scholarly journals Lattice structure of Weyl groups via representation theory of preprojective algebras

2018 ◽  
Vol 154 (6) ◽  
pp. 1269-1305 ◽  
Author(s):  
Osamu Iyama ◽  
Nathan Reading ◽  
Idun Reiten ◽  
Hugh Thomas
Keyword(s):  
Representation Theory ◽  
Weyl Group ◽  
Lattice Structure ◽  
Weak Order ◽  
Weyl Groups ◽  
Preprojective Algebra ◽  
Combinatorial Description ◽  
Irreducible Elements ◽  
Preprojective Algebras

This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\unicode[STIX]{x1D70F}$-rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$-rigid) modules and layers of $\unicode[STIX]{x1D6F1}$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$. We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\unicode[STIX]{x1D70F}^{-}$-rigid modules for type $A$ and $D$.

WEYL GROUPS AND BASIS ELEMENTS OF HECKE ALGEBRAS OF GELFAND–GRAEV REPRESENTATIONS

2011 ◽  
Vol 10 (05) ◽  
pp. 849-864 ◽  
Author(s):  
JULIANNE G. RAINBOLT
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Hecke Algebras ◽  
Initial Section ◽  
Weyl Groups ◽  
Group Of Lie Type ◽  
Combinatorial Description ◽  
Algorithmic Method ◽  
A Finite Group ◽  
Theoretic Description

The initial section of this article provides illustrative examples on two ways to construct the Weyl group of a finite group of Lie type. These examples provide the background for a comparison of the elements in the Weyl groups of GL(n, q) and U(n, q) that are used in the construction of the standard bases of the Hecke algebras of the Gelfand–Graev representations of GL(n, q) and U(n, q). Using a theorem of Steinberg, a connection between a theoretic description of bases of these Hecke algebras and a combinatorial description of these bases is provided. This leads to an algorithmic method for generating bases of the Hecke algebras of the Gelfand–Graev representations of GL(n, q) and U(n, q).

scholarly journals The Weak Order on Weyl Posets

2019 ◽  
Vol 72 (4) ◽  
pp. 867-899
Author(s):  
Joël Gay ◽  
Vincent Pilaud
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Coxeter Group ◽  
Lattice Structure ◽  
Root Systems ◽  
Weak Order ◽  
Generalized Associahedra

AbstractWe define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice induced by antisymmetric closed subsets of roots is again a lattice. We then study further subposets of this lattice that naturally correspond to the elements, the intervals, and the faces of the permutahedron and the generalized associahedra of the corresponding Weyl group. These results extend to arbitrary finite crystallographic root systems the recent results of G. Chatel, V. Pilaud, and V. Pons on the weak order on posets and its induced subposets.

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 On flag varieties, hyperplane complements and Springer representations of Weyl groups

1990 ◽  
Vol 49 (3) ◽  
pp. 449-485 ◽  
Author(s):  
G. I. Lehrer ◽  
T. Shoji
Keyword(s):  
Algebraic Group ◽  
Weyl Group ◽  
Nilpotent Element ◽  
Parabolic Type ◽  
Linear Algebraic Group ◽  
Weyl Groups ◽  
Complex Numbers ◽  
Flag Varieties ◽  
Group Representations ◽  
Definition Of

AbstractLet G be a connected reductive linear algebraic group over the complex numbers. For any element A of the Lie algebra of G, there is an action of the Weyl group W on the cohomology Hi(BA) of the subvariety BA (see below for the definition) of the flag variety of G. We study this action and prove an inequality for the multiplicity of the Weyl group representations which occur ((4.8) below). This involves geometric data. This inequality is applied to determine the multiplicity of the reflection representation of W when A is a nilpotent element of “parabolic type”. In particular this multiplicity is related to the geometry of the corresponding hyperplane complement.

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

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).

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.

scholarly journals An Order on Circular Permutations

10.37236/9982 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Antoine Abram ◽  
Nathan Chapelier-Laget ◽  
Christophe Reutenauer
Keyword(s):  
Symmetric Group ◽  
Weak Order ◽  
Rank Function ◽  
Weyl Groups ◽  
Eulerian Numbers ◽  
Affine Weyl Groups ◽  
Special Formula ◽  
Circular Permutations ◽  
Semidistributive Lattice ◽  
Young’S Lattice

Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

Admissible Sequences for Twisted Involutions in Weyl Groups

2011 ◽  
Vol 54 (4) ◽  
pp. 663-675 ◽  
Author(s):  
Ruth Haas ◽  
Aloysius G. Helminck
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Weyl Groups ◽  
One To One ◽  
Admissible Sequences

AbstractLetW be a Weyl group, Σ a set of simple reflections inW related to a basis Δ for the root system Φ associated with W and θ an involution such that θ(Δ) = Δ. We show that the set of θ- twisted involutions in W, = {w ∈ W | θ(w) = w–1} is in one to one correspondence with the set of regular involutions . The elements of are characterized by sequences in Σ which induce an ordering called the Richardson–Springer Poset. In particular, for Φ irreducible, the ascending Richardson–Springer Poset of , for nontrivial θ is identical to the descending Richardson–Springer Poset of .

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