# Quantum Schur-Weyl duality and projected canonical bases

Let \H r be the generic type A Hecke algebra defined over \ZZu, u^{-1}. The Kazhdan-Lusztig bases \{C w\} {w \in \S r} and \{C w\} {w \in \S r} of \H r give rise to two different bases of the Specht module M \lambda, \lambda \vdash r, of \H r. These bases are not equivalent and we show that the transition matrix S\lambda between the two is the identity at u = 0 and u = \infty. To prove this, we first prove a similar property for the transition matrices \tilde{T}, \tilde{T} between the Kazhdan-Lusztig bases and their projected counterparts \{\tilde{C} w\} {w \in \S r}, \{\tilde{C} w\} {w \in \S r}, where \tilde{C} w := C w p \lambda, \tilde{C} w := C w p \lambda and p \lambda is the minimal central idempotent corresponding to the two-sided cell containing w. We prove this property of \tilde{T},\tilde{T} using quantum Schur-Weyl duality and results about the upper and lower canonical basis of V^{\tsr r} V the natural representation of U q\gl n from \cite{GL, Brundan}. We also conjecture that the entries of S\lambda have a certain positivity property.

Author: Jonah Blasiak

Source: https://archive.org/