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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 ...

**25**

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**3**answers

729 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 ...

**24**

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**5**answers

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 ...

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**0**answers

171 views

### Uniform proof that a finite reflection group is determined by its degrees?

Given a finite (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 polynomial ...

**15**

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**2**answers

376 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**

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**3**answers

495 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 ...

**12**

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**2**answers

664 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
$$B_{i,i+1} = R \otimes_{i,i+1} R$$
...

**12**

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**1**answer

428 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 ...

**12**

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**2**answers

603 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 ...

**11**

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**3**answers

667 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 ...

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1k 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 ...

**11**

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668 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
...

**11**

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**1**answer

239 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 ...

**10**

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**3**answers

637 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 ...

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367 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 ...

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325 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**

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**3**answers

503 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

415 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**

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**2**answers

476 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 ...

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**1**answer

337 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 ...

**7**

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**1**answer

236 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**

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**1**answer

87 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 ...

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408 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 ...

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130 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**

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**1**answer

194 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**

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**1**answer

217 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**

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122 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 ...

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### 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 ...

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**4**answers

549 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

231 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 ...

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### 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, ...

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votes

**2**answers

576 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**

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158 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 ...

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703 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 ...

**3**

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**1**answer

253 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 ...

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81 views

### Topological obstruction to icosahedral symmetry?

Let $G$ be a compact simple lie group of rank $n$. Then the Poincaré series of $G$ is given by $$P(G,q)=\prod_{i=1}^n (1+q^{2d_i-1}),$$
where the integers $d_1\leq d_2\leq \cdots \leq d_n$ are the ...

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409 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 ...

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votes

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966 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 ...

**2**

votes

**1**answer

148 views

### Eigenvalues for elements of (infinite) Coxeter groups

My current research requires some knowledge on the eigenvectors of elements (of infinite order) of Coxeter groups view as reflections in their geometric representation.
After some reading, my ...

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**1**answer

324 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 ...

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223 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 ...

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**1**answer

122 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 ...

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**1**answer

154 views

### Which subgroups of a finite reflection group have distingushed coset representatives?

Let $W$ be a finite reflection group with length function $l$ and let $I$ be a set of simple reflections that generate $W$. Let $\phi$ be an automorphism of $W$ permuting $I$. Consider the orbits of ...

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108 views

### simple roots of a reflection subgroup

Consider a Hermitian symmetric pair of complex Lie algebras $(\mathfrak{g},\mathfrak{k})$ and split the set of roots into compact roots (i.e. roots of $\mathfrak{k}$) and noncomapt roots $\Delta = ...

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189 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 ...

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170 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. ...

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vote

**1**answer

231 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 ...

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449 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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**2**answers

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### quasi-minuscule representations

Wich representations of $F_{4}$, $E_{8}$ and $G_{2}$ are quasi-minuscule?

**0**

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**1**answer

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### Subgroups of finite reflection groups that do not fix a point

Let $(W,S)$ be a finite irreducible Coxeter-System of rank $n$ and $E$ be a real reflection representation of $W$. Let $x\in E$ and suppose that the isotropy group of $x$ is generated by one element ...