## Explicit matrices for irreducible representations of Weyl groups (2004)

Citations: | 6 - 2 self |

### BibTeX

@MISC{Stembridge04explicitmatrices,

author = {John R. Stembridge},

title = {Explicit matrices for irreducible representations of Weyl groups},

year = {2004}

}

### OpenURL

### Abstract

### Citations

417 | The representation theory of the symmetric groups - James - 1978 |

358 |
Representations of Coxeter groups and Hecke algebras
- Kazhdan, Lusztig
- 1979
(Show Context)
Citation Context ...] and [R]). Here, we are primarily concerned with the five exceptional groups. An alternative approach to the representations of a Weyl group involves the W -graph construction of Kazhdan and Lusztig =-=[KL1]-=-. In this approach, the representing matrices are encoded (mainly) by a single edge-weighted graph whose vertices correspond to basis elements of the representation. The original W -graphs in [KL1] pr... |

137 |
The Representation Theory of the Symmetric
- James, Kerber
- 1981
(Show Context)
Citation Context ...tations of a Weyl group W . For the symmetric groups, such matrices are well known, the most prominent being the seminormal and orthogonal matrix models constructed by Alfred Young [Y] (see also [G], =-=[JK]-=-, [OV], and [Ru]), and it is possible to extend these models to cover the remaining classical Weyl groups (e.g., see [F1] and [R]). Here, we are primarily concerned with the five exceptional groups. A... |

127 |
Characters of Finite Coxeter Groups and Iwahori-Hecke
- Geck, Pfeiffer
(Show Context)
Citation Context ...in [KL1] provide ZW -modules for each Kazhdan-Lusztig cell, although not all irreducible representations are afforded by such cells. Later work of Gyoja [Gy] (see also the discussion in Chapter 11 of =-=[GP]-=-) demonstrates that there is a W -graph affording every irreducible representation of every Weyl group, but knowing the existence of a W -graph is not the same as having explicit matrices. In order to... |

70 | A New Approach to the Representation Theory of the Symmetric Groups
- Vershik
(Show Context)
Citation Context ...s of a Weyl group W . For the symmetric groups, such matrices are well known, the most prominent being the seminormal and orthogonal matrix models constructed by Alfred Young [Y] (see also [G], [JK], =-=[OV]-=-, and [Ru]), and it is possible to extend these models to cover the remaining classical Weyl groups (e.g., see [F1] and [R]). Here, we are primarily concerned with the five exceptional groups. An alte... |

44 |
On the eigenvalues of representations of reflection groups and wreath products
- Stembridge
- 1989
(Show Context)
Citation Context ... restrict irreducibly to the same W -module (if µ ̸= ν), or to a sum of two distinct W -modules V ± µ,µ (if µ = ν and n is even). The latter pairs are clones, and it is known (e.g., by Theorem A.1 of =-=[S1]-=-) that the traces of the para-Coxeter element w = s2s4 ···sn on these two clones must differ. Via the branching rule for the B-series, one may deduce that the only other clones of an irreducible W -mo... |

42 |
A construction of representations of Weyl groups
- Springer
- 1978
(Show Context)
Citation Context ...[Y] (in particular, see QSA V for types B and D); the exceptional groups were settled by Kondo [K] and Benard [Be]. Later, Springer’s construction provided a more unified approach to the subject (see =-=[Sp]-=- and [KL2]). In view of Proposition 1.1, we may conclude the following. Theorem 2.1. Every representation of a Weyl group has a hereditary Q-basis that is seminormal, as well as a hereditary R-basis t... |

30 |
A unitarity criterion for p-adic groups
- Barbasch, Moy
- 1989
(Show Context)
Citation Context ...derstand the structure and classification of the unitary representations of real and p-adic semisimple Lie groups. For example, in the split p-adic case, it is known from the work of Barbasch and Moy =-=[BM]-=- that the unitarity of a spherical representation may be detected by testing an element of the group algebra RW for positive semi-definiteness in the regular representation. By passing to the simple c... |

29 | Seminormal representations of Weyl groups and Iwahori-Hecke algebras
- Ram
- 1997
(Show Context)
Citation Context ...thogonal matrix models constructed by Alfred Young [Y] (see also [G], [JK], [OV], and [Ru]), and it is possible to extend these models to cover the remaining classical Weyl groups (e.g., see [F1] and =-=[R]-=-). Here, we are primarily concerned with the five exceptional groups. An alternative approach to the representations of a Weyl group involves the W -graph construction of Kazhdan and Lusztig [KL1]. In... |

22 |
A topological approach to Springer’s representations
- Kazhdan, Lusztig
- 1980
(Show Context)
Citation Context ...articular, see QSA V for types B and D); the exceptional groups were settled by Kondo [K] and Benard [Be]. Later, Springer’s construction provided a more unified approach to the subject (see [Sp] and =-=[KL2]-=-). In view of Proposition 1.1, we may conclude the following. Theorem 2.1. Every representation of a Weyl group has a hereditary Q-basis that is seminormal, as well as a hereditary R-basis that is uni... |

16 | The unitary spherical spectrum for split classical groups
- Barbasch
(Show Context)
Citation Context ...educible W -representations, and in the split real cases, there is hope that these necessary conditions are sufficient. (In the split classical cases, recent work of Barbasch confirms the sufficiency =-=[B]-=-.) We plan to use the explicit matrix models reported on here to apply these tests for unitarity in the exceptional cases, with the ultimate goal being the classification of the spherical unitary dual... |

