Weyl alternants are skew-invariant

Given: The root system with Weyl group $W$ and length function $\mathrm{\ell }$, the completed character ring $\mathcal{R}$ with its action of $W$ on finite-support elements, the alternants $A(\nu )$, and elements $w\in W$, $\nu \in {\mathfrak{h}}^{\ast }$.

For finite-support $g={\sum }_{\mu }{c}_{\mu }{e}^{\mu }$ one has $w\cdot g={\sum }_{\mu }{c}_{\mu }{e}^{w\mu }$, and $A(\nu )={\sum }_{x\in W}(-1{)}^{\mathrm{\ell }(x)}{e}^{x\nu }$ is a finite-support element of $\mathcal{R}$ (The Weyl alternation operator, The completed formal character ring).

The sign $(-1{)}^{\mathrm{\ell }}$ is a homomorphism: for all $u,v\in W$, $(-1{)}^{\mathrm{\ell }(uv)}=(-1{)}^{\mathrm{\ell }(u)}(-1{)}^{\mathrm{\ell }(v)}$, and $(-1{)}^{\mathrm{\ell }(w^{-1})}=(-1{)}^{\mathrm{\ell }(w)}$ (The sign of the Weyl length is multiplicative).

The action of $W$ on ${\mathfrak{h}}^{\ast }$ is a group action by the root reflections $s_{\alpha }(\lambda )=\lambda -⟨\lambda ,{\alpha }^{\vee }⟩\alpha$; the simple reflection $s_i$ satisfies $s_i{\alpha }_i=-{\alpha }_i\mathrm{\ne }{\alpha }_i$, so $s_i$ is not the identity, and $\mathrm{\ell }$ is the least number of simple reflections in an expression for an element, so $\mathrm{\ell }(s_i)=1$ (Root reflections and the Weyl group action, The Weyl group is finite and faithful, Finite Weyl root system, lattice and chamber conventions).