2.1 Wedderburn decomposition [Reference Curtis and Reiner2, Reference Fulton and Harris4]

Let G be a finite group. Write $\widehat {G}$ for the index set of isomorphic classes of the irreducible representations of G over the complex numbers $\mathbb {C}$ and let $\{ V_{\lambda } \mid \lambda \in \widehat {G} \}$ be a complete set of representatives of the isomorphism classes of irreducible representations of G. Set $d_\lambda = \dim V_\lambda $ for each $\lambda \in \widehat {G}$ . Denote by $\mathrm {Mat}_{d_\lambda }(\mathbb {C})$ the set of all $d_\lambda \times d_\lambda $ matrices over $\mathbb {C}$ . Note that the representations $\pi _{\lambda } : G \rightarrow \mathrm {Aut} ( V_\lambda )$ extend uniquely to algebra homomorphisms $\widetilde {\pi }_{\lambda } : \mathbb {C}[G] \rightarrow \mathrm {Mat}_{d_\lambda }(\mathbb {C})$ for $\lambda \in \widehat {G}$ , where $\mathbb {C}[G]$ denotes the group algebra associated with G over $\mathbb {C}$ . These linear maps give a canonical algebra isomorphism called the Wedderburn decomposition:

(2.1) $$ \begin{align} \widetilde{\pi} : \mathbb{C}[G] \longrightarrow \bigoplus_{\lambda \in \widehat{G}} \mathrm{Mat}_{d_\lambda}(\mathbb{C}). \end{align} $$

2.2 Schur orthogonality relation [Reference Serre13]

Let $(\pi ^\lambda , V_\lambda )$ and $(\pi ^\mu , V_\mu )$ be irreducible representations of a finite group G. Consider any linear transformation $T: V_\mu \to V_\lambda $ and set

(2.2) $$ \begin{align} A_T = \sum_{g \in G} \pi^\lambda(g) T \pi^\mu(g^{-1}) : V_\mu \longrightarrow V_\lambda. \end{align} $$

Then, for $h \in G$ and $x \in V_\mu $ ,

$$ \begin{align*} \begin{aligned} \pi^\lambda(h) A_T(x) &= \sum_{g \in G} \pi^\lambda(hg) T \pi^\mu(g^{-1})(x) \\ &= \sum_{t \in G} \pi^\lambda(t) T \pi^\mu(t^{-1}h)(x) \quad (\text{set } t = hg) \\ &= \sum_{t \in G} \pi^\lambda(t) T \pi^\mu(t^{-1}) \pi^\mu(h)(x) = A_T ( \pi^\mu(h)x). \end{aligned} \end{align*} $$

This implies that the linear map $A_T$ is a G-module homomorphism from $V_\mu $ to $V_\lambda $ . It follows from Schur’s lemma that

(2.3) $$ \begin{align} A_T = \begin{cases} 0 & \text{if } V_\mu \not\simeq V_\lambda, \\ c_T\ \mathrm{id}_{V_\lambda} & \text{if } V_\mu \simeq V_\lambda \quad (c_T \in \mathbb{C}). \end{cases} \end{align} $$

In the case of $V_\mu \simeq V_\lambda $ , we see from (2.2) and (2.3) that

(2.4) $$ \begin{align} \mathrm{tr}\ A_T = \sum_{g \in G} \mathrm{tr}(\pi^\lambda(g) T \pi^\lambda(g^{-1})) = |G|\ \mathrm{tr}\ T = c_T \dim V_\lambda. \end{align} $$

It follows from (2.4) that $c_T = ({|G|}/{d_\lambda })\ \mathrm {tr}\ T$ and this yields

(2.5) $$ \begin{align} \sum_{g \in G} \pi^\lambda(g) T \pi^\lambda(g^{-1}) = \frac{|G|}{d_\lambda}\ \mathrm{tr}\ T \cdot \mathrm{id}_{V_\lambda}. \end{align} $$

