15
votes
2answers
398 views

Is there a natural notion of completion of a Coxeter system?

Let $(W,S)$ be a Coxeter system. Then any element of $W$ can be written as a finite products of elements of $S$. I want some notion of a "completion" of $W$, call it $\hat{W}$, whose elements are ...
2
votes
1answer
128 views

affine weyl group and affine schubert cells

Let $G$ a connected reductive split group over $k=\bar{k}$, $(B,T)$ a split Borel pair. Let $F:=k((t)))$. Let $\tilde{W}$ the extended Weyl group, $\tilde{W}=N_{G}(T(F))/T(O)$. By Iwasawa ...
1
vote
1answer
244 views

weights and exceptional root systems

Let $G$ a simple simply connected group over $\mathbb{C}$ and $W$ his Weyl group. Let $\lambda$ a minuscule or quasiminuscule weight. For which types and for which weights do we have that: $\forall ...
8
votes
3answers
533 views

Is there a list of Kazhdan-lusztig polynomials?

When studying the combinatorics and representation of Coxeter groups, I often find it irksome to compute KL- and R- polynomials from scratch on Maple or Sage. The time it consumes to generate a ...
6
votes
2answers
136 views

Irreducibility of Coxeter Graphs as a Function of Generating Sets

Given a Coxeter system $(W, S)$, we can form its Coxeter graph, and say that the system is irreducible if the graph is connected. Now, irreducibility is not solely a function of $W$; it depends also ...
34
votes
1answer
860 views

Implications of non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials?

In their seminal 1979 paper here, Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the corresponding Iwahori-Hecke algebra. In particular they showed how to pass from a standard ...
2
votes
2answers
419 views

elements in the weyl group

Let W be the Weyl a group of a semisimple simply connected group over C. Let I={1,...,r} the set of simple roots. For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple ...
6
votes
0answers
127 views

Classifying algebras with two idempotent generators and involution

Say that a finite-dimensional algebra $H$ over a field $K$ is dihedral if $H$ is generated by idempotents $P_1$ and $P_2$ and there is an algebra involution interchanging $P_1$ and $P_2$. For ...
12
votes
2answers
689 views

Intrinsic characterization of Soergel bimodules?

A Soergel bimodule (for $S_n$) is a bimodule over $R = \mathbb{Q}[x_1,\dots,x_n]$ which appears as a summand/grading shift of tensor products of the basic bimodules $$B_{i,i+1} = R \otimes_{i,i+1} R$$ ...
3
votes
1answer
261 views

Spectrum of adjacency matrix of simple Lie algebra.

Let $\mathfrak{g}$ be a finite dimensional, simple, complex Lie algebra. Define the Coxeter adjacency matrix to be the matrix $A=2I-C$ where $C$ is the Cartan matrix of $\mathfrak{g}$. Let ...
2
votes
2answers
1k views

weyl group representations

I am looking for some references about irreducible representations of the Weyl Group over simple Lie Groups, both classical and exceptional ones. In particular I want to know the dimensions and the ...
3
votes
2answers
739 views

Representations of finite Coxeter groups

What is reference for complex irreducible representations of Hecke algebra of finite Coxeter groups (say generic case q =1)? I am interested in knowing its Wedderburn decomposition. So want explicit ...
24
votes
5answers
4k views

Definitions of Hecke algebras

There is a definition of Iwahori-Hecke algebras for Coxeter groups in terms of generators and relations and there is a definition of Hecke algebras involving functions on locally compact groups. Are ...