Invariant theory for coincidental complex reflection groups
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Mathematische Zeitschrift
Invariant theory for coincidental complex reflection groups Victor Reiner2 · Anne V. Shepler1 · Eric Sommers3 Received: 8 October 2019 / Accepted: 1 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Reflection groups · Invariant theory · Weyl groups · Coxeter groups · F-vector · H-vector Mathematics Subject Classification 13A50 · 05Axx · 20F55
Victor Reiner partially supported by NSF Grant DMS-1601961; Anne V. Shepler partially supported by Simons Foundation Grant #429539.
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Anne V. Shepler [email protected] Victor Reiner [email protected] Eric Sommers [email protected]
1
Department of Mathematics, University of North Texas, Denton, Texas, USA
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School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA
3
Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts, USA
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V. Reiner et al.
1 Introduction Molchanov [24] hypothesized a formula for the dimensions of invariants of certain finite real reflection groups acting on skew-symmetric tensors and dual tensors with polynomial coefficients. His formula gives evidence that Solomon’s invariant theory [35] for differential forms may have an extension to mixed derivation differential forms. We examine these forms not just for real reflection groups, but for complex reflection groups in general and compute their Hilbert series. We reformulate Molchanov’s hypothesis in terms of exponents and coexponents of the group. Although his formula and this reformulation do not hold for all real reflection groups, they do hold for the important class of coincidental reflection groups. The invariant theory of reflection groups acting on V = Cn displays a wondrous numerology controlled by two sequences of positive integers, the exponents e1 ≤ e2 ≤ · · · ≤ en and coexponents e1∗ ≤ e2∗ ≤ · · · ≤ en∗ . Solomon’s Theorem [35] gives the dimensions of W -inv
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