Questions tagged [coxeter-groups]

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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A spin extension of a Coxeter group?

Consider a Coxeter group $W$ with simple generators $S$ and Coxeter matrix $\left( m_{s,t}\right) _{\left( s,t\right) \in S\times S}$. Let $\mathfrak{M}$ be the set of all pairs $\left(s, t\right) ...
darij grinberg's user avatar
19 votes
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What is the centralizer of a Coxeter element?

Let $(W,S)$ be a Coxeter system (of finite rank) and $c \in W$ a Coxeter element. If $W$ is finite, then the centralizer $C_W(c)$ is the cyclic group generated by $c$ (e.g. see the book "Reflection ...
P. Wegener's user avatar
13 votes
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A hard Lefschetz theorem for nilCoxeter algebras

Let $W$ be a finite Coxeter group and $\mathcal{N}(W)$ its nilCoxeter algebra (over the reals, say), as defined at https://en.wikipedia.org/wiki/Nil-Coxeter_algebra. $\mathcal{N}(W)$ has a natural ...
Richard Stanley's user avatar
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Writing an element of a free product of $C_2$'s as a product of order-$2$ elements

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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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” ...
Jim Humphreys's user avatar
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Recognizing reflection subgroups of Coxeter groups

Given a Coxeter system $(W,S)$ with reflections $T$, and any subset $A \subseteq T$, it is known that the reflection subgroup $W_A$ generated by $A$ has a canonical choice $S_A$ of generators so that $...
Christian Gaetz's user avatar
10 votes
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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 ...
Misha's user avatar
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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 ...
Chris Brav's user avatar
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Inversions for parity preserving presentations

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 ...
Ben Webster's user avatar
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Name for an involution associated to a Coxeter element

Let $(W,S)$ be a finite Coxeter system, and $c \in W$ a Coxeter element. There is an involution $g\in W$ for which the involutive map $w \mapsto gw^{-1}g$ fixes $c$. Is there a standard name for this ...
Sam Hopkins's user avatar
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Is the order complex of open Bruhat intervals polytopal?

Let $P$ be the Bruhat order of a Coxeter group, and let $s<t$ in $P$. The set $\Delta(s,t)$ of all chains of the open interval $(s,t)$ (called the order complex of $(s,t)$) is a simplicial complex. ...
Richard Stanley's user avatar
8 votes
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Number of occurrences of certain generators in expressions in Coxeter groups

Let $W$ be a Coxeter group (finite or infinite) with (finite) set $S$ of Coxeter generators, and let $I \subseteq S$ be some subset. If $w\in W$ then I call $m_I(w)$ the minimum total number of ...
Gro-Tsen's user avatar
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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 ...
Qiaochu Yuan's user avatar
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182 views

Chevalley-Solomon formula and Weyl character formula

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 $(\operatorname{Sym}^\bullet V)^W$...
Antoine Labelle's user avatar
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Generating cycles inside Tits' graph of words for a positive braid

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 ...
Allen Knutson's user avatar
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332 views

Why are fundamental weights denoted by omega?

In my field (and many others, I believe) the absolutely standard notation for the fundamental weights of a root system is lowercase omega: $\omega$. Recently I was taken aback to receive a copyedited ...
Igor Makhlin's user avatar
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Regarding $F_4$ and $G_2$ Lie algebras, do there exist $F_n$ or $G_n$ families of Lie algebras?

For example, $E_6$ exceptional Lie algebra is part of the $E_n$ series of Lie algebras (Kac-Moody algebras). Are $F_4$ or $G_2$ maybe also parts of some $F_n$ or $G_n$ series of Lie algebras or are ...
Domagoj Hranjec's user avatar
6 votes
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Indices of Coxeter groups in themselves

Every Euclidean Coxeter group ($P_n$, $Q_n$, $R_n$, $S_n$, $T_7$, $T_8$, $T_9$, $U_5$, $V_3$, $W_2$) contains infinitely many scaled copies of itself as subgroups. What are all the possible indices of ...
Daniel Sebald's user avatar
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366 views

Symmetry group and irreducible representation

Let $S$ be a bounded geometric shape in the Euclidean space $E=\mathbb{R^n}$. Assume that the origin of $E$ is a fixed point of every element of the symmetry group $G(S)$ of $S$, and assume that $G(S) ...
Sebastien Palcoux's user avatar
6 votes
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164 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 example,...
Jonah Blasiak's user avatar
5 votes
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191 views