Fix a basis $\mathcal {B}_\lambda = \{v_1, \ldots , v_{d_\lambda }\}$ for $V_\lambda $ and take the linear map T in (2.2) to be ${T_{jk}: V_\lambda \to V_\lambda }$ , which sends $v_k$ to $v_j$ and every other basis vector to $0$ . Then, (2.5) gives

(2.6) $$ \begin{align} \sum_{g \in G} \pi^\lambda(g) T_{jk} \pi^\lambda(g^{-1}) = \frac{|G|}{d_\lambda} \delta_{jk}\ \mathrm{id}_{V_\lambda}, \end{align} $$

where $\delta _{jk}$ denotes the Kronecker delta. Comparing the $(i, l)$ -entries on both sides of (2.6) gives

(2.7) $$ \begin{align} \sum_{g \in G} \pi^\lambda_{ij}(g) \pi^\lambda_{kl}(g^{-1}) = \frac{|G|}{d_\lambda} \delta_{jk} \delta_{il}, \end{align} $$

where $\pi ^\lambda _{ab}(g)$ denotes the $(a, b)$ -entry of the matrix of the linear map $\pi ^\lambda (g)$ with respect to the basis $\mathcal {B}_\lambda $ . Combining (2.3) and (2.7) yields the Schur orthogonality relations:

(2.8) $$ \begin{align} \sum_{g \in G} \pi^\lambda_{ij}(g) \pi^\mu_{kl}(g^{-1}) = \frac{|G|}{d_\lambda} \delta_{\lambda\mu} \delta_{il} \delta_{jk}. \end{align} $$

2.3 Partitions [Reference Macdonald6, Reference Sagan12]

A partition $\lambda $ of m is a weakly decreasing sequence ${\lambda = (\lambda _1, \ldots , \lambda _n)}$ of integers such that:

1. (1) $\lambda _1 \ge \cdots \ge \lambda _n \ge 0$ ;

2. (2) $\lambda _1 + \cdots + \lambda _n = m$ .

We write $\lambda \vdash m$ if $\lambda $ is a partition of m, and identify a partition with its Young diagram, which is a left-justified array of rows of unit boxes so that the first row contains $\lambda _1$ boxes, the second row contains $\lambda _2$ boxes and so on. A tableau $\mathcal {T}$ of shape $\lambda \vdash m$ is obtained from the Young diagram of $\lambda $ by filling in the boxes of the Young diagram with the numbers in a given set. In this case, we denote by $\text {sh}(\mathcal {T}) = \lambda $ the tableau $\mathcal {T}$ of shape $\lambda $ . A tableau of shape $\lambda \vdash m$ is called standard if the boxes are filled bijectively with the entries $1, \ldots , m$ and the entries strictly increase along both the rows from left to right and the columns from top to bottom. A tableau is also called semistandard if the entries weakly increase along each row from left to right and strictly increase in each column from top to bottom.

2.4 Symmetric functions [Reference Ceccherini-Silberstein, Scarabotti and Tolli1, Reference Macdonald6, Reference Sagan12]

Let $S_n$ be the symmetric group on the set $\{1, \ldots , n\}$ . The symmetric group $S_n$ acts on the polynomial ring $\mathbb {C}[x_1, \ldots , x_n]$ by permuting the variables. A polynomial invariant under this action is called symmetric and we denote by $\Lambda _n = \mathbb {C}[x_1, \ldots , x_n]^{S_n}$ the ring of symmetric polynomials. For each ${k \in \mathbb {Z}_{\ge 0}}$ , write $\Lambda _n^k$ for the set of homogeneous symmetric polynomials of degree k. The kth power-sum symmetric function is defined by ${p_k := \sum _{i=1}^n x_i^k}$ and we set ${p_\lambda := p_{\lambda _1} \cdots p_{\lambda _r}}$ for a partition $\lambda = (\lambda _1, \ldots , \lambda _r) \vdash m$ . Then, the set ${\{ p_\mu \mid \mu \vdash m, \, \ell (\mu ) \le n \}}$ forms a basis of $\Lambda _n^m$ , where $\ell (\mu )$ denotes the number of parts of the partition $\mu $ .

