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Enumerating traceless matrices over compact discrete valuation rings. (English) Zbl 1414.05024
Summary: We enumerate traceless square matrices over finite quotients of compact discrete valuation rings by their image sizes. We express the associated rational generating functions in terms of statistics on symmetric and hyperoctahedral groups, viz. Coxeter groups of types \(A\) and \(B\), respectively. These rational functions may also be interpreted as local representation zeta functions associated to the members of an infinite family of finitely generated class-2-nilpotent groups.
As a byproduct of our work, we obtain descriptions of the numbers of traceless square matrices over a finite field of fixed rank in terms of statistics on the hyperoctahedral groups.

05A15 Exact enumeration problems, generating functions
11M41 Other Dirichlet series and zeta functions
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20B30 Symmetric groups
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