The geometric representation on the simple-root basis over a common splitting field, and the root set

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.