Abacus-histories and the combinatorics of creation operators

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.