Let $\mathcal {T}$ be a semistandard tableau of shape $\mu \vdash m$ with entries in $\{1, \ldots , n\}$ . For each i, let $m_i$ denote the number of entries of $\mathcal {T}$ equal to i. Then, we call ${\text {wt}(\mathcal {T}) = (m_1, \ldots , m_n)}$ the weight of $\mathcal {T}$ . Write $x^\alpha $ for the monomial $x^\alpha = x_1^{\alpha _1} \cdots x_n^{\alpha _n}$ for $\alpha = (\alpha _1, \ldots , \alpha _n) \in \mathbb {Z}_{\ge 0}^n$ . The Schur function associated with the partition $\mu \vdash m$ is

$$ \begin{align*} S_\mu(x_1, \ldots, x_n) := \sum_{\text{sh}(\mathcal{T})=\mu} x^{\text{wt}(\mathcal{T})}, \end{align*} $$

where the sum is taken over all semistandard tableaux $\mathcal {T}$ of shape $\mu $ with $\ell (\mu ) \le n$ . The Schur functions $\{ S_\mu \mid \mu \vdash m \}$ are symmetric and form a basis of $\Lambda _n^m$ .

2.5 The characteristic map [Reference Ceccherini-Silberstein, Scarabotti and Tolli1, Reference Macdonald6, Reference Sagan12]

Assume that a permutation $\sigma \in S_m$ is written as a product of disjoint cycles $\sigma =\sigma _1\cdots \sigma _k,$ where the cycle $\sigma _i$ has length $l_i$ for ${i=1,\ldots ,k}$ . If $l_1\ge \cdots \ge l_k$ and $\mu =(l_1,\ldots ,l_k)$ is a partition of m, then $\sigma $ is said to be of type $\mu $ , and we write $\mathrm {type}(\sigma )=\mu $ . Observe that two permutations $\sigma $ and $\tau $ have the same type if and only if they are conjugate, that is, $\sigma =\rho \tau \rho ^{-1}$ for some $\rho \in S_m$ . Hence, the conjugacy classes of $S_m$ are naturally indexed by the partitions of m.

Now, let $z=\sum _{\sigma \in S_m}c_\sigma \sigma $ be an element of the centre $Z(\mathbb {C}[S_m])$ and define ${C_\mu =\sum _{\mathrm {type}(\sigma )=\mu }\sigma }$ . Thus, $C_\mu $ is the sum of all permutations belonging to the conjugacy class corresponding to the partition $\mu \vdash m$ . Since the coefficients $c_\sigma $ are constant on each conjugacy class of $S_m$ , we may express z as

(2.9) $$ \begin{align} z = \sum_{\mu \vdash m} a_\mu C_\mu. \end{align} $$

Let $R(S_m)$ denote the $\mathbb {Z}$ -module generated by the irreducible characters $\chi _\mu $ of $S_m$ , where $\chi _\mu $ is the character of the irreducible representation $V_\mu $ . The characteristic map $\mathit {ch}:R(S_m)\rightarrow \Lambda ^m_n$ is defined by

(2.10) $$ \begin{align} \mathit{ch}(\chi_\mu) = \frac{1}{m!} \sum_{\sigma \in S_m} \chi_{\mu}(\sigma) p_{\text{type}(\sigma)} \end{align} $$

and is extended linearly to all of $R(S_m)$ . The image $\mathit {ch}(\chi _\mu )$ is called the characteristic of $\chi _\mu $ . The $\mathbb {Z}$ -module $R(S_m)$ may be identified with the centre $Z(\mathbb {C}[S_m])$ , so the characteristic map can equally be regarded as a map on $Z(\mathbb {C}[S_m])$ . More precisely, let $\chi ^\nu _\mu $ denote the common value of the character $\chi _\mu $ on the conjugacy class corresponding to the partition $\nu \vdash m$ . Then, each character $\chi _\mu \in R(S_m)$ can be identified with the element $\sum _{\nu \vdash m}\chi ^\nu _\mu C_\nu $ of the centre by (2.9). Moreover, (2.10) can be rewritten as

