Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups

Let $S$ be a finite set and let $m:S\times S\to \{1,2,3,\dots \}\cup \{\mathrm{\infty }\}$ satisfy $m(s,s)=1$ for all $s\in S$ and $m(s,t)=m(t,s)\in \{2,3,\dots \}\cup \{\mathrm{\infty }\}$ for all $s\mathrm{\ne }t$. Such an $m$ is a Coxeter matrix on $S$.

The presented group. Let $R\subseteq F(S)$ be the set of relators $R:=\{s^2:s\in S\}\cup \{(st{)}^{m(s,t)}:s,t\in S,m(s,t)<\mathrm{\infty }\},$ where $F(S)$ is the free group on $S$ (Free group on a set of generators, Reduced words form the free group on an alphabet) and $(st{)}^k$ is the $k$-fold product (Powers $g^n$: natural exponents in a monoid and integer exponents in a group, with $g^0=e$), and let $N:=⟨\text{\hspace{0.17em}⁣}⟨R⟩\text{\hspace{0.17em}⁣}{⟩}_{F(S)}$ be the normal closure of $R$ in $F(S)$ (The normal closure of a subset of a group, The normal closure of $R$ is the set of finite products of conjugates of elements of $R$ and their inverses). Define $W:=F(S)\mathrm{/}N,$ and write $s$ (as well as $sN$) for the image of $s\in S$ in $W$ (The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$, Group and abelian group).

Universal property. For every group $G$ and every map $f:S\to G$ with $f(s{)}^2=1$ for all $s\in S$ and $(f(s)f(t){)}^{m(s,t)}=1$ in $G$ for all $s\mathrm{\ne }t$ with $m(s,t)<\mathrm{\infty }$, there is a unique group homomorphism $\phi :W\to G$ with $\phi (s)=f(s)$ for all $s\in S$ (Monoid homomorphism and group homomorphism, A homomorphism that kills a normal subgroup factors uniquely through the quotient group, Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group, Group presentation by generators and relations, Relators and relations; finitely generated, finitely related, and finite presentations, In $⟨X\mid R⟩$, the words $u$ and $v$ represent the same element if and only if $u^{-1}v\in ⟨\text{\hspace{0.17em}⁣}⟨R⟩\text{\hspace{0.17em}⁣}⟩$).

Length and reduced words. For $w\in W$ put $\mathrm{\ell }(w):=\min \{k\in \mathbb{N}:\text{ there exist }{s}_1,\dots ,s_k\in S\text{ with }w=s_1\cdots s_k\}\mathrm{.}$ The minimum exists: every element of the free group $F(S)$ is represented by a finite word in $S\cup S^{-1}$ (Words in an alphabet with formal inverses, elementary cancellation, and reduced words), and $s^{-1}=s$ in $W$ because $s^2\in R$, so every element of $W$ is the value of a finite word in $S$; the set of admissible $k\in \mathbb{N}$ is therefore nonempty and has a least element by the well-ordering principle (The well-ordering principle, The natural numbers $\mathbb{N}$ (von Neumann)). A word $(s_1,\dots ,s_k)$ in $S$ is a reduced expression of $w$ when $w=s_1\cdots s_k$ and $k=\mathrm{\ell }(w)$, and nonreduced otherwise. The empty word $()$ is a word in $S$ of length $0$, and $\mathrm{\ell }(1)=0$; it is the reduced expression of $1$.

Terminology. The pair $(W,S)$ is the Coxeter system presented by the Coxeter matrix $(S,m)$; when $s\mapsto s$ is understood, we also say that $W$ is the Coxeter group presented by $(S,m)$.

Standard parabolic subgroups. For $J\subseteq S$ put $W_J:=⟨\{s:s\in J\}⟩\le W$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups).

Conventions. $m(s,t)$ is declared to be the order of $st$ in $W$; a value $m(s,t)=\mathrm{\infty }$ imposes no relator on $s,t$. This definition asserts no finiteness, faithfulness, or completeness property of the presentation: the statements that the elements $s\in S$ are pairwise distinct in $W$ (so that $S$ may be viewed as a labelled subset of $W$), that $\mathrm{\ell }(s)=1$, that a word is reduced exactly when it admits no two-letter deletion, and that $(W_J,J)$ is the Coxeter system presented by the restricted matrix $m{\mathrm{\mid }}_J$ with intrinsic length equal to the ambient length are recorded with their justifiers below and are not used before those results.

The properties announced in the Conventions paragraph are supplied later in this pair: pairwise distinctness of the simple generators, $\mathrm{\ell }(s)=1$, and the deletion characterisation of reduced words in Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action ↗; the intrinsic presentation and length of the standard parabolic subgroups in Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification ↗. Until those results are available, $W$ and $\mathrm{\ell }$ are to be read exactly as constructed above.

No form of choice is used in the construction: $W$ is the quotient of the free group on the finite set $S$ by an explicitly generated normal closure, and the only minimisation in the definition is over a nonempty set of natural numbers.