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Invariant theory for coincidental complex reflection groups

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Abstract

V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and he speculated that it had a certain product formula involving the exponents of the group. We show that Molchanov’s speculation is false in general but holds for all coincidental complex reflection groups when appropriately modified using exponents and co-exponents. These are the irreducible well-generated (i.e., duality) reflection groups with exponents forming an arithmetic progression and include many real reflection groups and all non-real Shephard groups, e.g., the Shephard-Todd infinite family G(d, 1, n). We highlight consequences for the q-Narayana and q-Kirkman polynomials, giving simple product formulas for both, and give a q-analogue of the identity transforming the h-vector to the f-vector for the coincidental finite type cluster/Cambrian complexes of Fomin–Zelevinsky and Reading. We include the determination of the Hilbert series for the non-coincidental irreducible complex reflection groups as well.

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Notes

  1. Specifically, one applies the map \(\psi \mapsto d\psi \) from [28], which is the same as the map \(\psi \mapsto \tilde{\theta }_E(\psi )\) described in Sect. 4 below.

  2. The authors thank Dennis Stanton for pointing them to this identity.

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Authors and Affiliations

  1. Department of Mathematics, University of North Texas, Denton, Texas, USA

    Anne V. Shepler

  2. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Victor Reiner

  3. Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts, USA

    Eric Sommers

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  1. Victor Reiner
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  2. Anne V. Shepler
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  3. Eric Sommers
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Correspondence to Anne V. Shepler.

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Victor Reiner partially supported by NSF Grant DMS-1601961; Anne V. Shepler partially supported by Simons Foundation Grant #429539.

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Reiner, V., Shepler, A.V. & Sommers, E. Invariant theory for coincidental complex reflection groups. Math. Z. 298, 787–820 (2021). https://doi.org/10.1007/s00209-020-02592-8

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