Murnaghan-Type Representations for the Positive Elliptic Hall Algebra

TL;DR

This paper introduces a new family of graded representations for the positive elliptic Hall algebra, generalizing symmetric Macdonald functions and revealing combinatorial structures related to Young tableaux.

Contribution

It constructs novel representations indexed by Young diagrams, with explicit combinatorial rules and product-series identities extending Macdonald function theory.

Findings

Explicit combinatorial rule for operator actions

New product-series identities involving Young tableaux

Generalization of symmetric Macdonald functions

Abstract

We construct a new family of graded representations $\widetilde{W}_{\lambda}$ for the positive elliptic Hall algebra $\mathcal{E}^{+}$ indexed by Young diagrams $\lambda$ which generalize the standard $\mathcal{E}^{+}$ action on symmetric functions. These representations have homogeneous bases of eigenvectors for the action of the Macdonald element $P_{0,1} \in \mathcal{E}^{+}$ with distinct $\mathbb{Q}(q,t)$-rational spectrum generalizing the symmetric Macdonald functions. The analysis of the structure of these representations exhibits interesting combinatorics arising from the stable limits of periodic standard Young tableaux. We find an explicit combinatorial rule for the action of the multiplication operators $e_r[X]^{\bullet}$ generalizing the Pieri rule for symmetric Macdonald functions. We will also naturally obtain a family of interesting $(q,t)$ product-series identities which come from keeping track of certain combinatorial statistics associated to periodic standard Young tableaux.