Bounded Littlewood identities for cylindric Schur functions

TL;DR

This paper extends classical bounded Littlewood identities to affine cylindric Schur functions, revealing new combinatorial connections and providing determinantal formulas for these sums.

Contribution

It introduces affine analogs of bounded Littlewood identities for cylindric Schur functions and explores their combinatorial implications.

Findings

Derived determinantal formulas for cylindric Schur function sums.

Established connections between cylindric tableaux and noncrossing/nonnesting matchings.

Uncovered new combinatorial relationships involving affine symmetric functions.

Abstract

The identities which are in the literature often called ``bounded Littlewood identities" are determinantal formulas for the sum of Schur functions indexed by partitions with bounded height. They have interesting combinatorial consequences such as connections between standard Young tableaux of bounded height, lattice walks in a Weyl chamber, and noncrossing matchings. In this paper we prove affine analogs of the bounded Littlewood identities. These are determinantal formulas for sums of cylindric Schur functions. We also study combinatorial aspects of these identities. As a consequence we obtain an unexpected connection between cylindric standard Young tableaux and \( r \)-noncrossing and \( s \)-nonnesting matchings.