Let $k$ be a field of characteristic different from $2$.

Let $n\geq 1$ be an integer, and let $T$ be the maximal torus of the $k$-algebraic group $PGL_n$, namely the quotient of diagonal matrices by the diagonal action of $\mathbb{G}_{m,k}$.

Then let the symmetric group $S_n$ act on T by permuting the diagonal entries.

We know from general results that $H^2(S_n,T)$ is killed by $n!$, but can we do better ?

Is it killed by $n$, for example ? what is the exponent of $H^2(S_n,T)$?

Thanks in advance!


This is an interesting question indeed. I can not offer a full answer, only the following observation.

First consider the case $n=3$, i.e., $T_3$ is the maximal torus of $\operatorname{PGL}_3$ with the natural action of $S_3$. Let's use the Hochschild-Serre spectral sequence to compute $\operatorname{H}^\bullet(S_3,T_3)$.

First, let $G=\mathbb{Z}/3$ and use the standard resolution $$ \cdots\to \mathbb{Z}[G]\stackrel{\Delta}{\rightarrow} \mathbb{Z}[G]\stackrel{t-1}{\rightarrow}\mathbb{Z}[G]\to\mathbb{Z}\to 0 $$ where $t$ is a generator of $G$ and $\Delta=1+t+t^2$. Now $\operatorname{H}^\bullet(G,T_3)=\operatorname{H}^\bullet(\operatorname{Hom}_{\mathbb{Z}[G]}(\mathbb{Z}[G],T_3))$ is the cohomology of the complex $$ T_3\stackrel{(t-1)^\ast}{\rightarrow} T_3\stackrel{\Delta^\ast}{\rightarrow} T_3\to \cdots $$

To evaluate this, describe the action of $G$ on $T_3$. The maximal torus $T_3$ in $\operatorname{PGL}_3$ is given by diagonal matrices $\operatorname{diag}(a,b,c)$ modulo scalar matrices $\operatorname{diag}(a,a,a)$. Using representatives $\operatorname{diag}(1,a,b)$, the action of $t$ on $T_3$ is given by $\operatorname{diag}(1,a,b)\to \operatorname{diag}(b,1,a)\sim \operatorname{diag}(1,b^{-1},ab^{-1})$. Consequently, the action of $t-1$ on $T_3$ is $(a,b)\mapsto ((ab)^{-1},ab^{-2})$ and the action of $\Delta$ on $T_3$ is trivial. From this, we obtain $$ \operatorname{H}^i(G,T_3)=\left\{\begin{array}{ll} \mu_3 & i \textrm{ even}\\ k^\times/(k^\times)^3 & i \textrm{ odd} \end{array}\right. $$ where $\mu_3$ is the group of third roots of unity in $k$.

Next, we describe the action of $\mathbb{Z}/2$ on $\operatorname{H}^i(G,T_3)$. On $T_3$, the non-trivial element $\sigma$ acts as $\operatorname{diag}(1,a,b)\mapsto \operatorname{diag}(1,b,a)$. It follows, that $\sigma$ acts via $x\mapsto x^{-1}$ on $\operatorname{H}^i(G,T_3)$. Again, we use the standard resolution to compute the cohomology $\operatorname{H}^j(\mathbb{Z}/2,\operatorname{H}^i(G,T_3))$. The element $\sigma-1$ acts as identity on $\mu_3$, and the element $\sigma+1$ acts trivially on $\mu_3$. Similarly, $\sigma-1$ acts as identity on $k^\times/(k^\times)^3$, and $\sigma+ 1$ acts trivially. The result is that $$ \operatorname{H}^i(\mathbb{Z}/2,\operatorname{H}^j(\mathbb{Z}/3,T_3))=0 $$ for all $i,j$, so the Hochschild-Serre spectral sequence degenerates and shows that $\operatorname{H}^i(S_3,T_3)=0$ for all $i$.

I claim that the same argument shows that the $p$-torsion in $\operatorname{H}^i(S_p,T_p)$ is trivial. In fact, we only need to compute the cohomology of the normalizer $N_p$ of the $p$-Sylow with $T_p$-coefficients. This is done as above via the Hochschild-Serre spectral sequence. In the first step, $$ \operatorname{H}^i(\mathbb{Z}/p,T_p)\cong\left\{\begin{array}{ll} \mu_p & i \textrm{ even}\\ k^\times/(k^\times)^p & i \textrm{ odd}. \end{array}\right. $$ On these groups, the quotient $N_p/(\mathbb{Z}/p)\cong\mathbb{Z}/(p-1)$ acts via cyclic permutation of the powers. As above, the map $\tau-1$ is an isomorphism in the Hom-complex, showing the vanishing of cohomology $\operatorname{H}^\bullet(N_p,T_p)$ as above.

So, as a partial answer to your question: the cohomology in case $n$ prime is only annihilated by $n$ if it is completely trivial.

I would love to know the general statement, what is $\operatorname{H}^\bullet(S_n,T_n)$, and I would expect it to be known and written somewhere. A seemingly related question I would also like to see answered: what is a reference for cohomology of the normalizer of a maximal torus in an algebraic group. The above cohomology groups are part of the Hochschild-Serre spectral sequence computing the normalizer of the maximal torus in $\operatorname{PGL}_n$. Over algebraically closed fields, the homology of the normalizer of the maximal torus (with finite coefficients) can be understood in terms of \'etale cohomology, see the discussion in Chapter 5 of K. Knudson ``Homology of linear groups''. Can somebody take it from here?


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