{*} \otimes 𝐿_{𝜆}) \otimes 𝜌^{- 𝑑} \\ = & ⨁_{𝑑 ⩾ 0} ⨁_{\begin{matrix} 𝜆 ⊢ 𝑛 \\ 𝑙 (𝜆) ⩽ 𝑁 \end{matrix}} (𝐿_{𝜆}^{*} \otimes 𝐿_{𝜆}) \otimes {(Λ^{𝑁} 𝑉 \otimes Λ^{𝑁} 𝑉^{*})}^{\otimes 𝑑} \\ = & ⨁_{𝑑 ⩾ 0} ⨁_{\begin{matrix} 𝜆 ⊢ 𝑛 \\ 𝑙 (𝜆) ⩽ 𝑁 \end{matrix}} 𝐿_{𝜆 - (𝑑^{𝑁})}^{*} \otimes 𝐿_{𝜆 - (𝑑^{𝑁})} \end{aligned}

So far we have copies of the same $𝐿_{𝜆}^{*}$ appearing from various $𝑑$ . Recall that $𝑅$ is the quotient of $ℂ [{Mat}_{𝑁 \times 𝑁}] [𝑦]$ by $(𝑦 det - 1)$ . In this quotient, $(𝐿_{𝜆}^{*} \otimes 𝐿_{𝜆}) 𝑦^{𝑑}$ is identified with $(𝐿_{𝜆 + (1^{𝑁})}^{*} \otimes 𝐿_{𝜆 + (1^{𝑁})}) 𝑦^{𝑑 + 1}$ . Therefore, as $𝐺 \times 𝐺$ -representations,

𝑅 ≅ ⨁_{𝜆 \in Λ^{+}} 𝐿_{𝜆}^{*} \otimes 𝐿_{𝜆} .

COR 4.23

{𝐿_{𝜇} : 𝜇 \in Λ^{+}}

is a complete list of pairwise non-isomorphic irreducible algebraic representations of

GL (𝑉)

4.5 The Okounkov-Vershik Approach

We treat $𝔖_{𝑘}$ as the subgroup of $𝔖_{𝑘 + 1}$ fixing $𝑘 + 1$ .

Suppose $𝐴$ and $𝐵$ are semisimple algebras, $𝜏 : 𝐵 \to 𝐴$ an algebra homomorphism. Since $𝐴$ and $𝐵$ are semisimple, we can write them as $𝐴 = ⨁_{𝑉 \in Irr (𝐴)} End (𝑉)$ , $𝐵 = ⨁_{𝑈 \in Irr (𝐵)} End (𝑈)$ . For $𝑈 \in Irr (𝐵)$ and $𝑉 \in Irr (𝐴)$ , let $𝑀_{𝑉, 𝑈} = {Hom}_{𝐵} (𝑈, 𝑉)$ , where $𝑉$ is seen as an representation of $𝐵$ by composition $𝐵 \overset{𝜏}{\to} 𝐴 \overset{𝜌_{𝑉}}{\to} End (𝑉)$ . Then $𝑉 ≅ ⨁_{𝑈 \in Irr (𝐵)} 𝑈 \otimes 𝑀_{𝑉, 𝑈}$ as representations of $𝐵$ ( $𝑀_{𝑉, 𝑈}$ records the multiplicity of $𝑈$ and does not have a $𝐵$ -action).

Define the centralizer $𝑍_{𝐵} (𝐴) = {𝑎 \in 𝐴 : 𝑎 𝜏 (𝑏) = 𝜏 (𝑏) 𝑎, \forall 𝑏 \in 𝐵}$ . The centralizer is a subalgebra of $𝐴$ , and acts on $𝑀_{𝑉, 𝑈}$ by $(𝑎 \cdot 𝜑) (𝑢) = 𝑎 (𝜑 (𝑢))$ .

Since $𝐴$ can be written as $⨁_{𝑉 \in Irr (𝐴)} End (𝑉)$ , we can write $𝑍_{𝐵} (𝐴)$ as

\begin{aligned} 𝑍_{𝐵} (𝐴) = & 𝑍_{𝐵} (⨁_{𝑉 \in Irr (𝐴)} End (𝑉)) \\ = & ⨁_{𝑉 \in Irr (𝐴)} {Hom}_{𝐵} (𝑉, 𝑉) \\ = & ⨁_{𝑉 \in Irr (𝐴)} {Hom}_{𝐵} (⨁_{𝑈 \in Irr (𝐵)} 𝑈 \otimes 𝑀_{𝑉, 𝑈}, ⨁_{𝑈^{'} \in Irr (𝐵)} 𝑈 \otimes 𝑀_{𝑉, 𝑈^{'}}) \\ \overset{Schur lemma}{=} & ⨁_{\begin{matrix} 𝑉 \in Irr (𝐴) \\ 𝑈 \in Irr (𝐵) \\ 𝑀_{𝑉, 𝑈} \neq 0 \end{matrix}} End (𝑀_{𝑉, 𝑈}) . \end{aligned}

As a corollary, $𝑍_{𝐵} (𝐴)$ is commutative if and only if for all $𝑈 \in Irr (𝐵)$ and $𝑉 \in Irr (𝐴)$ , $dim {Hom}_{𝐵} (𝑈, 𝑉) ⩽ 1$ .

Suppose $𝐻 \subseteq 𝐺$ are finite groups. If $𝜏 : 𝐵 \to 𝐴$ is an inclusion of group algebras $ℂ [𝐻] ↪︎ ℂ [𝐺]$ , then $𝑍_{ℂ [𝐻]} (ℂ [𝐺])$ consists of elements $\sum_{𝑔 \in 𝐺} 𝑎_{𝑔} 𝑔$ where $𝑎_{𝑔} = 𝑎_{ℎ 𝑔 ℎ^{- 1}}$ for all $ℎ \in 𝐻$ . Hence, $𝑍_{ℂ [𝐻]} (ℂ [𝐺])$ has a basis $𝑏_{𝐶} = \sum_{𝑔 \in 𝐶} 𝑔$ where $𝐶$ is an $𝐻$ -conjugacy class in $𝐺$ .

We now take $𝜏 : 𝐵 \to 𝐴$ to be the inclusion $ℂ [𝔖_{𝑚}] ↪︎ ℂ [𝔖_{𝑛}]$ for $0 ⩽ 𝑚 ⩽ 𝑛$ and denote $𝑍_{𝑚} (𝑛) = 𝑍_{ℂ [𝔖_{𝑚}]} (ℂ [𝔖_{𝑛}])$ . The $𝔖_{𝑚}$ -conjugacy classes in $𝔖_{𝑛}$ are cycles with marked elements $𝑚 + 1, 𝑚 + 2, \dots, 𝑛$ . For example, when $𝑚 = 3$ , $𝑛 = 6$ , some marked cycles are

(* * 4) (5) (6), (* * 5) (4 6), (* * 5) (* 4) (6), \dots

For another example, if $𝑚 = 𝑛 - 1$ , then $(* 𝑛)$ is a conjugacy class, and the corresponding basis element $𝑏_{(* 𝑛)} = \sum_{𝑗 = 1}^{𝑛 - 1} (𝑗 𝑛)$ is called the $𝑛$ -th Jucys-Murphy element.

