Geometric Schur Duality of Classical Type

By Huanchen Bao, Jonathan R. Kujawa, Yiqiang Li, Weiqiang Wang. 2014

116 citations recorded by the source corpus.

This is a generalization of the classic work of Beilinson, Lusztig and MacPherson. In this paper (and an Appendix) we show that the quantum algebras obtained via a BLM-type stabilization procedure in the setting of partial Flag varieties of type B/C are two (modified) coideal subalgebras of the quantum general linear Lie algebra, U.$$ \overset{.}{\mathbf{U}} $$ℐ and U.$$ \overset{.}{\mathbf{U}} $$ʅ . We provide a geometric realization of the Schur-type duality of Bao–Wang between such a coideal algebra and Iwahori–Hecke algebra of type B. The monomial bases and canonical bases of the Schur algebras and the modified coideal algebra U.$$ \overset{.}{\mathbf{U}} $$ℐ are constructed.In an Appendix by three authors, a more subtle 2-step stabilization procedure leading to U.$$ \overset{.}{\mathbf{U}} $$ʅ is developed, and then monomial and canonical bases of U.$$ \overset{.}{\mathbf{U}} $$ʅ are constructed. It is shown that U.$$ \overset{.}{\mathbf{U}} $$ʅ is a subquotient of U.$$ \overset{.}{\mathbf{U}} $$ℐ with compatible canonical bases. Moreover, a compatibility between canonical bases for modified coideal algebras and Schur algebras is established.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6ff391787ae0052797e2015b27184ddeb0954d67

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Citations in the corpus

Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality (2013, methodology): “It is related to for instance [9] or [35] by using instead of convolution algebras of functions on partial flag varieties over finite fields the corresponding (graded version of) categories of perverse sheaves on partial flag varieties defined over the complex numbers, that is the uncategorified ver”
Affine flag varieties and quantum symmetric pairs (2016, unknown): “) was constructed in [BW13] for various tensor product modules of iUpglnq, and the ıcanonical basis for the idempotented form iU9 pgl nq was subsequently constructed in [BKLW14]. It has been shown in [FL14] that coideal like algebras together with their ıcanonical bases arise from partial ﬂag variet”
A new approach to Kazhdan-Lusztig theory of type ${\bf B}$ via quantum symmetric pairs (2013, methodology): “s natural to ask for a geometric interpretation of the type Bduality as well as ı-canonical basis developed algebraically and categorically in this paper. This turns out to have a classical answer in [BKLW], which settles another old open problem of understanding the quantum 8 HUANCHEN BAO AND WEIQI”