Coxeter Presentations, Exchange, and Reduced Word Theorems

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.

Let $(S,m)$ be a finite Coxeter matrix and let $W$ be the group presented by it as in Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups.

A common splitting field. Put $h:=\prod (X^{2m(s,t)}-1)\in \mathbb{Q}[X]$, the product over all unordered pairs $\{s,t\}\subseteq S$ with $s\mathrm{\ne }t$ and $m(s,t)<\mathrm{\infty }$ (an empty product when $\mathrm{\#}S\le 1$). Let $K$ be a splitting field of $h$ over $\mathbb{Q}$ (The rationals as equivalence classes of pairs of integers, The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, Over an integral domain, degrees add under multiplication of nonzero polynomials, For every field $F$, $F[x]$ is a unique factorisation domain, Every nonzero polynomial over a field has a splitting field, Every finite family of nonzero polynomials has a splitting field, obtained from their product, Polynomials that split and splitting fields of a polynomial or a family of polynomials). The prime subfield of $K$ is $\mathbb{Q}$ (A field's prime subfield is isomorphic to $\mathbb{Q}$ in characteristic zero and to ${\mathbb{F}}_p$ in characteristic $p$, The characteristic of a ring: the least $n\ge 1$ with $n\cdot 1_R=0$ when one exists, and $0$ otherwise, The rationals form a field), so $\mathrm{char}K=0$ (The characteristic of a field is zero or a prime number) and $X^n-1$ is separable over $K$ for every $n\ge 1$ ($t^n-1$ is separable over $K$ exactly when the characteristic does not divide $n$, and then a splitting field carries $n$ distinct $n$-th roots of unity). For each finite edge, $X^{2m(s,t)}-1$ is a factor of $h$ (Over an integral domain, degrees add under multiplication of nonzero polynomials), so all its roots lie in $K$ and, being separable of degree $2m(s,t)$, it has exactly $2m(s,t)$ distinct roots there ($t^n-1$ is separable over $K$ exactly when the characteristic does not divide $n$, and then a splitting field carries $n$ distinct $n$-th roots of unity, Polynomials that split and splitting fields of a polynomial or a family of polynomials); the group ${\mu }_{2m(s,t)}(K)$ of such roots is therefore finite of order $2m(s,t)$ and hence cyclic, so it contains a primitive $2m(s,t)$-th root of unity (${\mu }_n(K)$ is cyclic of order dividing $n$, and has a primitive $n$-th root of unity exactly when its order is $n$, The group ${\mu }_n(K)$ of $n$-th roots of unity in a field, and primitive $n$-th roots of unity). Fix one primitive $2m(s,t)$-th root of unity ${\zeta }_{st}\in K$ for each unordered finite edge (a finite selection, since $S$ is finite) and put $c_{st}:=c_{ts}:={\zeta }_{st}+{\zeta }_{st}^{-1}\in K(m(s,t)<\mathrm{\infty }),c_{st}:=c_{ts}:=2(m(s,t)=\mathrm{\infty })\mathrm{.}$

The representation on the simple-root basis. Let $E$ be the $K$-vector space with basis $({\alpha }_s{)}_{s\in S}$ (Vector space over a field, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis). For each $s\in S$ let ${\sigma }_s:E\to E$ be the unique $K$-linear map with ${\sigma }_s({\alpha }_s)=-{\alpha }_s,{\sigma }_s({\alpha }_t)={\alpha }_t+c_{st}{\alpha }_s(t\mathrm{\ne }s)\mathrm{.}$ Directly from the definition each ${\sigma }_s$ is an involution: ${\sigma }_s^2({\alpha }_s)={\alpha }_s$ and, for $t\mathrm{\ne }s$, ${\sigma }_s^2({\alpha }_t)={\sigma }_s({\alpha }_t+c_{st}{\alpha }_s)={\alpha }_t+c_{st}{\alpha }_s-c_{st}{\alpha }_s={\alpha }_t$. Hence every ${\sigma }_s$ is invertible with ${\sigma }_s^{-1}={\sigma }_s$, and the group these maps generate acts on $E$ (Linear map between vector spaces over the same field, Invertible linear maps, linear isomorphisms, and inverse linear maps, Identity maps and composites of linear maps are linear).

The root set. The root set of the construction is $\mathrm{\Phi }:=\{{\sigma }_{s_1}{\sigma }_{s_2}\cdots {\sigma }_{s_k}({\alpha }_t):k\ge 0,s_1,\dots ,s_k,t\in S\}\subseteq E,$ the orbit of the simple roots under the group generated by the ${\sigma }_s$ (each generator is an involution). Writing $\sigma :W\to {\mathrm{G}\mathrm{L}}_K(E)$ for the representation supplied by The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness ↗, one has $\mathrm{\Phi }=\{{\sigma }_w({\alpha }_s):w\in W,s\in S\}$, the $W$-orbit of the simple roots.