Let $𝔖_{[𝑚 + 1, 𝑛]} = {𝑔 \in 𝔖_{𝑛} : 𝑔 (𝑖) = 𝑖, \forall 1 ⩽ 𝑖 ⩽ 𝑚}$ . $𝑍_{𝑚} (𝑛)$ contains the following elements:

center of $ℂ [𝔖_{𝑚}]$ , which is $𝑍_{𝑚} (𝑚)$ ;
$ℂ [𝔖_{[𝑚 + 1, 𝑛]}]$ ;
$𝐽_{𝑚 + 1}, 𝐽_{𝑚 + 2}, \dots, 𝐽_{𝑛}$ ; they mutually commute.

THM 4.24

The algebra

𝑍_{𝑚} (𝑛)

is generated by the above elements.

As a corollary, $𝑍_{𝑛 - 1} (𝑛)$ is commutative, so any irreducible representation of $𝔖_{𝑛 - 1}$ appears at most once in the restriction of any irreducible representation of $𝔖_{𝑛}$ to $𝔖_{𝑛 - 1}$ , and according to Schur lemma, $𝐽_{𝑛}$ acts by scalar multiplication on each irreducible $𝔖_{𝑛 - 1}$ -subrepresentation of any irreducible $𝔖_{𝑛}$ -representation. (This is a consequence of the branching law: the restriction of $𝑉_{𝜆}$ from $𝔖_{𝑛}$ to $𝔖_{𝑛 - 1}$ decomposes into $𝑉_{𝜆 ∖ □}$ where $□$ ranges through all removable box of $𝜆$ .)

Based on this, we define the branching graph as an infinite directed tree with vertices isomorphism classes of irreducible representations of symmetric groups, and edges $𝑈 \to 𝑉$ where $𝑉 \in Irr (𝔖_{𝑛})$ , $𝑈 \in Irr (𝔖_{𝑛 - 1})$ for some $𝑛$ , such that $𝑈$ appears in ${Res}_{𝔖_{𝑛 - 1}}^{𝔖_{𝑛}} (𝑉)$ . This graph remains invariant if we tensor every representation with the sign representation, replacing $𝑈$ by $𝑈 \otimes {sign}_{𝑛}$ ; this makes it “symmetric”.

Suppose $𝑉^{𝑛} \in Irr (𝔖_{𝑛})$ for $𝑛 \in ℤ_{+}$ , and denote $Path (𝑉^{𝑚}, 𝑉^{𝑛})$ the set of all paths from $𝑉^{𝑚}$ to $𝑉^{𝑛}$ in the branching graph. Let $Path (𝑉^{𝑛}) = Path ({triv}_{1}, 𝑉^{𝑛})$ and ${Path}_{𝑛} = ⨆_{𝑉^{𝑛} \in Irr (𝔖_{𝑛})} Path (𝑉^{𝑛})$ . For $𝑃 = (𝑉^{𝑚} \to \dots \to 𝑉^{𝑛}) \in Path (𝑉^{𝑚}, 𝑉^{𝑛})$ , write $𝑉^{𝑚} (𝑃)$ for the copy of $𝑉^{𝑚}$ inside $𝑉^{𝑛}$ by the composition of inclusions corresponding to $𝑃$ . Then, as $𝔖_{𝑚}$ -representations,

𝑉^{𝑛} = ⨁_{𝑉^{𝑚} \in Irr (𝔖_{𝑚})} ⨁_{𝑃 \in Path (𝑉^{𝑚}, 𝑉^{𝑛})} 𝑉^{𝑚} (𝑃) .

Let $𝜑_{𝑃} : 𝑉^{𝑚} ↪︎ 𝑉^{𝑛}$ be the embedding corresponding to path $𝑃$ ; they form a basis for ${Hom}_{𝔖_{𝑚}} (𝑉^{𝑚}, 𝑉^{𝑛})$ .

$𝑍_{𝑚} (𝑛)$ acts on ${Hom}_{𝔖_{𝑚}} (𝑉^{𝑚}, 𝑉^{𝑛})$ by $(𝑎 \cdot 𝜑) (𝑢) = 𝑎 \cdot 𝜑 (𝑢)$ . Since $𝐽_{𝑘} \in 𝑍_{𝑚} (𝑛)$ , $𝐽_{𝑘}$ also acts on ${Hom}_{𝔖_{𝑚}} (𝑉^{𝑚}, 𝑉^{𝑛})$ . For every edge $𝑉^{𝑘 - 1} \to 𝑉^{𝑘}$ in the path, assign a number $𝑤_{𝑘}$ to it, defined as the scalar by which $𝐽_{𝑘}$ acts on $𝑉^{𝑘 - 1}$ : ${𝐽_{𝑘} |}_{𝑉^{𝑘 - 1}} = 𝑤_{𝑘} {id}_{𝑉^{𝑘 - 1}}$ . Then $𝐽_{𝑘}$ acts on $𝜑_{𝑃}$ as $𝐽_{𝑘} \cdot 𝜑_{𝑃} = 𝑤_{𝑘} 𝜑_{𝑃}$ . The weight vector $(𝑤_{𝑚 + 1}, \dots, 𝑤_{𝑛})$ is denoted by $𝑤_{𝑃}$ .

LEM 4.25

{𝜑_{𝑃} : 𝑃 \in Path (𝑉^{𝑚}, 𝑉^{𝑛})}

is a basis for

{Hom}_{𝔖_{𝑚}} (𝑉^{𝑚}, 𝑉^{𝑛})

Since $𝑉^{1} = {triv}_{1}$ , ${Hom}_{𝔖_{𝑛}} (𝑉^{1}, 𝑉^{𝑛}) = 𝑉^{𝑛}$ . Let $𝑣_{𝑃} = 𝜑_{𝑃}$ for $𝑃 \in Path (𝑉^{𝑛})$ . As a corollary, ${𝑣_{𝑃} : 𝑃 \in Path (𝑉^{𝑛})}$ is a basis for $𝑉^{𝑛}$ , and for any $1 ⩽ 𝑘 ⩽ 𝑛$ , $𝐽_{𝑘} 𝑣_{𝑃} = 𝑤_{𝑘} 𝑣_{𝑃}$ .

As another corollary, if $𝑃 \in Path (𝑉^{𝑚})$ and $𝑄 \in Path (𝑉^{𝑚}, 𝑉^{𝑛})$ , then $𝑣_{𝑃 𝑄}$ is proportional to $𝜑_{𝑄} (𝑣_{𝑃})$ .

THM 4.26

Suppose

𝑃, 𝑃^{'} \in {Path}_{𝑛}

. Then

𝑤_{𝑃} = 𝑤_{𝑃^{'}}

if and only if

𝑃 = 𝑃^{'}

Denote ${wt}_{𝑛} = {𝑤_{𝑃} : 𝑃 \in {Path}_{𝑛}}$ . We say $𝑤_{𝑃}$ and $𝑤_{𝑃^{'}}$ are r-equivalent if they are the weights of two paths leading to the same irreducible representation. This is an equivalence relation, therefore the study of $Irr (𝔖_{𝑛})$ reduces to the study of equivalence classes of r-equivalence.

Let $Path (𝑃, 𝑖)$ be the set of paths that differs from $𝑃$ only at level $𝑖$ . Denote $𝑉_{𝑃, 𝑖} = {Span}_{ℂ} {𝑣_{𝑃} : 𝑃 \in Path (𝑃, 𝑖)} \subseteq 𝑉^{𝑛}$ .

PROP 4.27

𝑉_{𝑃, 𝑖} \subseteq 𝑉^{𝑛}

is a

𝑍_{𝑖 - 1} (𝑖 + 1)

-submodule, and it’s an irreducible

𝑍_{𝑖 - 1} (𝑖 + 1)

-module.

