Journal article
Quantum Schur–Weyl duality and projected canonical bases
Journal of algebra, v 402, pp 499-532
15 Mar 2014
Abstract
Let Hr be the generic type A Hecke algebra defined over Z[u,u−1]. The Kazhdan–Lusztig bases {Cw}w∈Sr and {Cw′}w∈Sr of Hr give rise to two different bases of the Specht module Mλ, λ⊢r, of Hr. 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 {C˜w}w∈Sr, {C˜w′}w∈Sr, where C˜w:=Cwpλ, C˜w′:=Cw′pλ and pλ 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⊗r (V the natural representation of Uq(gln)) from [14,11,7]. We also conjecture that the entries of S(λ) have a certain positivity property.
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Details
- Title
- Quantum Schur–Weyl duality and projected canonical bases
- Creators
- Jonah Blasiak - Drexel University
- Publication Details
- Journal of algebra, v 402, pp 499-532
- Publisher
- Elsevier
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000331418500021
- Scopus ID
- 2-s2.0-84892647557
- Other Identifier
- 991019168854804721
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- Web of Science research areas
- Mathematics