From Schur-Weyl duality to quantum symmetric pairs

The famous Schur-Weyl duality is a double centralizer property between the symmetric group and the general linear group/Lie algebra bypassing Schur algebras. In various formulations, the duality plays a fundamental role in representation theory. Around 1985, Jimbo introduced a quantized duality between the Hecke algebra and the quantum group of type A bypassing q-Schur algebras. It is then made possible to construct quantum groups from the collection of q-Schur algebras by Beilinson, Lusztig and MacPherson (BLM). In a seemingly unrelated direction, Letzter and Kolb developed a theory of quantizing the symmetric pairs consisting of a Lie algebra and its fixed-point subalgebra. The results are known as the quantum symmetric pairs, including examples arising from the reflection equations, the Onsager algebras from Ising model, the twisted Yangians and the general intersection matrix Lie algebras. In this talk I will provide examples of certain Schur-type dualities beyond type A, and exhibit a new family of quantum symmetric pairs in terms of the algebras constructed a la BLM. I will summarize with applications in representation theory.