Affine Combinatorics

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Abstract:The main premise of this project is the development of affine combinatorics, which are combinatorial structures coming from representation theory, algebraic geometry, and mathematical physics, in particular affine crystal theory and affine Schubert calculus. Using the newly constructed combinatorial models for Kirillov-Reshetikhin crystals by the PI and her collaborators, it is proposed to tackle the X=M conjecture of Hatayama et al. which gives explicit fermionic formulas for configuration sums of certain statistical mechanical models. Affine crystal structures have recently also been found to be important in combinatorial expressions for Littlewood-Richardson and fusion coefficients, as well as charge formulas for Macdonald polynomials. In particular, Fomin and Greene's theory of noncommutative symmetric functions in the realm of Kirillov-Reshetikhin crystals, the affine nilCoxeter algebra, and the affine local plactic algebra will be used to find such formulas. In addition, the PI proposes to study the combinatorial representation theory of bi-Hecke monoids and algebras, and the quantum R-matrix to explore generalizations of affine crystals. Combinatorial methods are often amenable to computational investigations. The robust implementation of algorithms derived from the project will lead to the development of new packages for the open-source computer algebra system SAGE.The PI's research is in the area of algebraic combinatorics. This is the study of discrete objects and maps between them (combinatorics) 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). One of 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. Due to its concreteness, combinatorics is very amenable to computational investigations. The algorithms derived from this project will be implemented in the open-source computer algebra system SAGE.