Parabolic coordinate subspaces and fundamental faces

PDF p. 7: (3.1)–(3.2)

Use the Coxeter-system notation of Coxeter-system, roots, length and weak-order conventions: ${\mathrm{\Gamma }}_0$ is the vertex set, $V={⨁}_{i\in {\mathrm{\Gamma }}_0}\mathbb{R}{\alpha }_i$ has basis $({\alpha }_i{)}_{i\in {\mathrm{\Gamma }}_0}$, and $\mathrm{\Theta }=V^{\ast }$ is the space of real linear functionals on $V$. The vertex set may be infinite.

For $I\subseteq {\mathrm{\Gamma }}_0$, write $I^c={\mathrm{\Gamma }}_0\setminus I$. Define the parabolic coordinate subspace ${\mathrm{\Theta }}_I$ and the fundamental face $C_I$ by ${\mathrm{\Theta }}_I=\{\phi \in \mathrm{\Theta }:\phi ({\alpha }_i)=0\text{ for every }i\in I\}$ and $C_I=\{\phi \in {\mathrm{\Theta }}_I:\phi ({\alpha }_j)>0\text{ for every }j\in I^c\}$.

By Coxeter-system, roots, length and weak-order conventions, the ${\alpha }_i$ form a basis and $\mathrm{\Theta }$ is the algebraic dual. Its elements are linear maps with pointwise vector-space operations, as in Linear functionals and the algebraic dual V^*=L(V,F). The zero functional belongs to ${\mathrm{\Theta }}_I$. If $f,g\in {\mathrm{\Theta }}_I$, then $(f+g)({\alpha }_i)=f({\alpha }_i)+g({\alpha }_i)=0$ for every $i\in I$; for $r\in \mathbb{R}$, $(rf)({\alpha }_i)=rf({\alpha }_i)=0$ for every $i\in I$. Thus ${\mathrm{\Theta }}_I$ contains zero and is closed under addition and scalar multiplication, so it is a linear subspace by Linear subspace of a vector space. The strict inequalities select a well-defined subset $C_I$ of it. For nonemptiness, construct the functional explicitly. Write $v={\sum }_{j\in F}{a}_j{\alpha }_j$ with $F\subseteq {\mathrm{\Gamma }}_0$ finite, and set ${\phi }_I(v)={\sum }_{j\in F\setminus I}{a}_j$. The basis property makes the coefficients unique, and enlarging $F$ adds only zero coefficients, so this is a well-defined real number independent of the finite presentation.

For $u={\sum }_{j\in F}{a}_j{\alpha }_j$ and $v={\sum }_{j\in F}{b}_j{\alpha }_j$, use a common finite $F$ by adjoining zero coefficients. For $r,s\in \mathbb{R}$, finite-sum distributivity gives ${\phi }_I(ru+sv)={\sum }_{j\in F\setminus I}(r{a}_j+s{b}_j)=r{\phi }_I(u)+s{\phi }_I(v)$. Hence ${\phi }_I$ is linear and belongs to $\mathrm{\Theta }$. Its value on ${\alpha }_i$ is $0$ for $i\in I$ and $1$ for $i\mathrm{\notin }I$, so ${\phi }_I\in C_I$.

If $\psi \in \mathrm{\Theta }$ has these same prescribed basis values, linearity on each finite sum gives $\psi (v)={\sum }_{j\in F}{a}_j\psi ({\alpha }_j)={\sum }_{j\in F\setminus I}{a}_j={\phi }_I(v)$, proving uniqueness. If $I={\mathrm{\Gamma }}_0$, the formula gives the zero functional; it also applies when the vertex set is empty, with every sum empty. No choice of an additional basis or infinite summation is used.