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

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 $k[V] \otimes \Lambda(V)$ for all complex reflection groups, providing an analogue of a classic result of Solomon which describes the invariant super-polynomials in $k[V] \otimes \Lambda(V^*)$. Using our construction, we explicitly describe the alternating harmonics and coinvariants for all real reflection groups.