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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.

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The simplest possible complex reflection group is the cyclic group of order $n$ acting on $\mathbb C$ by rotations. This is also the orientation-preserving subgroup of the dihedral group of order $2n$,...
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Let $G=(V,E)$ be a simple (finite) graph, and form the associated right-angled Artin group $W =\langle x_v, v \in V \mid x_ux_v = x_vx_u\textrm{ for $\{u,v\}\notin E$}\rangle$. (Note that the edges ...
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Let $(W,S)$ be the affine Weyl group associated to a simple root system. For $x,y \in W$ corresponding to alcoves in the dominant Weyl chamber, there are parabolic Kazhdan–Lusztig polynomials $n_{y,x}$...
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Using cubical complexes, Carl Droms in A complex for right-angled Coxeter groups, Proc. Amer. Math. Soc. 131 (2003), 2305-2311, https://doi.org/10.1090/S0002-9939-02-06774-6 proved that given a ...
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For um…reasons, I have been poking around OEIS, looking for subsequences that occur frequently. I have found that the number of reduced words of length $n$ in a Coxeter group on $N$ generators with ...
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Here is my question: As we know, for a string-type Coxeter diagram such as $$\circ\overset{p}{---}\circ \overset{q}{---}\circ \overset{r}{---}\circ \cdots \circ $$ where $p,q,r,\ldots$ are integers ...
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Let $(W,S)$ be a Coxeter system. Let $W'$ be subgroup of $W$ generated by finite number of (say, $r$) reflections (elements in $\bigcup_{w\in W}wSw^{-1}$) such that $W'$ is infinite, but any $r-1$ of ...
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Let $(W,S)$ be arbitary Coxeter system, and $V$ be its geometric representation and $V^*$ be its dual representation. By a reflection, we mean a element of $\bigcup_{w\in W}wSw^{-1}$. Let $W'$ be a ...
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A basic question about finite type cluster algebras. Let $A$ denote the cluster algebra of type $\mathrm C_{n-1}$, $n\ge 3$. I will view $A$ as a $\mathbb Z$-algebra with the $n^2$ generators $\Delta_{...
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The symmetric group is the automorphism group of the Boolean lattice of an $n$-set. Question: Is there also a "canonical" nice lattice whose automorphism group is equal to the Coxeter group ...
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Let $W_a$ be the affine Weyl group for some semisimple Lie algebra $\mathfrak{g}$, equipped with the Bruhat order $w \le w'$. Fix some (large if you want) prime $p$ and divide weight space into "...
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Let $G$ be a group corresponding to a finite Coxeter system $(W,S)$. Does there exist an algorithm, which on input $(W,S)$ tells what is the minimal cardinality of a generating set for $G$? A weaker ...
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In the article Bolza surface, Wikipedia alludes to a connection between the Bolza curve (the complex algebraic curve of genus 2 with the largest symmetry group) and a certain quaternion algebra. But ...
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I am studying the article "Symmetrization operators in polynomial rings" by A. Lascoux and M.-P. Schützenberger (MSN). Specifically, I am trying to prove the following claim involving ...
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Let $(W,S)$ be a Coxeter system. For any $s,t\in S$, denote $m_{st}$ be the order of $st$. By a reflection, we mean the element in $W$ which conjugates to some simple reflection. My question is: Let $\...
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I have a question about the difference between two description of affine Weyl groups. Let me write two descriptions of affine Weyl groups: Let $\mathfrak{g}=\mathfrak{g}(A)$ be affine Lie algebras ...
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Let $(W,S)$ be the affine Weyl group associated to a simple root system. For $x,y \in W$ we have the usual Kazhdan--Lustig polynomials $h_{y,x} \in \mathbb{Z}[v]$ in Soergel's normalisation, and if ...
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Who first calculated the orders of the Weyl groups $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$? How were these orders calculated?
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Let $(W,S)$ be a Coxeter system and assume that $|S|$ is finite. Certainly, $W$ is generated by $|S|$ simple reflections. My question is: Can $W$ be generated by fewer reflections? (Including non-...
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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 ...
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Consider a (finite reduced irreducible crystallographic) root system $\Phi$ and four positive roots $\alpha,\beta,\gamma,\delta$ such that $\{\alpha,\beta\} \neq \{\gamma,\delta\}$ and $\alpha+\beta=\...
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The affine Weyl group of type $A_n$ can be described as follows. It is the group of all permutations $\sigma: \mathbb Z \to \mathbb Z$ such that $\sigma(i+n)=\sigma(i)+n$ and $\sum_{i=1}^n (\sigma(i)-...
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Let $S$ be a Coxeter diagram, i.e. an unoriented graph, edges labelled by weights $3,4,5,...$ . This includes (affine) Dynkin diagrams, Then Theorem 3.1.3 of Brouwer and Haemers' Spectral Graph ...
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Let $H$ be a regular hexagonal room centered at the origin. Let $W$ be the group generated by reflections about the six sides of $H$. It's well known that $W$ is the affine Weyl group of type $\...
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Let $(B,N,W,S)$ be a Tits system with $W$ a finite coxeter group. Let $w\in W$, consider $BwBw^{-1}B$, then by Bruhat decomposition, it is a disjoint union of some $BxB$, $x\in W$. My question: Let $X\...
