Semisimple modular reductions

Let $G$ be a finite group and let $\rho : G \rightarrow \mathrm{GL}_d(\mathbb{Q})$ be a representation of $G$ by $d \times d$ matrices with rational entries. We regard these matrices as acting on row vectors in an $d$-dimensional vector space $V$ with standard basis $e_1, \ldots, e_n$. The $\mathbb{Z}$-submodule of $V$ spanned by $\bigl\{e_i \rho(g) : i \in \{1,\ldots,d\}, g \in G\bigr\}$ is free of rank $d$, and so has a $\mathbb{Z}$-basis, say $u_1, \ldots, u_d$. In this new basis, the matrices representing elements of $G$ have integer entries. We may therefore assume that $\rho$ has image in $\mathrm{GL}_d(\mathbb{Z})$.

It is now easy to obtain a representation of $G$ in prime characteristic $p$: just think of the entries of each $\rho(g)$ as lying in $\mathbb{F}_p$ rather than $\mathbb{Z}$. A representation obtained in this way is called a $p$-modular reduction of $V$. The indefinite article is correct: $p$-modular reductions defined with respect to different $\mathbb{Z}$-bases need not be isomorphic. However, as in the Jordan–Hölder Theorem, they always have the same multiset of composition factors. (For a proof, see Theorem 32 in Serre, Linear representations of finite groups.)

I think of representations of finite groups as rather ‘rigid’ objects, so find it slightly surprising how much freedom there often is to obtain different $p$-modular reductions. Here is an example from the symmetric group. Switching to the language of modules, let $n \in \mathbb{N}$, let $p$ be a prime dividing $n$, and let $M = \langle e_1, \ldots, e_n \rangle_\mathbb{Q}$ be the natural permutation module for $\mathbb{Q} S_n$. The subspace

$U = \langle e_i - e_j : 1 \le i < j \le n \rangle_\mathbb{Q}$

of $M$ is a $\mathbb{Q}S_n$-submodule. One obvious $\mathbb{Z}$-basis for $U$ is $e_1-e_n, \ldots, e_{n-1}-e_n$. Provided $n \ge 4$, in this basis, the matrices representing the generators $(1,2)$ and $(1,2,\ldots, n)$ of $S_n$ are

$\left( \begin{matrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \end{matrix}\right) \quad, \left( \begin{matrix} -1 & 1 & 0 & \ldots & 0 & 0 \\ -1 & 0 & 1 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -1 & 0 & 0 & \ldots & 0 & 1 \\ -1 & 0 & 0 & \ldots & 0 & 0 \end{matrix} \right)$

respectively. Consider the $\mathbb{Q}$-basis for $U$ in which $e_{n-1} - e_n$ is replaced with

$v = e_1 + \cdots + e_{n-1} - (n-1)e_n = (e_1-e_n) + \cdots + (e_{n-1} - e_n).$

Since the coefficient of $e_{n-1} - e_n$ in $v$ is $1$, the change of basis matrix is triangular with $1$s on its diagonal. Therefore $e_1-e_n, \ldots, e_{n-2}-e_n, v$ is a $\mathbb{Z}$-basis for $U$. In this basis the matrices representing the chosen generators are conjugated to

$\left( \begin{matrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \end{matrix}\right) ,\quad \left( \begin{matrix} -1 & 1 & 0 & \ldots & 0 & 0 \\ -1 & 0 & 1 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ -2 & -1 & -1 & \ldots & -1 & 1 \\ -n & 0 & 0 & \ldots & 0 & 1 \end{matrix} \right)$

respectively.

Let $W$ be the $\mathbb{F}_pS_n$-module obtained by $p$-modular reduction. Since $p$ divides $n$, we see from the final row of the matrices immediately above that $\langle v \rangle_{\mathbb{F}_p}$ is a $1$-dimensional trivial $\mathbb{F}_pS_n$-submodule of $W$. The quotient $W / \langle v \rangle$ is a simple $\mathbb{F}_pS_n$-module, $D$ say. By the result on composition factors mentioned earlier, any $p$-modular reduction of $U$ has composition factors $\mathbb{F}_p$ and $D$.

