Project Summary: quantizing Schur functors

Geometric complexity theory (GCT) is an approach to P vs. NP and related problems in complexity theory using algebraic geometry and representation theory. A fundamental problem in representation theory, believed to be important for this approach, is the Kronecker problem, which asks for a positive combinatorial formula for the multiplicity gλμν of an irreducible representation Mν of the symmetric group in the tensor product Mλ ⊗Mμ. Blasiak, Mulmuley, and Sohoni have been developing an approach to this problem using quantum groups and canonical bases. The main focus of this proposal is to push this approach further. Canonical bases were first defined by Kazhdan and Lusztig through their study of singularities of Schubert varieties. Of particular interest for this project is their remarkable ability to connect combinatorics and representation theory. For instance, canonical bases beautifully connect the RSK correspondence with quantum Schur-Weyl duality, give a Littlewood-Richardson rule for all types, and have been used by Blasiak to explain the appearance of the combinatorial operations cyclage and catabolism in the graded characters of Garsia-Procesi modules. The nonstandard quantum group and nonstandard Hecke algebra were defined by Mulmuley and Sohoni to study the Kronecker problem. Recently, Mulmuley, Sohoni, and Blasiak have obtained the beginnings of a theory of canonical bases for these nonstandard objects. They construct a canonical basis of a certain representation of the nonstandard quantum group and use this to solve the Kronecker problem in the case of two two-row shapes. The Uq(sl2)-graphical calculus is used to organize and count the resulting crystal components, which is believed to be the beginnings of a new kind of graphical calculus in this setting. Blasiak, together with Mulmuley, Sohoni, and others, plans to push this approach further, in the following ways.