Explicit matrices for irreducible representations of Weyl groups
Author:
John R. Stembridge
Journal:
Represent. Theory 8 (2004), 267289
MSC (2000):
Primary 20F55, 20C40; Secondary 05E15, 2004
Posted:
July 8, 2004
Erratum:
Represent. Theory 10 (2006), 48
MathSciNet review:
2077483
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: We present algorithms for constructing explicit matrices for every irreducible representation of a Weyl group, with particular emphasis on the exceptional groups. The algorithms we present will produce representing matrices in either of two forms: real orthogonal, with matrix entries that are square roots of rationals, or rational and seminormal. In both cases, the matrices are ``hereditary'' in the sense that they behave well with respect to restriction along a chosen chain of parabolic subgroups.
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 J. S. Frame, Orthogonal group matrices of hyperoctahedral groups, Nagoya Math. J. 27 (1966), 585590. MR 0197583 (33:5748)
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 [KL1]
 D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165184. MR 0560412 (81j:20066)
 [KL2]
 D. Kazhdan and G. Lusztig, A topological approach to Springer's representations, Adv. in Math. 38 (1980), 222228. MR 597198 (82f:20076)
 [K]
 T. Kondo, The characters of the Weyl group of type , J. Fac. Sci. Univ. Tokyo Sect. I 11 (1965), 145153. MR 0185018 (32:2488)
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 A. Ram, Seminormal representations of Weyl groups and IwahoriHecke algebras, Proc. London Math. Soc. (3) 75 (1997), 99133. MR 1444315 (98d:20007)
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 D. E. Rutherford, Substitutional Analysis, University Press, Edinburgh, 1948. MR 0027272 (10:280i)
 [Sp]
 T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), 279293. MR 0491988 (58:11154)
 [S1]
 J. R. Stembridge, On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math. 140 (1989), 353396. MR 1023791 (91a:20022)
 [S2]
 J. R. Stembridge, A Maple package for root systems and finite Coxeter groups, available electronically at www.math.lsa.umich.edu/jrs/maple.html.
 [OV]
 A. Okounkov and A. Vershik, A new approach to representation theory of symmetric groups, Selecta Math.(N.S.) 2 (1996), 581605. MR 1443185 (99g:20024)
 [Y]
 A. Young, The collected papers of Alfred Young, University of Toronto Press, Toronto, 1977. 0439548 (55:12438)
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Additional Information
John R. Stembridge
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1109
Email:
jrs@umich.edu
DOI:
http://dx.doi.org/10.1090/S1088416504002365
PII:
S 10884165(04)002365
Received by editor(s):
March 12, 2004
Posted:
July 8, 2004
Additional Notes:
This work was supported by NSF grants DMS–0070685 and DMS–0245385
Article copyright:
© Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
