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On the computation of Kronecker coefficients Tewari, Vasu Vineet


A major open problem in algebraic combinatorics is to find a combinatorial rule to compute the Kronecker product of two Schur functions. This is the same as decomposing the inner tensor product of two irreducible characters of the symmetric group as a sum of irreducible characters. Given that there is a combinatorial rule, namely the Littlewood-Richardson rule, which describes a way to compute the outer tensor product of two irreducible characters of the symmetric group, one would expect an algorithm which achieves the same purpose in the case of the inner tensor product. Jeffrey Remmel and Tamsen Whitehead first came up with a description of the Kronecker coefficients occurring in the Kronecker product of two Schur functions, both indexed by partitions of length at most 2. Mercedes Rosas later arrived at the same result using a different approach. The solution of the general problem would have implications in Complexity Theory and Quantum Information Theory. Our goal in this thesis is to derive formulae for computing the Kronecker product in certain cases where the Schur functions are indexed by partitions which are nearly rectangular. In particular, we study s{(n,n-1,1)}*s{(n,n)}, s{(n-1,n-1,1)}*s{(n,n-1)}, s{(n-1,n-1,2)}*s{(n,n)}, s{(n-1,n-1,1,1)}*s{(n,n)} and s{(n,n,1)}*s{(n,n,1)}. Our approach relies mainly on the fruitful interplay between manipulation of symmetric functions and the representation theory of the symmetric group. As a consequence of these formulae, we also derive an expression enumerating certain standard Young tableaux of bounded height.

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