Suppose we scale the final basis vector $v$ by $\gamma \in \mathbb{Q}$. If the coefficient of $e_{i} - e_n$ in $v \rho(g)$ is $\alpha$, then the coefficient of $e_i - e_n$ in $\gamma v \rho(g)$ is $\gamma \alpha$. Similarly, if the coefficient of $v$ in $(e_i - e_n) \rho(g)$ is $\beta$ then the coefficient of $\gamma v$ in $(e_i - e_n) \rho(g)$ is $\gamma^{-1} \beta$. The effect on matrices is to scale the final column by $\gamma$, and the final row by $\gamma^{-1}$, leaving the entry in the bottom right unchanged. The new matrices are therefore

$\left( \begin{matrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \end{matrix}\right) ,\quad \left( \begin{matrix} -1 & 1 & 0 & \ldots & 0 & 0 \\ -1 & 0 & 1 & \ldots & 0 & 0 \\ -1 & 0 & 0 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ -2 & -1 & -1 & \ldots & -1 & \gamma \\ -n/\gamma & 0 & 0 & \ldots & 0 & 1 \end{matrix} \right)$

respectively. Let $p^a$ be the highest power of $p$ dividing $n$. Taking $\gamma = p^a$ we see that $U$ has a $p$-modular reduction in which $D$ appears as the unique bottom composition factor, and $\mathbb{F}_p$ as the unique top composition factor. Moreover, if $a \ge 2$ then by taking $\gamma = p$ we get a semisimple $p$-modular reduction of $U$.

This trick for moving composition factors up or down in the radical series of a module generalizes. In fact, over a suitable ring, one may move composition factors around almost at will. One striking application is Lemma 18.2 in Feit, Finite linear groups. The special case for rational representations is proved below.

Theorem. Let $G$ be a finite group and let $U$ be a $\mathbb{Q}G$-module. There is a field $K$ containing $\mathbb{Q}$ such that the $KG$-module $U \otimes_\mathbb{Q} K$ has a semisimple $p$-modular reduction.

Proof. Suppose that $U$ has a $p$-modular reduction having exactly $e$ composition factors. There is an $\mathbb{Z}$-basis of $U$ in which the representing matrices have the block form

$\left( \begin{matrix} X_{11} & pX_{12} & \ldots & pX_{1e} \\ X_{21} & X_{22} & \ldots & pX_{2e} \\ \vdots & \vdots & \ddots & \vdots \\ X_{e1} & X_{e2} & \ldots & X_{ee} \end{matrix} \right).$

Let $K = \mathbb{Q}[p^{1/e}]$. Let $t = p^{1/e}$. Scaling the basis vectors corresponding to the row containing $X_{kk}$ by $t^{k-1}$ for each $k$ gives a $\mathbb{Z}[t]$-basis of $U$ in which the representing matrices are

$\left( \begin{matrix} X_{11} & t^{e-1}X_{12} & \ldots & tX_{ee} \\ tX_{21} & X_{22} & \ldots & t^2X_{2e} \\ \vdots & \vdots & \ddots & \vdots \\ t^{e-1}X_{e1} & t^{e-2}X_{e2} & \ldots & X_{ee} \end{matrix} \right).$

The corresponding $p$-modular reduction, defined by quotienting $\mathbb{Z}[t]$ by its maximal ideal $\langle t \rangle$, is semisimple. $\Box$

The following theorem gives another generalization of the example above. Given a commutative ring $R$ and a partition $\lambda$, let $S_R^\lambda$ denote the corresponding Specht module for $RS_n$.

Theorem. Let $\lambda$ be a partition. There is a $p$-modular reduction of $S^\lambda_\mathbb{Q}$ isomorphic to $(S^\lambda_{\mathbb{F}_p})^\star$.

Proof. Let $n = |\lambda|$. Let $t$ be a tableau of shape $\lambda$ with distinct entries from $\{1,\ldots, n\}$. Let $a_\lambda = \sum_{\sigma} \sigma$, respectively $b_\lambda = \sum_\sigma \mathrm{sgn}(\sigma)$, where the sums are over all permutations $\sigma$ that permute amongst themselves the entries of each row, respectively column, of $t$. There are isomorphisms $S^\lambda_\mathbb{Z} \cong a_\lambda b_\lambda \mathbb{Z}S_n$ and $(S^\lambda_\mathbb{Z})^\star \cong b_\lambda a_\lambda \mathbb{Z}S_n$. Moreover, $a_\lambda b_\lambda$ is a pseudo-idempotent, i.e. $(a_\lambda b_\lambda)^2 = \delta a_\lambda b_\lambda$ for some non-zero $\delta \in \mathbb{Z}$. (For proofs of these facts see Lemma 5 and Exercise 20 in Chapter 7 of Fulton, Young tableaux.) Hence $b_\lambda a_\lambda \sigma \mapsto a_\lambda b_\lambda a_\lambda \sigma$ defines an injective homomorphism $(S^\lambda_\mathbb{Z})^\star \rightarrow S^\lambda_\mathbb{Z}$. Therefore $S^\lambda_\mathbb{Z}$ has a $\mathbb{Z}$-basis defining a reduction modulo $p$ of $S^\lambda_\mathbb{Q}$ isomorphic to $(S^\lambda_{\mathbb{F}_p})^\star$. $\Box$

An alternative description of the inclusion homomorphism in this proof is given by Fayers in Proposition 4.7 of On the structure of Specht modules. The results on Schaper layers in his paper can be used to give many further examples of semisimple $p$-modular reductions of Specht modules that do not require a field extension.