Tropical Weyl group action via point configurations and \(\tau\)-functions of the \(q\)-Painlevé equations
The aim of this letter is to present an algebraic approach to a tropical representation of Weyl groups over the field of \(\tau\)-functions through intersection theory on rational surfaces. It is known that point sets in the projective space are closely related to Weyl groups. Let \(X_{m,n}\) be the configuration space of \(n\) points of \(P^{m-1}\) in general position. Then, the Weyl group corresponding to the Dynkin diagram \(T_{2,m,n-m}\) acts birationally on \(X_{m,n}\) and is generated by permutations of \(n\) points and the standard Cremona transformation with respect to each \(m\) points. In particular, the construction includes \(E_8^{(1)}\), \(E_7^{(1)}\), \(E_6^{(1)}\) and \(E_5^{(1)}\) as affine cases; each of them gives rise to the \(q\)-difference Painlevé equation. The paper consists of 4 sections; it contains an interesting relation between tropical representations of the Weyl groups and \(q\)-difference Painlevé equation.
scientific article; zbMATH DE number 5127087