Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.


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!

share|cite|improve this question

1 Answer 1

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?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.