Recall that $𝑍_{𝑖 - 1} (𝑖 + 1)$ is generated by $𝑍_{𝑖 - 1} (𝑖 - 1)$ , $𝑋_{1} = 𝐽_{𝑖}$ , $𝑋_{2} = 𝐽_{𝑖 + 1}$ and $𝑇 = (𝑖 𝑖 + 1)$ . Observe that $𝑋_{1}$ , $𝑋_{2}$ and $𝑇$ has the following relations:

𝑋_{1} 𝑋_{2} = 𝑋_{2} 𝑋_{1}, 𝑇^{2} = 1, 𝑇 𝑋_{1} = 𝑋_{2} 𝑇 - 1 .

The algebra generated by the above three relations is called a degenerate affine Hecke algebra $ℋ︀ (2)$ . Then there is an algebra homomorphism $ℋ︀ (2) \to 𝑍_{𝑖 - 1} (𝑖 + 1)$ .

PROP 4.28

𝑀

is an irreducible

𝑍_{𝑖 - 1} (𝑖 + 1)

-module, then

𝑀

is also irreducible as an

ℋ︀ (2)

-module.

THM 4.29

The finite-dimensional irreducible representations of $ℋ︀ (2)$ are classified by $(𝑎, 𝑏) \in ℂ^{2} \mapsto 𝐿 (𝑎, 𝑏)$ , where $(𝑎, 𝑏)$ are simultaneous eigenvalues of $𝑋_{1}$ and $𝑋_{2}$ in $𝐿 (𝑎, 𝑏)$ :

If $𝑏 = 𝑎 + 1$ , then $𝐿 (𝑎, 𝑏) = ℂ$ , $𝑇 \mapsto 1$ , $𝑋_{1} \mapsto 𝑎$ and $𝑋_{2} \mapsto 𝑏$ .
If $𝑏 = 𝑎 - 1$ , then $𝐿 (𝑎, 𝑏) = ℂ$ , $𝑇 \mapsto - 1$ , $𝑋_{1} \mapsto 𝑎$ and $𝑋_{2} \mapsto 𝑏$ .
If $𝑏 \neq 𝑎 \pm 1$ , then $𝐿 (𝑎, 𝑏) = ℂ^{2}$ , $𝑇 \mapsto (\begin{matrix} 1 \\ 1 \end{matrix})$ , $𝑋_{1} \mapsto (\begin{matrix} 𝑎 & - 1 \\ 𝑏 \end{matrix})$ and $𝑋_{2} \mapsto (\begin{matrix} 𝑏 & 1 \\ 𝑎 \end{matrix})$ .
$𝑋_{1}$ and $𝑋_{2}$ are diagonalizable in $𝐿 (𝑎, 𝑏)$ if and only if $𝑎 \neq 𝑏$ .

$𝑉_{𝑃, 𝑖}$ is isomorphic to ${Hom}_{𝔖_{𝑖 - 1}} (𝑉^{𝑖 - 1}, 𝑉^{𝑖 + 1})$ as a $𝑍_{𝑖 - 1} (𝑖 + 1)$ -module, and this is an irreducible representation for $𝑍_{𝑖 - 1} (𝑖 + 1)$ . Since there is an algebra homomorphism from $ℋ︀ (2)$ to $𝑍_{𝑖 - 1} (𝑖 + 1)$ , $𝑉_{𝑃, 𝑖}$ is also an irreducible $ℋ︀ (2)$ -module. Therefore, $𝑉_{𝑃, 𝑖}$ is isomorphic to one of the $𝐿$ defined above. By the classification, we obtain

THM 4.30

Path (𝑃, 𝑖)

theorem

Let $𝑤_{𝑃} = (𝑤_{1}, \dots, 𝑤_{𝑛}) \in {wt}_{𝑛}$ . Then

$𝑤_{𝑖} \neq 𝑤_{𝑖 + 1}$ ;
If $𝑤_{𝑖 + 1} = 𝑤_{𝑖} \pm 1$ , then $Path (𝑃, 𝑖) = {𝑃}$ ;
If $𝑤_{𝑖 + 1} \neq 𝑤_{𝑖} \pm 1$ , then $Path (𝑃, 𝑖) = {𝑃, 𝑃^{'}}$ and
$𝑤_{𝑃^{'}} = (𝑤_{1}, \dots, 𝑤_{𝑖 - 1}, 𝑤_{𝑖 + 1}, 𝑤_{𝑖}, 𝑤_{𝑖 + 2}, \dots, 𝑤_{𝑛});$
If $𝑖 < 𝑛 - 1$ , $𝑤_{𝑖} = 𝑤_{𝑖 + 1} \pm 1$ implies that $𝑤_{𝑖 + 2} \neq 𝑤_{𝑖}$ .

According to this theorem, we define an admissible transposition on $ℂ^{𝑛}$ as a transposition $(𝑤_{1}, \dots, 𝑤_{𝑛}) \mapsto (\dots, 𝑤_{𝑖 - 1}, 𝑤_{𝑖 + 1}, 𝑤_{𝑖}, 𝑤_{𝑖 + 2}, \dots)$ where $𝑤_{𝑖 + 1} \neq 𝑤_{𝑖} \pm 1$ . We say two elements of $ℂ^{𝑛}$ are c-equivalent, denoted by $\underset{c}{\sim}$ , if one is obtained from the other by a sequence of admissible transpositions. A combinatorial weight is an element in $ℂ^{𝑛}$ such that every element $𝑤$ c-equivlent to it satisfies $𝑤_{1} = 0$ , $𝑤_{𝑖} \neq 𝑤_{𝑖 + 1}$ for all $1 ⩽ 𝑖 ⩽ 𝑛 - 1$ , and $𝑤_{𝑖 + 1} = 𝑤_{𝑖} \pm 1$ implies $𝑤_{𝑖 + 2} \neq 𝑤_{𝑖}$ for all $1 ⩽ 𝑖 ⩽ 𝑛 - 2$ . Let ${cwt}_{𝑛}$ be the set of combinatorial weights.

COR 4.31

${wt}_{𝑛} \subseteq {cwt}_{𝑛}$ , and ${wt}_{𝑛}$ is a union of c-equivalence classes;
c-equivalence implies r-equivalence in ${wt}_{𝑛}$ .

Therefore, the number of r-equivalence classes is no larger than the number of c-equivalence classes in ${wt}_{𝑛}$ , and is in turn no larger the that in ${cwt}_{𝑛}$ . On the other hand, the number of r-equivalence classes is equal to the number of irreducible representations of $𝔖_{𝑛}$ , which is $𝑝 (𝑛)$ , the partition number of $𝑛$ . We now prove the final combinatorial lemma, that

LEM 4.32

Every c-equivalence class in ${cwt}_{𝑛}$ contains an element of the form

\begin{aligned} ( \\ 0, 1, 2, \dots, 𝑛_{1} - 1, \\ - 1, 0, 1 \dots, 𝑛_{2} - 2, \\ - 2, - 1, 0, \dots, 𝑛_{3} - 3, \\ \dots, \\ 1 - 𝑘, 2 - 𝑘, 3 - 𝑘, \dots, 𝑛_{𝑘} - 𝑘 \\ ) \end{aligned}

for some positive integers $𝑛_{1} ⩾ 𝑛_{2} ⩾ \dots ⩾ 𝑛_{𝑘} ⩾ 1$ such that $\sum_{𝑖} 𝑛_{𝑖} = 𝑛$ .

