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arXiv · 1908.00196

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

Joshua P Swanson
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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.

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Joshua P Swanson. 2019-08-01. Alternating super-polynomials and super-coinvariants of finite reflection groups. https://arxiv.org/abs/1908.00196

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