Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank

Finite-generator parabolic embedding. The image of ${\varphi }_J$ is $⟨J⟩$, since its image is generated by the named generators. To prove injectivity, let $g\in W_J^{\ast }$ map to $1$, and represent $g$ by a word $u$ in $J$. In the free group $F(S)$, the word $u$ lies in the normal closure $N$ of the Coxeter relators by [F1]. By [F7], $N$ is the subgroup generated by all conjugates of those relators and their inverses: that subgroup is normal and contains the relators, and every normal subgroup containing the relators contains it. Therefore $u$ is a finite product of conjugates of relators and their inverses. Let $K$ contain $J$ and every letter appearing in these finitely many conjugating words and relators. The inclusion $F(K)\to F(S)$ has a retraction sending generators outside $K$ to $1$, so it is injective by the free-group universal property [F6]. Thus the same finite product identity holds in $F(K)$, and $u$ is the identity in the Coxeter group presented on $K$. Applying [F2] to the finite system on $K$ and its subset $J$ shows that the map from $W_J^{\ast }$ into this group is injective. Hence $g=1$.

Intrinsic and ambient lengths. Every $J$-word is an $S$-word, so ambient length is at most intrinsic $J$-length. Conversely, take $w\in W_J^{\ast }$ and a reduced ambient expression for ${\varphi }_J(w)$, of length $r$. Let $K$ contain $J$ and the letters in that expression. By step 1.1 applied to $K$, the presentation on $K$ embeds in $W$; thus the equality between a $J$-word for $w$ and the chosen ambient word already holds in the group on $K$. By [F2], intrinsic $J$-length equals $K$-length on $W_J^{\ast }$. The chosen expression gives $K$-length at most $r$, while every $K$-word is also an $S$-word and so has length at least $r$. Hence both lengths equal $r$, proving intrinsic and ambient length equality.

Simple-generator length changes. Fix $s\in S$ and $w\in W$. Choose a finite $K$ containing $s$ and the letters of a word for $w$. Steps 1.1 and 2.1 embed the group on $K$ and preserve lengths. Applying [F3] in that finite-generator system gives both asserted $\pm 1$ formulas in $W$.

Bruhat comparison implies a subword. Suppose $y\le w$, and fix a reduced expression of $w$. Choose a finite Bruhat chain from $y$ to $w$. Each element in this chain has finite word support, and each reflection used in an edge is a conjugate of a simple generator by a finite word. The union of these supports and the letters of the fixed expression of $w$ is contained in a finite $K\subseteq S$. By steps 1.1 and 2.1, the chain lies in the embedded finite-generator group on $K$ with all lengths preserved; its edges are still right multiplications by reflections. The finite-generator subword theorem [F5] now gives a reduced expression of $y$ as a subword of the fixed expression of $w$.

A reduced subword implies Bruhat comparison. Suppose a reduced expression of $y$ occurs as a subword of a fixed reduced expression of $w$. Let $J$ be the finite support of that expression of $w$. The subword theorem [F5] gives $y\le w$ in the finite-generator Coxeter group on $J$. By steps 1.1 and 2.1 its presentation embeds in $W$ and preserves lengths; each reflection in the finite Bruhat chain maps to a reflection in $W$. The chain therefore gives $y\le w$ in the global reflection-chain order.

Every element below $w$ is represented by a subword of a fixed reduced expression of $w$, by step 3.2. There are at most $2^{\mathrm{\ell }(w)}$ such subwords, so the lower interval is finite with the stated bound. A strict comparison has a nonempty length-increasing chain by [F1], hence $\mathrm{\ell }(y)<\mathrm{\ell }(w)$ when $y<w$. Steps 3.1–3.3 prove all remaining clauses. All supports, products in the normal closure, and chains used above are finite.