Bitableaux and Zero Sets of Dual Canonical Basis Elements

Abstract

Abstract. We state new results concerning the zero sets of polynomials belonging to the dual canonical basis of C[x 1,1 , . . . , x n,n ]. As an application, we show that this basis is related by a unitriangular transition matrix to the simpler bitableau basis popularized by Désarménien-Kung-Rota. It follows that spaces spanned by certain subsets of the dual canonical basis can be characterized in terms of products of matrix minors, or in terms of their common zero sets.

Relations between the Clausen and Kazhdan–Lusztig representations of the symmetric group

“…3.1-3.2].) We define the shape of v to be the partition sh(v) = sh(P(v)) = sh(Q (v)) of n. Following [13], we use the map v → (P(v), Q (v)) to transfer the iterated dominance order on standard bitableaux of size n to S n . To be precise, we define the iterated dominance order on S n by declaring v ≤ I w if we have (P(v), Q (v)) I (P(w), Q (w)).…”

“…For each permutation v in S n , we follow [13] in defining the bideterminant R v (x) by A natural S n -action on C[x] is given by…”

“…Like the McDonough-Pallikaros construction, our new construction of the Kazhdan-Lusztig representations does not rely upon preorders. (See also [13] for an earlier appearance of the transition matrix A, and [14], [15] for previous related work on the dual canonical basis.) In Sections 2-3, we review basic definitions related to the symmetric group, Hecke algebra, and Kazhdan-Lusztig modules.…”

“…This leads to our main result that this last family of S n -modules gives a new, preorder-free construction of the Kazhdan-Lusztig representations of S n . We finish by showing that the relationship between the bideterminant and Kazhdan-Lusztig immanant bases studied in [13,Sec. 5] leads to unitriangular transition matrices relating Clausen's irreducible representations of S n to those of Kazhdan and Lusztig.…”

“…, x n,n ]. Borrowing ideas from Clausen, and applying vanishing properties of Kazhdan-Lusztig immanants obtained in [13], we then modify our specialized modules to eliminate all quotients. This leads to our main result that this last family of S n -modules gives a new, preorder-free construction of the Kazhdan-Lusztig representations of S n .…”