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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Igusa-type functions associated to finite formed spaces and their functional equations
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by Benjamin Klopsch and Christopher Voll PDF
Trans. Amer. Math. Soc. 361 (2009), 4405-4436

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
  • MR Author ID: 658412
  • 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
  • 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.
  • © Copyright 2009 Benjamin Klopsch and Christopher Voll
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 4405-4436
  • MSC (2000): Primary 05E15; Secondary 15A63, 20F55
  • DOI: https://doi.org/10.1090/S0002-9947-09-04671-6
  • MathSciNet review: 2500892