Alternating super-polynomials and super-coinvariants of finite reflection groups

2019/08/01 by Swanson, Joshua P
#05E10 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1908.00196

Abstract

Motivated by a recent conjecture of Zabrocki, Wallach described the alternants in the super-coinvariant algebra of the symmetric group in one set of commuting and one set of anti-commuting variables under the diagonal action. We give a type-independent generalization of Wallach's result to all real reflection groups G. As an intermediate step, we explicitly describe the alternating super-polynomials in k[V] ⊗ Λ(V) for all complex reflection groups, providing an analogue of a classic result of Solomon which describes the invariant super-polynomials in k[V] ⊗ Λ(V^*). Using our construction, we explicitly describe the alternating harmonics and coinvariants for all real reflection groups.

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