17,484 research outputs found
Littlewood-Richardson Coefficients via Yang-Baxter Equation
The purpose of this paper is to present an interpretation for the
decomposition of the tensor product of two or more irreducible representations
of GL(N) in terms of a system of quantum particles. Our approach is based on a
certain scattering matrix that satisfies a Yang-Baxter type equation. The
corresponding piecewise-linear transformations of parameters give a solution to
the tetrahedron equation. These transformation maps are naturally related to
the dual canonical bases for modules over the quantum enveloping algebra
. A byproduct of our construction is an explicit description for the
cone of Kashiwara's parametrizations of dual canonical bases. This solves a
problem posed by Berenstein and Zelevinsky. We present a graphical
interpretation of the scattering matrices in terms of web functions, which are
related to honeycombs of Knutson and Tao.Comment: 24 page
Lecture notes: Semidefinite programs and harmonic analysis
Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th
International Workshop on High Performance Optimization Techniques (Algebraic
Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg
University, The Netherlands.Comment: 31 page
On the existence of 0/1 polytopes with high semidefinite extension complexity
In Rothvo\ss{} it was shown that there exists a 0/1 polytope (a polytope
whose vertices are in \{0,1\}^{n}) such that any higher-dimensional polytope
projecting to it must have 2^{\Omega(n)} facets, i.e., its linear extension
complexity is exponential. The question whether there exists a 0/1 polytope
with high PSD extension complexity was left open. We answer this question in
the affirmative by showing that there is a 0/1 polytope such that any
spectrahedron projecting to it must be the intersection of a semidefinite cone
of dimension~2^{\Omega(n)} and an affine space. Our proof relies on a new
technique to rescale semidefinite factorizations
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