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**34**

votes

**1**answer

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### 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 ...

**31**

votes

**5**answers

5k 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 ...

**25**

votes

**3**answers

983 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 ...

**22**

votes

**2**answers

460 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 ...

**16**

votes

**2**answers

543 views

### why most of the angles are right

The Coxeter–Dynkin diagrams tell us that in a spherical Coxeter simplex most of the dihedral angles are right. Say among $\tfrac{n{\cdot}(n+1)}2$ dihedral angles we can have at most $n$ angles which ...

**15**

votes

**3**answers

806 views

### Actions on Sⁿ with quotient Sⁿ

What is known about isometric actions on $\mathbb S^n$ such that the quotient space is homeomorphic to $\mathbb S^n$?
Comments.
I am mostly interested in (maybe trivial) properties of such ...

**15**

votes

**2**answers

442 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 ...

**15**

votes

**3**answers

574 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 ...

**14**

votes

**2**answers

2k 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 ...

**14**

votes

**1**answer

589 views

### Coxeter Arrangements and an Identity

Let $\{A_i\}$ be a collection of $m$ hyperplanes in $\mathbb{C}^n$ which all pass through the origin (a central hyperplane arrangement). Such an arrangement is called Coxeter if reflecting across any ...

**13**

votes

**3**answers

738 views

### Is {6,3,7} an 'ultrahyperbolic' Coxeter group?

These pictures, drawn by Roice Nelson, are attempts to visualize a geometry having as symmetries the {6,3,7} Coxeter group, by which I mean the one coming from the Coxeter diagram
...

**13**

votes

**2**answers

843 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
$$B_{i,i+1} = R \otimes_{i,i+1} R$$
...

**13**

votes

**2**answers

332 views

### Coxeter exchanges in non-reduced words

Let $Q$ be a word in the generators of some Coxeter group, and consider a subword $R$ (not necessarily reduced, though I might want $Q$ to be).
Define the greedy or Demazure product of $R$ as ...

**12**

votes

**2**answers

692 views

### Does the Poincare series of a Coxeter group always describe a “flag variety”?

Let $W$ be a Coxeter group and let $P_W(q) = \sum_{w \in W} q^{\ell(w)}$ be its Poincare series. When $W$ is the Weyl group of a simple algebraic group $G$ (hence $W$ is finite), $P_W(q)$ is the ...

**12**

votes

**3**answers

459 views

### actions of the hyperoctahedral group

I am looking for actions (i.e., permutation representations) of the hyperoctahedral group $H_n$ (also known as the group of signed permutations) studied in the literature, i.e., homomorphisms from ...

**11**

votes

**3**answers

780 views

### A special tessellation

Let $P$ be a convex $n$-gon. Suppose that we have an infinite number of $P$s, and that each of them is colored either red or blue. Here, let us consider the following operations :
Operation 1 : Place ...

**11**

votes

**2**answers

923 views

### Random walks on Coxeter groups

Let $G_N$ be the group generated by elements $a_1,\ldots,a_N$ subject to the relations $a_i^2=1$ and $(a_ia_j)^3=1$. The growth function of $G_N$ is then
...

**10**

votes

**3**answers

506 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**

votes

**1**answer

110 views

### Cross between the nil-Hecke ring and the group ring of a Coxeter group

A Coxeter system $(W,S)$ has a set of generators $S=\{s_1,s_2,\ldots\}$ and the Coxeter group $W$ is determined by relations of the form $(s_is_j)^{m_{ij}}=1$ for some integers $m_{ij}$, where ...

**9**

votes

**0**answers

141 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” ...

**9**

votes

**0**answers

351 views

### Reflection groups in O(n+1,n) arising `in nature'?

For a while a friend and I have been thinking about a family of integral symmetric bilinear forms of signature $(n+1,n)$. Such lattices in our case arise 'in nature' (in a certain problem about vector ...

**8**

votes

**3**answers

757 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 ...

**8**

votes

**2**answers

453 views

### Symmetric functions in type B and type D

It is well known that the symmetric groups have a very nice and explicit representation theory. This is in particular true when one works with the collection of all symmetric groups simultaneously, in ...

**8**

votes

**2**answers

586 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 ...

**8**

votes

**1**answer

270 views

### Demazure product in Coxeter and Artin groups

As a follow-up of Allen's question Coxeter exchanges in non-reduced words, I wonder whether it is known that the Demazure product is well-defined in Artin groups. This is:
Let $(W,S)$ be a Coxeter ...

**8**

votes

**0**answers

79 views

### Ideals in strong Bruhat order

Recall that a (lower) ideal in a poset $(P, \le)$ is a subset $I\subset P$ such that $x\in I \Rightarrow y\in I$ for all $y\le x$. In am interested in ideals in finite Coxeter groups $W$ equipped ...

**8**

votes

**0**answers

266 views

### An angle-doubling trick of Kirillov and Berenstein [closed]

Kirillov and Berenstein, in their article "Groups generated by involutions, Gelfand-Tsetlin patterns and combinatorics of Young tableaux" (available at math.uoregon.edu/~arkadiy/bk1.pdf), present a ...

**7**

votes

**4**answers

398 views

### Is there a non-explicit characterization of the Specht modules?

It is a basic fact about the symmetric group $S_n$ that its irreducible representations are indexed by partitions of $n$.
My question is, can the association between partitions and irreps be ...

**7**

votes

**2**answers

297 views

### Are there infinitely many commensurable classes of finite-covolume hyperbolic Coxeter groups?

Allcock(2006) proved that
there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space $H^n$ for every $n\le 19$ (resp. $n\le 6$).
His main technique of ...

