2022-11-21

## General representation theory

### Representation of a group

Let $G$ be a group and $V$ a vector space. A representation is a map $\rho$ that sends elements in $G$ to linear maps on $V,$ with the condition that it is compatible with the group structure. To be precise, let $\psi_g : V \to V$ be the linear map on $V$ we associate with $g \in G.$ Then

\[ \psi_{g_1 g_2} = \psi_{g_1} \circ \psi_{g_2} \text{ for all } g_1,g_2 \in G. \]Thus, a representation is a group homomorphism $\rho : G \to GL(V).$

**Example.**

Let $G = \symS_n$ and $V=\setC[x_1,\dotsc,x_n].$ Note that $V$ is not finite-dimensional. Then we let $G$ acts on $V$ by permuting the indices of the variables. By picking a basis of monomials of $V,$ then $\rho$ sends a permutation $g \in G$ to a permutation matrix in $GL(V).$

### Irreducible representation

Let $W \subset V$ be a subspace and $\rho:G \to GL(V)$ be a representation. If $\rho(g)w \in W$ for all $w \in W,$ then we say that $W$ is $G$-invariant.

If the only $G$-invariant subspaces of $V$ is $V$ itself and $\{0\},$ then we say that $\rho$ is an irreducible representation.

Every representation can be decomposed as a direct sum irreducible representations. That is, if $\rho: G \to V$ is a representation, we can write $V = V_1 \oplus \dotsb \oplus V_k$ such that for every $i,$ we have

\[ \rho(g) v \in V_i \; \text{ for all }\; v \in V_i \]and $\rho:G \to V_i$ is an irreducible representation. Here, one needs to be a bit more formal and restrict the maps in an appropriate fashion.

### Tensor products

If $\rho_1$ and $\rho_2$ are two representations of $G,$ sending group elements to elements in $GL(V)$ and $GL(W),$ then we define the tensor product of representations as the map sending $g$ to the tensor product of the individual images:

\[ (\rho_1 \otimes \rho_2)(g) \coloneqq (\rho_1(g) \otimes \rho_2(g)) \]### Character

Let $\rho: G \to GL(V)$ be a representation of $G.$ Then the character is the map $\chi_{\rho}:G \to \setC,$ defined as $\chi_{\rho}(g) \coloneqq tr( \rho(g)).$

A character is irreducible if it is the character of an irreducible representation.

## Representation theory of $\symS_n$

It is very common to consider $\symS_n$ acting on some vector space. This is then an $\symS_n$-module, and a representation of $\symS_n.$

As a typical example of $\symS_n$-modules, we have the Specht modules.

### Frobenius characteristic

Let $\chi$ be a class function on $\symS_n.$ We then define the Frobenius characteristic (or Frobenius image) as the map

\[ \frobChar(\chi) \coloneqq \frac{1}{n!} \sum_{\sigma \in \symS_n} \chi(\sigma) \powerSum_\sigma(\xvec) = \sum_{\mu} \chi_\mu \frac{\powerSum_\mu(\xvec)}{z_\mu}. \]This map has the property that for the irreducible characters $\chi^\lambda$ of $\symS_n,$ which are class functions, we have

\[ \frobChar(\chi^\lambda) = \schurS_\lambda(\xvec). \]Note that the Hilbert series can be recovered from the Frobenius characteristic. If $M = \bigoplus_k M^k$ is a graded $\symS_n$-module, and

\[ \frobChar(M) = \sum_{k \geq 0} \sum_{\lambda \vdash k} a(\lambda,k) \schurS_\mu(\xvec) \]then the Hilbert series is given by

\[ \sum_{k \geq 0} t^k \sum_{\lambda \vdash k} a(\lambda,k) f^{\mu} \]where $f^\mu$ is the number of standard Young tableaux of shape $\mu.$ The Hilbert series can alternatively described simply as $\langle \frobChar(M), \completeH_{1^n} \rangle.$

### Computing characters of $\symS_n$

You have an $\symS_n$-module $M$ and wish to compute its Frobenius characteristic $\frobChar_M(\xvec).$ This is the step-by-step guide (inspired by lecture notes by M. Zabrocki).

The same method can be adapted to compute characters for arbitrary finite groups.

- We assume that $M$ is the span of some set of polynomials in $\setC[x_1,\dotsc,x_n].$
- Use linear algebra and find a basis for $M = \langle \bvec_1,\bvec_2,\dotsc,\bvec_d \rangle.$
- For each $\sigma \in \symS_n,$ we compute its character $\chi_M(\sigma)$ as follows. For each $i \in [d],$ solve the linear system \[ \sigma(\bvec_i) = c_{i1} \bvec_1 + c_{i2} \bvec_2 + \dotsb + c_{id} \bvec_d. \] Then $\chi_M(\sigma) \coloneqq c_{11} + c_{22} + \dotsb + c_{dd}.$ This is a type of trace and independent of the choice of basis. It is in fact sufficient to compute $\chi_M(\sigma)$ for each type $\mu,$ as $\chi_M(\sigma)=\chi_M(\tau)$ whenever $\sigma$ and $\tau$ have the same cycle type.
- Now that we have computed $\chi_M(\sigma),$ we use the above formula and get that \[ \frobChar_M(\xvec) = \frac{1}{n!} \sum_{\sigma \in \symS_n} \chi_M(\sigma) \powerSum_\sigma(\xvec). \]
We then have the following relation: \[ M = \bigoplus_{\lambda \vdash n} \left(S^{\lambda} \right)^{\oplus c_\lambda} \qquad \Longleftrightarrow \qquad \frobChar_M(\xvec) = \sum_{\lambda \vdash n} c_\lambda \schurS_\lambda(\xvec). \] Here, we have expressed $M$ as a sum of irreducible $\symS_n$-modules, denoted $S^{\lambda}.$ In particular, we must have that the coefficients $c_\lambda$ are non-negative integers. Note that $\dim(S^{\lambda}) = f^\lambda,$ the number of standard Young tableaux of shape $\lambda.$

In all examples below, $\symS_n$ act on the space by permuting the variable indices.

