Abacus-histories and the combinatorics of creation operators

TL;DR

This paper introduces abacus-histories as combinatorial models to understand the Schur expansions of symmetric functions generated by creation operators, providing new insights and bijective proofs for related properties.

Contribution

It develops novel combinatorial models called abacus-histories for symmetric functions generated by creation operators, enabling explicit Schur expansion formulas and bijective proofs.

Findings

New combinatorial models for symmetric functions

Explicit Schur expansion formulas using abacus-histories

Bijective proofs of properties of Bernstein operators and Hall-Littlewood polynomials

Abstract

Creation operators act on symmetric functions to build Schur functions, Hall--Littlewood polynomials, and related symmetric functions one row at a time. Haglund, Morse, Zabrocki, and others have studied more general symmetric functions $H_{\alpha}$, $C_{\alpha}$, and $B_{\alpha}$ obtained by applying any sequence of creation operators to $1$. We develop new combinatorial models for the Schur expansions of these and related symmetric functions using objects called abacus-histories. These formulas arise by chaining together smaller abacus-histories that encode the effect of an individual creation operator on a given Schur function. We give a similar treatment for operators such as multiplication by $h_m$, $h_m^{\perp}$, $\omega$, etc., which serve as building blocks to construct the creation operators. We use involutions on abacus-histories to give bijective proofs of properties of the Bernstein creation operator and Hall-Littlewood polynomials indexed by three-row partitions.