Polynomiality of Subdimensions of Diagonal Harmonics and a Sharp Stability Bound

2025/06/19 by Xinxuan Wang, Wang, Xinxuan
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2506.16566

openalex publication_date 2025/06/19 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

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 \(a + b\).

Citations

• Monomial stability of Frobenius images
• Schedules and the Delta Conjecture
• A proof of the shuffle conjecture
• Multivariate Diagonal Coinvariant Spaces for Complex Reflection Groups

Related

• Stability theorems for multiplicities in graded S_n-modules
• Polynomial Hilbert-Schmidt stability of the lamplighter group
• Triangular Diagonal Harmonics Conjectures
• Universal graded characters and limit of Lusztig q-analogues
• Representation theory and the diagonal coinvariant ring of the type B Weyl group