Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice

Let $S$ be finite, let $m$ be a Coxeter matrix on $S$, let $W$ be the presented Coxeter group (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), let $\mathrm{\Gamma }$ be its Coxeter diagram (Coxeter diagrams: edges, labels, components and finite type), and let $V={\mathbb{R}}^S$ carry the real Coxeter form $B$ (The real Coxeter form, its radical, reflections, and form-preserving maps): $B(e_s,e_s)=1$, $B(e_s,e_t)=-\cos (\pi \mathrm{/}m(s,t))$ for finite labels, and $B(e_s,e_t)=-1$ when $m(s,t)=\mathrm{\infty }$. Put $\mathrm{rad}(B):=\{v\in V:B(v,w)=0\text{ for every }w\in V\}$ (The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space).

(1) Affine form type. The system $(W,S)$, and with it $m$, $\mathrm{\Gamma }$ and $B$, is of affine form type when $\mathrm{\Gamma }$ is connected and $B$ is positive semidefinite (meaning $B(v,v)\ge 0$ for every $v\in V$) of corank one, that is, ${\dim }_{\mathbb{R}}\mathrm{rad}(B)=1$ (Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis). By Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions ↗ (1), this is equivalent to requiring that $\mathrm{\Gamma }$ be connected, $B$ positive semidefinite, and $B$ not positive definite. This is the only notion called affine on this page. It is a condition on the Coxeter form, not a synonym for an infinite abstract Coxeter group. Disconnected systems are outside this definition; the companion examples page treats reducible forms separately.

(2) Positive radical vector. A positive radical vector for affine form type is a vector $\delta ={\sum }_{s\in S}{\delta }_s{e}_s\in \mathrm{rad}(B)$ with ${\delta }_s>0$ for every $s\in S$. For affine form type such a vector exists and $\mathrm{rad}(B)=\mathbb{R}\delta$, by Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions ↗ (1); this clause names the object and records the property supplied by that lemma.

(3) Radical quotient. Let $U:=V\mathrm{/}\mathrm{rad}(B)$ (The quotient vector space $V\mathrm{/}W$ and its canonical projection), and define $b(\bar{v},\bar{w}):=B(v,w)$. The form $b$ is well-defined, symmetric and positive definite. Thus $U$ is a Euclidean vector space (Real and complex inner-product spaces and their induced length) of dimension $\mathrm{\mid }S\mathrm{\mid }-1$ (Linear subspace of a vector space, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis).

(4) Affine slice and walls. For a positive radical vector $\delta$, put $E_{\delta }:=\{\phi \in V^{\ast }:\phi (\delta )=1\},$ where $V^{\ast }$ is the algebraic dual (Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$). This is an affine subspace (Affine subspaces as translates $x+U$ of linear subspaces) with direction space $K_{\delta }:=\{\phi \in V^{\ast }:\phi (\delta )=0\}=\mathrm{Ann}(\mathrm{rad}(B))\mathrm{.}$ Precomposition with the quotient projection identifies $K_{\delta }$ with $U^{\ast }$. Let $b^{\mathrm{♭}}:U\to U^{\ast }$ be $b^{\mathrm{♭}}(u):=b(u,\cdot )$ (Bilinear forms on $V$ correspond linearly and bijectively to linear maps $V\to V^{\ast }$); it is an isomorphism, as verified in the remarks, and the dual form is $b^{\ast }(\alpha ,\beta ):=b((b^{\mathrm{♭}}{)}^{-1}\alpha ,(b^{\mathrm{♭}}{)}^{-1}\beta )\mathrm{.}$ Transporting $b^{\ast }$ to the direction space $K_{\delta }$ makes $E_{\delta }$ a Euclidean affine space of dimension $\mathrm{\mid }S\mathrm{\mid }-1$. For $s\in S$, its wall is $H_s:=\{\phi \in E_{\delta }:\phi (e_s)=0\}$, its open alcove is $A:=\{\phi \in E_{\delta }:\phi (e_s)>0\text{ for every }s\in S\}$, and $\bar{A}$ denotes the closure of $A$ in $E_{\delta }$.