**Example.**

Let $M = \langle x_1 + x_2 + x_3 \rangle.$ This is a one-dimensional vector space, and all elements in $\symS_3$ fixes $M.$ Thus $\chi_M(\sigma)=1$ for all $\sigma \in \symS_3$ and

\[ \frobChar_M(\xvec) = \frac{1}{3!} \sum_{\sigma \in \symS_3} \powerSum_\sigma(\xvec) = \schurS_{3}(\xvec). \]The same thing happens for $M = \langle x_1 x_2 x_3 \rangle.$

**Example.**

Let $M = \langle (x_1-x_2)(x_1-x_3)(x_2-x_3) \rangle.$ This is also a one-dimensional vector space, but odd permutations change the sign of the basis vector. Thus $\chi_M(\sigma)=\sign(\sigma)$ and

\[ \frobChar_M(\xvec) = \frac{1}{3!} \sum_{\sigma \in \symS_3} \sign(\sigma) \powerSum_\sigma(\xvec) = \schurS_{111}(\xvec). \]**Example.**

Let $M = \langle x_1 , x_2 , x_3 \rangle.$ Note that $\langle x_1 + x_2 + x_3 \rangle$ is an invariant subspace.

We have that $\frobChar_M(\xvec) = \schurS_{3}(\xvec) + \schurS_{21}(\xvec).$

**Example.**

Let $\Delta = \prod_{1\leq i < j \leq n} (x_j-x_i).$ Let $M$ be the module spanned by $\Delta$ and all partial derivatives of all orders of $\Delta.$ Then the graded Frobenius characteristic is given by

\[ \frobChar_M(\xvec) = \sum_{\lambda \vdash n} \schurS_\lambda(\xvec) \sum_{T \in SYT(\lambda)} q^{\maj(T)}. \]In particular, $M$ has dimension $n!$ â€” this is a result by M. Haiman.

**Example.**

Symmetric group action on set-partitions and non-crossing set partitions: https://www.youtube.com/watch?v=IJ3tTGJTofQ. See also $SL_3$-webs.

## Representation theory of $\GL_n$

We can also study representation theory of infinite groups, such as the group of invertible $n\times n$-matrices with complex entries. This is the general linear group, $\GL_n(\setC).$ For a brief introduction to representation theory of $\GL_n(\setC),$ see for example [Chap. 7, App. 2, Sta01].

A representation of $\GL_n(\setC)$ is a group homomorphism $\phi : \GL_n(\setC) \to \GL_m(\setC)$ for some $m.$ We will only study the cases when $\phi$ is homogeneous and rational, meaning that $\phi(\alpha A) = \alpha^k \phi(A)$ for some $k \in \setZ$ and all $\alpha \in \setC \setminus \{0\}.$ The integer $k$ is called the degree of $\phi,$ and if $k\geq 0,$ we say that $\phi$ is a polynomial representation.

As a concrete example, $\phi : \GL_n(\setC) \to \GL_1(\setC)$ defined as $\phi(A) \to \det(A)$ is a homogeneous polynomial representation of degree $n.$

Let $\phi$ be a rational representation $\phi$ of $\GL_n(\setC).$ If $A$ has eigenvalues $x_1,\dotsc,x_n,$ there is a Laurent polynomial

\[ \glChar(\phi)(\xvec) = \sum_{x^\alpha \in \setZ^n} m_\alpha x^\alpha \]called the character of $\phi,$ such that $\phi(A)$ has eigenvalue $x^\alpha$ with multiplicity $m_\alpha.$ Moreover, if $\phi$ is polynomial, then $\glChar(\phi)(\xvec)$ is a polynomial in $x_1,\dotsc,x_n.$

**Theorem.**

Every irreducible homogeneous polynomial representation $\phi$ of $\GL_n(\setC)$ is given as

\[ \glChar(\phi)(\xvec) = \schurS_\lambda(x_1,\dotsc,x_n) \]for some $\lambda \vdash n,$ where $\schurS_\lambda$ is a Schur polynomial.

Given two characters, $\phi,$ $\varphi,$ we can define the tensor product $\phi \otimes \varphi.$ We then have

\[ \glChar(\phi \otimes \varphi)(\xvec) = \glChar(\phi)(\xvec) \cdot \glChar(\varphi)(\xvec). \]In particular, if $\phi$ and $\varphi$ are irreducible, then the character $\glChar(\phi \otimes \varphi)(\xvec)$ decompose into irreducible representations via the Littlewoodâ€“Richardson rule.

If $\phi : \GL(U) \to \GL(V)$ and $\varphi : \GL(V) \to \GL(W)$ are representations, then the composition $\phi\varphi$ is a representation. The character of $\phi\varphi$ is related to the individual characters as

\[ \glChar(\phi \varphi) = \glChar(\phi)[ \glChar(\varphi) ], \]where we use the pletystm operation.

## References

- [Sta01] Richard P. Stanley. Enumerative Combinatorics: Volume 2. Cambridge University Press, First edition, 2001.