12 |
E.: Substitutional Analysis
- Rutherford
- 1948
(Show Context)
Citation Context ...l group W . For the symmetric groups, such matrices are well known, the most prominent being the seminormal and orthogonal matrix models constructed by Alfred Young [Y] (see also [G], [JK], [OV], and =-=[Ru]-=-), and it is possible to extend these models to cover the remaining classical Weyl groups (e.g., see [F1] and [R]). Here, we are primarily concerned with the five exceptional groups. An alternative ap... |

10 | A Maple package for root systems and finite Coxeter groups
- Stembridge
- 1992
(Show Context)
Citation Context ...asily generate these character tables starting from a permutation representation of the group; the Maple package coxeter provides character tables and fusion maps for all of the finite Coxeter groups =-=[S2]-=-. It is well known that branching from a classical Weyl group to the previous Weyl group in the same series is multiplicity-free. Less well known is that multiplicity-free branching is also found amon... |

9 |
The classes and representations of the groups of 27 lines and 28
- Frame
- 1951
(Show Context)
Citation Context ...roups of types A, B, D (and G2), these are well known and easy to compute. Among the exceptional groups, the character table of W (F4) was first obtained by Kondo [K], and W (En) (n=6, 7, 8) by Frame =-=[F2]-=-, [F3]. Modern computer algebra packages such as GAP and Magma can easily generate these character tables starting from a permutation representation of the group; the Maple package coxeter provides ch... |

5 |
A rational-function identity related to the Murnaghan-Nakayama formula for the characters of Sn
- Greene
- 1992
(Show Context)
Citation Context ...resentations of a Weyl group W . For the symmetric groups, such matrices are well known, the most prominent being the seminormal and orthogonal matrix models constructed by Alfred Young [Y] (see also =-=[G]-=-, [JK], [OV], and [Ru]), and it is possible to extend these models to cover the remaining classical Weyl groups (e.g., see [F1] and [R]). Here, we are primarily concerned with the five exceptional gro... |

3 | Characters of Coxeter groups and IwahoriHecke algebras - Geck, Pfeiffer - 2000 |

3 |
On the existence of a W -graph for an irreducible representation of a Coxeter group
- Gyoja
- 1984
(Show Context)
Citation Context ... of the representation. The original W -graphs in [KL1] provide ZW -modules for each Kazhdan-Lusztig cell, although not all irreducible representations are afforded by such cells. Later work of Gyoja =-=[Gy]-=- (see also the discussion in Chapter 11 of [GP]) demonstrates that there is a W -graph affording every irreducible representation of every Weyl group, but knowing the existence of a W -graph is not th... |

3 |
The collected papers of Alfred Young
- Young
- 1977
(Show Context)
Citation Context ...rreducible representations of a Weyl group W . For the symmetric groups, such matrices are well known, the most prominent being the seminormal and orthogonal matrix models constructed by Alfred Young =-=[Y]-=- (see also [G], [JK], [OV], and [Ru]), and it is possible to extend these models to cover the remaining classical Weyl groups (e.g., see [F1] and [R]). Here, we are primarily concerned with the five e... |

3 |
The characters of the Weyl group E8
- Frame
- 1967
(Show Context)
Citation Context ...of types A, B, D (and G2), these are well known and easy to compute. Among the exceptional groups, the character table of W (F4) was first obtained by Kondo [K], and W (En) (n=6, 7, 8) by Frame [F2], =-=[F3]-=-. Modern computer algebra packages such as GAP and Magma can easily generate these character tables starting from a permutation representation of the group; the Maple package coxeter provides characte... |

1 |
On the Schur indices of characters of the exceptional Weyl groups
- Benard
- 1971
(Show Context)
Citation Context ...-by-case basis. For the classical Weyl groups, it can be traced back to the work of Young [Y] (in particular, see QSA V for types B and D); the exceptional groups were settled by Kondo [K] and Benard =-=[Be]-=-. Later, Springer’s construction provided a more unified approach to the subject (see [Sp] and [KL2]). In view of Proposition 1.1, we may conclude the following. Theorem 2.1. Every representation of a... |

1 |
Orthogonal group matrices of hyperoctahedral groups
- Frame
- 1966
(Show Context)
Citation Context ...al and orthogonal matrix models constructed by Alfred Young [Y] (see also [G], [JK], [OV], and [Ru]), and it is possible to extend these models to cover the remaining classical Weyl groups (e.g., see =-=[F1]-=- and [R]). Here, we are primarily concerned with the five exceptional groups. An alternative approach to the representations of a Weyl group involves the W -graph construction of Kazhdan and Lusztig [... |

1 | The characters of the Weyl group - Frame - 1967 |

1 | The characters of the Weyl group of type F 4 - Kondo - 1965 |

1 | On the eigenvalues of representations of re groups and wreath products - Stembridge - 1989 |

1 | A Maple package for root systems and Coxeter groups, available electronically at hwww.math.lsa.umich.edu/~jrs/maple.htmli - Stembridge |

1 |
The characters of the Weyl group of type F4, J.Fac.Sci.Univ.TokyoSect.I
- Kondo
- 1965
(Show Context)
Citation Context ...ained on a case-by-case basis. For the classical Weyl groups, it can be traced back to the work of Young [Y] (in particular, see QSA V for types B and D); the exceptional groups were settled by Kondo =-=[K]-=- and Benard [Be]. Later, Springer’s construction provided a more unified approach to the subject (see [Sp] and [KL2]). In view of Proposition 1.1, we may conclude the following. Theorem 2.1. Every rep... |