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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.

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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[1] E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957. · Zbl 0077.02101
[2] Peter J. Cameron, Projective and polar spaces, QMW Maths Notes, vol. 13, Queen Mary and Westfield College, School of Mathematical Sciences, London, 199?. · Zbl 0473.51002
[3] Jan Denef and Diane Meuser, A functional equation of Igusa’s local zeta function, Amer. J. Math. 113 (1991), no. 6, 1135 – 1152. · Zbl 0749.11053
[4] Marcus du Sautoy, Zeta functions of groups: the quest for order versus the flight from ennui, Groups St. Andrews 2001 in Oxford. Vol. I, London Math. Soc. Lecture Note Ser., vol. 304, Cambridge Univ. Press, Cambridge, 2003, pp. 150 – 189. · Zbl 1195.11123
[5] Marcus du Sautoy and Fritz Grunewald, Analytic properties of zeta functions and subgroup growth, Ann. of Math. (2) 152 (2000), no. 3, 793 – 833. · Zbl 1006.11051
[6] Marcus du Sautoy and Fritz Grunewald, Zeta functions of groups and rings, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 131 – 149. · Zbl 1112.20024
[7] Marcus P. F. du Sautoy and Alexander Lubotzky, Functional equations and uniformity for local zeta functions of nilpotent groups, Amer. J. Math. 118 (1996), no. 1, 39 – 90. · Zbl 0864.20020
[8] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Matematyka konkretna, 2nd ed., Wydawnictwo Naukowe PWN, Warsaw, 1998 (Polish, with Polish summary). Translated from the second English (1994) edition by P. Chrząstowski, A. Czumaj, L. Gąsieniec and M. Raczunas.
[9] Larry C. Grove, Classical groups and geometric algebra, Graduate Studies in Mathematics, vol. 39, American Mathematical Society, Providence, RI, 2002. · Zbl 0990.20001
[10] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. · Zbl 0725.20028
[11] Jun-ichi Igusa, Universal \?-adic zeta functions and their functional equations, Amer. J. Math. 111 (1989), no. 5, 671 – 716. · Zbl 0707.14016
[12] B. Klopsch and C. Voll, Counting subgroups of the higher Heisenberg groups, unpublished. · Zbl 1193.11083
[13] John Milnor and Dale Husemoller, Symmetric bilinear forms, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. · Zbl 0292.10016
[14] Albrecht Pfister, Quadratic forms with applications to algebraic geometry and topology, London Mathematical Society Lecture Note Series, vol. 217, Cambridge University Press, Cambridge, 1995. · Zbl 0847.11014
[15] F. Buekenhout , Handbook of incidence geometry, North-Holland, Amsterdam, 1995. Buildings and foundations. · Zbl 0821.00012
[16] Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. · Zbl 0889.05001
[17] Richard P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57 – 83. · Zbl 0384.13012
[18] C. Voll, Functional equations for local normal zeta functions of nilpotent groups, Geom. Funct. Anal. 15 (2005), no. 1, 274 – 295. With an appendix by A. Beauville. · Zbl 1135.11331
[19] Christopher Voll, Counting subgroups in a family of nilpotent semi-direct products, Bull. London Math. Soc. 38 (2006), no. 5, 743 – 752. · Zbl 1120.11040
[20] -, Functional equations for zeta functions of groups and rings, Ann. of Math., to appear.
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