New Applications of Combinatorics to Representation Theory and Schubert Calculus

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The main goal of the research is to use combinatorial structures and methods in order to perform computations in algebra (more specifically, representation theory), geometry, topology, and number theory; hidden connections between various areas are also revealed in this process. Combinatorics studies various discrete structures (such as permutations, partially ordered sets, and graphs), which are well suited for encoding complex mathematical objects and for the related computations. Representation theory is a fundamental tool for studying symmetry, by realizing the elements of abstract groups/algebras as linear transformations (of some vector spaces). The PI studies representations of Lie algebras and quantum groups, which have many applications to physics, such as calculating the probability of a particle system being in a given state at a particular time. In geometry, the PI focuses on Schubert calculus, which has its origins in enumerative geometry (e.g., counting the lines or planes satisfying a number of generic intersection conditions), but is currently related to modern areas such as quantum cohomology. Some of the models developed by the PI have been or will be implemented in the open source computer algebra system SAGE.The proposed research consists of the following main projects, to be pursued with several collaborators. (1) The PI will extend his work on uniform combinatorial models for Kirillov-Reshetikhin (KR) crystals of affine Lie algebras from the single column KR crystals to the arbitrary ones. (2) Uniform combinatorial formulas are sought for the (non-metaplectic and metaplectic) Iwahori Whittaker functions, which are a basic tool in the theory of automorphic forms. Connections with Schubert calculus will also be investigated. (3) The combinatorics of the geometric Satake correspondence (realizing geometrically the irreducible representations of reductive groups) is studied via a combinatorial decomposition of Lusztig's q-analogue of the Weyl character. (4) In Schubert calculus, the PI has several projects related to various cohomologies of generalized flag varieties. In particular, an extension of Schubert calculus beyond K-theory is pursued based on the so-called Kazhdan-Lusztig Schubert classes in hyperbolic cohomology (which gives a stalk version of the elliptic cohomology of Ginzburg-Kapranov-Vasserot); these classes were defined by the PI in previous joint work.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.