Lectures on dualities ABC in representation theory

Abstract

In these lecture notes for a summer mini-course, we provide an exposition on quantum groups and Hecke algebras, including (quasi) R-matrix, canonical basis, and $q$-Schur duality. Then we formulate their counterparts in the setting of $ı$quantum groups arising from quantum symmetric pairs, including (quasi) K-matrix, $ı$-canonical basis, and $ı$Schur duality. As an application, the ($ı$-)canonical bases are used to formulate Kazhdan-Lusztig theories and character formulas in the BGG categories for Lie (super)algebras of type A-D. Finally, geometric constructions for $q$-Schur and $ı$Schur dualities are provided.

57 pages

References in corpus (3)

• Classical weight modules over $ı$quantum groups
• Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik-Zamolodchikov equations and Letzter-Kolb coideals
• Cells in affine q-Schur algebras

Cited by in corpus (3)

• $ı$Schur duality and Kazhdan-Lusztig basis expanded
• $ı$Hall algebra of Jordan quiver and $ı$Hall-Littlewood functions
• Duplex Hecke Algebras and Related Quantum Schur Duality