In particular, the number of c-equivalence classes in ${cwt}_{𝑛}$ is no larger than $𝑝 (𝑛)$ , completing the cycle of inequalities.

Consider the lexicographic order on an equivalence class in

{cwt}_{𝑛}

. Let

(𝑤_{1}, \dots, 𝑤_{𝑛})

be the maximal element in its equivalence classes. This element has the desired form.

This implies ${cwt}_{𝑛} = {wt}_{𝑛}$ , c-equivalence is r-equivalence, and the $(𝑛_{1}, \dots, 𝑛_{𝑘})$ in the lemma is unique; this establishes a bijection between irreducible representations of $𝔖_{𝑛}$ with a partition $(𝑛_{1}, \dots, 𝑛_{𝑘})$ of $𝑛$ .

To see the relation with Young diagrams, suppose $𝜆$ be a partition of $𝑛$ and $𝑇$ is a standard Young tableaux of of shape $𝜆$ . Its content is defined by $𝑐 (𝑇) = (𝑥_{1} - 𝑦_{1}, \dots, 𝑥_{𝑛} - 𝑦_{𝑛}) \in ℤ^{𝑛}$ , where $(𝑥_{𝑖}, 𝑦_{𝑖})$ is the coordinate of the box labelled by $𝑖$ . The content $𝑇 \mapsto 𝑐 (𝑇)$ gives a bijection between standard Young tableaux with $𝑛$ boxes and ${cwt}_{𝑛}$ , and the shape of the tableaux is equal to the partition defined in Lemma 4.32. As a corollary, $𝑉_{𝜆}$ has a basis given by $𝑣_{𝑇}$ , where $𝑇$ iterates over all standard Young tableaux of shape $𝜆$ , and the Jucys-Murphy elements act on this basis as $𝐽_{𝑖} 𝑣_{𝑇} = (𝑥_{𝑖} - 𝑦_{𝑖}) 𝑣_{𝑇}$ , where $(𝑥_{𝑖}, 𝑦_{𝑖})$ is the coordinate of the box labelled $𝑖$ in $𝑇$ . As another corollary, we obtain the branching rule ${Res}_{𝔖_{𝑛 - 1}}^{𝔖_{𝑛}} (𝑉_{𝜆}) = ⨁_{𝜇 = 𝜆 ∖ □} 𝑉_{𝜇}$ , and $𝐽_{𝑛}$ acts on $𝑉_{𝜇}$ by scalar multiplication by the content of the removed box.

Section 5 Representations of ${GL}_{2} (𝔽_{𝑞})$

Let $𝐺 = {GL}_{2} (𝔽_{𝑞})$ , with $| 𝐺 | = (𝑞^{2} - 1) (𝑞^{2} - 𝑞)$ . Suppose $𝐴 \in 𝐺$ with eigenvalues $𝜆_{1}$ , $𝜆_{2}$ . Let $𝐶_{𝐴}$ be the conjugacy class of $𝐴$ .

Case a: parabolic, $𝜆_{1} = 𝜆_{2} = 𝑥 \in 𝔽_{𝑞}^{\times}$ . If $𝐴$ is diagonalizable, then $| 𝐶_{𝐴} | = 1$ . If not, then $𝐴 \sim (\begin{matrix} 𝑥 & 1 \\ 𝑥 \end{matrix})$ , and $| 𝐶_{𝐴} | = \frac{| 𝐺 |}{(𝑞 - 1) 𝑞} = 𝑞^{2} - 1$ .

Case b: hyperbolic, $𝜆_{1} \neq 𝜆_{2} \in 𝔽_{𝑞}^{\times}$ . Then $| 𝐶_{𝐴} | = \frac{| 𝐺 |}{{(𝑞 - 1)}^{2}} = 𝑞^{2} + 𝑞$ .

Case c: elliptic, $𝜆_{1} \neq 𝜆_{2} \notin 𝔽_{𝑞}$ . Consider a quadratic extension of $𝔽_{𝑞}$ , $𝔽_{𝑞^{2}} = 𝔽_{𝑞} [\sqrt{𝜀}]$ . Then $𝜆_{1} = 𝛼_{1} + 𝛼_{2} \sqrt{𝜀}$ , $𝜆_{2} = 𝛼_{1} - 𝛼_{2} \sqrt{𝜀} = 𝜆_{1}$ , where $𝛼_{𝑖} \in 𝔽_{𝑞}^{*}$ , and $𝐴 \sim (\begin{matrix} 𝑥 & 𝜀 𝑦 \\ 𝑦 & 𝑥 \end{matrix})$ . $| 𝐶_{𝐴} | = 𝑞^{2} - 𝑞$ .

Represen- tatives $𝐴$	$\| 𝐶_{𝐴} \|$	number of classes
$(\begin{matrix} 𝑥 \\ 𝑥 \end{matrix})$	$1$	$𝑞 - 1$
$(\begin{matrix} 𝑥 & 1 \\ 𝑥 \end{matrix})$	$𝑞^{2} - 1$	$𝑞 - 1$
$(\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}), 𝑥 \neq 𝑦$	$𝑞^{2} + 𝑞$	$\frac{1}{2} (𝑞 - 1) (𝑞 - 2)$
$(\begin{matrix} 𝑥 & 𝜀 𝑦 \\ 𝑦 & 𝑥 \end{matrix})$	$𝑞^{2} - 𝑞$	$\frac{1}{2} 𝑞 (𝑞 - 1)$

We start with $1$ -dimensional representations. Any $1$ -dimensional representation factors through $𝐺 / [𝐺, 𝐺]$ .

LEM 5.1

[𝐺, 𝐺] = {SL}_{2} (𝔽_{𝑞})

; the quotient map

𝐺 \to 𝐺 / [𝐺, 𝐺]

is given by

𝑔 \mapsto det 𝑔

Therefore, $𝐺 / [𝐺, 𝐺] ≅ 𝔽_{𝑞}^{*} ≅ 𝐶_{𝑞 - 1}$ . Therefore, any $1$ -dimensional representation of $𝐺$ is of the form $𝜌 (𝑔) = 𝜉 (det 𝑔)$ , where $𝜉$ is a $1$ -dimensional representation of $𝔽_{𝑞}^{*}$ . This representation will be denoted $ℂ_{𝜆}$ . This gives us $𝑞 - 1$ $1$ -dimensional representations.

We can also construct the principal series representations. Let $𝐵$ be the subgroup of upper diagonal matrices (the Borel subgroup), $𝑈$ the subgroup of upper diagonal matrices with diagonals equal to $1$ , and $𝑇$ the subgroup of diagonal matrices. Then $𝐵 / [𝐵, 𝐵] = 𝐵 / 𝑈 = 𝑇$ , and if $𝜆_{𝑖}$ are representations of $𝔽_{𝑞}^{\times}$ , then $(\begin{matrix} 𝑎 \\ 𝑑 \end{matrix}) \mapsto 𝜆_{1} (𝑎) 𝜆_{2} (𝑏)$ gives a representation of $𝑇$ . The composition $𝜌_{𝜆_{1} 𝜆_{2}} : 𝐵 \to 𝑇 \to ℂ^{\times}$ gives a representation of $𝐵$ , denoted $ℂ_{𝜆_{1} 𝜆_{2}}$ , and its induces a representation of $𝐺$ , denoted $𝑉_{𝜆_{1} 𝜆_{2}} = {Ind}_{𝐵}^{𝐺} ℂ_{𝜆_{1} 𝜆_{2}}$ .

