Explicit matrices for irreducible representations of Weyl groups

Stembridge, John

doi:10.1090/S1088-4165-04-00236-5

Explicit matrices for irreducible representations of Weyl groups
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by John R. Stembridge

Represent. Theory 8 (2004), 267-289

DOI: https://doi.org/10.1090/S1088-4165-04-00236-5

Published electronically: July 8, 2004

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Erratum: Represent. Theory 10 (2006), 48-48.

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.

References

Dan Barbasch and Allen Moy, A unitarity criterion for $p$-adic groups, Invent. Math. 98 (1989), no. 1, 19–37. MR 1010153, DOI 10.1007/BF01388842

Unitary spherical spectrum for split classical groups

Mark Benard, On the Schur indices of characters of the exceptional Weyl groups, Ann. of Math. (2) 94 (1971), 89–107. MR 297887, DOI 10.2307/1970736
J. S. Frame, Orthogonal group matrices of hyperoctahedral groups, Nagoya Math. J. 27 (1966), 585–590. MR 197583, DOI 10.1017/S0027763000026404
J. S. Frame, The classes and representations of the groups of $27$ lines and $28$ bitangents, Ann. Mat. Pura Appl. (4) 32 (1951), 83–119. MR 47038, DOI 10.1007/BF02417955
J. S. Frame, The characters of the Weyl group $E_{8}$, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 111–130. MR 0269751, DOI 10.1016/B978-0-08-012975-4.50017-5
Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
Brian D. O. Anderson and Michael Green, Hilbert transform and gain/phase error bounds for rational functions, IEEE Trans. Circuits and Systems 35 (1988), no. 5, 528–535. MR 936289, DOI 10.1109/31.1780

On the existence of a $W$-graph for an irreducible representation of a Coxeter group

86

Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
David Kazhdan and George Lusztig, A topological approach to Springer’s representations, Adv. in Math. 38 (1980), no. 2, 222–228. MR 597198, DOI 10.1016/0001-8708(80)90005-5
Takeshi Kondo, The characters of the Weyl group of type $F_{4}$, J. Fac. Sci. Univ. Tokyo Sect. I 11 (1965), 145–153 (1965). MR 185018
Arun Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), no. 1, 99–133. MR 1444315, DOI 10.1112/S0024611597000282
Daniel Edwin Rutherford, Substitutional Analysis, Edinburgh, at the University Press, 1948. MR 0027272
T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no. 3, 279–293. MR 491988, DOI 10.1007/BF01403165
John R. Stembridge, On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math. 140 (1989), no. 2, 353–396. MR 1023791, DOI 10.2140/pjm.1989.140.353

A Maple package for root systems and finite Coxeter groups

www.math.lsa.umich.edu/$^\sim$jrs/maple.html

Andrei Okounkov and Anatoly Vershik, A new approach to representation theory of symmetric groups, Selecta Math. (N.S.) 2 (1996), no. 4, 581–605. MR 1443185, DOI 10.1007/PL00001384

The collected papers of Alfred Young