Combinatorial representation theory applied to Schubert calculus and Markov chains

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The award supports PI's research in the area of algebraic combinatorics. This is the study of discrete objects and maps between them together with algebraic properties that govern the structures of these objects. The combinatorial problems which arise are related to representation theory (study of symmetries), physics (energy levels of particles and their structure coefficients), and geometry (intersections of curves in space). Among the main tools used in this research are crystal graphs, which describe these structures in the "zero temperature limit" and yet encapture all of the important properties. Since combinatorics is concrete, it is very amenable to computational investigations. The robust implementation of algorithms derived from the project will lead to the development of new code for the open-source computer algebra system Sage. The dissemination of this new software through Sage will not only advance the proposed research program, but has already led to cross-fertilization between various areas in mathematics and computer science (for example through a semester program at ICERM), and this is expected to continue. This project involves the training of graduate students.The combinatorial study of affine Schubert calculus has led to new tableaux related to weak and strong Bruhat order in analogy to semistandard Young tableaux. Crystal base theory has a long tradition of answering representation theoretic questions in combinatorial terms, such as multiplicities of tensor products or positive character expansions. Applying crystal bases to the theory of weak tableaux, the PI and her collaborators were able to answer certain longstanding questions about structure constants of flag Gromov--Witten invariants and quantum Schubert structure coefficients. The PI seeks to complete this theory and apply it to strong tableaux. This would prove, for example, that k-Schur functions, which are affine versions of the usual Schur functions and form a set of representatives for the Schubert classes of the cohomology of the affine Grassmannian, expand positively in terms of Schur functions. The complete theory will facilitate the study of combinatorial expressions for generalizations of Littlewood--Richardson coefficients, fusion coefficients, and their q-analogues as well as one-dimensional sums, using algebras motivated by geometry and physics and relations to Macdonald theory, Demazure crystals, and Kirillov--Reshetikhin crystals. In addition, the PI and her collaborators have recently successfully developed a theory of Markov chains governed by R-trivial monoids. This has led to an abundance of new conjectures about eigenvalues, multiplicities, and mixing times as well as new Markov chains (for example, on linear extensions of posets) that will also be studied further.