Type B fermionic coinvariant rings

2026/08/03 by Yuhan Jiang, John Lentfer
Mathematics · #math.CO #math.RT #msc:05E10 #msc:05E05 #msc:20C33 #msc:20C30

paper · pdf

34 pages, 4 tables, 1 figure. Comments are welcome

arxiv created 2026/08/03 · arxiv updated 2026/08/05

Abstract

Let \mathfrakB_n denote the hyperoctahedral group. The type B coinvariant rings R_\mathfrakB_n^(k,j) are quotients of the ring of polynomials in k sets of n commuting variables and j sets of n anticommuting variables by the ideal generated by the diagonal \mathfrakB_n-invariants without constant term. Building upon the work of Kim--Rhoades (2022), we give an explicit formula for the bigraded Frobenius series of R_\mathfrakB_n^(0,2): the bigraded multiplicity of each irreducible \mathfrakB_n-character is a single Schur polynomial, so R_\mathfrakB_n^(0,2) is multiplicity-free as a GL₂ × \mathfrakB_n-module. We then determine that the trigraded multiplicity of the sign character of R_\mathfrakB_n^(0,3) is given by a single Schur function. Finally, for all k and j, we determine the multiplicity of the standard character in the type A coinvariant ring R_n^(k,j), as well as the multiplicities of the characters indexed by the bipartitions ((n-1),(1)) and ((n-1,1),\varnothing) in R_\mathfrakB_n^(k,j). These are the first nontrivial characters established for all (k,j) in either of types A or B.

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