THM 5.2

If $𝜆_{1} \neq 𝜆_{2}$ , then $𝑉_{𝜆_{1} 𝜆_{2}}$ is irreducible. Moreover, $𝑉_{𝜆_{1} 𝜆_{2}}$ are distinct for different unordered pairs ${𝜆_{1}, 𝜆_{2}}$ ( $𝜆_{1} \neq 𝜆_{2}$ ).
If $𝜆_{1} = 𝜆_{2} = 𝜇$ , then $𝑉_{𝜇 𝜇} ≅ ℂ_{𝜇} \oplus 𝑊_{𝜇}$ , where $𝑊_{𝜇}$ is an irreducible representation of $𝐺$ . Moreover, $𝑊_{𝜇}$ are distinct for different $𝜇$ ’s.

By $𝐺 = 𝐵 \cup 𝐵 𝑠 𝐵$ where $𝑠 = (\begin{matrix} 1 \\ 1 \end{matrix})$ , and $𝑠 𝐵 𝑠^{- 1} \cap 𝐵 = 𝑇$ . By Theorem 3.17, ${Res}_{𝐵}^{𝐺} (𝑉_{𝜆_{1} 𝜆_{2}}) = 𝜌_{𝜆_{1} 𝜆_{2}} \oplus {Ind}_{𝑇}^{𝐵} (𝜌_{𝜆_{1} 𝜆_{2}}^{𝑠})$ , where $𝜌_{𝜆_{1} 𝜆_{2}}^{𝑠} : (\begin{matrix} 𝑎 \\ 𝑑 \end{matrix}) \mapsto 𝜌_{𝜆_{1} 𝜆_{2}} (𝑠^{- 1} (\begin{matrix} 𝑎 \\ 𝑑 \end{matrix}) 𝑠) = 𝜆_{2} (𝑎) 𝜆_{1} (𝑑)$ . When $𝜆_{1} \neq 𝜆_{2}$ , $𝜌_{𝜆_{1}, 𝜆_{2}}^{𝑠}$ and ${Res}_{𝑇}^{𝐵} (𝜌_{𝜆_{1} 𝜆_{2}})$ are then disjoint, so $𝑉_{𝜆_{1} 𝜆_{2}}$ are irreducible. Moreover, if $𝜆_{1} \neq 𝜆_{2}$ , $𝜆_{1}^{'} \neq 𝜆_{2}^{'}$ ,

\begin{aligned} {Hom}_{𝐺} (𝑉_{𝜆_{1} 𝜆_{2}}, 𝑉_{𝜆_{1}^{'}, 𝜆_{2}^{'}}) ≅ & {Hom}_{𝐵} (𝜌_{𝜆_{1} 𝜆_{2}} \oplus {Ind}_{𝑇}^{𝐵} (𝜌_{𝜆_{1} 𝜆_{2}}^{𝑠}), 𝜌_{𝜆_{1}^{'} 𝜆_{2}^{'}}) \\ = & {Hom}_{𝐵} (𝜌_{𝜆_{1} 𝜆_{2}}, 𝜌_{𝜆_{1}^{'} 𝜆_{2}^{'}}) \oplus {Hom}_{𝑇} (𝜌_{𝜆_{1} 𝜆_{2}}^{𝑠}, Res 𝜌_{𝜆_{1}^{'} 𝜆_{2}^{'}}) \\ = & {Hom}_{𝐵} (𝜌_{𝜆_{1} 𝜆_{2}}, 𝜌_{𝜆_{1}^{'} 𝜆_{2}^{'}}) \oplus {Hom}_{𝑇} (Res 𝜌_{𝜆_{2} 𝜆_{1}}, Res 𝜌_{𝜆_{1}^{'} 𝜆_{2}^{'}}) \\ = & ℂ iff {𝜆_{1}, 𝜆_{2}} = {𝜆_{1}^{'}, 𝜆_{2}^{'}} . \end{aligned}

When $𝜆_{1} = 𝜆_{2} = 𝜇$ , ${Res}_{𝐵}^{𝐺} (ℂ_{𝜇}) = 𝜌_{𝜇 𝜇}$ , so ${Hom}_{𝐺} (𝐶_{𝜇}, 𝑉_{𝜇 𝜇}) ≅ {Hom}_{𝐵} ({Res}_{𝐵}^{𝐺} (ℂ_{𝜇}), 𝜌_{𝜇 𝜇}) ≅ ℂ$ , so $ℂ_{𝜇} \subseteq 𝑉_{𝜇 𝜇}$ . Therefore, $𝑉_{𝜇} 𝜇 = 𝐶_{𝜇} \oplus 𝑊_{𝜇}$ . Since ${Hom}_{𝐺} (𝑉_{𝜇 𝜇}, 𝑉_{𝜇 𝜇}) = ℂ^{2}$ by the same calculation as Equation 70, we deduce ${Hom}_{𝐺} (𝑊_{𝜇}, 𝑊_{𝜇}) = ℂ$ , so $𝑊_{𝜇}$ is irreducible of dimension $𝑞$ . Moreover, ${Hom}_{𝐺} (𝑉_{𝜇 𝜇}, 𝑉_{𝜈 𝜈}) = {Hom}_{𝐵} (𝜌_{𝜇 𝜇}, 𝜌_{𝜈 𝜈}) \oplus {Hom}_{𝑇} (Res 𝜌_{𝜇 𝜇}, Res 𝜌_{𝜈 𝜈}) = ℂ^{2}$ iff $𝜇 = 𝜈$ , so $𝑊_{𝜇} ≅ 𝑊_{𝜈}$ iff $𝜇 = 𝜈$ .

Therefore $𝑉_{𝜆_{1} 𝜆_{2}}$ gives us $\frac{1}{2} (𝑞 - 1) (𝑞 - 2)$ many $(𝑞 + 1)$ -dimensional irreducible representations, and $𝑊_{𝜇}$ gives us $𝑞 - 1$ many $𝑞$ -dimensional irreducible representations. By Proposition 4.11,

Conjugacy class	$𝜒_{𝑉_{𝜆_{1} 𝜆_{2}}} (𝑔)$
$(\begin{matrix} 𝑥 \\ 𝑥 \end{matrix})$	$(𝑞 + 1) 𝜆_{1} (𝑥) 𝜆_{2} (𝑥)$
$(\begin{matrix} 𝑥 & 1 \\ 𝑥 \end{matrix})$	$𝜆_{1} (𝑥) 𝜆_{2} (𝑥)$
$(\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}), 𝑥 \neq 𝑦$	$𝜆_{1} (𝑥) 𝜆_{2} (𝑦) + 𝜆_{2} (𝑥) 𝜆_{1} (𝑦)$
$(\begin{matrix} 𝑥 & 𝜀 𝑦 \\ 𝑦 & 𝑥 \end{matrix})$	$0$

Another family of representations are the complementary series representations. We can identify $𝐺$ with the group of invertible $𝔽_{𝑞}$ -linear automorphisms of $𝔽_{𝑞^{2}}$ . This way, $𝐺$ contains the cyclic subgroup $𝔽_{𝑞^{2}}^{\times} = {𝑥 + 𝑦 \sqrt{𝜀} : 𝑥, 𝑦 \in 𝔽_{𝑞}, (𝑥, 𝑦) \neq (0, 0)}$ , or in ${GL}_{2} (𝔽_{𝑞})$ , ${(\begin{matrix} 𝑥 & 𝜀 𝑦 \\ 𝑦 & 𝑥 \end{matrix}) : (𝑥, 𝑦) \neq (0, 0)}$ . Denote the subgroup by $𝐾$ . Then every character $𝜈$ of $𝐾$ can be induced to a representation of $𝐺$ , denoted $𝑌_{𝜈}$ . By Proposition 4.11,

