Dirac cohomology, the projective supermodules of the symmetric group and the Vogan morphism

Kieran Calvert

doi:10.1093/qmath/hay057

Dirac cohomology, the projective supermodules of the symmetric group and the Vogan morphism

The Quarterly Journal of Mathematics ◽

10.1093/qmath/hay057 ◽

2018 ◽

Vol 70 (2) ◽

pp. 535-563 ◽

Cited By ~ 1

Author(s):

Kieran Calvert

Keyword(s):

Symmetric Group ◽

Hecke Algebras ◽

Explicit Description ◽

Weyl Groups ◽

Explicit Model ◽

Dirac Theory ◽

Dirac Cohomology ◽

Graded Modules ◽

Projective Representations ◽

Spectrum Data

Abstract We derive an explicit description of the genuine projective representations of the symmetric group Sn using Dirac cohomology and the branching graph for the irreducible genuine projective representations of Sn. Ciubotaru and He [D. Ciubotaru and X. He, Green polynomials of Weyl groups, elliptic pairings, and the extended index. Adv. Math., 283:1–50, 2015], using the extended Dirac index, showed that the characters of the projective representations of Sn are related to the characters of elliptic-graded modules. We derive the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties ℬe of g and are able to use Dirac cohomology to construct an explicit model for the projective representations. We also describe Vogan’s morphism for Hecke algebras in type A using spectrum data of the Jucys–Murphy elements.

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Murnaghan-Nakayama Rules for Characters of Iwahori-Hecke Algebras of the Complex Reflection Groups G(r, p, n)

Canadian Journal of Mathematics ◽

10.4153/cjm-1998-009-x ◽

1998 ◽

Vol 50 (1) ◽

pp. 167-192 ◽

Cited By ~ 11

Author(s):

Tom Halverson ◽

Arun Ram

Keyword(s):

Symmetric Group ◽

Infinite Series ◽

Hecke Algebras ◽

Hecke Algebra ◽

Diagonal Matrix ◽

Weyl Groups ◽

Reflection Groups ◽

Irreducible Characters ◽

Complex Reflection ◽

Complex Reflection Groups

AbstractIwahori-Hecke algebras for the infinite series of complex reflection groups G(r, p, n) were constructed recently in the work of Ariki and Koike [AK], Broué andMalle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in whichwe derivedMurnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of G(r, p, n) given by Ariki and Koike [AK] and Ariki [Ari].

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Rouquier blocks of the cyclotomic Hecke algebras of G(de, e, r)

Nagoya Mathematical Journal ◽

10.1017/s0027763000009880 ◽

2010 ◽

Vol 197 ◽

pp. 175-212

Author(s):

Maria Chlouveraki

Keyword(s):

Infinite Series ◽

Hecke Algebras ◽

Weyl Groups ◽

Reflection Groups ◽

Complex Reflection ◽

Complex Reflection Groups ◽

Cyclotomic Hecke Algebras

The Rouquier blocks of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the families of characters defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite seriesG(de, e, r), thus completing their calculation for all complex reflection groups.

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The noncommutativity of Hecke algebras associated to Weyl groups

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1995-1254829-9 ◽

1995 ◽

Vol 123 (8) ◽

pp. 2363-2363 ◽

Cited By ~ 1

Author(s):

Michael R. Anderson

Keyword(s):

Hecke Algebras ◽

Weyl Groups

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Projective representations of rotation subgroups of Weyl groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410006638x ◽

1987 ◽

Vol 101 (1) ◽

pp. 21-35 ◽

Cited By ~ 3

Author(s):

D. Theo

Keyword(s):

Recent Work ◽

Ad Hoc ◽

Weyl Groups ◽

Orthogonal Groups ◽

Central Extensions ◽

Projective Representations ◽

Rotation Groups ◽

Spin Representations ◽

Double Coverings

By exploiting the well known spin representations of the orthogonal groups O(l), Morris [12] was able to give a unified construction of some of the projective representations of Weyl groups W(Φ) which had previously only been available by ad hoc means [5]. The principal purpose of the present paper is to give a corresponding construction for projective representations of the rotation subgroups W+(Φ) of Weyl groups. Thus we construct non-trivial central extensions of W+(Φ) via the well-known double coverings of the rotation groups SO(l). This adaptation allows us to give a unified way of obtaining the basic projective representations of W+(Φ) from those of W(Φ) determined in [12]. Hence our work is a development of the recent work of Morris, and is an extension of Schur's work on the alternative groups [15].

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Characters of projective representations of the infinite generalized symmetric group

Sbornik Mathematics ◽

10.1070/sm2008v199n10abeh003966 ◽

2008 ◽

Vol 199 (10) ◽

pp. 1421-1450

Author(s):

A V Dudko ◽

N I Nessonov

Keyword(s):

Symmetric Group ◽

Projective Representations

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Some combinatorial results involving shifted Young diagrams

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410006388x ◽

1986 ◽

Vol 99 (1) ◽

pp. 23-31 ◽

Cited By ~ 18

Author(s):

A. O. Morris ◽

A. K. Yaseen

Keyword(s):

Symmetric Group ◽

Block Structure ◽

Young Diagrams ◽

Linear Representations ◽

Projective Representations

In [6] the first author introduced some combinatorial concepts involving Young diagrams corresponding to partitions with distinct parts and applied them to the projective representations of the symmetric group Sn. A conjecture concerning the p-block structure of the projective representations of Sn was formulated in terms of these concepts which corresponds to the well-known, but long proved, Nakayama ‘conjecture’ for the p-block structure of the linear representations of Sn. This conjecture has recently been proved by Humphreys [1].

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On bases of centres of Iwahori–Hecke algebras of the symmetric group

Journal of Algebra ◽

10.1016/j.jalgebra.2005.03.030 ◽

2005 ◽

Vol 289 (1) ◽

pp. 42-69 ◽

Cited By ~ 3

Author(s):

Andrew Francis ◽

Lenny Jones

Keyword(s):

Symmetric Group ◽

Hecke Algebras

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Irreducible Modular Representations of the Reflection GroupG(m,1,n)

Journal of Mathematics ◽

10.1155/2015/808520 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9

Author(s):

José O. Araujo ◽

Tim Bratten ◽

Cesar L. Maiarú

Keyword(s):

Symmetric Group ◽

Weyl Group ◽

Reflection Group ◽

Geometric Construction ◽

Weyl Groups ◽

Modular Representations ◽

Complex Reflection ◽

Complex Reflection Group ◽

One Year ◽

Combinatorial Methods

In an article published in 1980, Farahat and Peel realized the irreducible modular representations of the symmetric group. One year later, Al-Aamily, Morris, and Peel constructed the irreducible modular representations for a Weyl group of typeBn. In both cases, combinatorial methods were used. Almost twenty years later, using a geometric construction based on the ideas of Macdonald, first Aguado and Araujo and then Araujo, Bigeón, and Gamondi also realized the irreducible modular representations for the Weyl groups of typesAnandBn. In this paper, we extend the geometric construction based on the ideas of Macdonald to realize the irreducible modular representations of the complex reflection group of typeG(m,1,n).

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