Diagonal Supersymmetry for Coinvariant Rings

TL;DR

This paper establishes a new algebraic structure on bosonic-fermionic coinvariant rings for finite groups, proving a conjecture related to diagonal supersymmetry and character series decomposition.

Contribution

It introduces a natural module structure involving super Lie algebras and symmetric groups, proving the diagonal supersymmetry conjecture for symmetric groups.

Findings

Character series are sums of super Schur functions times irreducible characters.

The module structure is universal, independent of parameters k and j.

Proves the diagonal supersymmetry conjecture of Bergeron (2020).

Abstract

For finite groups $G$, we show that bosonic-fermionic coinvariant rings have a natural $U(\mathfrak{gl}(k|j)) \otimes \mathbb{C}[G]$-module structure. In particular, we show that their character series are a sum of super Schur functions $s_\lambda(\mathbf{q}/\mathbf{u})$ times irreducible characters of $G$ with universal coefficients, which do not depend on $k,j$. In the case where $G$ is the symmetric group with diagonal action, this proves the "Diagonal Supersymmetry" conjecture of Bergeron (2020).