Conjugacy class	$𝜒_{𝑌_{𝜈}} (𝑔)$
$(\begin{matrix} 𝑥 \\ 𝑥 \end{matrix})$	$𝑞 (𝑞 - 1) 𝜈 (𝑥)$
$(\begin{matrix} 𝑥 & 1 \\ 𝑥 \end{matrix})$	$0$
$(\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}), 𝑥 \neq 𝑦$	$0$
$(\begin{matrix} 𝑥 & 𝜀 𝑦 \\ 𝑦 & 𝑥 \end{matrix})$	$𝜈 (𝑥 + 𝑦 \sqrt{𝜀}) + 𝜈 (𝑥 - 𝑦 \sqrt{𝜀})$

Notice that $𝜈 (𝑥 - 𝑦 \sqrt{𝜀}) = {𝜈 (𝑥 + 𝑦 \sqrt{𝜀})}^{𝑞}$ via the Frobenius automorphism $𝑥 \mapsto 𝑥^{𝑞}$ . Also via the Frobenius automorphism, $𝑌_{𝜈} ≅ 𝑌_{𝜈^{𝑞}}$ .

Consider the virtual representation $𝑋_{𝜈} = 𝑊_{1} \otimes 𝑉_{𝛼 1} - 𝑉_{𝛼 1} - 𝑌_{𝜈}$ , where $𝛼 = {𝜈 |}_{𝔽_{𝑞}^{\times}}$ .

Conjugacy class	$𝜒_{𝑋_{𝜈}} (𝑔)$
$(\begin{matrix} 𝑥 \\ 𝑥 \end{matrix})$	$(𝑞 - 1) 𝛼 (𝑥)$
$(\begin{matrix} 𝑥 & 1 \\ 𝑥 \end{matrix})$	$- 𝛼 (𝑥)$
$(\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}), 𝑥 \neq 𝑦$	$0$
$(\begin{matrix} 𝑥 & 𝜀 𝑦 \\ 𝑦 & 𝑥 \end{matrix})$	$- 𝜈 (𝑥 + 𝑦 \sqrt{𝜀}) - {𝜈 (𝑥 + 𝑦 \sqrt{𝜀})}^{𝑞}$

LEM 5.3

Assume

𝜈^{𝑞} \neq 𝜈

. Then

(𝜒_{𝑋_{𝜈}}, 𝜒_{𝑋_{𝜈}}) = 1

and

𝜒_{𝑋_{𝜈}} (1) = 𝑞 - 1 > 0

, so

𝑋_{𝜈}

is an irreducible representation.

𝑋_{𝜈} ≅ 𝑋_{𝜈^{'}}

iff

𝜈^{'} \neq 𝜈, 𝜈^{𝑞}

$𝑋_{𝜈}$ thus gives us $\frac{1}{2} 𝑞 (𝑞 - 1)$ many $(𝑞 - 1)$ -dimensional irreducible representations.

$𝑉_{𝜆_{1} 𝜆_{2}}$ ( $𝜆_{1} \neq 𝜆_{2}$ ), $𝑊_{𝜇}$ , $ℂ_{𝜇}$ and $𝑋_{𝜈}$ ( $𝜈^{𝑞} \neq 𝜈$ ) thus gives all irreducible representations of ${GL}_{2} (𝔽_{𝑞})$ .

Section 6 Representations of Quivers

Suppose $Γ = (𝐼, 𝐸)$ is a connected graph with no self loops. Denote its adjacency matrix by $𝑅_{Γ}$ . $Γ$ is called a Dynkin diagram if the quadratic form on $ℝ^{𝑁}$ defined by $𝐴_{Γ} = 2 𝐼 - 𝑅_{Γ}$ is positive definite.

THM 6.1

All Dynkin diagrams are classified by three families:

𝐴_{𝑁}

(shown here

𝐴_{5}

)

𝐷_{𝑁}

(shown here

𝐷_{6}

)

𝐸_{6}

𝐸_{7}

𝐸_{8}

Suppose $Γ$ is a Dynkin diagram, define the bilinear form $𝐵 (𝑥, 𝑦) = 𝑥^{⊤} 𝐴_{Γ} 𝑦$ on $ℤ^{𝑛}$ . Then $𝐵$ is positive definite, and $𝐵 (𝑥, 𝑥) = 2 \sum 𝑥_{𝑖}^{2} - \sum_{𝑖 \neq 𝑗} 𝑟_{𝑖 𝑗} 𝑥_{𝑖} 𝑥_{𝑗}$ , where $𝑟_{𝑖 𝑗}$ is $1$ or $0$ determined by whether $𝑖$ and $𝑗$ are connected or not. In particular, $𝐵 (𝑥, 𝑥)$ is always even. A root with respect to a certain positive definite inner product $𝐵$ is a shortest non-zero vector in $ℤ^{𝑛}$ , and in this case, $𝑣 \in ℤ^{𝑛}$ such that $𝐵 (𝑣, 𝑣) = 2$ . Obviously there are only finitely many roots. Define the simple roots $𝛼_{𝑖} = (0, \dots, 0, 1, 0, \dots, 0)$ with $1$ on the $𝑖$ -th component.

LEM 6.2

𝛼

is a root,

𝛼 = \sum_{𝑖} 𝑘_{𝑖} 𝛼_{𝑖}

, then either all

𝑘_{𝑖}

are non-negative, or all

𝑘_{𝑖}

are non-positive.

𝛼

is called a positive root or a negative root correspondingly.

Argue by contradiction. Assume that

𝑘_{𝑖} > 0

𝑘_{𝑗} < 0

and

𝑘_{𝑠} = 0

for all

𝑠

in between

𝑖

and

𝑗

. Let

𝜀

be the edge connecting

𝑖

with the other vertex

𝑖^{'}

towards

𝑗

. Delete

𝜀

and

Γ

breaks into two parts,

Γ_{1}

containing

𝑖

and

Γ_{2}

containing

𝑗

. Suppose

𝛽

and

𝛾

are vectors obtained by restricting the indices of

𝛼

to vertices of

Γ_{1}

and

Γ_{2}

, respectively, then

𝛼 = 𝛽 + 𝛾

𝐵 (𝛽, 𝛽) ⩾ 2

𝐵 (𝛾, 𝛾) ⩾ 2

, and

2 = 𝐵 (𝛼, 𝛼) = 𝐵 (𝛽, 𝛽) + 𝐵 (𝛾, 𝛾) + 2 𝐵 (𝛽, 𝛾)

. However,

𝐵 (𝛽, 𝛾) = - 𝑘_{𝑖} 𝑘_{𝑖^{'}} ⩾ 0

, contradiction.

Let $𝑅$ be the set of roots, $𝑅^{+}$ the set of positive roots, and $𝑅^{-}$ the set of negative roots ( $𝑅^{-} = - 𝑅^{+}$ ).

If $𝛼$ is a root, the reflection $𝑠_{𝛼}$ on $ℝ^{𝑛}$ is $𝑠_{𝛼} (𝑣) = 𝑣 - 𝐵 (𝑣, 𝛼) 𝛼$ . It is equivalent as the reflection along the hyperplain $𝐻_{𝛼}$ orthogonal to $𝛼$ . If $𝛼 = 𝛼_{𝑖}$ is a simple root, $𝑠_{𝛼} = 𝑠_{𝑖}$ is called a simple reflection, and the subgroup of $O (ℝ^{𝑛})$ generated by $𝑠_{𝑖}$ is called the Weyl group $𝑊$ . If the number of roots is finite, then $𝑊$ is finite. If we fix a labelling $1, 2, \dots, 𝑛$ of the vertices of $𝑄$ , then $𝑐 = 𝑠_{1} 𝑠_{2} \dots 𝑠_{𝑛} \in 𝑊$ is called the Coxeter element.

