Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order

Let $S$ be any set, possibly infinite, and let $m:S\times S\to \{1,2,3,\dots \}\cup \{\mathrm{\infty }\}$ be a Coxeter matrix: $m(s,s)=1$, and for distinct $s,t$, $m(s,t)=m(t,s)\in \{2,3,\dots \}\cup \{\mathrm{\infty }\}$. Let $F(S)$ be the free group on $S$, let $N$ be the normal closure of

$$\{s^2:s\in S\}\cup \{(st{)}^{m(s,t)}:s\mathrm{\ne }t,m(s,t)<\mathrm{\infty }\},$$

and define $W=F(S)\mathrm{/}N$. The image of $s\in S$ in $W$ is also denoted $s$. This is the Coxeter group presented by $(S,m)$.

For $w\in W$, define its length by

$$\mathrm{\ell }(w)=\min \{k\in \mathbb{N}:w=s_1\cdots s_k\text{ for some }{s}_1,\dots ,s_k\in S\}\mathrm{.}$$

The set is nonempty because every element of the quotient has a finite word in $S\cup S^{-1}$ and the relators give $s^{-1}=s$; its minimum exists by the well-ordering principle. A word of length $\mathrm{\ell }(w)$ for $w$ is a reduced expression; the empty word is the reduced expression of $1$.

Define the reflection set $T=\{xs{x}^{-1}:x\in W,s\in S\}$. Write $u\to v$ if $v=ut$ for a reflection $t\in T$ and $\mathrm{\ell }(v)>\mathrm{\ell }(u)$. Define the reflection-chain Bruhat order by $u\le v$ if there is a finite chain $u=u_0\to u_1\to \cdots \to u_k=v$; the empty chain is allowed. Length strictly increases on each nonempty chain, so this relation is a partial order. The exact reduced-subword description and finite-interval consequence needed below are established in Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank ↗.

When $S$ is finite, the presentation and length agree with Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, and the reflection-chain order agrees with The Bruhat graph by length-increasing reflection chains, the Bruhat order, inversion symmetry, and reflection parity. No exchange, standard-parabolic, reduced-subword, or interval-finiteness property is included in this definition; those properties are justified separately.