(2.11) $$ \begin{align} \mathit{ch}(\chi_\mu) = \frac{1}{m!} \sum_{\nu \vdash m} \chi_\mu^\nu n_\nu p_\nu, \end{align} $$

where $n_\nu $ denotes the cardinality of the conjugacy class corresponding to $\nu \vdash m$ . If $z_\nu $ is the order of the centraliser $Z_{S_m}(\sigma )$ for any permutation $\sigma $ satisfying ${\mathrm {type}(\sigma )=\nu }$ , then $n_\nu ={m!}/{z_\nu }$ . Therefore, (2.11) becomes $ch(\chi _\mu )=\sum _{\nu \vdash m}{\chi ^\nu _\mu }/{z_\nu }p_\nu $ . Identifying $\chi _\mu $ with $\sum _{\nu \vdash m}\chi ^\nu _\mu C_\nu $ yields the equivalent description

$$ \begin{align*} \mathit{ch}: Z(\mathbb{C}[S_m]) \to \Lambda_n^m \end{align*} $$

given by $ch(C_\nu )={p_\nu }/{z_\nu }$ for every conjugacy class $C_\nu $ .

4.1 Matrix units in $\mathbb {C}[S_m]$

By the representation theory of $S_m$ , the irreducible representations of $S_m$ over $\mathbb {C}$ can be parametrised by the partitions of m [Reference Fulton and Harris4, Reference Sagan12]. For a partition $\lambda \vdash m$ , write $V_\lambda $ for the corresponding irreducible representation of $S_m$ . The representation space $V_\lambda $ is equipped with an $S_m$ -invariant inner product $( \ | \ )$ , and there is an orthonormal basis $\{v_{\mathcal {U}}\}$ parametrised by the set of standard tableaux $\mathcal {U}$ of shape $\lambda $ [Reference Ceccherini-Silberstein, Scarabotti and Tolli1, Reference Macdonald6]. We call this orthonormal basis the Young basis. Applying (2.1) to the case of $S_m$ , we have an algebra isomorphism

(4.1) $$ \begin{align} \widetilde{\pi} : \mathbb{C}[S_m] \to \bigoplus_{\lambda \vdash m} \text{Mat}_{f_\lambda}(\mathbb{C}), \end{align} $$

where $f_{\lambda }$ is the dimension of $V_{\lambda }$ . For standard tableaux $\mathcal {U}$ and $\mathcal {V}$ of shape $\lambda $ , let $E_{\mathcal {U}\mathcal {V}}^\lambda $ be the matrix unit in $\text {Mat}_{f_\lambda }(\mathbb {C})$ whose $( \mathcal {U},\mathcal {V} )$ -entry is $1$ and all other entries are $0$ . Then, under the isomorphism (4.1), the matrix unit $E_{\mathcal {U}\mathcal {V}}^\lambda \in \text {Mat}_{f_\lambda }(\mathbb {C})$ corresponds to

$$ \begin{align*}e_{\mathcal{U}\mathcal{V}}^\lambda = \frac{f_\lambda}{m!} \sum_{s \in S_m} (s\, v_{\mathcal{U}} \mid v_{\mathcal{V}}) s^{-1} \in \mathbb{C}[S_m]\end{align*} $$

by Theorem 3.4.

4.2 Centre of $\mathbb {C}[S_m]$

Recall from Section 2.5 that $\{ C_\mu \mid \mu \vdash m \}$ forms a basis of the centre $Z(\mathbb {C}[S_m])$ . However, from Section 4.1, $\{ \sum _{\text {sh}(\mathcal {U})=\lambda } E_{\mathcal {U}\mathcal {U}}^\lambda \mid \lambda \vdash m \}$ forms a basis of the centre of $\bigoplus _{\lambda \vdash m} \text {Mat}_{f_\lambda }(\mathbb {C})$ , where the sum is taken over the standard tableaux of shape $\lambda $ . Thus, by (4.1), we obtain another basis $\{ \sum _{\text {sh}(\mathcal {U})=\lambda } e_{\mathcal {U}\mathcal {U}}^\lambda \mid \lambda \vdash m \}$ of $Z(\mathbb {C}[S_m])$ . Set $I_\lambda = \sum _{\text {sh}(\mathcal {U})=\lambda } e_{\mathcal {U}\mathcal {U}}^\lambda $ for each $\lambda \vdash m$ .

