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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Igusa-type functions associated to finite formed spaces and their functional equations


Authors: Benjamin Klopsch and Christopher Voll
Journal: Trans. Amer. Math. Soc. 361 (2009), 4405-4436
MSC (2000): Primary 05E15; Secondary 15A63, 20F55
Published electronically: March 13, 2009
MathSciNet review: 2500892
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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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Additional Information

Benjamin Klopsch
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany
Address at time of publication: Department of Mathematics, Royal Holloway, University of London, Egham TW20 0EX, United Kingdom
Email: Benjamin.Klopsch@rhul.ac.uk

Christopher Voll
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany
Address at time of publication: School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom
Email: C.Voll.98@cantab.net

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04671-6
PII: S 0002-9947(09)04671-6
Keywords: Finite formed spaces, Coxeter groups, zeta functions, functional equations
Received by editor(s): August 7, 2006
Received by editor(s) in revised form: October 25, 2007
Published electronically: March 13, 2009
Additional Notes: The results in this paper form part of the first author’s Habilitation thesis at the University of Düsseldorf. The second author acknowledges support by the Deutsche Forschungsgemeinschaft and the Max-Planck-Gesellschaft. He gratefully acknowledges the hospitality of the Heinrich-Heine-Universität in Düsseldorf and the Max-Planck-Institut für Mathematik in Bonn during the writing of this paper. This paper forms part of his Habilitation thesis at the University of Düsseldorf.
Article copyright: © Copyright 2009 Benjamin Klopsch and Christopher Voll