Asymptotics of generalized Galois numbers via affine Kac-Moody algebras

TL;DR

This paper investigates the exponential growth of generalized Galois numbers, connecting their asymptotics to theta functions and affine Kac-Moody algebra representations, and applies results to q-ary code enumeration.

Contribution

It introduces a novel asymptotic enumeration method using Demazure module limits of affine Kac-Moody algebra representations, linking algebraic structures to combinatorial counts.

Findings

Derived asymptotic formulas for generalized Galois numbers

Expressed initial values using theta functions and partition generating functions

Applied results to estimate the number of q-ary codes

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 q-ary codes, and conclude with some final remarks about possible future research concerning asymptotic enumerations via limit constructions for affine Kac-Moody algebras.