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 $W\subset\operatorname{GL}(V)$ is a finite reflection group, and let $S\subset W$ be a set of simple reflections so that $(W,S)$ is a finite … Read more

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 combinatorics it is common to study the properties of reduced wiring diagrams, where no two wires cross twice (one could “reduce” such a diagram … Read more

Coxeter groups generated by one finite conjugacy class

Let (W,S) be an arbitrary Coxeter system. We consider the following scenario: Let O be a conjugacy class of an element w in W which is finite and which generates the whole group W. The only example of the above scenario I know arises in simply laced irreducible finite Coxeter groups (i.e. all Coxeter integers … Read more

What are the normal subgroups of the finite Coxeter Groups of type Bn?

Let Bn=⟨ρ0,ρ1,…,ρn−1⟩ subject to the relations that (ρiρj)mi,j=id with mi,i=1, mi,j=2 for |i−j|≥2, mi,i+1=3 for 0≤i<n−1 and finally mn−1,n=4. What are the normal subgroups of Bn and where might I find a reference? Computationally, I see that for all n>4 there are 9 of them and I would like to find some source which describes … Read more

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 C–rank≥2 and let H be a normal subgroup of G(Z) with a finite index (so H is Zariski dense in G). Let T a maximal torus in G and B a Borel subgroup containing T, let wα1,…,wαn represent the simple reflections of the … Read more

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 where most of the infinite families of finite or affine diagrams come from. Given a positive root of … Read more

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 {Cw} be the Kazhdan-Lusztig C-basis (an answer in terms of the C′w-basis is of course equally useful, though). Let H be the corresponding Hecke algebra over Z[q,q−1], and write hx,y,z … Read more

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 of elements of the symmetric group $\mathbb{S}_m$ if $1,s_{a_r},s_{a_{r-1}}s_{a_r},\ldots,s_{a_2}\cdots s_{a_r},s_{a_1}\cdots s_{a_r}$ are all distinct elements of the symmetric group $\mathbb{S}_m$. For example, $32323132323132323$ gives rise … Read more

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 other words, if P(V) is the polynomial ring on V, then P(V)W is a (finitely generated) polynomial ring. I would like … Read more