Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with coxeter-groups
Search options not deleted
user 4231
A Coxeter group is a group defined by a presentation by involutions $r_i$ with relators $(r_ir_j)^{m_{ij}}=1$ for certain family $(m_{ij})$ of integers greater than 1.
41
votes
2
answers
2k
views
Implications of non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials?
In their seminal 1979 paper Representations of Coxeter groups and Hecke algebras (Invent. Math. 53, doi:10.1007/BF01390031),
Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the corr …
- 52.9k
35
votes
3
answers
2k
views
Is there any need to study Coxeter systems (W,S) with S infinite?
In their treatise Groupes et algebres de Lie, Bourbaki (no doubt heavily influenced by Tits) devoted Chapter IV (1968) to the general theory of what they dubbed "Coxeter systems" $(W,S)$ along with "T …
- 52.9k
24
votes
2
answers
609
views
Uniform proof that a finite (irreducible real) reflection group is determined by its degrees?
Given a finite (irreducible real) group $G$ generated by reflections acting on euclidean $n$-space, it was shown by Chevalley in the 1950s that the algebra of invariants of $G$ in the associated polyn …
- 52.9k
13
votes
Accepted
Coxeter groups - Parabolic subgroups
The history is definitely somewhat convoluted. Note first that the term "Coxeter group" itself was introduced by Bourbaki in their 1968 volume containing chapters 4-6 of Groupes et algebres de Lie. …
- 52.9k
12
votes
0
answers
416
views
Reference for class of involutions containing longest element of finite Coxeter group?
Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” reflectio …
- 52.9k
12
votes
Accepted
Is Soergel's proof of Kazhdan-Lusztig positivity for Weyl groups independent of other proofs?
My understanding is that Soergel's approach applies just to finite Weyl groups and not directly to other finite Coxeter groups (or more generally), since what he can actually prove depends on some of …
- 52.9k
10
votes
Accepted
Representations of finite Coxeter groups
There are many relevant papers, but the most convenient book to consult is:
MR1778802 (2002k:20017) 20C15 (20C08 20F55),
Geck, Meinolf (F-LYON-GD); Pfeiffer,G¨otz (IRL-GLWY)
Characters of finite Coxet …
- 52.9k
9
votes
2
answers
687
views
Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs)?
There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl group …
- 52.9k
8
votes
Accepted
How to think about the simple reflection $s_0$ in the affine Weyl group?
You are asking several questions here, so it may be useful to separate out what is going on first in the setting of affine reflection groups. This is independent of the application to algebraic grou …
- 52.9k
7
votes
Accepted
weyl group representations
Let me add a useful reference book, probably no longer in print but found in many libraries: R.W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley-Interscience, 1985. …
- 52.9k
7
votes
1
answer
470
views
Centralizer of longest element in a finite irreducible Weyl group: related to folding of ADE...
Say $(W,S)$ is a finite Coxeter group, such as a Weyl group (which satisfies an additional crystallographic condition). Assume also that $W$ is irreducible. Then it has a longest element $w_o$ r …
- 52.9k
6
votes
Embed one Coxeter System into another
It's not clear exactly what your first sentence is asking for (there are a variety of surveys and books). For instance, are you only interested in finite Coxeter groups?
The embeddings of the non …
- 52.9k
5
votes
Accepted
Eigenvalues for elements of (infinite) Coxeter groups
The eigenvalues of elements of infinite order are certainly not trivial to study,
and as far as I know little has been determined about them.
Keep in mind that arbitrary infinite Coxeter groups are qu …
- 52.9k
5
votes
Accepted
Recursive formula for inverse Kazhdan-Lusztig polynomials
The case of an affine Weyl group is apparently the only one which has been looked at closely. But it may be hard to answer your specific question. As far as I know, there are two relevant papers, …
- 52.9k
5
votes
Accepted
Spectrum of adjacency matrix of simple Lie algebra.
The formulation is somewhat out of focus, starting with the notation $a_n(x)$ for characteristic polynomial (what is $n$?). The roots indicated do occur in Coxeter's formulation, but not as the eig …
- 52.9k