All Questions
Tagged with hecke-algebras coxeter-groups
18 questions
38
votes
5
answers
11k
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 ...
- 5,243
20
votes
0
answers
394
views
A spin extension of a Coxeter group?
Consider a Coxeter group $W$ with simple generators $S$ and Coxeter matrix $\left( m_{s,t}\right) _{\left( s,t\right) \in S\times S}$.
Let $\mathfrak{M}$ be the set of all pairs $\left(s, t\right) ...
- 33.8k
19
votes
4
answers
973
views
Are there Hamilton paths in Cayley graphs of Coxeter groups?
Hi everyone.
I want to optimize certain computation on finite Coxeter groups $(W,S)$. Basically I compute the matrices $\rho(T_w)$ for all $w\in W$ of a matrix representation $H\to K^{d\times d}$ of ...
- 9,650
18
votes
2
answers
3k
views
Is Soergel's proof of Kazhdan-Lusztig positivity for Weyl groups independent of other proofs?
Let $(W, S)$ be a Coxeter system. Soergel defined a category of bimodules $B$ over a polynomial ring whose split Grothendieck group is isomorphic to the Hecke algebra $H$ of $W$. Conjecturally, the ...
- 118k
15
votes
3
answers
799
views
Kazhdan-Luzstig Polynomials and Lower Intervals in the Bruhat Order
I have read in a number of places that the lower Bruhat interval $[e, w]$ is rank-symmetric if and only if the KL-polynomial $P_{e, w}(q) = 1$. All of the proofs I've come across use "rationally ...
- 341
9
votes
2
answers
911
views
Subexpressions of reduced words in Coxeter groups
Let $\underline{w} = [s_1, s_2, \dots ,s_n]$ be a reduced expression in a Coxeter group $W$. Given $x$ in $W$ one can consider the set $\Pi(\underline{w},x)$ consisting of all subexpressions of $\...
- 6,162
8
votes
3
answers
635
views
Motivation for the Kazhdan-Lusztig involution
I would like to know about the motivation behind the Kazhdan–Lusztig involution on an Iwahori–Hecke algebra.
I'll borrow the conventions from Libedinsky's Gentle introduction to Soergel bimodules I: ...
- 721
6
votes
1
answer
400
views
Efficient enumeration of Bruhat intervals
Hi everyone.
I'm currently programming some stuff for Hecke algebras. My current implementations have several bottlenecks and I'd like to improve that as much as I can so that I can use stuff like $...
- 9,650
6
votes
1
answer
587
views
When are parabolic Kazhdan-Lusztig polynomials nonzero?
Let $W$ be a Coxeter group with simple reflections $S$ and let $J \subseteq S$. Let $P^J_{\tau, \sigma}$ be the parabolic Kazhdan-Lusztig polynomials in the case $u = q$ in the sense of On Some ...
- 1,021
5
votes
2
answers
2k
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 ...
- 899
5
votes
0
answers
207
views
Kazhdan-Lusztig basis elements appearing in product with distinguished involution
My apologies if the below is too malformed to make sense.
Let $(W,S)$ be the affine Weyl group of a reductive group $G$, and let $\{C_w\}$ be the Kazhdan-Lusztig $C$-basis (an answer in terms of the $...
- 413
4
votes
1
answer
241
views
Number of pairs of permutation in $S_n$ whose $\mu$-coefficient (of their Kazhdan Lusztig polynomial) is non-zero
I am interested in how many pairs of permutations $(u,w)$ in $S_n$, such that the $\mu$-coefficient of its Kazhdan-Lusztig polynomial $P_{u,w}(q)$ is non-zero? where $\mu_{u,w}=[q^{\frac{l(w)-l(u)-1}{...
4
votes
0
answers
202
views
Shifts in the decomposition of Bott-Samelson bimodules
Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that $W\subset\...
- 868
4
votes
0
answers
94
views
Highest (short) roots and commutation relations in (twisted) DAHA
I am trying to understand, explicitly, the commutation relation between $X_\vartheta$ and $Y_\vartheta = T_0T_{s_\vartheta}$ in the (twisted) DAHA for a root system $R$, where $\vartheta$ is the ...
- 41
3
votes
1
answer
237
views
Reference request: Finite (multi-parameter) Iwahori-Hecke algebras are pairwise isomorphic
Let $(W,S)$ be a Coxeter system. Let $q=(q_s)_{s\in S} \in \mathbb{R}^{\text{#}S}$ be a tuple of positive real numbers with $q_s=q_t$ whenever $s$ and $t$ are conjugate to each other. Follwing Davis', ...
- 741
3
votes
0
answers
414
views
Kazhdan-Lusztig basis of the symmetric group algebra $\mathbb{K}S_n$
Let $S_n$ be the symmetric group on the set $\{1,2,\ldots,n\}$ and $\mathbb{K}$ a field. Denote by $\mathbb{K}S_n$ the symmetric group algebra.
What's Kazhdan-Lusztig basis of $\mathbb{K}S_n$? I ...
- 331
2
votes
0
answers
72
views
Does this finitely-generated algebra have a name?
I've been led to consider certain finitely-generated algebras that arise from some Coxeter groups (finite and affine Weyl groups at least). As a very concrete example, consider the infinite dihedral ...
- 13.3k
1
vote
0
answers
32
views
Cellular basis of $KW(B_2)$
Take $K$ a field. Let $W(B_2)$ be the coxeter group of type $B_2$ with a set $\{s_0,s_1\}$ of generators. Then $W(B_2)=\{1, s_0,s_1,s_0s_1,s_1s_0,s_0s_1s_0, s_1s_0s_1,s_1s_0s_1s_0\}$. I know that the ...
- 331