Igusa-type functions associated to finite formed spaces and their functional equations
Igusa-type functions associated to finite formed spaces and their functional equations
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.
finite formed spaces, coxeter groups, zeta functions, functional
equations
4405-4436
Klopsch, Benjamin
3556ffc9-0748-4eee-aecc-d1dc70a30638
Voll, Christopher
b7bd2890-38ac-4e05-adb7-3b376916ff79
13 March 2009
Klopsch, Benjamin
3556ffc9-0748-4eee-aecc-d1dc70a30638
Voll, Christopher
b7bd2890-38ac-4e05-adb7-3b376916ff79
Klopsch, Benjamin and Voll, Christopher
(2009)
Igusa-type functions associated to finite formed spaces and their functional equations.
Transactions of the American Mathematical Society, 361, .
(doi:10.1090/S0002-9947-09-04671-6).
Abstract
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.
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More information
Submitted date: 7 August 2006
Published date: 13 March 2009
Keywords:
finite formed spaces, coxeter groups, zeta functions, functional
equations
Organisations:
Pure Mathematics
Identifiers
Local EPrints ID: 55274
URI: http://eprints.soton.ac.uk/id/eprint/55274
ISSN: 0002-9947
PURE UUID: 38629107-73e1-43e8-a03c-bd11fb777e6a
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Date deposited: 19 Aug 2008
Last modified: 15 Mar 2024 10:53
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Contributors
Author:
Benjamin Klopsch
Author:
Christopher Voll
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