The coxeter-groups tag has no usage guidance.

**15**

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**3**answers

567 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 ...

**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 ...

**13**

votes

**2**answers

825 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
$$B_{i,i+1} = R \otimes_{i,i+1} R$$
...

**10**

votes

**3**answers

479 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 ...

**2**

votes

**0**answers

260 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 ...

**3**

votes

**1**answer

296 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 ...

**5**

votes

**4**answers

628 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 ...

**4**

votes

**2**answers

826 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 ...

**5**

votes

**1**answer

272 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 ...

**8**

votes

**2**answers

577 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 ...

**2**

votes

**1**answer

367 views

### Orthogonal group of the lattice $I_{p,q}$?

Here $I_{p,q}$ is the unique-up-to-isometry unimodular lattice of signature $(p,q)$, whose Gram matrix is diagonal with $p$ 1s and $q$ -1s.
In his paper "ON GROUPS OF UNIT ELEMENTS OF CERTAIN ...

**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 ...

**2**

votes

**1**answer

233 views

### When are root hyperplanes locally finite?

For an even integral lattice $L$ (of arbitrary signature, non-degenerate but not necessarily unimodular), consider roots (vectors of square $2$) and their orthogonal hyperplanes.
First, I'd like to ...

**9**

votes

**0**answers

350 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 ...

**2**

votes

**2**answers

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 ...

**5**

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**0**answers

514 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, ...

**8**

votes

**2**answers

447 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 ...

**3**

votes

**0**answers

180 views

### How linearly independent are the obvious combinatorial invariants of a Bruhat interval?

Let $[u, v]$ be a Bruhat interval in some Coxeter group. Let $I$ be the set of all Bruhat intervals. I am interested in functions $I \to \mathbb{Z}$ which are invariant under poset isomorphisms. ...

**3**

votes

**2**answers

987 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 ...

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votes

**1**answer

295 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

**0**answers

242 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 ...

**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 ...

**12**

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**2**answers

674 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 ...

**1**

vote

**0**answers

584 views

### A question on a Davis complex of a Coxeter group

Let us have a look at p. 64 of M. Davis book "The Geometry and Topology of Coxeter Groups". The discussion preceeding Definition 5.1.3. shows that $\mathcal{U}(G, X)/G$ is homeomorphic to $X$. Theorem ...

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votes

**1**answer

198 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 ...

**14**

votes

**1**answer

558 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 ...

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### 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 ...