Polynomiality of Subdimensions of Diagonal Harmonics and a Sharp Stability Bound

TL;DR

This paper proves that the dimensions of certain diagonal harmonic spaces stabilize polynomially with respect to n, providing explicit formulas and a sharp stability bound based on combinatorial methods.

Contribution

It derives an explicit combinatorial polynomial for the dimension of bigraded diagonal harmonic spaces and establishes a sharp stability bound of a + b.

Findings

Dimension polynomial degree is a + b.

Explicit combinatorial formula for dimensions.

Sharp stability bound of a + b.

Abstract

A sequence of representations \(V_n\) of the symmetric group \(S_n\) is called representation (multiplicity) stable if, after some \(n\), the irreducible decomposition of \(V_n\) stabilizes. In particular, Church, Ellenburg and Farb (2015) showed that for fixed \(a\) and \(b\), the space of diagonal harmonics \(DH_n^{a,b}\) exhibits this behavior, with its dimension eventually stabilizing to a polynomial in \(n\). Building on this result, we use the Schedules Formula by Haglund and Loehr (2005) to obtain an explicit combinatorial polynomial for the dimension of the bigraded spaces \(DH_n^{a,b}\). This derivation not only yields the dimension formula but also produces a new sharp stability bound of \(a + b\), and determines the exact degree of the dimension polynomial, which is also \(a + b\).