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Coinvariant algebras and fake degrees for spin Weyl groups of classical type

Published online by Cambridge University Press:  09 September 2013

CONSTANCE BALTERA
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, U.S.A. e-mails: cgb2k@virginia.edu, ww9c@virginia.edu
WEIQIANG WANG
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, U.S.A. e-mails: cgb2k@virginia.edu, ww9c@virginia.edu
Corresponding

Abstract

The coinvariant algebra of a Weyl group plays a fundamental role in several areas of mathematics. The fake degrees are the graded multiplicities of the irreducible modules of a Weyl group in its coinvariant algebra, and they were computed by Steinberg, Lusztig and Beynon–Lusztig. In this paper we formulate a notion of spin coinvariant algebra for every Weyl group. Then we compute all the spin fake degrees for each classical Weyl group, which are by definition the graded multiplicities of the simple modules of a spin Weyl group in the spin coinvariant algebra. The spin fake degrees for the exceptional Weyl groups are given in a sequel.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

[BW]Baltera, C. and Wang, W. Coinvariant algebras and fake degrees for spin Weyl groups of exceptional type. Preprint (2013), arXiv:1306.1290.Google Scholar
[BL]Beynon, W.M. and Lusztig, G.Some numerical results on the characters of exceptional Weyl groups. Math. Proc. Camb. Phil. Soc. 84 (1978), 417426.CrossRefGoogle Scholar
[BCT]Barbasch, D., Ciubotaru, D. and Trapa, P.Dirac cohomology for graded affine Hecke algebras. Acta Math. 209 (2012), 197227.CrossRefGoogle Scholar
[BMM]Broué, M., Malle, G. and Michel, J. Split Spetses for primitive reflection groups. arXiv:1204.5846.Google Scholar
[Ca]Carter, R.Conjugacy classes in the Weyl group. Comp. Math. 25 (1972), 159.Google Scholar
[CW]Cheng, S.-J. and Wang, W.Dualities and Representations of Lie Superalgebras. Graduate Studies in Math. 144 (Amer. Math. Soc., 2012).Google Scholar
[Hu]Humphreys, J.E.Reflection Groups and Coxeter Groups (Cambridge University Press, 1990).CrossRefGoogle Scholar
[IY]Ihara, S. and Yokonuma, T.On the second cohomology groups (Schur multipliers) of finite reflection groups. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 11 (1965), 155171.Google Scholar
[Joz1]Józefiak, T.Semisimple superalgebras. In Algebra–Some Current Trends (Varna, 1986), pp. 96113. Lect. Notes in Math. 1352 (Springer-Berlag, Berlin-New York, 1988).CrossRefGoogle Scholar
[Joz2]Józefiak, T.Characters of projective representations of symmetric groups. Expo. Math. 7 (1989), 193247.Google Scholar
[Joz3]Józefiak, T.A class of projective representations of hyperoctahedral groups and Schur Q-functions. Topics in Algebra, Banach Center Publ. 26, Part 2 (PWN-Polish Scientific Publishers, Warsaw 1990), 317326.CrossRefGoogle Scholar
[Kar]Karpilovsky, G.The Schur multiplier. London Math. Soc. Monogr. (N.S.) (Oxford University Press, 1987).Google Scholar
[KW1]Khongsap, T. and Wang, W.Hecke–Clifford algebras and spin Hecke algebras I: the classical affine type. Transform. Groups 13 (2008), 389412.CrossRefGoogle Scholar
[KW2]Khongsap, T. and Wang, W.Hecke–Clifford algebras and spin Hecke algebras II: the rational double affine type. Pacific J. Math. 238 (2008), 73103.CrossRefGoogle Scholar
[KW3]Khongsap, T. and Wang, W.Hecke–Clifford algebras and spin Hecke algebras IV: odd double affine type. Special Issue on Dunkl Operators and Related Topics, SIGMA 5 (2009), 012, 27 pages, arXiv:0810.2068.Google Scholar
[Kle]Kleshchev, A.Linear and Projective Representations of Symmetric Groups (Cambridge University Press, 2005).CrossRefGoogle Scholar
[Lu1]Lusztig, G.Irreducible representations of finite classical groups. Invent. Math. 43 (1977), 125175.CrossRefGoogle Scholar
[Lu2]Lusztig, G.Characters of reductive groups over a finite field. Ann. of Math Stud. 107 (Princeton University Press, 1984).Google Scholar
[Mac]Macdonald, I.G.Symmetric functions and Hall polynomials, Second edition (Clarendon Press, Oxford, 1995).Google Scholar
[Mo1]Morris, A.Projective characters of exceptional Weyl groups. J. Algebra 29 (1974), 567586.CrossRefGoogle Scholar
[Mo2]Morris, A.Projective representations of reflection groups. Proc. London Math. Soc. 32 (1976), 403420.CrossRefGoogle Scholar
[Re1]Read, E.Projective characters of the Weyl group of type F 4. J. London Math. Soc. (2), 8 (1974), 8393.CrossRefGoogle Scholar
[Re2]Read, E.On projective representations of the finite reflection groups of type Bl and Dl. J. London Math. Soc. (2), 10 (1975), 129142.CrossRefGoogle Scholar
[Sch]Schur, I.Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. J. Reine Angew. Math. 139 (1911), 155250.Google Scholar
[Se]Sergeev, A.The Howe duality and the projective representation of symmetric groups. Represent Theory 3 (1999), 416434.CrossRefGoogle Scholar
[Stm]Stembridge, J.The projective representations of the hyperoctahedral group. J. Algebra 145 (1992), 396453.CrossRefGoogle Scholar
[Stn]Steinberg, R.A geometric approach to the representations of the full linear group over a Galois field. Trans. Amer. Math. Soc. 71 (1951), 274282.CrossRefGoogle Scholar
[WW1]Wan, J. and Wang, W.Spin invariant theory for the symmetric group. J. Pure Appl. Algebra 215 (2011), 15691581.CrossRefGoogle Scholar
[WW2]Wan, J. and Wang, W.Lectures on spin representation theory of symmetric groups. Bull. Inst. Math. Acad. Sinica (N.S.) 7 (2012), 91164.Google Scholar
[WW3]Wan, J. and Wang, W.Frobenius character formula and spin generic degrees for Hecke-Clifford algebra. Proc. London Math. Soc. 106 (2013), 287317.CrossRefGoogle Scholar
[Ya]Yamaguchi, M.A duality of the twisted group algebra of the symmetric group and a Lie superalgebra. J. Algebra 222 (1999), 301327.CrossRefGoogle Scholar

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