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Let $(W,S)$ be a finite Coxeter group, $W_\gamma,W_\nu,W_\mu$ be three parabolic subgroups. For $w,v\in W$, let us consider the set $W_\gamma wW_\nu vW_\mu$. Does there exixt a unique longest element (...
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I am looking for a reference for the following statement: Theorem. Let $\Gamma$ be a finite labelled graph and $C(\Gamma)$ the corresponding Coxeter group. For every $\Xi \subset V(\Gamma)$, the ...
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Recall that the nilCoxeter algebra $\mathcal{N}_W$ for a Coxeter group $W$ is given by the $\mathbf{k}$-basis $x_w$ for each $w\in W$ and multiplication $x_ux_v=x_{uv}$ if $\ell(uv)=\ell(u)+\ell(v)$ ...
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Let $\Phi\subset V$ be a root system of rank $r$ with Weyl group $W$, a choice of positive roots $\Phi_+$ and exponents $d_1, \ldots, d_r$ (i.e. the invariant algebra $(\operatorname{Sym}^\bullet V)^W$...
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Let $H^2=\{(x,y)\mid y>0\}$ be the hyperbolic upper-half plane. Let $K=Q(\sqrt{d})$ be a quadratic number field, and $\mathcal{O}_K$ be the ring of algebraic integers in it. Let $\Gamma=\Delta(p,q,...
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I've gotten stuck on a slightly random combinatorial question, and I'm doing a bit of a shot in the dark here to see if someone else has thoughts about it. I'm interested in studying a permutation of ...
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Consider an affine Weyl group $\hat W$ of a simple Lie type. Let $w \in \hat W$ and let $C^L(w)$ denote the left cell in $\hat W$ containing $w$. Is there a good criterion to test whether $C^L(w)$ has ...
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I searched for a finite-volume, hyperbolic Coxeter polytope of dimension $n \geq 4$ with the following properties $a$ and $b$. $a$) It has exactly one ideal vertex; $b$) if a bounded facet and an ...
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Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers. The Weyl ...
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Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers. The Weyl ...
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I have a question about the proof of lemma 2.14 in Macdonald's paper The Poincaré Series of a Coxeter Group [1], where he used induction on $l(w)$ to prove that if $|E|=|R(w)|$, then $E=R(w)$. The ...
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Let $W_n$ be a Coxeter group of type $B_n$ with $n\geq 1$. Concretely, it is generated by a set of simple reflexions $S = \{s_1,\ldots ,s_n\}$ which satisfy the relations $s_i^2 = 1, s_is_j=s_js_i$ as ...
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Consider $S_\infty$ as a Coxeter group with Coxeter generators the adjacent transpositions $s_i$, $i\geq 1$. We view elements of $S_\infty$ as functions $u:\mathbb{N}\to\mathbb{N}$. Recall the Lehmer ...
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Let $G(4, 7)$ be an abstract group with the presentation $$\langle a,b,c | a^2 = b^2 = c^2 = 1, (ab)^4 = (bc)^4 = (ca)^4 = 1, (acbc)^7 = (baca)^7 = (cbab)^7 = 1 \rangle $$ Richard Schwartz considered ...
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I'm reading part of a proof that says that Weyl groups of apartments of buildings are Coxeter groups. Let $\Delta$ be a building and let $\Sigma$ be a fixed apartment of $\Delta$. Let $C$ be a fixed ...
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I have a question about affine Coxeter groups when reading Humphreys's book: Let $(W,S)$ be an irreducible affine Coxeter group, $M=(m_{ij})$ be its Coxeter matrix, and $\{\alpha_s\}_{s\in S}\in V$ be ...
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Let $(W,S)$ be a Coxeter system of type $A_n$, with $$S=\{s_1,\ldots,s_n\}$$ satisfying the usual relations, and let $R=\mathbb{Z}[x_1,\ldots,x_{n+1}]$ be a polynomial ring. $W$ acts on $R$ by ...
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Consider the root system $R$ for a Coxeter system $(W,S)$ of type $A_n$ with a choice of simple roots. Denote by $I(w)$ for $w\in W$ the set of positive roots $\beta\in R^+$ such that $w(\beta)$ is a ...
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My question is simple: Suppose that $G$ is isomorphic to the free product of finitely many copies of $C_2$. Is it true that any element $g \in G$ can be written as a product $g = s_1 \dotsm s_m$ such ...
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Platonic groups of symmetry are Weyl groups for the exceptional Lie algebra E6->E8, as root systems. These can be viewed as mirrors in a kaleidoscope (Goodman). I would like to know if one can ...
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For the symmetric group on $n$ objects $S_n$ the question of how to write its longest element $w_0$ as a reduced decomposition is an important combinatorical problem. As example, in this question the ...
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Let $Br_n$ be the braid group and consider words in its generators (not in the inverses). Two such words define the same "positive" braid if one can be obtained from the other by commuting ...
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$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Lift{Lift}$The subgroup of $ \SO(n) $ of signed permutations has order $ n!2^{n-1} $. I will call ...
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Let $W$ be the Weyl group of a semisimple algebraic group $G$. $I$ be the simple roots. $J\subset I$ generate a parabolic subgroup of $W$ denote by $W_J$. $w^J$ is the shortest representative of $w$ ...
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Take a triangle in the (Euclidean or hyperbolic) plane, and consider the group of isometries generated by the reflections in the three sides of the triangle. If the angles between adjacent sides are ...
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