Combinatorics Seminar: Kronecker coecients and bipartite ribbon graphs

Wednesday, March 31 at 11:00am to 12:00pm

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Sanjaye Ramgoolam, Queen Mary University

Abstract: Bipartite ribbon graphs with n edges have a combinatoric characterisation in terms of pairs of permutations in the symmetric group Sn , subject to an equivalence relation. There is an associative algebra K(n) spanned by these equivalence classes, which uses Sn group multiplications. The algebra admits a decomposition into matrix blocks, one for every triple of Young diagrams with non-vanishing Kronecker coefficient C, forming
subspaces of K(n) of dimension equal to C2. For each Young diagram triple, the block can be identified as the null space of an operator, built from central elements of the algebra K(n), associated with cyclic permutations. These operators are represented, on the ribbon graph basis of the algebra, by integer matrices constructed using permutation group multiplications and integer normalized characters of symmetric groups. Integer null vectors spanning the block define a sub-lattice of the lattice of integer linear combinations of ribbon graphs. They can be obtained using Hermite normal form algorithms for the integer matrices. A similar construction works for C, upon using an involution on the algebra.

This talk is based on the paper


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