(5) Abstentions and conventions. Beyond the constructions and properties above, this definition does not assert that the $H_s$ are affine hyperplanes, that $A$ is nonempty, bounded or a simplex, that $W$ acts on $E_{\delta }$ as a group generated by affine reflections, or that the translates $w\bar{A}$ cover $E_{\delta }$. The phrase affine form type refers only to the Coxeter-matrix condition in (1); extended affine Weyl groups and the directed affine Dynkin diagrams used in Lie theory are separate conventions and are not defined here. No choice principle is used in these finite-dimensional constructions.

The quotient-form assertions in (3) follow directly from positive semidefiniteness. If $r\in \mathrm{rad}(B)$, then replacing either representative by one differing by a radical vector does not change $B(v,w)$, so $b$ is well-defined; symmetry is inherited from $B$. If $b(\bar{v},\bar{v})=0$, then for every $w\in V$ and every $t\in \mathbb{R}$, $0\le B(v+tw,v+tw)=2tB(v,w)+t^{2}B(w,w)\mathrm{.}$ Both signs of arbitrarily small $t$ force $B(v,w)=0$. This holds for every $w$, so $v\in \mathrm{rad}(B)$ and $\bar{v}=0$.

The dimension claim in (3) is also explicit. Since the radical has dimension one, choose a nonzero $r={\sum }_s{r}_s{e}_s$ in it and an index $s_0$ with $r_{s_0}\mathrm{\ne }0$. The classes ${\bar{e}}_s$ for $s\mathrm{\ne }{s}_0$ span $U$, because ${\bar{e}}_{s_0}=-{\sum }_{s\mathrm{\ne }{s}_0}(r_s\mathrm{/}{r}_{s_0}){\bar{e}}_s$. They are independent: if ${\sum }_{s\mathrm{\ne }{s}_0}{a}_s{e}_s$ lies in $\mathbb{R}r$, its $s_0$ coordinate forces that multiple of $r$ to be zero, and then every $a_s=0$. They are therefore a basis of size $\mathrm{\mid }S\mathrm{\mid }-1$.

For (4), $E_{\delta }$ is nonempty: choose $s_0$ with ${\delta }_{s_0}>0$ and use the coordinate functional ${\phi }_0(v):=v_{s_0}\mathrm{/}{\delta }_{s_0}$. Its direction is $K_{\delta }$, which equals the annihilator of $\mathrm{rad}(B)$ because that radical is the line $\mathbb{R}\delta$. The map $U^{\ast }\to K_{\delta }$, $\lambda \mapsto \lambda \circ \pi$, with $\pi :V\to U$ the quotient projection, is injective since $\pi$ is onto; it is surjective because each functional in $K_{\delta }$ vanishes on the radical and therefore factors through $\pi$. A basis of $U$ gives dual coordinate functionals: each functional is determined by its values on that basis, and arbitrary values extend linearly. Thus they form a basis of $U^{\ast }$ and $\dim U^{\ast }=\dim U=\mathrm{\mid }S\mathrm{\mid }-1$. Finally, $b^{\mathrm{♭}}$ is injective: if $b^{\mathrm{♭}}(u)=0$, then $b(u,u)=0$, hence $u=0$ by positive definiteness. The images under $b^{\mathrm{♭}}$ of a basis of $U$ are therefore $\mathrm{\mid }S\mathrm{\mid }-1$ independent vectors in the $\mathrm{\mid }S\mathrm{\mid }-1$ dimensional space $U^{\ast }$, so they form a basis and $b^{\mathrm{♭}}$ is an isomorphism. The displayed formula therefore defines a positive-definite inner product on $U^{\ast }$ and hence on $K_{\delta }$. All selections above are single finite-dimensional constructions; no arbitrary choice principle is used.