Diagonal supersymmetry for coinvariant rings

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 sums of super Schur functions $s_{\hat{I}}»(\mathbf{q}/\mathbf{u})$ 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 F. Bergeron (2020).

18 pages, minor revisions, updated references, corrected statement of Corollary 3.2 (previously Corollary 3.1)