Quantum Schur–Weyl duality and projected canonical bases

Let HrHr be the generic type A   Hecke algebra defined over Z[u,u−1]Z[u,u−1]. The Kazhdan–Lusztig bases {Cw}w∈Sr{Cw}w∈Sr and {Cw′}w∈Sr of HrHr give rise to two different bases of the Specht module MλMλ, λ⊢rλ⊢r, of HrHr. These bases are not equivalent and we show that the transition matrix S(λ)S(λ) between the two is the identity at u=0u=0 and u=∞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 {C˜w}w∈Sr, {C˜w′}w∈Sr, where C˜w:=Cwpλ, C˜w′:=Cw′pλ and pλpλ is 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⊗rV⊗r (V   the natural representation of Uq(gln)Uq(gln)) from [14], [11] and [7]. We also conjecture that the entries of S(λ)S(λ) have a certain positivity property.