## Explicit matrices for irreducible representations of Weyl groups

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- by John R. Stembridge PDF
- Represent. Theory
**8**(2004), 267-289 Request permission

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
[B]B D. Barbasch, - 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
[Gy]Gy A. Gyoja, - 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
[S2]S2 J. R. Stembridge, - 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
[Y]Y A. Young,

*Unitary spherical spectrum for split classical groups*, preprint.

*On the existence of a $W$-graph for an irreducible representation of a Coxeter group*, J. Algebra

**86**(1984), 422–438.

*A Maple package for root systems and finite Coxeter groups*, available electronically at www.math.lsa.umich.edu/$^\sim$jrs/maple.html.

*The collected papers of Alfred Young*, University of Toronto Press, Toronto, 1977. 0439548 (55:12438)

## Additional Information

**John R. Stembridge**- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1109
- Email: jrs@umich.edu
- Received by editor(s): March 12, 2004
- Published electronically: July 8, 2004
- Additional Notes: This work was supported by NSF grants DMS–0070685 and DMS–0245385
- © Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**8**(2004), 267-289 - MSC (2000): Primary 20F55, 20C40; Secondary 05E15, 20-04
- DOI: https://doi.org/10.1090/S1088-4165-04-00236-5
- MathSciNet review: 2077483