Multiparameter quantum Schur duality of type B

Bao, Huanchen; Wang, Weiqiang; Watanabe, Hideya

doi:10.1090/proc/13749

Cited by 31 publications

(38 citation statements)

References 13 publications

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“…Section 6 is dedicated to the counterparts of Theorems B and C for the case ‚ " ı (see Theorems 6.2.2 and 6.3.8). In Section 7 we show that the stabilization algebras coincide with the gl-variants U  , U ı of the multiparameter quantum symmetric pair coideal subalgebras studied by Bao-Wang-Watanabe in [BWW18] (referred as U  , U ı therein). The argument is made bypassing the idempotented (or modified) quantum algebras.…”

Section: A New Directionmentioning

confidence: 93%

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Schur Algebras and Quantum Symmetric Pairs With Unequal Parameters

Lai

Luo

2019

International Mathematics Research Notices

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We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson-Lusztig-MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type AIII/AIV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated to any weight function. These bases are the counterparts of Lusztig's bar-invariant basis for Hecke algebras with unequal parameters. In the appendix we provide an algebraic version of a type D Beilinson-Lusztig-MacPherson construction which is first introduced by Fan-Li from a geometric viewpoint.

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Section: A New Directionmentioning

confidence: 93%

“…The (multiparameter) quantum symmetric pairs pU, U  q in this paper are the gl-variant of the quantum symmetric pairs in[BWW18].…”

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confidence: 99%

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Schur Algebras and Quantum Symmetric Pairs With Unequal Parameters

Lai

Luo

2019

International Mathematics Research Notices

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“…Work by H. Bao and W. Wang on type AIII/AIV quantum symmetric pairs, see Table 1, shows such quantum symmetric pairs admit a canonical basis [BW16]. Furthermore, they have a geometric interpretation through the geometry of partial flag varieties of type B/C [BSWW16], and are Schur-Weyl dual to Hecke algebras of type B [BWW16].…”

Section: Introductionmentioning

confidence: 99%

A topological origin of quantum symmetric pairs

Weelinck

2019

Sel. Math. New Ser.

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It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are Z 2 -orbifold versions of the little disks operads. We classify their categorical algebras and describe these explicitly in terms of a finite list of functors, natural isomorphisms and coherence equations. In dimension two, the categorical algebras are braided monoidal categories with an anti-involution together with a pointed module category carrying a universal solution to the (twisted) reflection equation. Main examples are obtained from the categories of representations of a ribbon Hopf algebra with an involution and a quasi-triangular coideal subalgebra, such as a quantum group and a quantum symmetric pair coideal subalgebra. Remark 1.10. The reflection equation algebra O q (G) is an (equivariant) quantization of the Semenov-Tian-Shansky Poisson bracket on O(G) [Mud06]. This algebra goes by many names: Majid's braided dual of U q (g) [Maj95], the quantum-loop algebra [AS96] and is isomorphic to the locally finite part of U q (g) via the Rosso form [KS09, Proposition 2.8].6

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“…One of the key ingredients for the construction of the j-canonical bases is the intertwiner (also known as the quasi-K-matrix) Υ. Using Υ, Bao The multi-parameter version of U j was considered in [BWW18]. Thanks to the integrality of the intertwiner Υ, the notion of j-canonical bases can be de- (pQ[p, q, q −1 ]⊕qQ[q])b .…”

Section: Introductionmentioning

confidence: 99%