Shuffle tableaux

Elements of Lusztig’s dual canonical bases are Schur-positive when evaluated on (generalized) Jacobi-Trudi matrices. This deep property was proved by Rhoades and Skandera, relying on a result of Haiman, and ultimately on the (proof of) Kazhdan-Lusztig conjecture. For a particularly tractable part of the dual canonical basis – called Temperley-Lieb immanants – we give a generalization of Littlewood-Richardson rule using shuffle tableaux. We then use our new rule to prove a special case of a Schur log-concavity conjecture by Lam-Postnikov-Pylyavskyy.

For information about the Richard P. Stanley Seminar in Combinatorics, visit https://math.mit.edu/combin/