Diagonal torus characters and the Weyl action

Let $n\ge 1$, let $q$ be a prime power, and put $G={\mathrm{GL}}_n({\mathbb{F}}_q)$ with diagonal torus $T$, upper triangular Borel $B=T⋉U$, monomial subgroup $N$ and Weyl group $W=N\mathrm{/}T\cong S_n$ (Standard subgroups of finite general linear groups, Permutation Weyl group and inversion length). For $w\in S_n$ let $\dot{w}:=P_w\in G$ be the permutation matrix of $w$, so that ${\dot{w}}^{-1}t\dot{w}$ is diagonal for every $t\in T$. A character of $T$ is a group homomorphism $\chi :T\to {\mathbb{C}}^{\times }$; write $\widehat{T}:=\mathrm{Hom}(T,{\mathbb{C}}^{\times })$ for the group of characters, with pointwise multiplication and the trivial character $1_{\widehat{T}}$.

Coordinates. Since $T\cong ({\mathbb{F}}_q^{\times }{)}^n$ via $t=\mathrm{diag}(t_1,\dots ,t_n)\mapsto (t_1,\dots ,t_n)$, the assignment $\chi \mapsto ({\chi }_1,\dots ,{\chi }_n)$ with $\chi (\mathrm{diag}(t_1,\dots ,t_n))={\prod }_{i=1}^n{\chi }_i(t_i)$ is a bijection from $\widehat{T}$ onto the set of $n$-tuples of characters of ${\mathbb{F}}_q^{\times }$; the ${\chi }_i$, given by ${\chi }_i(a)=\chi (\mathrm{diag}(1,\dots ,a,\dots ,1))$ with $a$ in the $i$-th position, are the coordinates of $\chi$, and the tuple of coordinates determines $\chi$ by the displayed product formula.

The Weyl action and equal-character blocks. The group $S_n$ acts on $\widehat{T}$ by $(w\cdot \chi )(t):=\chi ({\dot{w}}^{-1}t\dot{w})(w\in S_n,\chi \in \widehat{T},t\in T),$ and for the coordinates this reads $(w\cdot \chi {)}_j={\chi }_{w^{-1}(j)}(1\le j\le n):$ the action permutes the coordinates. For $\chi \in \widehat{T}$ define the Weyl stabiliser $W_{\chi }:=\{w\in S_n:w\cdot \chi =\chi \}$. A permutation $w$ lies in $W_{\chi }$ exactly when ${\chi }_{w^{-1}(j)}={\chi }_j$ for all $j$, that is, exactly when $w$ preserves every level set $\{j:{\chi }_j=a\}$, $a\in \mathrm{Hom}({\mathbb{F}}_q^{\times },{\mathbb{C}}^{\times })$. These level sets, the orbits of $W_{\chi }$ on $\{1,\dots ,n\}$, are the equal-character blocks of $\chi$; they are the maximal subsets on which $\chi$ is constant. If they have sizes $n_1,\dots ,n_k$, with $n_1+\cdots +n_k=n$, then $W_{\chi }$ is the Young subgroup $S_{n_1}\times \cdots \times S_{n_k}\le S_n$ of permutations preserving each block, so that $\mathrm{\mid }{W}_{\chi }\mathrm{\mid }={\prod }_{r=1}^k{n}_r!$.

Intrinsic Coxeter system. The intrinsic Coxeter generators of $W_{\chi }$ are the transpositions of successive elements of each equal-character block, listed in increasing order. They need not be adjacent transpositions of $S_n$: for $\chi =(a,b,a)$ with $a\mathrm{\ne }b$ the block $\{1,3\}$ produces the generator $(13)$. To record this system choose a permutation $\sigma \in S_n$ for which $\eta :=\sigma \cdot \chi$ has each equal-character block contiguous, the blocks being ordered by the first occurrence of their character in $\chi$ and the order inside each block preserved; such a $\sigma$ is obtained by listing the blocks in that order. Write $S_{\eta }$ for the set of ambient adjacent transpositions $s_i$ with $i,i+1$ inside one block of $\eta$, and put $S_{\chi }:={\sigma }^{-1}{S}_{\eta }\sigma$. Then $(W_{\chi },S_{\chi })$ is the Coxeter system obtained by transporting the product system of the standard contiguous Young subgroup $W_{\eta }$. The intrinsic length on $W_{\chi }$ is transported along this identification from the length of the product system; it is not the ambient inversion length $\mathrm{\ell }$ of $S_n$, and $S_{\chi }$ need not consist of simple reflections of $W$.

Regular characters. Call $\chi$ regular when its coordinates are pairwise distinct, that is, when every equal-character block has size $1$; then $W_{\chi }=1$, $k=n$ and $S_{\chi }=\mathrm{\varnothing }$. At the other extreme, since ${\mathbb{F}}_2^{\times }$ is the trivial group, for $q=2$ there is exactly one character of $T$, and then $W_{\chi }=S_n$ and $k=1$.