Young's Orthogonal Form of Irreducible Projective Representations of the Symmetric Group

1990 ◽  
Vol s2-42 (3) ◽  
pp. 437-451 ◽  
Author(s):  
M. L. Nazarov
Keyword(s):  
Symmetric Group ◽  
Projective Representations ◽  
Orthogonal Form

Some combinatorial results involving shifted Young diagrams

1986 ◽  
Vol 99 (1) ◽  
pp. 23-31 ◽  
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].

Fischer Matrices for Projective Representations of Generalized Symmetric Groups

2009 ◽  
Vol 16 (03) ◽  
pp. 449-462
Author(s):  
Mohammed S. Almestady ◽  
Alun O. Morris
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Symmetric Groups ◽  
Type B ◽  
Projective Representations ◽  
Ordinary Case ◽  
Covering Groups

The aim of this work is to calculate the Fischer matrices for the covering groups of the Weyl group of type Bn and the generalized symmetric group. It is shown that the Fischer matrices are the same as those in the ordinary case for the classes of Sn which correspond to partitions with all parts odd. For the classes of Sn which correspond to partitions in which no part is repeated more than m times, the Fischer matrices are shown to be different from the ordinary case.

scholarly journals The α-regular classes of the generalized symmetric group

1976 ◽  
Vol 17 (2) ◽  
pp. 144-150 ◽  
Author(s):  
E. W. Read
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Analogous Problem ◽  
Normal Subgroups ◽  
Projective Representations ◽  
Regular Classes ◽  
Factor Set

The α-regular classes of any finite group G are important since they are those classes on which the projective characters of G with factor set α take non-zero value, and thus a knowledge of the α-regular classes gives the number of irreducible projective representations of G with factor set α (see [4]). Here we look at the particular case of the generalized symmetric group Cm wr Sl. The analogous problem of constructing the irreducible projective representations of Cm wr Sl has been dealt with in [6] by generalizing Clifford's theory of inducing from normal subgroups, but unfortunately, it is not in general possible to determine the irreducible projective characters (and hence the α-regular classes) by this method.

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

2018 ◽  
Vol 70 (2) ◽  
pp. 535-563 ◽  
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.

Real Projective Representations of SN and AN

1994 ◽  
Vol 46 (3) ◽  
pp. 543-573
Author(s):  
John Q. Huang
Keyword(s):  
Symmetric Group ◽  
Structure Theorem ◽  
Alternating Group ◽  
The Real ◽  
The Third ◽  
Clifford Modules ◽  
Projective Representations

AbstractThree main results are obtained in this paper: one generalizes the Atiyah-Bott-Shapiro periodicity equivalence on the category of real Clifford modules, (Theorem 2.2); another establishes the existence of two algebras for real projective representations of the symmetric group Sn and the alternating group An, (Theorem 3.2) and determines their structure, (Theorem 6.1); the third describes all the real projective representations of Sn and An except for some small numbers n, (Theorem 7.2).

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