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Transactions of the American Mathematical Society

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Restricting invariants of unitary reflection groups

Authors: Nils Amend, Angela Berardinelli, J. Matthew Douglass and Gerhard Röhrle
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 20F55; Secondary 13A50
Published electronically: February 14, 2018
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Abstract: Suppose that $ G$ is a finite, unitary reflection group acting on a complex vector space $ V$ and $ X$ is the fixed point subspace of an element of $ G$. Define $ N$ to be the setwise stabilizer of $ X$ in $ G$, $ Z$ to be the pointwise stabilizer, and $ C=N/Z$. Then restriction defines a homomorphism from the algebra of $ G$-invariant polynomial functions on $ V$ to the algebra of $ C$-invariant functions on $ X$. Extending earlier work by Douglass and Röhrle for Coxeter groups, we characterize when the restriction mapping is surjective for arbitrary unitary reflection groups $ G$ in terms of the exponents of $ G$ and $ C$ and their reflection arrangements. A consequence of our main result is that the variety of $ G$-orbits in the $ G$-saturation of $ X$ is smooth if and only if it is normal.

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Additional Information

Nils Amend
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Address at time of publication: Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Gottfried Wilhelm Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany

Angela Berardinelli
Affiliation: Department of Mathematics and Information Technology Mercyhurst University Erie, Pennsylvania 16546

J. Matthew Douglass
Affiliation: Department of Mathematics University of North Texas Denton, Texas 76203
Address at time of publication: Division of Mathematical Sciences, National Science Foundation, 2415 Eisenhower Ave, Alexandria, Virginia 22314

Gerhard Röhrle
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany

Keywords: Reflection arrangements, unitary reflection groups, invariants, smooth orbit variety
Received by editor(s): September 27, 2015
Received by editor(s) in revised form: October 24, 2016, and November 6, 2016
Published electronically: February 14, 2018
Additional Notes: This work was partially supported by a grant from the Simons Foundation (Grant #245399 to the third author). The third author would like to acknowledge that some of this material is based upon work supported by (while serving at) the National Science Foundation.
The authors acknowledge support from the DFG-priority program SPP1489 “Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory”.
Dedicated: To the memory of Robert Steinberg
Article copyright: © Copyright 2018 American Mathematical Society

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