Real coroot signs, word length and inversion sets

Use the minimal realization of Realization of a generalized cartan matrix and the dual action of Simple reflections and the kac moody weyl group. The real roots are ${\mathrm{\Delta }}^{\mathrm{r}\mathrm{e}}=\{w{\alpha }_i:w\in W,1\le i\le n\}$ and the real coroots are ${\mathrm{\Delta }}^{\vee ,\mathrm{r}\mathrm{e}}=\{w{h}_i:w\in W,1\le i\le n\}$. A real coroot is positive if its coordinates in the independent family $(h_i)$ are nonnegative integers; it is negative if its negative is positive. The corresponding root signs use $({\alpha }_i)$.

These vectors are nonzero, since each $w$ is invertible. The reflection formulas preserve the two integral spans, so the coordinates exist uniquely and are integral. The assertion that each such vector has exactly one of these signs is proved in Reduced words, root signs and finite coroot inversions ↗, rather than presumed in this definition.

The length $\mathrm{\ell }(w)$ is the least nonnegative length of a word in the simple reflections representing $w$. Such lengths form a nonempty subset of $\mathbb{N}$ by the definition of $W$. The identity has length zero. Define the coroot inversion set and the negative coroot set of an integral weight by $\mathrm{Inv}(w)=\{\beta \in {\mathrm{\Delta }}_{+}^{\vee ,\mathrm{r}\mathrm{e}}:w\beta \in -{\mathrm{\Delta }}_{+}^{\vee ,\mathrm{r}\mathrm{e}}\},N(\lambda )=\{\beta \in {\mathrm{\Delta }}_{+}^{\vee ,\mathrm{r}\mathrm{e}}:\lambda (\beta )<0\}\mathrm{.}$ Here $\lambda \in P$ as in Kac moody integral and dominant integral weights, so every pairing in the second set is a real integer, regardless of the complex values on complementary Cartan directions. Finiteness of $\mathrm{Inv}(w)$ is supplied by the sign lemma; no finiteness of $N(\lambda )$ for arbitrary integral $\lambda$ is asserted.

For applying general GCM results to coroots, transpose the realization: the Cartan is ${\mathfrak{h}}^{\ast }$, its simple coroots are ${\alpha }_i$, and its simple roots are the evaluations $h_i\in ({\mathfrak{h}}^{\ast }{)}^{\ast }$. Their pairing is ${\alpha }_i(h_j)=a_{ji}$; both families are independent and $\dim {\mathfrak{h}}^{\ast }=2n-\mathrm{rank}(A^T)$. Thus this is a minimal realization of $A^T$. Its Weyl action on $\mathfrak{h}$ is exactly the original dual action. Dual inverse actions are faithful and satisfy exactly the same word relations, so corresponding elements have the same minimal word lengths. The empty simple index set gives no real roots or coroots, the trivial group and empty inversion sets. No choice axiom is used.