Conventions and limits. (i) $c_{st}=c_{ts}$; $c_{st}=0$ exactly when $m(s,t)=2$; $c_{st}=2$ when $m(s,t)=\mathrm{\infty }$; replacing ${\zeta }_{st}$ by ${\zeta }_{st}^{-1}$ leaves $c_{st}$ unchanged, so the construction depends on the chosen primitive roots only through the numbers $c_{st}$. (ii) No positivity of $c_{st}$, no integrality of roots over $\mathbb{Z}$, no identification of roots with reflections, and no faithfulness or definiteness of any form is asserted; those are separate matters, not part of this construction. (iii) $S$ is finite throughout, so the basis $({\alpha }_s{)}_{s\in S}$ is finite and every linear map is specified by finitely many values.

The single recorded justifier of this definition is The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness ↗, which proves ${\sigma }_s^2=\mathrm{i}\mathrm{d}$, the exact order of ${\sigma }_s{\sigma }_t$, and the induced homomorphism $\sigma :W\to {\mathrm{G}\mathrm{L}}_K(E)$; only after that lemma is $\mathrm{\Phi }$ literally the $W$-orbit of the simple roots.

No choice is used. The splitting field and the maps ${\sigma }_s$ are constructions, and one primitive $2m$-th root is fixed for each of finitely many edges; the convention $c_{st}=2$ for $m(s,t)=\mathrm{\infty }$ introduces no root of unity at all.

Let $(S,m)$ be a finite Coxeter matrix, and let $K$, $E$, $({\alpha }_s{)}_{s\in S}$, the constants $c_{st}$ and the linear maps ${\sigma }_s$ be as in The geometric representation on the simple-root basis over a common splitting field, and the root set; let $W$ and $\mathrm{\ell }$ be as in Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups. The involution, representation, signed-action and word assertions below hold for every finite $S$, including $S=\mathrm{\varnothing }$ and $\mathrm{\#}S=1$. Only the rank-two assertions require distinct generators.