LEM 6.3

Let

𝛽 = \sum 𝑘_{𝑖} 𝛼_{𝑖} \neq 0

with

𝑘_{𝑖} ⩾ 0

for all

𝑖

. Then there is some

𝑁 > 0

, such that

𝑐^{𝑁} 𝛽

has at least one strictly negative coefficient.

Suppose $𝑄$ is a quiver, or equivalently a directed graph. For any $𝑖 \in 𝐼$ , define the representation $𝑆 (𝑖)$ by assigning to each vertex $𝑗$ the vector space ${𝑆 (𝑖)}_{𝑗} = 0$ if $𝑗 \neq 𝑖$ and $ℂ$ if $𝑗 = 𝑖$ , and for any $ℎ \in 𝐸$ , the linear map assigned to $ℎ$ is $𝑥_{ℎ} = 0$ . $𝑆 (𝑖)$ is an irreducible representation, and are non-isomorphic for different $𝑖$ .

THM 6.4

Assume

𝑄

has no oriented cycles. Then

𝑆 (𝑖)

are all irreducible representations of

𝑄

Suppose

𝑉

is a irreducible representation of

𝑄

, let

𝐼^{'} = {𝑖 \in 𝐼 : 𝑉_{𝑖} \neq 0}

. We want to find an

𝑖 \in 𝐼^{'}

such that

dim 𝑉_{𝑖} \cdot 𝑆 (𝑖)

is a subrepresentation of

𝑉

. A sufficient condition for such an

𝑖

if that for any

𝑗 \in 𝐼^{'}

, there is no edge from

𝑖

𝑗

. This is ensured by the assumption that

𝑄

has no oriented cycles.

For a representation $𝑉$ of $𝑄$ , define its dimension vector by

dim 𝑉 = (dim 𝑉_{1}, \dots, dim 𝑉_{𝑛}) \in ℤ_{⩾ 0}^{𝑁} .

For $𝑣 \in ℤ_{⩾ 0}^{𝐼}$ , let the representation space be $Rep (𝑄, 𝑣)$ be the space of representations $𝑉$ of $𝑄$ such that $dim 𝑉 = 𝑣$ , Equivalently,

Rep (𝑄, 𝑣) = ⨁_{ℎ \in 𝐸} {Hom}_{ℂ} (ℂ^{𝑣_{ℎ^{'}}}, ℂ^{𝑣_{ℎ^{″}}}) .

$GL (𝑣) ≔ \prod_{𝑖 \in 𝐼} GL (𝑣_{𝑖})$ acts on $Rep (𝑄, 𝑣)$ by conjugation: $𝑥_{ℎ} \mapsto 𝑔_{ℎ^{″}} 𝑥_{ℎ} 𝑔_{ℎ^{'}}^{- 1}$ . Two representations in $Rep (𝑄, 𝑣)$ are isomorphic if and only if they are in the same $GL (𝑣)$ -orbit.

A quiver $𝑄$ is of finite type if the number of indecomposable representations is finite up to isomorphisms. A non-example is the Jordan quiver consisting of one vertex and one self-loop.

THM 6.5 Gabriel

A connected quiver $𝑄$ is of finite type, if and only if the underlying unoriented graph $Γ$ is a Dynkin diagram.

Moreover, isomorphism classes of indecomposable representations of $𝑄$ are in one-to-one correspondence with the positive roots associated to $Γ$ , by $𝑉 ⟷ dim 𝑉$ .

For the only if part we refer to the homework (it requires a bit algebraic geometry). We now prove the if part.

If $𝑄$ is a quiver, call $𝑖 \in 𝐼$ a sink or source if all edges connected to $𝑖$ points towards / away from $𝑖$ . If $𝑖$ is a sink or a source, let $𝑄_{𝑖}$ be the quiver obtained from $𝑄$ by reversing all arrows connected to $𝑖$ .

We define the reflection functor, as follows. If $𝑖$ is a sink, define $𝐹_{𝑖}^{+} : Rep (𝑄) \to Rep (𝑄_{𝑖})$ by

{(𝐹_{𝑖}^{+} (𝑉))}_{𝑗} = {\begin{cases} 𝑉_{𝑗} & 𝑗 \neq 𝑖 \\ ker (⨁_{𝑘 \to 𝑖} 𝑉_{𝑘} \to 𝑉_{𝑖}) & 𝑗 = 𝑖 \end{cases}

and for an edge $𝑗 \overset{ℎ}{\to} 𝑘$ , if $𝑘 \neq 𝑖$ , $𝑥_{ℎ}$ stays the same, and if $𝑘 = 𝑖$ , then $𝑗 \to 𝑖$ becomes $𝑗 \leftarrow 𝑖$ and the map is the composition

ker (⨁_{𝑙 \to 𝑖} 𝑉_{𝑙} \to 𝑉_{𝑖}) ↪︎ ⨁_{𝑙 \to 𝑖} 𝑉_{𝑙} ↠ 𝑉_{𝑗} .

Similarly, if $𝑖$ is a source, define $𝐹_{𝑖}^{-} : Rep (𝑄) \to Rep (𝑄_{𝑖})$ by

{(𝐹_{𝑖}^{-} (𝑉))}_{𝑗} = {\begin{cases} 𝑉_{𝑗} & 𝑗 \neq 𝑖 \\ coker (𝑉_{𝑖} \to ⨁_{𝑖 \to 𝑘} 𝑉_{𝑘}) & 𝑗 = 𝑖 \end{cases}

and for an edge $𝑗 \overset{ℎ}{\to} 𝑘$ , if $𝑗 \neq 𝑖$ , $𝑥_{ℎ}$ stays the same, and if $𝑗 = 𝑖$ , then $𝑖 \to 𝑘$ becomes $𝑖 \leftarrow 𝑘$ and the map is the composition

𝑉_{𝑘} ↪︎ ⨁_{𝑖 \to 𝑙} 𝑉_{𝑙} ↠ coker (𝑉_{𝑖} \to ⨁_{𝑖 \to 𝑙} 𝑉_{𝑙})

PROP 6.6

Suppose $𝑉$ is an indecomposable representation of $𝑄$ .

If $𝑖$ is a sink, then either $𝑉 ≅ 𝑆 (𝑖)$ , or $⨁_{𝑙 \to 𝑖} 𝑉_{𝑙} \to 𝑉_{𝑖}$ is surjective.
If $𝑖$ is a source, then either $𝑉 ≅ 𝑆 (𝑖)$ , or $𝑉_{𝑖} \to ⨁_{𝑖 \to 𝑙} 𝑉_{𝑙}$ is injective.

