Restricting invariants of unitary reflection groups
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- by Nils Amend, Angela Berardinelli, J. Matthew Douglass and Gerhard Röhrle PDF
- Trans. Amer. Math. Soc. 370 (2018), 5401-5424 Request permission
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.References
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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
- MR Author ID: 1076282
- Email: amend@math.uni-hannover.de
- Angela Berardinelli
- Affiliation: Department of Mathematics and Information Technology, Mercyhurst University, Erie, Pennsylvania 16546
- Email: aberardinelli@mercyhurst.edu
- 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
- Email: mdouglas@nsf.gov
- Gerhard Röhrle
- Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
- MR Author ID: 329365
- Email: gerhard.roehrle@rub.de
- 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”. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 5401-5424
- MSC (2010): Primary 20F55; Secondary 13A50
- DOI: https://doi.org/10.1090/tran/7129
- MathSciNet review: 3812110
Dedicated: To the memory of Robert Steinberg