The inversion set of a Weyl group element

Let $\mathrm{\Phi }$ be a reduced crystallographic root system with positive system ${\mathrm{\Phi }}^{+}$ and Weyl group $W$. The standing conventions are those of Finite Weyl root system, lattice and chamber conventions and Positive systems and simple roots: ${\mathrm{\Phi }}^{+}$ is the positive system attached to a chamber, each $w\in W$ permutes $\mathrm{\Phi }$ and acts on the ambient Euclidean space, and $\mathrm{\ell }$ is the length function of Length and longest Weyl-group element, which is also the minimal word length in simple reflections (Weyl length equals inversion number). The Lie-algebraic realization of the same Weyl action is the one of Root reflections and the Weyl group action inside The root set is a reduced crystallographic root system. Throughout, $\rho$ denotes the Weyl vector of The Weyl vector rho for a chosen positive system.

The inversion set. For $w\in W$ put

${\mathrm{\Phi }}_w=\{\alpha \in {\mathrm{\Phi }}^{+}:w^{-1}\alpha <0\}\mathrm{.}$

This is the set that indexes the exterior root covectors of the extremal cochains constructed from this page. It is computed from the inverse action, not from the action of $w$ itself.

Identification with the published inversion sets. The published inversion set of Finite Weyl root system, lattice and chamber conventions, written $N(w)$ in Length and longest Weyl-group element, is $\mathrm{Inv}(w)=N(w)=\{\alpha \in {\mathrm{\Phi }}^{+}:w\alpha \in {\mathrm{\Phi }}^{-}\},$ that is, the positive roots sent by $w$ itself to negative roots. With this convention ${\mathrm{\Phi }}_w=\mathrm{Inv}(w^{-1})=N(w^{-1}),$ since $w^{-1}\alpha \in {\mathrm{\Phi }}^{-}$ says exactly that $\alpha$ is a positive root sent by $w^{-1}$ to a negative root. Consequently, by the identification of length with inversion number and the equality of the lengths of inverse elements, $\mathrm{\mid }{\mathrm{\Phi }}_w\mathrm{\mid }=\mathrm{\mid }\mathrm{Inv}(w^{-1})\mathrm{\mid }=\mathrm{\ell }(w^{-1})=\mathrm{\ell }(w)\mathrm{.}$ The middle equality is the definition $\mathrm{\ell }(v)=\mathrm{\mid }N(v)\mathrm{\mid }$ of Length and longest Weyl-group element; the last equality holds because reversing an expression of $w$ in simple reflections expresses $w^{-1}$ with the same number of letters, so the minimal word lengths agree, and both equal the inversion numbers (Weyl length equals inversion number; Finite Weyl strong exchange and deletion, which states that word length equals inversion length for these simple reflections).

Extremal values and the half-sum identity. For the unit $1\in W$ nothing is inverted, ${\mathrm{\Phi }}_1=\mathrm{\varnothing }$; for the longest element $w_0$, whose existence and uniqueness are proved in Weyl length equals inversion number and which satisfies $w_0({\mathrm{\Phi }}^{+})={\mathrm{\Phi }}^{-}$, one has $w_0^{-1}({\mathrm{\Phi }}^{+})={\mathrm{\Phi }}^{-}$ and hence ${\mathrm{\Phi }}_{w_0}={\mathrm{\Phi }}^{+}$. Moreover ${\sum }_{\alpha \in {\mathrm{\Phi }}_w}\alpha =\rho -w\rho \mathrm{.}$ Indeed, $w{\mathrm{\Phi }}^{+}=\{\beta \in \mathrm{\Phi }:w^{-1}\beta \in {\mathrm{\Phi }}^{+}\}$, and this set is the disjoint union of the positive roots with $w^{-1}\alpha >0$ and the negatives of the roots in ${\mathrm{\Phi }}_w$: a root $\beta$ with $w^{-1}\beta >0$ is either positive, or of the form $-\alpha$ with $\alpha \in {\mathrm{\Phi }}_w$, and no other possibility occurs. Since $\rho =\frac{1}{2}{\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}\alpha$ and $w\rho =\frac{1}{2}{\sum }_{\beta \in w{\mathrm{\Phi }}^{+}}\beta$, the difference is $\rho -w\rho =\frac{1}{2}({\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}\alpha -{\sum }_{\alpha :w^{-1}\alpha >0}\alpha +{\sum }_{\alpha \in {\mathrm{\Phi }}_w}\alpha )=\frac{1}{2}({\sum }_{\alpha \in {\mathrm{\Phi }}_w}\alpha +{\sum }_{\alpha \in {\mathrm{\Phi }}_w}\alpha )={\sum }_{\alpha \in {\mathrm{\Phi }}_w}\alpha \mathrm{.}$ The first sum is over all positive roots, the second over the positive roots with $w^{-1}\alpha >0$, and their difference is the sum over ${\mathrm{\Phi }}_w$. This identity is the sign-normalized half-sum statement used, in the form ${\sum }_{\alpha \in {\mathrm{\Phi }}_w}\alpha =\rho -w\rho$, in the extremal-cochain lemma below.