Now, we compute the transition matrix between the two bases $\{ C_\mu \mid \mu \vdash m \}$ and ${\{ I_\lambda \mid \lambda \vdash m \}}$ of $Z(\mathbb {C}[S_m])$ .

For each $\lambda \vdash m$ and $\sigma \in S_m$ , we fix the Young basis $\{v_{\mathcal {U}}\}$ on the irreducible representation space $V_{\lambda }$ of $S_m$ . Write $\pi ^{\lambda }_{\mathcal {U} \mathcal {V}} ( \sigma )$ for the $( \mathcal {U}, \mathcal {V} )$ -entry of the matrix $\pi ^{\lambda } ( \sigma )$ associated with the representation $( \pi ^{\lambda }, V_{\lambda } )$ . Since $C_\mu = \sum _{\text {type}(\sigma )=\mu } \sigma $ , it follows from Corollary 3.5 that

(4.2) $$ \begin{align} C_\mu = \sum_{\text{type}(\sigma)=\mu} \sum_{\lambda, \mathcal{U}, \mathcal{V}} \pi_{\mathcal{U}\mathcal{V}}^\lambda(\sigma) e_{\mathcal{U}\mathcal{V}}^\lambda = \sum_{\lambda, \mathcal{U}, \mathcal{V}} \bigg( \sum_{\text{type}(\sigma)=\mu} \pi_{\mathcal{U}\mathcal{V}}^\lambda(\sigma) \bigg) e_{\mathcal{U}\mathcal{V}}^\lambda, \end{align} $$

where:

1. (1) $\lambda $ runs over the partitions of m; and

2. (2) $\mathcal {U}$ and $\mathcal {V}$ are taken over the standard tableaux of shape $\lambda $ .

Now, consider the algebra homomorphism $\widetilde {\pi }^\lambda : \mathbb {C}[S_m] \to \mathrm {End} ( V_{\lambda } )$ associated with the irreducible representation $( \pi ^{\lambda }, V_{\lambda } )$ . Since $C_\mu \in Z(\mathbb {C}[S_m])$ , the map $\widetilde {\pi }^\lambda (C_\mu ) : V_\lambda \to V_\lambda $ is an $S_m$ -module homomorphism. So, Schur’s lemma and the definition of $C_{\mu }$ yield

(4.3) $$ \begin{align} \widetilde{\pi}^\lambda(C_\mu) = a(\lambda, \mu) \, \text{id}_{V_\lambda} = \sum_{\text{type}(\sigma)=\mu} \pi^\lambda(\sigma) \end{align} $$

for some $a(\lambda , \mu ) \in \mathbb {C}$ . Thus, from (4.3),

$$ \begin{align*}\sum_{\text{type}(\sigma)=\mu} \pi_{\mathcal{U}\mathcal{V}}^\lambda(\sigma) = a(\lambda, \mu) \, \delta_{\mathcal{U}\mathcal{V}}.\end{align*} $$

Hence, (4.2) gives rise to

(4.4) $$ \begin{align} C_\mu = \sum_{\lambda, \mathcal{U}, \mathcal{V}} a(\lambda, \mu) \, \delta_{\mathcal{U}\mathcal{V}} \, e_{\mathcal{U}\mathcal{V}}^\lambda = \sum_{\lambda, \mathcal{U}} a(\lambda, \mu) \, e_{\mathcal{U}\mathcal{U}}^\lambda. \end{align} $$

Taking the trace of both sides of (4.3),

(4.5) $$ \begin{align} \text{tr } \widetilde{\pi}^\lambda(C_\mu) = \sum_{\text{type}(\sigma)=\mu} \text{tr } \pi^\lambda(\sigma) = a(\lambda, \mu) \, f_\lambda. \end{align} $$

