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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.

References.

[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.

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  • $\begingroup$ 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? $\endgroup$ – Francesco Polizzi Dec 12 '18 at 6:19
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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.

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

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