Kazhdan-Lusztig basis elements appearing in product with distinguished involution

My apologies if the below is too malformed to make sense. Let $(W,S)$ be the affine Weyl group of a reductive group $G$, and let $\{C_w\}$ be the Kazhdan-Lusztig $C$-basis (an answer in terms of the $...
Stefan  Dawydiak's user avatar
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113 views

Progress on the result about montonicity of Kazhdan Lustzig polynomials

I am reading the paper Masato Kobayashi---Combinatorics on Bruhat Graphs and Kazhdan-Lusztig Polynomials. Let $P_{x,w}$ be the Kazhdan Lusztig polynomial of $W$. There is a result about ...
James Cheung's user avatar
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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, ...
Jim Humphreys's user avatar
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117 views

A basis for the 0-Hecke ring

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 ...
Matt Samuel's user avatar
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Group generated by Coxeter groups of $B_8$ and $E_8$

Note: $B_8$, $D_8$, and $E_8$ are referring to the Coxeter groups. What is the smallest group $G$ with the following properties? $G$ has a subgroup isomorphic to $E_8$. $G$ has a subgroup isomorphic ...
Daniel Sebald's user avatar
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142 views

Polynomial invariants of infinite reflection groups

It is a famous theorem of Shepard-Todd that the ring of invariant polynomials of a finite complex reflection group $W$ acting on a complex vector space $V$ is actually itself a polynomial ring. In ...
K Goldman's user avatar
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180 views

The Weyl group of Kac-Moody algebra and Coxeter group

Let $\mathfrak{g}$ be a Kac-Moody algebra, $W$ be its Weyl group, generate by fundmental reflections {$r_1,...,r_n$}. For $i,j$, we have relation $(r_ir_j)^{m_{ij}}=1$, $1\le i,j\le n$. Let $W^{'}$ be ...
fool rabbit's user avatar
4 votes
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141 views

Words that give rise to an enumeration of elements of the symmetric group

Let $\mathbb{S}_m$ be the symmetric group on $m$ letters. Let $n=m-1$. Let $\mathbf{w}=a_1\cdots a_r$ be a word on the alphabet $\{1,\ldots,n\}$. We say that $\mathbf{w}$ gives rise to an enumeration ...
Christoph Mark's user avatar
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Stability of infinite root systems with a long path in their Coxeter diagrams

Given a Cartan matrix associated to a Coxeter diagram, I can modify it by replacing one of the edges in the diagram with a long chain of vertices connected by simply laced edges; for example, this is ...
Will Dana's user avatar
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Do these Zariski-dense subgroups of complex Chevalley group have non-empty intersection with this Bruhat cell?

Let $G$ be a complex Chevalley group (not necessarily adjoint type) with $\operatorname{\mathbb{C}-rank}\geq2$ and let $H$ be a normal subgroup of $G(\mathbb Z)$ with a finite index (so $H$ is Zariski ...
Ami's user avatar
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What are the normal subgroups of the finite Coxeter Groups of type Bn?

Let $B_n = \langle \rho_0,\rho_1,\ldots,\rho_{n-1} \rangle$ subject to the relations that $(\rho_i\rho_j)^{m_{i,j}} = id$ with $m_{i,i} = 1$, $m_{i,j} = 2$ for $|i-j|\ge 2$, $m_{i,i+1} =3$ for $0 \le ...
Rob Nicolaides's user avatar
4 votes
0 answers
95 views

Ref. request: Enumerating elements of Bruhat cells

Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then $$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$ where we embed the ...
Dirk's user avatar
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Reference request for generalized root systems

Where can I find information on root systems where the inner product is other than the standard (all positive) signature?
unknown's user avatar
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Have wiring diagrams been generalized to arbitrary digraphs?

