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## Minimal matrices in algebra and combinatorics

### Ernesto Vallejo

Department of Mathematics, MIT and Instituto de Matematicas, UNAM.

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Thursday, March 27, 2003

102 Bradley Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: **
A minimal matrix is a matrix $A$ with non-negative integer entries such that the sequence of its entries, arranged in non-increasing order, is minimal (in the dominance order of partitions) among all sequences obtained from matrices that have the same row and column sum vectors as $A$. Minimal matrices first appeared in a characterization of 3-dimensional (0,1)-matrices that are uniquely determined by its plane sums.

It is a major open problem to give a satisfactory description of the multiplicity of a complex irreducible character of the symmetric group $S_n$ in the Kronecker product of other two irreducible characters of $S_n$. In this talk we explain how to use minimal matrices to produce minimal components in Kronecker products of irreducible characters. A theorem due to E. Snapper plays an important role in some results presented in this talk.

This talk will be accessible to graduate students.