UC Berkeley

The (k, j)-bosonic-fermionic coinvariant ring R (k,j) n , defined for k sets of commuting variables and j sets of anticommuting variables, generalizes many previously-studied coinvariant rings. As special cases, R (k,j) n includes the classical coinvariant ring R (1,0) n , the diagonal coinvariant ring R (2,0) n studied by Haiman, the superspace coinvariant ring R (1,1) n , Zabrocki’s module for the Delta conjecture R (2,1) n , and the diagonal fermionic coinvariant ring R (0,2) n . In this dissertation, we prove results on R (k,j) n for specific small values of k, j, along with structural results for all k, j.We give the first conjectural construction of a monomial basis for the coinvariant ring R (1,2) n , for the symmetric group Sn acting on one set of commuting and two sets of anticommuting variables. Our construction interpolates between the modified Motzkin path basis for R (0,2) n of Kim–Rhoades (2022) and the super-Artin basis for R (1,1) n conjectured by Sagan–Swanson (2024) and proven by Angarone et al. (2025). We prove that our proposed basis has cardinality 2 n−1n!, aligning with a conjecture of Zabrocki (2020) on the dimension of R (1,2) n , and show how it gives a combinatorial expression for the Hilbert series.We also conjecture a Frobenius series for R (1,2) n . We show that these proposed Hilbert and Frobenius series on R (1,2) n are equivalent to conjectures of Iraci, Nadeau, and Vanden Wyngaerd (2024) in terms of segmented Smirnov words, by exhibiting a weight-preserving bijection between our proposed basis and their segmented permutations. We extend some of their results on the sign character to hook characters, and give a formula for the mµ coefficients of the conjectural Frobenius series.We conjecture a monomial basis for R (1,2) Bn , the analogous ring for the hyperoctahedral group Bn, and show that it has cardinality 4nn!.We determine the trigraded multiplicity of the sign character of the triagonal fermionic coinvariant ring R (0,3) n . As a corollary, this proves a conjecture of Bergeron (2020) that the multiplicity of the sign character of R (0,3) n is n 2 − n + 1.We also give an explicit formula for double hook characters in the diagonal fermionic coinvariant ring R (0,2) n , and discuss methods towards calculating the sign character of R (0,4) n .We give a multigraded refinement of a conjecture of Bergeron (2020) that the multiplicity of the sign character of the (1, 3)-bosonic-fermionic coinvariant ring R (1,3) n is 1/2 F3n, where Fn is the n-th Fibonacci number.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 sums of super Schur functions sλ(q/u) times irreducible characters of G with universal coefficients (independent of k, j). In the case where G is the symmetric group with diagonal action, this proves the “Diagonal Supersymmetry” conjecture of Bergeron (2020). Alfano (1994) determined a monomial basis, bigraded Hilbert series, and bigraded Frobenius series for the ring of diagonal dihedral group coinvariants R (2,0) I2(n) . Using diagonal supersymmetry, we determine a universal multigraded character series, universal multigraded Hilbert series, and monomial basis for the generalization to any k sets of bosonic variables and j sets of fermionic variables R (k,j) I2(n) .We determine that the standard character of all (k, j)-bosonic-fermionic coinvariant rings R (k,j) n has multigraded multiplicity s(1)(q/u) + s(2)(q/u) + · · · + s(n−1)(q/u).We note a conjectural relationship between multichains in the Tamari lattice and the multiplicity of the sign character in multivariate coinvariant rings R (k,0) n . We extend the operator conjecture of Haiman (1994) to bosonic-fermionic coinvariant rings R (k,j) n .