Diagonal Supersymmetry for Coinvariant Rings
John Lentfer
TL;DR
The paper introduces $(k|j)$-bosonic-fermionic coinvariant rings $R_G^{(k,j)}$ for a finite group $G$ and shows they carry a natural $U( rak{gl}(k|j)) \otimes \mathbb{C}[G]$-module structure. Using Howe duality, the authors decompose $R_G^{(k,j)}$ into simple factors with universal coefficients $c_{\lambda\mu}$, yielding a character formula $\mathrm{Char}(R_G^{(k,j)};\mathbf{q};\mathbf{u}) = \sum_{\lambda,\mu} c_{\lambda\mu} s_\lambda(\mathbf{q}/\mathbf{u}) \chi^{\mu}$ that is independent of $(k,j)$. For $G=\mathfrak{S}_n$ acting diagonally, this provides a universal, $(k,j)$-independent description of the multigraded Frobenius series, proving Bergeron's Diagonal Supersymmetry conjecture. The paper also develops a universal Frobenius series $\mathrm{Frob}(R_n^{(\infty,\infty)}; \mathbf{q};\mathbf{u})$, studies its restrictions to finite $(k,j)$, and analyzes the corresponding Hilbert series via cancellation, connecting to Delta-operator conjectures and known results in Macdonald theory.
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 $s_λ(\mathbf{q}/\mathbf{u})$ 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).