Rank and Permutation Representation

It comes from chapter 7, Steinberg_2012_Representation theory of finite groups.pdf.

permutation representation

Let $\sigma :G\to S_X$ be a group action. Define a representation $\tilde{\sigma }:G\to \mathrm{GL}(\mathbb{C}X)$ by setting

$${\tilde{\sigma }}_g(\sum _{x\in X}{c}_{x}x)=\sum _{x\in X}{c}_x{\sigma }_g(x)=\sum _{y\in X}{c}_{{\sigma }_{g^{-1}}}y.$$

One calls $\tilde{\sigma }$ the permutation representation associated to $\sigma$.

fixed subspace

Let $\phi :G\to \mathrm{GL}(V)$ be a representation. Then

$$V^G=\{v\in V:{\phi }_g(v)=v\text{ for all }g\in G\}$$

is the fixed subspace of $G$.

We can prove that (see Proposition 7.2.7), $V^G$ is the direct sum of all the copies of the trivial representation in $\phi$. In the other word, for a representation $\phi :G\to \mathrm{GL}(V)$ and the trivial character ${\chi }_1$ of $G$, there is

$$⟨{\chi }_{\phi },{\chi }_1⟩=\dim V^G.$$

Proposition

Let $\sigma :G\to S_X$ be a group action. Let ${\mathcal{O}}_1,\cdots {\mathcal{O}}_m$ be the orbits of $G$ on $X$ and define $v_i={\sum }_{x\in {\mathcal{O}}_i}x$. Then $v_1,\cdots ,v_m$ is a basis for $\mathbb{C}{X}^G$, the fixed subspace of $\tilde{\sigma }$, and hence $\dim \mathbb{C}{X}^G$ is the number of orbits of $G$ on $X$.

By ^a5ygsc, we know that for every permutation representation $\tilde{\sigma }$, it has at least $m$ trivial representation as a constituent, where $m$ is the number of orbits of $\sigma$. Thus $\tilde{\sigma }$ is not irreducible if its degree is not $1$. As a corollary of this proposition, we prove the Burnside’s lemma.

Burnside's lemma

Let $\sigma :G\to S_X$ be a group action and let $m$ be the number of orbits of $G$ on $X$. Then $$ m=\frac{1}{|G|}\sum_{g\in G}|\mathrm{Fix}(g)|.

Consider the group action $\tau :G\to S_{X\times X}$, then we get the following corollary.

Corollary

Let $\sigma :G\to S_X$ be a transitive group action. Then the equalities

$$\mathrm{rank}(\sigma )=\frac{1}{|G|}\sum _{g\in G}|\mathrm{Fix}(g){|}^2=⟨{\chi }_{\tilde{\sigma }},{\chi }_{\tilde{\sigma }}⟩.$$

Definition

For a transitive group action $\sigma :G\to S_X$, by ^a5ygsc, it has only one trivial sub-representation, denoted by $\mathbb{C}{v}_0$. Since $\sigma (g)$ is a permutation matrix for all $g$, the representation of $\sigma$ is unitary and so it is decomposable by ^0m7u7u. Then $\mathbb{C}X$ can be decomposed as $\mathbb{C}X=\mathbb{C}{v}_0\oplus V_0$ where $V_0=\mathbb{C}{v}_0^{\perp }$.

Let ${\tilde{\sigma }}^{\prime }$ be the restriction of $\tilde{\sigma }$ to $V_0$, then it is called the augmentation representation associated to $\sigma$.

By ^8a0152, we can use the augmentation representation ${\tilde{\sigma }}^{\prime }$ to describe the rank of $\sigma$, which deduces the following theorem.

Theorem

Let $\sigma :G\to S_X$ be a transitive group action. Then the argumentation representation ${\tilde{\sigma }}^{\prime }$ is irreducible iff $G$ is $2$-transitive on $X$.

\begin{proof} Since $\sigma$ is transitive, by ^a5ygsc, $\tilde{\sigma }$ has exactly one trivial sub-representation and so $⟨{\chi }_1,{\chi }_{{\tilde{\sigma }}^{\prime }}⟩=0$. Then ${\tilde{\sigma }}^{\prime }$ is irreducible iff $⟨{\chi }_{{\tilde{\sigma }}^{\prime }},{\chi }_{{\tilde{\sigma }}^{\prime }}⟩=1$ iff $\mathrm{rank}(\sigma )=2$, because

$$⟨{\chi }_{\sigma },{\chi }_{\sigma }⟩=⟨{\chi }_1,{\chi }_1⟩+2\mathrm{Re}⟨{\chi }_1,{\chi }_{{\tilde{\sigma }}^{\prime }}⟩+⟨{\chi }_{{\tilde{\sigma }}^{\prime }},{\chi }_{{\tilde{\sigma }}^{\prime }}⟩=1+0+1=2.$$

Now we finish the proof. \end{proof}

Remark. Similarly, one can prove that if ${\tilde{\sigma }}^{\prime }$ can be expressed as sum of two irreducible sub-representations, it is of rank $3$. It is used in Kantor_Liebler_1982_The Rank 3 Permutation Representations of the Finite Classical Groups.pdf.