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Singular structure of Toda lattices and cohomology of certain compact Lie groups
We study the singularities (blow-ups) of the Toda lattice associated with a
real split semisimple Lie algebra . 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 related to the maximal
compact subgroup of the group with over the finite field . Here is the Langlands dual of . The blow-ups of the Toda lattice
are given by the zero set of the -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
-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.Comment: 23 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
On the difficulty of finding spines
We prove that the set of symplectic lattices in the Siegel space
whose systoles generate a subspace of dimension at least 3 in
does not contain any -equivariant
deformation retract of
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