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Igusa-type functions associated to finite formed spaces and their functional equations. (English) Zbl 1229.05288
Summary: We study symmetries enjoyed by the polynomials enumerating non-degenerate flags in finite vector spaces, equipped with a non-degenerate alternating bilinear, Hermitian or quadratic form. To this end we introduce Igusa-type rational functions encoding these polynomials and prove that they satisfy certain functional equations.
Some of our results are achieved by expressing the polynomials in question in terms of what we call parabolic length functions on Coxeter groups of type $$A$$. While our treatment of the orthogonal case exploits combinatorial properties of integer compositions and their refinements, we formulate a precise conjecture how in this situation, too, the polynomials may be described in terms of parabolic length functions.

##### MSC:
 05E15 Combinatorial aspects of groups and algebras (MSC2010) 15A63 Quadratic and bilinear forms, inner products 20F55 Reflection and Coxeter groups (group-theoretic aspects)
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