Schur dualities arising from quantum symmetric pairs

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Shen, Yaolong, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Wang, Weiqiang, AS-Mathematics, University of Virginia

The Schur-Jimbo duality is one of the most fundamental topics in representation theory, bridging the irreducible representations of a Hecke algebra with those of a Drinfeld-Jimbo quantum group. Evolving alongside the advancements in the field, the Schur-Jimbo duality has been extended in tandem with the emergence of $\imath$quantum groups, which are a natural generalization of quantum groups arising from the theory of quantum symmetric pairs.

In this dissertation, we construct various $\imath$Schur dualities stemming from quantum symmetric pairs of types AI, AII, and AIII. Particularly, the $\imath$Schur duality of type AIII, accommodating black nodes in its Satake diagram, presents a unified extension of Jimbo-Schur duality and Bao-Wang's quasi-split $\imath$Schur duality.

Moreover, expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possibly non-parabolic) reflection subgroups of the Weyl group of type B. The quasi-parabolic KL bases on quasi-permutation Hecke modules are shown to match with the $\imath$canonical basis on the tensor space.

Finally, we establish two specific families of quantum supersymmetric pairs, denoted as type AIII and type AI-II, respectively. We elucidate their fundamental properties, including the coideal algebra property and the quantum Iwasawa decomposition, which ensure that the $\imath$quantum supergroups attain the expected sizes. Within the framework of these quantum supersymmetric pairs, we provide super generalizations of the aforementioned dualities.

PHD (Doctor of Philosophy)
Quantum symmetric pair, Schur duality, Quantum group, Hecke algebra
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