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This research demonstrates a module structure in coinvariant rings, showing implications for finite groups and Diagonal Supersymmetry.
For finite groups G, we show that bosonic-fermionic coinvariant rings have a natural U(gl(k|j)) ⊗ C[G]-module structure. In particular, we show that their character series are a sum of super Schur functions s_λ(q/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).
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John Lentfer (2025) studied this question.
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