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Asymptotics of generalized Galois numbers via affine Kac-Moody algebras

Author: Stavros Kousidis
Journal: Proc. Amer. Math. Soc. 141 (2013), 3313-3326
MSC (2010): Primary 05A16, 06B15; Secondary 94B05
Published electronically: June 10, 2013
MathSciNet review: 3080154
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Abstract: Generalized Galois numbers count the number of flags in vector spaces over finite fields. Asymptotically, as the dimension of the vector space becomes large, we give their exponential growth and determine their initial values. The initial values are expressed analytically in terms of theta functions and Euler's generating function for the partition numbers. Our asymptotic enumeration method is based on a Demazure module limit construction for integrable highest weight representations of affine Kac-Moody algebras. For the classical Galois numbers that count the number of subspaces in vector spaces over finite fields, the theta functions are Jacobi theta functions. We apply our findings to the asymptotic number of linear $ q$-ary codes and conclude with some final remarks about possible future research concerning asymptotic enumerations via limit constructions for affine Kac-Moody algebras and modularity of characters of integrable highest weight representations.

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Additional Information

Stavros Kousidis
Affiliation: Institute for Theoretical Physics, ETH Zürich, Wolfgang–Pauli–Strasse 27, CH-8093 Zürich, Switzerland – and – Institute of Physics, University of Freiburg, Rheinstrasse 10, 79104 Freiburg, Germany
Address at time of publication: Schenkendorfstr. 11, 53173 Bonn, Germany

Keywords: Asymptotic enumeration, Galois number, affine Kac-Moody algebra, Demazure module, linear code
Received by editor(s): September 23, 2011
Received by editor(s) in revised form: December 7, 2011
Published electronically: June 10, 2013
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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