**7**

votes

**1**answer

307 views

### Monotonic maximal chains in a Coxeter group

Let $(W, S)$ be a Coxeter system, and let $T = \bigcup_{w \in W, s \in S} wsw^{-1}$. Associated to every element $t \in T$ is a unique positive root $\alpha_t \in \Phi^{+}$ considered as a vector in ...

**7**

votes

**1**answer

135 views

### Quantum Cartan matrices and Coxeter elements

Let $\Gamma$ be a bipartite graph, with the vertices partitioned into disjoint sets $\Pi_1$ and $\Pi_2$. Let $W$ be the associated Weyl group, with Coxeter generators $\{s_i\}_{i\in \Gamma}$. Let ...

**7**

votes

**0**answers

243 views

### Do p-compact groups have a sufficiently good notion of “flag variety” and “intersection cohomology”?

This is mostly an idle question, since I don't think I'd be able to do anything with a positive answer. But a positive answer would still be interesting, I think.
Background
An outstanding problem ...

**6**

votes

**2**answers

737 views

### Algorithm for reducing words in a Coxeter group

Let $W$ be a Coxeter group with set of simple reflections $S$. Suppose that I have chosen a preferred reduced decomposition for every element of $W$. Given an arbitrary word in the alphabet $S$, is ...

**6**

votes

**1**answer

413 views

### Quotient of Coxeter group

Since the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$ seems to have resisted attacks of some powerful programs, I will turn to a group that seems to be a bit easier to ...

**6**

votes

**2**answers

165 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 ...

**6**

votes

**1**answer

229 views

### Origin of the numbers game

The numbers game is a (one-player) game played on a finite graph with an initial assignment of numbers to its vertices, studied by Alon, Bj\"orner, Brenti, Donnelly, Eriksson, Krasikov, Mozes, Peres, ...

**6**

votes

**1**answer

200 views

### Reference for representation of Weyl group using r_alpha + c partial_alpha

Take $W = S_n$ for simplicity, though other Weyl groups work too. Let $r_i$ denote the $i$th simple reflection acting on ${\mathbb A}^n$, and $\partial_i = 1/(x_i - x_{i+1}) (Id - r_i)$ denote the ...

**6**

votes

**1**answer

98 views

### Bruhat order of reflection subgroups

Let $(W,S)$ be a Coxeter group, $T=\bigcup_{w\in W}wSw^{-1}$ its set of reflections, and $A\subseteq T$. From results of Dyer and Deodhar, we know that the subgroup $W_A$ generated by the elements of ...

**6**

votes

**1**answer

187 views

### Generators of pure braid groups of arbitrary Coxeter groups

Let $W$ be an arbitrary Coxeter group, and let $A$ be the associated Artin-Tits braid group, with standard Coxeter generators $\sigma_i\in A$. Let $P$ be the "pure braid group", the kernel of the ...

**6**

votes

**1**answer

299 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 ...

**6**

votes

**0**answers

141 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 ...

**5**

votes

**4**answers

655 views

### Infinite Coxeter groups with a non-trivial finite conjugacy class?

Let $(W,S)$ be a Coxeter system, where $S$ is finite. Assume that $W$ has an infinite number of elements.
Is it true that conjugacy classes of elements of non-central elements of $S$ have always an ...

**5**

votes

**1**answer

105 views

### Quotient of Coxeter complex in terms of double cosets?

In Victor Reiner's Quotients of Coxeter Complexes and $P$-Partitions, we have the below definition for the quotient complex of a Coxeter complex by a finite subgroup of the Coxeter group. I think ...

**5**

votes

**1**answer

276 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 ...

**5**

votes

**0**answers

521 views

### Applications (and source) of Bourbaki exercise on root systems with two root lengths?

In Chapters 4-6 of Bourbaki's Groupes et algebres de Lie, Exercise 20 for Section
VI.1 concerns irreducible (reduced) root systems with roots of two lengths: in other
words, systems of types $B_\ell, ...

**4**

votes

**1**answer

467 views

### Coxeter groups - Parabolic subgroups

In the theory of Coxeter groups there is the notion of so called parabolic subgroups. I'm wondering is this term just a random name, or there are some historical reasons? Why parabolic?
Thanks.

**4**

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**2**answers

244 views

### The action of the center on the extended Dynkin diagram

Let $R$ be an irreducible root system with a basis $\Pi$.
We obtain the Dynkin diagram $D$ and the extended Dynkin diagram ${\widetilde{D}}$ of $R$ with respect to $\Pi$.
Let $Q^\vee\subset P^\vee$ ...

**4**

votes

**2**answers

851 views

### Presentation of the pure Artin groups

Let $W$ be a Coxeter group attached to a Coxeter matrix with entries $m_{ij}$ . The presentation of $W$ is given by
$$W= < T_1, \dots, T_n | T_i^2=1, T_iT_jT_i \ldots = T_jT_iT_j \ldots, i \neq ...

**4**

votes

**2**answers

192 views

### Centralizers of reflections in special subgroups of Coxeter groups

Let us consider a (not necessarily finite) Coxeter group $W$ generated by a finite set of involutions $S=\{s_1,...,s_n\}$ subject (as usual) to the relations $(s_is_j)^{m_{i,j}}$ with ...

**4**

votes

**1**answer

174 views

### When do reflection groups act discretely on quadrics in indefinite/semi-Riemannian situation?

The hyperbolic case seems to be well understood after work of Vinberg. Given a lattice $L$ with quadratic form $Q$ of signature $(1,n)$, the orthogonal group of $L$ acts discretely on the affine ...