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arXiv · 1102.1453

Quantum Schur-Weyl duality and projected canonical bases

Jonah Blasiak
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Abstract

Let \H_r be the generic type A Hecke algebra defined over \ZZ[u, u^{-1}]. The Kazhdan-Lusztig bases \{C_w\}_{w \in §_r} and \{C'_w\}_{w \in §_r} of \H_r give rise to two different bases of the Specht module M_λ, λ\vdash r, of \H_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 = \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 §_r}, \{\tilde{C}'_w\}_{w \in §_r}, where \tilde{C}_w := C_w p_λ, \tilde{C}'_w := C'_w p_λand p_λis 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, FKK, Brundan}. We also conjecture that the entries of S(λ) have a certain positivity property.

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Jonah Blasiak. 2013-12-10. Quantum Schur-Weyl duality and projected canonical bases. https://arxiv.org/abs/1102.1453

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