# Tagged Questions

The tag has no usage guidance.

41 views

### Orbit spaces of Coxeter groups and singularities

I have often seen in the literature the statement that the orbit spaces of irreducible finite Coxeter groups are equivalent to unfoldings of singularities. For instance, taken from Dubrovin, ...
471 views

### Can you prove Givental's conjecture on wavefronts and the icosahedron?

In his remarkable book The Theory of Singularities and its Applications, Vladimir Arnol'd discussed a conjecture of A. B. Givental, which asserts that the symmetry group of the icosahedron is secretly ...
118 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 ...
108 views

129 views

### Torsion-free, normal subgroups of certain Coxeter groups

Let $G$ be the reflection group of a regular, 4-dimensional, hyperbolic honeycomb. I would like to find a family $H_i < G$ of finite-index, torsion-free subgroups of $G$, so that I can represent ...
115 views

197 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 ...
86 views

142 views

### A question of braid words

Let $(W,S)$ be a Coxeter group, let $B(W,S)$ be the corresponding braid or Artin-Tits group. Set $S=\{s_1,\dots, s_n\}$ and denote by $\bf{S}=\{\sigma_1,\dots, \sigma_n\}$ the corresponding generators ...
120 views

### Coxeter Subgroups of Coxeter Groups

Is there an algorithm to determine all the Coxeter subgroups of a given Coxeter group? If we only want the Coxeter subgroups of finite index does that make the question easier? If we only want a ...
193 views

### Are the orders of the generators of a group and the product of pairs of thereof enough for this group to be isomorphic to a Coxeter group?

Let's say we have $n$ generators $x_1, x_2, \cdots, x_n$ along with the following facts concerning their orders: \begin{eqnarray*} ord(x_i) &=& 2 \text{ for } i = 1, 2, \cdots, n \\ ord(x_i ...
148 views

### On the vertices of a Coxeter complex

Let $A$ be a Coxeter complex which is euclidean, so I assume that $A$ is an affine space over the reals on which a Coxeter group $(W,S)$ acts, the elements of $S$ are reflections and I assume the ...
78 views

### Coxeter Isometry groups whose center has torsion

I'm looking for examples of Riemannian manifolds M of dimension $\geq 2$ such that the isometry group $Isom(M)$ contains as subgroup a finite Coxeter group $G$ such that $Tor(Z(G))$, the torsion group ...
146 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” ...
404 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 ...
193 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 $m_{i,j}=m_{j,i}$...
253 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 $Q^\vee\subset P^\vee$ ...
75 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 ...
258 views

### How to think about the simple reflection s_0 in the affine Weyl group?

Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...
82 views

### Reducing enumeration of reduced words in general Coxeter groups to another #P-complete problem

There is a polynomial time algorithm that takes as inputs a Coxeter system $(W,S)$ with $S$ finite (but with $W$ not necessarily finite), say encoded as a Coxeter matrix, as well as two reduced words ...
231 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, ...
257 views

### Fundamental invariants for root subsystems

Let $\Phi$ be an irreducible root system of rank $\ell$. The fundamental invariants of $\Phi$ is a set of $\ell$ integers $d_1, \cdots, d_\ell$ canonically attached to $\Phi$. Now suppose $\Psi$ is ...
154 views

Is the following statement true for finite reflection groups? Let $G$ be a finite reflection group acting on $\mathbb{R}^n$, let $x, y\in \mathbb{R}^n$ and let $z$ be in the orbit of $y$. If $\... 0answers 266 views ### An angle-doubling trick of Kirillov and Berenstein [closed] Kirillov and Berenstein, in their article "Groups generated by involutions, Gelfand-Tsetlin patterns and combinatorics of Young tableaux" (available at math.uoregon.edu/~arkadiy/bk1.pdf), present a ... 2answers 257 views ### Embed one Coxeter System into another What is a good reference that explains all the braid relations and diagrams for Coxeter systems concisely? In particular, how do I embed$H_3$inside$D_6$, or$H_4$inside$E_8$? Any hints? 1answer 99 views ### Double quotients of Coxeter groups have the chain property? Let$(W,S)$be a Coxeter group with Bruhat order$\leq$and length function$\ell(w)$. Definition: a subset$X \subset W$has the chain property if whenever$x,y \in X$with$x < y$, there ... 1answer 497 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. 3answers 752 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 $$\circ-6-\circ-3-\... 2answers 306 views ### Are there infinitely many commensurable classes of finite-covolume hyperbolic Coxeter groups? Allcock(2006) proved that there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n\le 19 (resp. n\le 6). His main technique of ... 2answers 445 views ### Is there a natural notion of completion of a Coxeter system? Let (W,S) be a Coxeter system. Then any element of W can be written as a finite products of elements of S. I want some notion of a "completion" of W, call it \hat{W}, whose elements are (... 1answer 189 views ### Which subgroups of a finite reflection group have distingushed coset representatives? Let W be a finite reflection group with length function l and let I be a set of simple reflections that generate W. Let \phi be an automorphism of W permuting I. Consider the orbits of ... 1answer 137 views ### Quantum Cartan matrices and Coxeter elements Let \Gamma be a bipartite graph, with the vertices partitioned into disjoint sets \Pi_1 and \Pi_2. Let W be the associated Weyl group, with Coxeter generators \{s_i\}_{i\in \Gamma}. Let \{... 3answers 786 views ### A special tessellation Let P be a convex n-gon. Suppose that we have an infinite number of Ps, and that each of them is colored either red or blue. Here, let us consider the following operations : Operation 1 : Place ... 0answers 96 views ### Topological obstruction to icosahedral symmetry? Let G be a compact simple lie group of rank n. Then the Poincaré series of G is given by$$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 ... 1answer 127 views ### 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 ... 2answers 466 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 ... 1answer 417 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 analyse.... 2answers 544 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 ... 1answer 301 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 ... 1answer 228 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 ... 1answer 176 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 ... 1answer 295 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 ...
Wich representations of $F_{4}$, $E_{8}$ and $G_{2}$ are quasi-minuscule?
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 = \...