Brink–Howlett groupoid

PDF p. 15: §4.1, definition before Remark 4.1

Let $\mathrm{\Gamma }$ be a Coxeter diagram with vertex set ${\mathrm{\Gamma }}_0$, Coxeter group $W$ with simple generators $s_i$, and geometric representation on the real vector space with basis $({\alpha }_i{)}_{i\in {\mathrm{\Gamma }}_0}$. Fix $J\subseteq {\mathrm{\Gamma }}_0$. For $I\subseteq {\mathrm{\Gamma }}_0$ write $W_I=⟨s_i:i\in I⟩$; an associate of $J$ is a subset $I$ for which $W_I$ is conjugate to $W_J$ in $W$.

The Brink–Howlett groupoid $\mathcal{B}\mathcal{H}(J)$ has the associates of $J$ as objects. For associates $I,K$, its morphisms $I\to K$ are tagged by their source and target and have labels

${\mathrm{Hom}}_{\mathcal{B}\mathcal{H}(J)}(I,K)=\{x\in W:\{{\alpha }_i:i\in I\}=\{x{\alpha }_k:k\in K\}\}\mathrm{.}$

For $x:I\to K$ and $y:K\to L$, their left-to-right composite is $xy:I\to L$. The identity at $I$ has label $1\in W$.

Associates of a parabolic subset

Coxeter-system, roots, length and weak-order conventions

Connected small groupoids and vertex groups

Set-sized groupoids, functors and composition conversion

The associates form a set of subsets of ${\mathrm{\Gamma }}_0$, and the tagged arrows form a set of triples $(I,x,K)$. The geometric representation is a group action. Thus if $x:I\to K$ and $y:K\to L$, then $xy$ sends the simple-root set of $L$ to that of $I$. The identity sends each simple-root set to itself, and the inverse of $x$ sends the simple-root set of $I$ to that of $K$. Associativity and the identity and inverse laws follow from those of $W$. Therefore these data form a small groupoid with the indicated left-to-right convention, as in Set-sized groupoids, functors and composition conversion and Connected small groupoids and vertex groups.

It is also connected. Let $I,K$ be associates of $J$, and choose $x$ with $W_I=x{W}_K{x}^{-1}$. Here $\mathrm{\ell }$ is the nonnegative integer word length defined in Coxeter-system, roots, length and weak-order conventions. Repeatedly multiply the current representative on the left by a generator in $I$ or on the right by one in $K$ whenever this decreases its nonnegative integer length. This terminates at $d\in W_{I}x{W}_K$ having no such descent. By Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank, part (3), multiplication by a simple generator changes length by exactly one. Thus $\mathrm{\ell }(s_{i}d)>\mathrm{\ell }(d)$ for every $i\in I$ and $\mathrm{\ell }(d{s}_k)>\mathrm{\ell }(d)$ for every $k\in K$, giving the strict inequalities needed for the two-sided minimum hypothesis below. Since the multiplying factors lie in the respective parabolics, $W_I=d{W}_K{d}^{-1}$.

For each $i\in I$, the element $d^{-1}{s}_{i}d$ has a finite expression in generators indexed by $K$. Choose a finite set of indices containing $i$, the support of a word for $d$, and the indices in that expression. By Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank, the Coxeter group on the chosen finite set of generators embeds in $W$ with lengths preserved. Applying The parabolic intersection W_I∩ dW_Jd⁻¹ for d∈^IW^J, part (3), to the intersections of $I$ and $K$ with that finite set shows that $d^{-1}{\alpha }_i={\alpha }_k$ for some $k\in K$. The same argument with $d^{-1}$ and the roles of $I,K$ reversed shows that every ${\alpha }_k$ arises this way. Hence $d^{-1}\{{\alpha }_i:i\in I\}=\{{\alpha }_k:k\in K\}$, so $d:I\to K$ is a morphism. The object $J$ exists, since $W_J$ is conjugate to itself; consequently the groupoid is nonempty and connected.