Combinatorial Representation Theory

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Algebraic objects such as groups are used to measure symmetry in a similar way as numbers are used to measure size. In combinatorial representation theory we seek to make algebraic objects more accessible by relating abstract algebraic objects to combinatorial objects such as graphs and matrices. The combinatorial objects are often easier to understand and more importantly they lead to more efficient computation. In this proposal we are interested in using combinatorial objects to understand products of representations. One product investigated in this proposal is the tensor product of representations of symmetry groups. The goal is to devise an algorithm that uses combinatorial objects to understand the decomposition of this product into simpler representations. The decomposition of tensor products is an important problem that has applications to a plethora of fields such as algebraic combinatorics, complexity theory, and statistics, and has applications in medicine, computer vision, physics, chemistry, and fast matrix multiplication. Essentially, it is the problem of recovering individual signals from a mixture of signals. There are three longstanding unsolved problems in combinatorial representation theory that seek to decompose representations into irreducible representations. These include the Kronecker problem, the Plethysm problem and the Restriction problem. These problems are interrelated and making progress in the understanding of any will lead to breakthroughs on the others. Zabrocki and the PI introduced a new basis of symmetric functions that arose from connections to the partition algebra and led to the introduction of new combinatorial objects in the study of the Kronecker problem. This new basis of symmetric functions has provided a better understanding of the connection between the three open problems and the combinatorial objects introduced have made the problems more accessible. In this proposal the PI and collaborators, including graduate students, will continue to develop algorithms using diagram algebras and symmetric functions that we hope will lead to advances in the understanding of the Kronecker problem.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.