Let $\mathbb{K}$ be a field of characteristic $\neq 2$ and let $(V, \omega)$ be a symplectic vector space. Then the Heisenberg group $\mathsf{Heis}(V, \, \omega)$ is the central extension of the additive group $V$ \begin{equation} 1 \to \mathbb{K} \to \mathsf{Heis}(V, \, \omega) \to V \to 1 \quad \quad (*) \end{equation} given as follows: the underlying set of $ \mathsf{Heis}(V, \, \omega)$ is $V \times \mathbb{K}$, endowed with the group law \begin{equation} (v_1, \, t_1)\,(v_2, \, t_2) = \left(v_1+v_2, \, t_1+t_2 + \frac{1}{2} \omega(v_1, \, v_2)\right). \end{equation}

If $\mathbb{K}= \mathbb{F}_p$, then $V$ is an elementary abelian $p$-group and $\mathsf{Heis}(V, \, \omega)$ is a an extra-special finite $p$-group. Moreover, since the extension $(*)$ is central, it defines a structure of trivial $V$-module on $\mathbb{F}_p$.

On the other hand, the cohomology algebra $H^*(V, \, \mathbb{F}_p)$, when the action is trivial, can be described as follows, see the Introduction to [AG09]:

Theorem. Let $V$ be an elementary abelian $p$-group, and let $\mathbb{F}_p$ be endowed with the structure of trivial $V$-module. Then there is an isomorphism of graded algebras $$H^*(V, \, \mathbb{F}_p) \simeq \Lambda(V^{\vee}) \otimes_{\mathbb{F}_p} S(V^{\vee}),$$ where the exterior copy of the dual space is $H^1(V, \mathbb{F}_p)$ and the polynomial copy lives in $H^2( V, \, \mathbb{F}_p)$; specifically, the polynomial copy is the image of the exterior copy under the Bockstein boundary map $\beta \colon H^1(V, \mathbb{F}_p) \to H^2( V, \, \mathbb{F}_p)$.

Now it seems (at least to me) raisonable to state the following

Conjecture. The cohomology class of the extension $(*)$ corresponds, under the above identification of the cohomology algebra $H^*(V, \, \mathbb{F}_p)$, to the element $\omega \otimes 1 \in H^2(V, \, \mathbb{F}_p)$, where $\omega \in \Lambda^2(V^{\vee})$ represents the non-degenerate alternating form on $V$ via the natural duality $\Lambda^2(V^{\vee}) \simeq \mathrm{Alt}^2(V)$.

So here is my

Question. Is the above conjecture true?

I am by no means an expert in group cohomology theory, so I apologize in advance if the answer turns out to be trivial for the experts in the field. Every reference to the relevant literature will be highly appreciated.


[AG09] F. A. Aksu, D. J. Green: Essential cohomology for elementary abelian $p$-groups, Journal of Pure and Applied Algebra 213, Issue 12 (2009), 2238-2243.

  • $\begingroup$ When V is 2-dimensional, I think that this is contained in Proposition 9.1 of the paper: I. Efrat and J. Minac, "On the descending central sequence of absolute Galois groups", Amer. J. Math. 133 (2011), 1503-1532. $\endgroup$
    – user05811
    Dec 11 '18 at 18:55
  • $\begingroup$ This follows from carefully tracking the universal coefficient theorem for group cohomology. See, e.g., groupprops.subwiki.org/wiki/… $\endgroup$
    – Marty
    Dec 12 '18 at 4:32
  • $\begingroup$ @Marty: could you please elaborate your comment into an answer? $\endgroup$ Dec 12 '18 at 6:19

I believe that your conjecture is equivalent to Theorem 3.5 in the paper Locally Compact Abelian Groups with Symplectic Self-duality, Advances in Mathematics, volume 225, pages 2429-2454, 2010.

  • 1
    $\begingroup$ Thank you for the answer. I will check the details. $\endgroup$ Dec 13 '18 at 6:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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