1. Involutions. ${\sigma }_s^2={\mathrm{i}\mathrm{d}}_E$ for every $s\in S$; each ${\sigma }_s$ is invertible, and $\mathrm{\ell }(s)=1$.
2. The rank-two block. For any distinct $s,t\in S$, put $m:=m(s,t)$, $c:=c_{st}$, $P:=\mathrm{span}({\alpha }_s,{\alpha }_t)\subseteq E$, $A:={\sigma }_s{\sigma }_t$ and $B:=A{\mathrm{\mid }}_P$. Then $A(P)\subseteq P$, and in the basis $({\alpha }_s,{\alpha }_t)$, $B({\alpha }_s)=(c^2-1){\alpha }_s+c{\alpha }_t,B({\alpha }_t)=-c{\alpha }_s-{\alpha }_t;\mathrm{tr}(B)=c^2-2,\det (B)=1.$ Moreover $A(v)-v\in P$ for every $v\in E$. Consequently $E=P\oplus Q$ with $Q:=\mathrm{span}\{{\alpha }_u:u\in S,u\mathrm{\notin }\{s,t\}\}$, and for every $v=p+q$ with $p\in P$, $q\in Q$ and every $k\ge 1$, $A^k(p+q)=B^{k}p+({\mathrm{i}\mathrm{d}}_P+B+\cdots +B^{k-1})(A(q)-q)+q\mathrm{.}$
3. Exact order of the rank-two product. For any distinct $s,t$ and the notation of (2), when $m<\mathrm{\infty }$ put $\zeta :={\zeta }_{st}$, of order $2m$, so $c=\zeta +{\zeta }^{-1}$; when $m=\mathrm{\infty }$ use $c=2$, with no $\zeta$ stipulated. Then: (a) if $2<m<\mathrm{\infty }$, the characteristic polynomial of $B$ is $X^2-(c^2-2)X+1=(X-{\zeta }^2)(X-{\zeta }^{-2})$ with distinct roots; $B$ is diagonalisable, $B^m={\mathrm{i}\mathrm{d}}_P$, ${\mathrm{i}\mathrm{d}}_P+B+\cdots +B^{m-1}=0$, and $B^k\mathrm{\ne }{\mathrm{i}\mathrm{d}}_P$ for $0<k<m$; (b) if $m=2$, then $c=0$ and $B=-{\mathrm{i}\mathrm{d}}_P$; (c) if $m=\mathrm{\infty }$, then $c=2$, $B\mathrm{\ne }{\mathrm{i}\mathrm{d}}_P$, $(B-{\mathrm{i}\mathrm{d}}_P{)}^2=0$, and $B^k\mathrm{\ne }{\mathrm{i}\mathrm{d}}_P$ for every $k\ge 1$. Hence ${\sigma }_s{\sigma }_t=A$ has order exactly $m$ when $m<\mathrm{\infty }$, and infinite order when $m=\mathrm{\infty }$; in particular $A^m={\mathrm{i}\mathrm{d}}_E$ and $A^k\mathrm{\ne }{\mathrm{i}\mathrm{d}}_E$ for $0<k<m$ in the finite case.
4. The representation and the exact dihedral orders. The assignment $s\mapsto {\sigma }_s$ respects every defining relator of $W$ and induces a unique homomorphism $\sigma :W\to {\mathrm{G}\mathrm{L}}_K(E)$ with $\sigma (s)={\sigma }_s$. Consequently, for any distinct $s,t\in S$, one has $s\mathrm{\ne }t$ in $W$ and $st$ has order exactly $m(s,t)$ in $W$ (infinite when $m(s,t)=\mathrm{\infty }$).
5. The signed reflection action. Let $T:=\{ws{w}^{-1}:w\in W,s\in S\}\subseteq W$. For $s\in S$ define $U_s:\{\pm 1\}\times T\to \{\pm 1\}\times T$ by $U_s(\epsilon ,r):=(\epsilon \cdot (-1{)}^{\delta (s,r)},srs),$ where $\delta (s,r)=1$ if $r=s$ and $\delta (s,r)=0$ otherwise. Then $U_s^2=\mathrm{i}\mathrm{d}$ for every $s$, and the assignment $(\epsilon ,r)\cdot s:=U_s(\epsilon ,r)$ extends to a well-defined right action of $W$ on $\{\pm 1\}\times T$ (so $(\epsilon ,r)\cdot 1=(\epsilon ,r)$ and $(\epsilon ,r)\cdot (wv)=((\epsilon ,r)\cdot w)\cdot v$). Indeed the only relations to check are $s^2=1$ and, for distinct $s,t$ with $m=m(s,t)<\mathrm{\infty }$, $(st{)}^m=1$: $U_s^2=\mathrm{i}\mathrm{d}$ because for $r=s$ the two signs cancel while for $r\mathrm{\ne }s$ one has $srs\mathrm{\ne }s$ and $U_s(\epsilon ,srs)=(\epsilon ,r)$; and the alternating word $stst\cdots$ of length $2m$ acts trivially because its $2m$ prefix reflections $r_i:=w_{i-1}{s}_i{w}_{i-1}^{-1}$ (with $w_j:=s_1\cdots s_j$) satisfy $r_{i+m}=r_i$ for $1\le i\le m$ and $r_1,\dots ,r_m$ are pairwise distinct, so each reflection of the dihedral subgroup $⟨s,t⟩$ occurs exactly twice and every accumulated sign is even while the conjugating coordinate returns to $r$ because $(st{)}^m=1$.
6. Prefix reflections, deletion and expression independence. Let $(s_1,\dots ,s_k)$ be a word in $S$ with value $w=s_1\cdots s_k$ and prefix reflections $r_i=s_1\cdots s_{i-1}{s}_i{s}_{i-1}\cdots s_1$, and let $n(r):=\mathrm{\#}\{i:r_i=r\}$. (a) If $r_i=r_j$ for some $i<j$, then $s_1\cdots \widehat{s_i}\cdots \widehat{s_j}\cdots s_k=w$: the two letters $s_i,s_j$ can be deleted. (b) $(-1{)}^{n(r)}=:\eta (r,w)$ depends only on $w$ and $r$, and the right action satisfies $(\epsilon ,r)\cdot w=(\epsilon \eta (r,w),w^{-1}rw)$. (c) If the word is reduced ($k=\mathrm{\ell }(w)$), then $n(r)\in \{0,1\}$ for every $r$, the map $i\mapsto r_i$ is injective, and the set $\mathrm{\Phi }(w):=\{r_1,\dots ,r_k\}=\{r\in T:\eta (r,w)=-1\}$ is independent of the reduced expression chosen, with $\mathrm{\#}\mathrm{\Phi }(w)=\mathrm{\ell }(w)$.
7. Ambient reducedness of dihedral words. Let $s\mathrm{\ne }t$, $m:=m(s,t)$, and for $q\ge 1$ let $w_q$ be the value of the alternating word of length $q$ beginning with $s$. If $m<\mathrm{\infty }$ and $q\le m$, or if $m=\mathrm{\infty }$ and $q\ge 1$, then $\mathrm{\ell }(w_q)=q$, every word in $S$ representing $w_q$ has length at least $q$, and $w_1,\dots ,w_q$ are pairwise distinct. Thus dihedral alternating words are reduced in the ambient group $W$, not merely inside $⟨s,t⟩$.