The Robinson-Schensted-Knuth algorithm from the symmetric group to pairs of Young tableaux is well-known, and has many applications. In this talk, I'll be discussing its application to the study of infinite dimensional representations of simple Lie groups. In this context, the symmetric group is the Weyl group for simple Lie groups of type A_n, such as SL(n + 1, C). I will present some background material, and then show how the Robinson-Schensted-Knuth algorithm is used to determine Kazhdan-Lusztig cells, or equivalently, primitive ideals in the universal enveloping algebra of the Lie algebra associated to the group. This proof uses the notions of descent set, Knuth transform, and generalized descent set.
I will then discuss the generalization of this situation to Lie groups of type B_n, C_n, and D_n. Here, the Weyl group is the hyperoctahedral group, that is, signed permutations. (For type D_n, the Weyl group is a subgroup of index 2 of the hyperoctahedral group.) For this situation, we need a version of RSK which produces pairs of domino tableaux. We also need another procedure, related to Lusztig's notion of special representation. I will describe the two procedures. I will show how the notions of descent set, Knuth transform, and generalized descent set are defined and used in this situation, to prove theorems analogous to those in the type A_n situation.
This talk is expository, and has no prerequisites except an acquaintance with RSK.