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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.

MSC:
05A15 Exact enumeration problems, generating functions
11M41 Other Dirichlet series and zeta functions
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20B30 Symmetric groups
Software:
Zeta
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References:
[1] Adin, R. M.; Brenti, F.; Roichman, Y., Descent representations and multivariate statistics, Transactions of the American Mathematical Society, 357, 3051-3082, (2005) · Zbl 1059.05105
[2] Belkale, P.; Brosnan, P., Matroids, motives, and a conjecture of Kontsevich, Duke Mathematical Journal, 116, 147-188, (2003) · Zbl 1076.14026
[3] Bender, E. A., On buckhiester’s enumeration of n × n matrices, Journal of Combinatorial Theory. Series A, 17, 273-274, (1974) · Zbl 0284.05008
[4] Björner, A.; Brenti, F., Combinatorics of Coxeter groups, (2005), New York · Zbl 1110.05001
[5] Brenti, F.; Carnevale, A., Proof of a conjecture of klopsch-voll on Weyl groups of type A, Transactions of the American Mathematical Society, 369, 7531-7547, (2017) · Zbl 1368.05007
[6] Sautoy, M. P. F.; Woodward, L., Zeta functions of groups and rings, (2008), Berlin
[7] Dung, D. H.; Voll, C., Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups, Transactions of the American Mathematical Society, 369, 6327-6349, (2017) · Zbl 06730687
[8] Lewis, J. B.; Liu, R. I.; Morales, A. H.; Panove, G.; Sam, S. V.; Zhang, Y. X., Matrices with restricted entries and q-analogues of permutations, Journal of Combinatorics, 2, 355-395, (2011) · Zbl 1247.05027
[9] Reiner, V., Signed permutation statistics, European Journal of Combinatorics, 14, 553-567, (1993) · Zbl 0793.05005
[10] T. Rossmann, Zeta, Version 0.3.2, https://doi.org/www.math.uni-bielefeld.de/ rossmann/Zeta/.
[11] Rossmann, T., Topological representation zeta functions of unipotent groups, Journal of Algebra, 448, 210-237, (2016) · Zbl 1409.11074
[12] Schein, M. M.; Voll, C., Normal zeta functions of the Heisenberg groups over number rings. I—the unramified case, Journal of the London Mathematical Society, 91, 19-46, (2015) · Zbl 1315.20023
[13] Seeley, C., 7-dimensional nilpotent Lie algebras, Transactions of the American Mathematical Society, 335, 479-496, (1993) · Zbl 0770.17003
[14] Stasinski, A.; Voll, C., A new statistic on the hyperoctahedral groups, Electronic Journal of Combinatorics, 20, 23, (2013) · Zbl 1295.05038
[15] Stasinski, A.; Voll, C., Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B, American Journal of Mathematics, 136, 501-550, (2014) · Zbl 1286.11140
[16] Stasinski, A.; Voll, C., Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces, Forum Mathematicum, 29, 717-734, (2017) · Zbl 1366.22009
[17] Stembridge, J. R.; Waugh, D. J., A Weyl group generating function that ought to be better known, Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae, 9, 451-457, (1998) · Zbl 0927.20020
[18] Theisinger, L., Bemerkung über die harmonische reihe, Monatshefte für Mathematik und Physik, 26, 132-134, (1915) · JFM 45.0419.01
[19] Voll, C., Functional equations for zeta functions of groups and rings, Annals of Mathematics, 172, 1181-1218, (2010) · Zbl 1314.11057
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