|
Igusa-type functions associated to finite formed spaces and their functional equations
Author(s):
Benjamin
Klopsch;
Christopher
Voll
Journal:
Trans. Amer. Math. Soc.
361
(2009),
4405-4436.
MSC (2000):
Primary 05E15;
Secondary 15A63, 20F55
Posted:
March 13, 2009
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
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  . 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.
References:
-
- 1.
- E. Artin, Geometric algebra, Interscience Publishers, New York, 1957. MR 0082463 (18:553e)
- 2.
- P. Cameron, Projective and polar spaces, QMW Maths Notes 13, Queen Mary and Westfield College, London, 1991. MR 1153019 (93c:51011)
- 3.
- J. Denef and D. Meuser, A functional equation for Igusa's local zeta function, Amer. J. Math. 113 (1991), no. 6, 1135-1152. MR 1137535 (93e:11145)
- 4.
- M.P.F. du Sautoy, Zeta functions of groups: The quest for order versus the flight from ennui, Groups St. Andrews 2001 in Oxford, Cambridge Univ. Press, Cambridge, 2003. MR 2051525 (2005i:20035)
- 5.
- M.P.F. du Sautoy and F. Grunewald, Analytic properties of zeta functions and subgroup growth, Ann. of Math. (2) 152 (2000), 793-833. MR 1815702 (2002h:11084)
- 6.
- M.P.F. du Sautoy and F. Grunewald, Zeta functions of groups and rings, International Congress of Mathematicians. Vol. II, 131-149, Eur. Math. Soc., Zürich, 2006. MR 2275592 (2008d:11102)
- 7.
- M.P.F. du Sautoy and A. Lubotzky, Functional equations and uniformity for local zeta functions of nilpotent groups, Amer. J. Math. 118 (1994), no. 1, 39-90. MR 1375303 (97g:11137)
- 8.
- R.L. Graham, D.E. Knuth, and O. Patashnik, Concrete mathematics, Addison Wesley, Reading, MA, 1998. MR 1678922 (99m:68002)
- 9.
- L.C. Grove, Classical groups and geometric algebra, Graduate Studies in Mathematics 39, AMS, Providence, RI, 2002. MR 1859189 (2002m:20071)
- 10.
- J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge Univ. Press, Cambridge, 1990. MR 1066460 (92h:20002)
- 11.
- J.-I. Igusa, Universal
 -adic zeta functions and their functional equations, Amer. J. Math. 111 (1989), 671-716. MR 1020825 (91e:11142) - 12.
- B. Klopsch and C. Voll, Counting subgroups of the higher Heisenberg groups, unpublished.
- 13.
- J. Milnor and D. Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73, Springer-Verlag, Berlin, 1973. MR 0506372 (58:22129)
- 14.
- A. Pfister, Quadratic forms with applications in algebraic geometry and topology, London Mathematical Society Lecture Note Series, vol. 217, Cambridge Univ. Press, Cambridge, 1995. MR 1366652 (97c:11046)
- 15.
- R. Scharlau, Buildings, Handbook of incidence geometry (F. Buekenhout, ed.), North-Holland, Amsterdam, 1995. MR 1360715 (96e:51002)
- 16.
- R.P. Stanley, Enumerative combinatorics, Cambridge Studies in Advanced Mathematics 49, vol. I, Cambridge Univ. Press, Cambridge, 1997. MR 1442260 (98a:05001)
- 17.
- R.P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1987), no. 1, 57-83. MR 0485835 (58:5637)
- 18.
- C. Voll, Functional equations for local normal zeta functions of nilpotent groups, Geom. Funct. Anal. (GAFA) 15 (2005), 274-295, with an appendix by A. Beauville. MR 2140633 (2006k:20065)
- 19.
- -, Counting subgroups in a family of semi-direct nilpotent products, Bull. London Math. Soc. 38 (2006), no. 5, 743-752. MR 2268358 (2007i:11123)
- 20.
- -, Functional equations for zeta functions of groups and rings, Ann. of Math., to appear.
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
05E15,
15A63, 20F55
Retrieve articles in all Journals with MSC
(2000):
05E15,
15A63, 20F55
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:
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
Posted:
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 of article:
Copyright
2009,
Benjamin Klopsch and Christopher Voll
|
Comments: webmaster@ams.org