Quantum Schur-Weyl duality and projected canonical bases

Original · EN

Let ᵣ be the generic type A Hecke algebra defined over [u, u⁻¹]. The Kazhdan-Lusztig bases {Cw}w ∈ §ᵣ and {C'w}w ∈ §ᵣ of ᵣ give rise to two different bases of the Specht module Mλ, λ r, of ᵣ. These bases are not equivalent and we show that the transition matrix S(λ) between the two is the identity at u = 0 and u = ∞. To prove this, we first prove a similar property for the transition matrices T, T' between the Kazhdan-Lusztig bases and their projected counterparts {Cw}w ∈ §ᵣ, {C'w}w ∈ §ᵣ, where Cw:= Cw pλ, C'w:= C'w pλand pλis the minimal central idempotent corresponding to the two-sided cell containing w. We prove this property of T,T' using quantum Schur-Weyl duality and results about the upper and lower canonical basis of V r (V the natural representation of Uq(ₙ)) from GL, FKK, Brundan. We also conjecture that the entries of S(λ) have a certain positivity property.