scholarly journals The Garnir Relations for Weyl Groups of Type $C_n$

10.37236/797 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Himmet Can
Keyword(s):  
Symmetric Groups ◽  
Root Systems ◽  
Weyl Groups ◽  
Specht Modules ◽  
Combinatorial Description ◽  
Combinatorial Constructions

The Garnir relations play a very important role in giving combinatorial constructions of representations of the symmetric groups. For the Weyl groups of type $C_n$, having obtained the alternacy relation, we give an explicit combinatorial description of the Garnir relation associated with a $\Delta$-tableau in terms of root systems. We then use these relations to find a $K$-basis for the Specht modules of the Weyl groups of type $C_n$.

scholarly journals Submodules of Specht modules for Weyl groups

1996 ◽  
Vol 39 (1) ◽  
pp. 43-50
Author(s):  
Saіt Halicioğlu
Keyword(s):  
Characteristic Zero ◽  
Symmetric Groups ◽  
Root Systems ◽  
Weyl Groups ◽  
Arbitrary Field ◽  
Specht Modules ◽  
Irreducible Modules

The construction of all irreducible modules of the symmetric groups over an arbitrary field which reduce to Specht modules in the case of fields of characteristic zero is given by G. D. James. Halicioğlu and Morris describe a possible extension of James' work for Weyl groups in general, where Young tableux are interpreted in terms of root systems. In this paper we show how to construct submodules of Specht modules for Weyl groups.

scholarly journals Specht modules for finite reflection groups

1995 ◽  
Vol 37 (3) ◽  
pp. 279-287 ◽  
Author(s):  
S. HalicioǦlu
Keyword(s):  
Present Author ◽  
Symmetric Groups ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Arbitrary Field ◽  
Young Tableaux ◽  
Reflection Groups ◽  
Specht Modules ◽  
Irreducible Modules ◽  
Original Approach

Over fields of characteristic zero, there are well known constructions of the irreducible representations, due to A. Young, and of irreducible modules, called Specht modules, due to W. Specht, for the symmetric groups Sn which are based on elegant combinatorial concepts connected with Young tableaux etc. (see, e.g. [13]). James [12] extended these ideas to construct irreducible representations and modules over an arbitrary field. Al-Aamily, Morris and Peel [1] showed how this construction could be extended to cover the Weyl groups of type Bn. In [14] Morris described a possible extension of James' work for Weyl groups in general. Later, the present author and Morris [8] gave an alternative generalisation of James' work which is an extended improvement and extension of the original approach suggested by Morris. We now give a possible extension of James' work for finite reflection groups in general.

scholarly journals Zeta-functions of root systems and Poincaré polynomials of Weyl groups

2020 ◽  
Vol 72 (1) ◽  
pp. 87-126
Author(s):  
Yasushi Komori ◽  
Kohji Matsumoto ◽  
Hirofumi Tsumura
Keyword(s):  
Zeta Functions ◽  
Root Systems ◽  
Weyl Groups

scholarly journals Specht modules and symmetric groups

1975 ◽  
Vol 36 (1) ◽  
pp. 88-97 ◽  
Author(s):  
M.H Peel
Keyword(s):  
Symmetric Groups ◽  
Specht Modules

INVERSIONS IN CLASSICAL WEYL GROUPS

2007 ◽  
Vol 09 (01) ◽  
pp. 1-20
Author(s):  
KEQUAN DING ◽  
SIYE WU
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Root Systems ◽  
Flag Manifold ◽  
Length Function ◽  
Weyl Groups ◽  
Flag Manifolds ◽  
A Finite Group

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cnand Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

scholarly journals Free Lie algebras as modules for symmetric groups

1999 ◽  
Vol 67 (2) ◽  
pp. 143-156 ◽  
Author(s):  
R. M. Bryant ◽  
L. G. Kovács ◽  
Ralph Stöhr
Keyword(s):  
Lie Algebras ◽  
Symmetric Groups ◽  
Free Lie Algebra ◽  
Prime Characteristic ◽  
Specht Modules ◽  
Generating Set ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Free Lie Algebras ◽  
Direct Decompositions

AbstractLet r be a positive integer, F a field of odd prime characteristic p, and L the free Lie algebra of rank r over F. Consider L a module for the symmetric group , of all permutations of a free generating set of L. The homogeneous components Ln of L are finite dimensional submodules, and L is their direct sum. For p ≤ r ≤ 2p, the main results of this paper identify the non-porojective indecomposable direct summands of the Ln as Specht modules or dual Specht modules corresponding to certain partitions. For the case r = p, the multiplicities of these indecomposables in the direct decompositions of the Ln are also determined, as are the multiplicities of the projective indecomposables. (Corresponding results for p = 2 have been obtained elsewhere.)

The Vertices of a Class of Specht Modules and Simple Modules for Symmetric Groups in Characteristic 2

2012 ◽  
Vol 19 (spec01) ◽  
pp. 987-1016 ◽  
Author(s):  
Susanne Danz ◽  
Karin Erdmann
Keyword(s):  
Symmetric Groups ◽  
Specht Modules ◽  
Simple Modules ◽  
Characteristic 2

We study Specht modules S(n-2,2) and simple modules D(n-2,2) for symmetric groups 𝔖n of degree n over a field of characteristic 2. In particular, we determine the vertices of these modules, and also provide some information on their sources.

scholarly journals Representations of Weyl groups of type B induced from centralisers of involutions

1991 ◽  
Vol 44 (2) ◽  
pp. 337-344 ◽  
Author(s):  
Philip D. Ryan
Keyword(s):  
Weyl Group ◽  
Symmetric Groups ◽  
Conjugacy Classes ◽  
Weyl Groups ◽  
Type B ◽  
Complex Character ◽  
Linear Character ◽  
Complex Linear

Let G be a Weyl group of type B, and T a set of representatives of the conjugacy classes of self-inverse elements of G. For each t in T, we construct a (complex) linear character πt of the centraliser of t in G, such that the sum of the characters of G induced from the πt contains each irreducible complex character of G with multiplicity precisely 1. For Weyl groups of type A (that is, for the symmetric groups), a similar result was published recently by Inglis, Richardson and Saxl.

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