Singular structure of Toda lattices and cohomology of certain compact Lie groups

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dc.creatorCasian, L.
dc.creatorKodama, Y.
dc.date2006-02-10
dc.date.accessioned2026-07-07T07:03:18Z
dc.date.available2026-07-07T07:03:18Z
dc.descriptionWe study the singularities (blow-ups) of the Toda lattice associated with a real split semisimple Lie algebra $\mathfrak g$. It turns out that the total number of blow-up points along trajectories of the Toda lattice is given by the number of points of a Chevalley group $K({\mathbb F}_q)$ related to the maximal compact subgroup $K$ of the group $\check G$ with $\check{\mathfrak g}={\rm Lie}(\check G)$ over the finite field ${\mathbb F}_q$. Here $\check{\mathfrak g}$ is the Langlands dual of ${\mathfrak g}$. The blow-ups of the Toda lattice are given by the zero set of the $τ$-functions. For example, the blow-ups of the Toda lattice of A-type are determined by the zeros of the Schur polynomials associated with rectangular Young diagrams. Those Schur polynomials are the $τ$-functions for the nilpotent Toda lattices. Then we conjecture that the number of blow-ups is also given by the number of real roots of those Schur polynomials for a specific variable. We also discuss the case of periodic Toda lattice in connection with the real cohomology of the flag manifold associated to an affine Kac-Moody algebra.
dc.description23 pages, 12 figures, To appear in the proceedings "Topics in Integrable Systems, Special Functions, Orthogonal Polynomials and Random Matrices: Special Volume, Journal of Computational and Applied Mathematics"
dc.identifierhttps://arxiv.org/abs/math/0602229
dc.identifierhttp://arxiv.org/abs/math/0602229
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/108925
dc.subjectAlgebraic Geometry
dc.subjectAlgebraic Topology
dc.subjectExactly Solvable and Integrable Systems
dc.titleSingular structure of Toda lattices and cohomology of certain compact Lie groups
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