Séminaire Lotharingien de Combinatoire, 93B.22 (2025), 12 pp.
Joshua Maglione and Christopher Voll
Hall-Littlewood Polynomials,
Affine Schubert Series, and Lattice Enumeration
Abstract.
We introduce multivariate rational generating series called
Hall-Littlewood-Schubert (HLSn) series. They are defined in terms of
polynomials related to Hall-Littlewood polynomials and semistandard Young
tableaux. We show that HLSn 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 HLSn series yield new formulae for Hecke series and p-adic integrals
associated with symplectic p-adic groups, and combinatorially defined quiver
representation zeta functions. HLSn series are q-analogs of Hilbert
series of Stanley-Reisner rings associated with posets arising from parabolic
quotients of Coxeter groups of type B with the Bruhat order.
Special values of coarsened HLSn series yield analogs of the classical
Littlewood identity for the generating functions of Schur polynomials.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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