Invariants of Weyl Group Action and q-characters of Quantum Affine Algebras

Abstract

Let W be the Weyl group corresponding to a finite dimensional simple Lie algebra $$\mathfrak {g}$$ of rank $$\ell $$ and let $$m>1$$ be an integer. In Inoue (Lett. Math. Phys. 111(1):32, 2021), by applying cluster mutations, a W-action on $$\mathcal {Y}_m$$ was constructed. Here $$\mathcal {Y}_m$$ is the rational function field on $$cm\ell $$ commuting variables, where $$c \in \{ 1, 2, 3 \}$$ depends on $$\mathfrak {g}$$ . This was motivated by the q-character map $$\chi _q$$ of the category of finite dimensional representations of quantum affine algebra $$U_q(\hat{\mathfrak {g}})$$ . We showed in Inoue (Lett. Math. Phys. 111(1):32, 2021) that when q is a root of unity, $$\textrm{Im} \chi _q$$ is a subring of the W-invariant subfield $$\mathcal {Y}_m^W$$ of $$\mathcal {Y}_m$$ . In this paper, we give more detailed study on $$\mathcal {Y}_m^W$$ ; for each reflection $$r_i \in W$$ associated to the ith simple root, we describe the $$r_i$$ -invariant subfield $$\mathcal {Y}_m^{r_i}$$ of $$\mathcal {Y}_m$$ .