Combinatorics and Complexity of Kronecker coefficients

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This research project concerns studies of fundamental symmetries in mathematics, and how they interact with each other, creating larger symmetries. The aim is to give a quantitative analysis of the number of times some symmetries appear as a result of such interactions. The analysis will help elucidate the nature of symmetries in general and will apply to other spaces and families of discrete objects exhibiting such symmetries. The work has important applications in probability and statistics.The proposal is concerned with the study of Kronecker coefficients of the symmetric group, which are some of the most classical objects in Algebraic Combinatorics. It aims at applications of enumerative and computational complexity nature, for estimating the coefficients and deciding their positivity. On a combinatorics side, the proposal aims to establish new unimodality results for various classes of partitions. The employed tools are largely intrinsic in these fields, involve group representation theory, Schur functions, and combinatorics of Young tableaux, with some bijective and analytic methods in partition theory added to the mix.