Richard P. Stanley Seminar in Combinatorics: Diagonal supersymmetry for coinvariant rings

The classical coinvariant ring was generalized by Haiman (1994) to the diagonal coinvariant ring, which consists of a polynomial ring in two sets of variables quotiented by the ideal generated by polynomials invariant under the diagonal action of the symmetric group, without constant term. In this talk, we will survey several recent extensions of the diagonal coinvariant ring to (k,j)-bosonic-fermionic coinvariant rings, which are defined analogously for k sets of commuting (bosonic) and j sets of anticommuting (fermionic) variables. We prove the “diagonal supersymmetry” conjecture of Bergeron (2020), which asserts that the multigraded Frobenius series of a (k,j)-bosonic-fermionic coinvariant ring can be expressed in terms of universal coefficients, super Schur functions, and Frobenius characters. Finally, we compute some of these universal series coefficients and discuss further applications and connections with other open problems in algebraic combinatorics.

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