Geometric Complexity Theory IV: quantum group for the Kronecker problem

Abstract

A fundamental problem in representation theory is to find an explicit positive rule, akin to the Littlewood-Richardson rule, for decomposing the tensor product of two irreducible representations of the symmetric group (Kronecker problem). In this paper a generalization of the Drinfeld-Jimbo quantum group, with a compact real form, is constructed, and also an associated semisimple algebra that has conjecturally the same relationship with the generalized quantum group that the Hecke algebra has with the Drinfeld-Jimbo quantum group. In the sequel [24] it is observed that an explicit positive decomposition rule for the Kronecker problem exists assuming that the coordinate ring of the generalized quantum group has a basis analogous to the canonical basis for the coordinate ring of the Drinfeld-Jimbo quantum group, as per Kashiwara and Lusztig [12, 17], or in the dual setting–the associated algebra has a basis analogous to the Kazhdan-Lusztig basis for the Hecke algebra [13], as suggested by the experimental and theoretical results therein. In the other sequel [23], similar quantum group and algebra are constructed for the generalized plethysm problem, of which the Kronecker problem studied here is a special case. These problems play a central role in geometric complexity theory– an approach to the P vs. NP and related problems. Visiting faculty member