Tropical Weyl group action via point configurations and \(\tau\)-functions of the \(q\)-Painlevé equations
type: Item
instance of: scholarly article
title: Tropical Weyl group action via point configurations and \(\tau\)-functions of the \(q\)-Painlevé equations
zbMATH Open document ID: 1118.39007
author: Teruhisa Tsuda
published in: Letters in Mathematical Physics
publication date: 2007-02-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 39A13
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 37K20
Mathematics Subject Classification ID: 37K35
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 20F55
zbMATH DE Number: 5127087
zbMATH Keywords: tropical representations
zbMATH Keywords: Weyl groups
zbMATH Keywords: \(q\)-Painlevé equation
zbMATH Keywords: \(\tau\)-functions
reviewed by: Ahmed Hegazi
MaRDI profile type: Publication
full work available at URL: s11005-006-0052-z
OpenAlex ID: W2037525289
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DOI: 10.1007/S11005-006-0052-Z
description: scientific article; zbMATH DE number 5127087