COR 6.7

Let

𝑉

be an indecomposable representation of

𝑄

. If

𝑖

is a sink or a source, then

𝐹_{𝑖}^{+} (𝑉)

𝐹_{𝑖}^{-} (𝑉)

is either zero or indecomposable. If it is zero, then

𝑉

𝑆 (𝑖)

. If it is indecomposable,

dim 𝐹_{𝑖}^{\pm} (𝑉) = 𝑠_{𝑖} \cdot dim 𝑉

Suppose $𝑄$ is a Dynkin diagram. Fix a labelling by $1, 2, \dots, 𝑛$ of $𝑄$ wuch that $𝑖 < 𝑗$ if one can reach $𝑗$ from $𝑖$ . This labelling has the property that if $𝑛$ is a sink of $𝑄$ , then $𝑛 - 1$ is a sink of $𝑄_{𝑛}$ . Therefore, we can consider a sequence of representations

\begin{matrix} 𝑉^{(0)} = 𝑉, 𝑉^{(1)} = 𝐹_{𝑛}^{+} (𝑉^{(0)}), 𝑉^{(2)} = 𝐹_{𝑛 - 1}^{+} (𝑉^{(1)}), \dots, 𝑉^{(𝑛)} = 𝐹_{1}^{+} (𝑉^{(𝑛 - 1)}) . \end{matrix}

Notice, however, that $𝑉^{(𝑛)}$ is again a representation of $𝑄$ because every arrow is reversed twice. We have $dim 𝑉^{(𝑛)} = 𝑐 \cdot dim 𝑉$ , where $𝑐$ is the Coxeter element. Therefore we can repeat the above process, replacing $𝑉^{(0)}$ by $𝑉^{(𝑛)}$ . This produces an infinite list of representations

\begin{aligned} 𝑉^{(0)}, 𝑉^{(1)}, \dots, 𝑉^{(𝑛 - 1)}; \\ 𝑉^{(𝑛)}, 𝑉^{(𝑛 + 1)}, \dots, 𝑉^{(2 𝑛 - 1)}; \\ \dots \end{aligned}

This list of indecomposable representations must have a zero somewhere, since otherwise $dim 𝑉^{(𝑀 𝑛)} = 𝑐^{𝑀} \cdot dim 𝑉$ will have a strictly negative component for some $𝑀$ . So there must be some $𝑚$ such that $𝑉^{(𝑚)} = 𝑆 (𝑝)$ (Lemma 6.3); take the smallest such $𝑚$ . Suppose $𝐹_{𝑖_{1}}^{+} \dots 𝐹_{𝑖_{𝑚}}^{+} (𝑉) = 𝑆 (𝑝)$ , then $𝑉 = 𝐹_{𝑖_{𝑚}}^{-} \dots 𝐹_{𝑖_{1}}^{-} (𝑆 (𝑝))$ , so $dim 𝑉 = 𝑠_{𝑖_{𝑚}} \dots 𝑠_{𝑖_{1}} (𝛼_{𝑝}) \in 𝑅^{+}$ . We have then produced a map from the indecomposable representations of $𝑄$ to the positive roots, mapping $𝑉$ to $dim 𝑉 \in 𝑅^{+}$ . This map is injective since it only depends on the initial dimension vector. To prove surjectivity suppose $𝛼 \in 𝑅^{+}$ . Analogously we apply simple reflections to $𝛼$ :

\begin{aligned} 𝛼, 𝑠_{𝑛} 𝛼, 𝑠_{𝑛 - 1} 𝑠_{𝑛} 𝛼, \dots, 𝑠_{2} \dots 𝑠_{𝑛} 𝛼; \\ 𝑐 𝛼, 𝑠_{𝑛} 𝑐 𝛼, 𝑠_{𝑛 - 1} 𝑠_{𝑛} 𝑐 𝛼, \dots, 𝑠_{2} \dots 𝑠_{𝑛} 𝑐 𝛼; \\ \dots \end{aligned}

This list of roots must have a negative root somewhere, since the first column contains an element with a negative component. Therefore, the last term to the first negative root in this list must be a simple root, since simple reflections $𝑠_{𝑖}$ permute positive roots except for $𝛼_{𝑖}$ itself. Suppose $𝑠_{𝑖_{1}} \dots 𝑠_{𝑖_{𝑙}} 𝛼 = 𝛼_{𝑝}$ . Let $𝑉 = 𝐹_{𝑖_{𝑙}}^{-} \dots 𝐹_{𝑖_{1}}^{-} (𝑆 (𝑝))$ , then $dim 𝑉 = 𝛼$ and $𝑉$ is an indecomposable representation.

Mathematical NotesRepresentation TheoryPDF

0 Section 0 Introduction

1 Section 1 Basic Notions of Representation Theory

● 1.1 Representation of Algebras

▾ 1.1.1 Associative Algebras

▾ 1.1.2 Representations

● 1.2 Representation of Quivers

● 1.3 Representation of Lie Algebras

● 1.4 Example: Representations of 𝔰𝔩2

2 Section 2 General Results on Representation Theory

● 2.1 Semisimple Representations

● 2.2 Semisimple Algebras

▾ 2.2.1 The Matrix Algebra

▾ 2.2.2 Classification of Semisimple Algebras

● 2.3 Characters

● 2.4 Jordan-Hölder Theorem and Krull-Schmidt Theorem

● 2.5 Representations of Tensor Products

3 Section 3 Representations of Finite Groups

● 3.1 Characters

● 3.2 Complex Representations

▾ 3.2.1 The Character Table

▾ 3.2.2 Frobenius Determinant

▾ 3.2.3 Frobenius-Schur Indicator

▾ 3.2.4 Frobenius Divisibility

▾ 3.2.5 Burnside’s Theorem

● 3.3 Virtual Representations

● 3.4 Restriction and Induction

▾ 3.4.1 Mackey’s Irreducibility Criterion

▾ 3.4.2 Artin’s Theorem

4 Section 4 Representations of Symmetric Groups

● 4.1 Young Tableux and Symmetric Functions

● 4.2 Specht Modules

● 4.3 Ring of Representations

● 4.4 Representations of GL(𝑉)

▾ 4.4.1 Schur-Weyl Duality

▾ 4.4.2 Algebraic Representations of GL(𝑉)

● 4.5 The Okounkov-Vershik Approach

5 Section 5 Representations of GL2(𝔽𝑞)

6 Section 6 Representations of Quivers

Section 0 Introduction

Section 1 Basic Notions of Representation Theory

1.1 Representation of Algebras

1.1.1 Associative Algebras

1.1.2 Representations

1.2 Representation of Quivers

1.3 Representation of Lie Algebras

1.4 Example: Representations of ${𝔰𝔩}_{2}$

Section 2 General Results on Representation Theory

2.1 Semisimple Representations

2.2 Semisimple Algebras

2.2.1 The Matrix Algebra

2.2.2 Classification of Semisimple Algebras

2.3 Characters

2.4 Jordan-Hölder Theorem and Krull-Schmidt Theorem

2.5 Representations of Tensor Products

Section 3 Representations of Finite Groups

3.1 Characters

3.2 Complex Representations

3.2.1 The Character Table

3.2.2 Frobenius Determinant

3.2.3 Frobenius-Schur Indicator

3.2.4 Frobenius Divisibility

3.2.5 Burnside’s Theorem

3.3 Virtual Representations

3.4 Restriction and Induction

3.4.1 Mackey’s Irreducibility Criterion

3.4.2 Artin’s Theorem

Section 4 Representations of Symmetric Groups

4.1 Young Tableux and Symmetric Functions

4.2 Specht Modules

4.3 Ring of Representations

4.4 Representations of $GL (𝑉)$

4.4.1 Schur-Weyl Duality

4.4.2 Algebraic Representations of $GL (𝑉)$

4.5 The Okounkov-Vershik Approach

Section 5 Representations of ${GL}_{2} (𝔽_{𝑞})$

Section 6 Representations of Quivers