Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration

TL;DR

This paper introduces multivariate generating series called Hall-Littlewood-Schubert series, which unify various enumeration problems in algebraic combinatorics, representation theory, and p-adic analysis through their versatile substitutions.

Contribution

It defines the $ ext{HLS}_n$ series and demonstrates their application to solve enumeration problems, derive new formulas, and connect to classical identities in algebraic combinatorics.

Findings

Provides solutions to lattice enumeration problems using $ ext{HLS}_n$ series.

Derives new formulas for Hecke series and p-adic integrals.

Establishes connections to classical identities like Littlewood's identity.

Abstract

We introduce multivariate rational generating series called Hall-Littlewood-Schubert ($\mathsf{HLS}_n$) series. They are defined in terms of polynomials related to Hall-Littlewood polynomials and semistandard Young tableaux. We show that $\mathsf{HLS}_n$ series provide solutions to a range of enumeration problems upon judicious substitutions of their variables. These include the problem to enumerate sublattices of a $p$-adic lattice according to the elementary divisor types of their intersections with the members of a complete flag of reference in the ambient lattice. This is an affine analog of the stratification of Grassmannians by Schubert varieties. Other substitutions of $\mathsf{HLS}_n$ series yield new formulae for Hecke series and $p$-adic integrals associated with symplectic $p$-adic groups, and combinatorially defined quiver representation zeta functions. $\mathsf{HLS}_n$ series are $q$-analogs of Hilbert series of Stanley-Reisner rings associated with posets arising from parabolic quotients of Coxeter groups of type $\mathsf{B}$ with the Bruhat order. Special values of coarsened $\mathsf{HLS}_n$ series yield analogs of the classical Littlewood identity for the generating functions of Schur polynomials.