A "combinatorial wiring diagram" is a way to define a permutation by a drawing of a particular planar digraph. For example, this wiring diagram corresponds to the permutation $(3412)$: In Coxeter ...
GMB's user avatar
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4 votes
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195 views

Shifts in the decomposition of Bott-Samelson bimodules

Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that $W\subset\...
Chris McDaniel's user avatar
4 votes
0 answers
92 views

Highest (short) roots and commutation relations in (twisted) DAHA

I am trying to understand, explicitly, the commutation relation between $X_\vartheta$ and $Y_\vartheta = T_0T_{s_\vartheta}$ in the (twisted) DAHA for a root system $R$, where $\vartheta$ is the ...
Matteo's user avatar
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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 ...
Drew Armstrong's user avatar
3 votes
0 answers
93 views

Does there exist a finite-volume hyperbolic Coxeter polytope with these properties?

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 ...
Edoardo Rizzi's user avatar
3 votes
0 answers
291 views

Is G(4,7) a Coxeter group

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 ...
Shijie Gu's user avatar
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Question on the Tits cone of an irreducible affine Coxeter group

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 the system of simple roots in the standard geometric realization, so the $\{\...
zemora's user avatar
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114 views

Why inherit the Tits systems structure by a $B$-adapted homomorphism?

Let $(G,B,N,S)$ be a Tits system and $\phi\colon G\longrightarrow \hat{G}$ a $B$-adapted in the sense of the paper Groupes réductifs sur un corps local: I of Bruhat–Tits. They said that $\phi$ is a $B$...
M masa's user avatar
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Algebraic hypersurfaces and Coxeter groups

What is the minimum degree of an algebraic hypersurface (not necessarily smooth) having each Coxeter group as its symmetry group?
Daniel Sebald's user avatar
3 votes
0 answers
169 views

The group of fixed points of an involution of a Weyl group

Let $R$ be a reduced root system in a vector space $V$ over $\mathbb Q$. Let $W=W(R)$ denote its Weyl group. Let $S\subset R$ be a basis of $R$ (a system of simple roots). Let $D=D(R,S)$ denote the ...
Mikhail Borovoi's user avatar
3 votes
0 answers
116 views

What total orders have people studied on Coxeter Groups?

I'm aware of the ShortLex total order that gives rise to the usual normal form. But are there any others that have naturally arose and people have studied?
Rob Nicolaides's user avatar
3 votes
0 answers
64 views

Parabolic invariants of coinvariant algebras for reflection groups

Let $W$ be a finite reflection group and $V$ its reflection representation (over $\mathbb{C}$). Let $S$ be the symmetric algebra on $V^*$, $I_W\subseteq S$ the ideal generated by the non-constant $W$-...
Alexander Woo's user avatar
3 votes
0 answers
101 views

Kazhdan-Lusztig polynomials and the defect of a Bruhat interval

Let $(W,S)$ be a Coxeter system with length function $\ell$ and $T=\bigcup_{w\in W}wSw^{-1}$. Set $N(u,v):=\{t\in T: u< tu \le v\}$, $\overline{\ell}(u,v):=|N(u,v)|$, $\ell(u,v):=\ell(v)-\...
James Cheung's user avatar
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3 votes
0 answers
236 views

The maximal order of an element in a Coxeter group

Let $W$ be a finite Coxeter group. Let $$ N_W=\operatorname{max}_{g\in W}\operatorname{ord}(g) $$ where $\operatorname{ord}(g)$ denotes the order of an element $g$. By Fermat's little theorem, we ...
Christoph Mark's user avatar
3 votes
0 answers
61 views

Directed galleries of the building of type $\widetilde{A}_{n}$

Let $X$ be the affine building of $GL_n(\mathbb{Q}_{p})$. We call oreinetd chamber of $X$ every sequence $\overrightarrow{C}=(s_1,...,s_n)$ of vertices such that $C=\{s_1,...,s_n\}$ is a chamber of $X$...
Rajkarov's user avatar
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Reference request: which elements in a Coxeter group has longest reflection length?

Let $V$ be a vector space over $\mathbb{R}$. An element $s \in GL(V)$ is a reflection if $H_s:=\ker(s-1)$ is a hyperplane and $s^2=1$. The eigenvalues of a reflection $s$ are $1, -1$. Every reflection ...
Jianrong Li's user avatar
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3 votes
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Question about Ext group in $\mathcal{O}^\mathfrak{p}$?

Let $W$ be a Weyl group, let $\mu$ be an antidominant weight. Let $W_I$ be the Weyl group generated by $I$ and $w_I$ be the longest element in $W_I$. Denote ${}^IW$ the set of minimal length coset ...
James Cheung's user avatar
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