Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/tran/7129
Published electronically: February 14, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461
  • [2] M. Auslander and D. A. Buchsbaum, On ramification theory in noetherian rings, Amer. J. Math. 81 (1959), 749-765. MR 0106929, https://doi.org/10.2307/2372926
  • [3] D. J. Benson, Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series, vol. 190, Cambridge University Press, Cambridge, 1993. MR 1249931
  • [4] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • [5] J. Denef and F. Loeser, Regular elements and monodromy of discriminants of finite reflection groups, Indag. Math. (N.S.) 6 (1995), no. 2, 129-143. MR 1338321, https://doi.org/10.1016/0019-3577(95)91238-Q
  • [6] J. Matthew Douglass and Gerhard Röhrle, Invariants of reflection groups, arrangements, and normality of decomposition classes in Lie algebras, Compos. Math. 148 (2012), no. 3, 921-930. MR 2925404, https://doi.org/10.1112/S0010437X11007512
  • [7] Meinolf Geck, Gerhard Hiss, Frank Lübeck, Gunter Malle, and Götz Pfeiffer, CHEVIE--a system for computing and processing generic character tables, Computational methods in Lie theory (Essen, 1994), Appl. Algebra Engrg. Comm. Comput. 7 (1996), no. 3, 175-210. MR 1486215, https://doi.org/10.1007/BF01190329
  • [8] Torsten Hoge and Gerhard Röhrle, Reflection arrangements are hereditarily free, Tohoku Math. J. (2) 65 (2013), no. 3, 313-319. MR 3102536, https://doi.org/10.2748/tmj/1378991017
  • [9] Robert B. Howlett, Normalizers of parabolic subgroups of reflection groups, J. London Math. Soc. (2) 21 (1980), no. 1, 62-80. MR 576184, https://doi.org/10.1112/jlms/s2-21.1.62
  • [10] G. I. Lehrer and T. A. Springer, Intersection multiplicities and reflection subquotients of unitary reflection groups. I, Geometric group theory down under (Canberra, 1996) de Gruyter, Berlin, 1999, pp. 181-193. MR 1714845
  • [11] Jean Michel, The development version of the CHEVIE package of GAP3, J. Algebra 435 (2015), 308-336. MR 3343221, https://doi.org/10.1016/j.jalgebra.2015.03.031
  • [12] M. Krishnasamy and D. E. Taylor, Normalisers of parabolic subgroups in finite unitary reflection groups, arXiv:1712.09563, preprint 2017.
  • [13] Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488
  • [14] R. W. Richardson, Normality of $ G$-stable subvarieties of a semisimple Lie algebra, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 243-264. MR 911144, https://doi.org/10.1007/BFb0079242
  • [15] M. Schonert et al,
    GAP - Groups, Algorithms, and Programming - version 3 release 4,
    Lehrstuhl D fur Mathematik, Rheinisch Westfalische Technische Hochschule, Aachen, Germany, 1997.
  • [16] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274-304. MR 0059914
  • [17] T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159-198. MR 0354894, https://doi.org/10.1007/BF01390173
  • [18] Robert Steinberg, Invariants of finite reflection groups, Canad. J. Math. 12 (1960), 616-618. MR 0117285, https://doi.org/10.4153/CJM-1960-055-3
  • [19] Robert Steinberg, Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc. 112 (1964), 392-400. MR 0167535, https://doi.org/10.2307/1994152
  • [20] D. E. Taylor, Reflection subgroups of finite complex reflection groups, J. Algebra 366 (2012), 218-234. MR 2942652, https://doi.org/10.1016/j.jalgebra.2012.04.033

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20F55, 13A50

Retrieve articles in all journals with MSC (2010): 20F55, 13A50


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
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
Email: gerhard.roehrle@rub.de

DOI: https://doi.org/10.1090/tran/7129
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

American Mathematical Society