From Section 2.5,

(4.6) $$ \begin{align} \sum_{\text{type}(\sigma)=\mu} \text{tr } \pi^\lambda(\sigma) = \sum_{\text{type}(\sigma)=\mu} \chi_\lambda(\sigma) = n_\mu \, \chi\,^\mu_\lambda. \end{align} $$

It follows from (4.5) and (4.6) that $a(\lambda , \mu ) = ({n_\mu }/{f_\lambda }) \chi ^\mu _\lambda $ , and (4.4) implies that

(4.7) $$ \begin{align} C_\mu = \sum_{\substack{\lambda \vdash m \\ \text{sh}(\mathcal{U})=\lambda}} \frac{n_\mu}{f_\lambda} \chi^\mu_\lambda \, e_{\mathcal{U}\mathcal{U}}^\lambda = \sum_{\lambda \vdash m} \frac{n_\mu}{f_\lambda} \chi^\mu_\lambda \sum_{\text{sh}(\mathcal{U})=\lambda} e_{\mathcal{U}\mathcal{U}}^\lambda = \sum_{\lambda \vdash m} \frac{n_\mu}{f_\lambda} \chi^\mu_\lambda \, I_\lambda. \end{align} $$

This yields the following result.

4.3 Frobenius characteristic and Schur functions

Recall from Section 2.5 that the characteristic map is the linear isomorphism defined by

$$ \begin{align*} \mathit{ch} : Z(\mathbb{C}[S_m]) \longrightarrow \Lambda^m_{n}, \quad C_\mu \mapsto \frac{p_\mu}{z_\mu} \quad\text{for each } \mu \vdash m. \end{align*} $$

By the Frobenius character formula (see [Reference Ceccherini-Silberstein, Scarabotti and Tolli1, Theorem 4.2.8] for details),

$$ \begin{align*} p_\mu = \sum_{\lambda \vdash m} \chi^\mu_\lambda S_\lambda. \end{align*} $$

Applying the characteristic map $\mathit {ch}$ ,

(4.8) $$ \begin{align} \mathit{ch}(C_\mu) = \sum_{\lambda \vdash m} \frac{\chi^\mu_\lambda}{z_\mu} S_\lambda. \end{align} $$

However, from (4.1), $Z(\mathbb {C}[S_m]) \simeq \bigoplus _{\lambda \vdash m} \mathbb {C} \, \text {id}_{V_\lambda }$ . Motivated by (4.7), we define a linear map

$$ \begin{align*}\widetilde{\mathit{ch}} : Z(\mathbb{C}[S_m]) \longrightarrow \bigoplus_{\lambda \vdash m} \mathbb{C} \, \text{id}_{V_\lambda}, \quad \widetilde{\mathit{ch}}\bigg(\frac{1}{m!} C_\mu\bigg) = \frac{1}{m!} \sum_{\lambda \vdash m} \frac{n_\mu}{f_\lambda} \chi^\mu_\lambda \, \text{id}_{V_\lambda}.\end{align*} $$

Using the fact that $n_\mu = {m!}/{z_\mu }$ , the linear map $\widetilde {\mathit {ch}}$ becomes

(4.9) $$ \begin{align} \widetilde{\mathit{ch}}\bigg(\frac{1}{m!} C_\mu\bigg) = \sum_{\lambda \vdash m} \frac{\chi^\mu_\lambda}{z_\mu} \frac{\text{id}_{V_\lambda}}{f_\lambda}. \end{align} $$

Recall that $\{ S_\lambda \mid \lambda \vdash m \}$ forms a basis of $\Lambda ^m_{n}$ and identify $\Lambda ^m_{n}$ with $\bigoplus _{\lambda \vdash m} \mathbb {C} \, \text {id}_{V_\lambda }$ via the correspondence $S_\lambda \mapsto {\text {id}_{V_\lambda }}/{f_\lambda }$ . Comparing (4.8) with (4.9), (4.9) can be understood as the (normalised) realisation of the characteristic map through the identity matrices. This realisation shows that, with respect to the characteristic map, the basis of $\Lambda ^m_{n}$ consisting of Schur functions is more natural than that consisting of power-sum symmetric functions.