612 research outputs found
Trivial Meet and Join within the Lattice of Monotone Triangles
The lattice of monotone triangles ordered by
entry-wise comparisons is studied. Let denote the unique minimal
element in this lattice, and the unique maximum. The number of
-tuples of monotone triangles with minimal infimum
(maximal supremum , resp.) is shown to
asymptotically approach as . Thus, with
high probability this event implies that one of the is
(, resp.). Higher-order error terms are also discussed.Comment: 15 page
Toda lattice, cohomology of compact Lie groups and finite Chevalley groups
In this paper, we describe a connection that exists among (a) the number of
singular points along the trajectory of Toda flow, (b) the cohomology of a
compact subgroup , and (c) the number of points of a Chevalley group
related to over a finite field . The Toda
lattice is defined for a real split semisimple Lie algebra , and
is a maximal compact Lie subgroup of associated to .
Relations are also obtained between the singularities of the Toda flow and the
integral cohomology of the real flag manifold with the Borel subgroup
of (here we have with a finite group ). We also compute the
maximal number of singularities of the Toda flow for any real split semisimple
algebra, and find that this number gives the multiplicity of the singularity at
the intersection of the varieties defined by the zero set of Schur polynomials.Comment: 28 pages, 5 figure
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
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