»ã±¨±êÌâ (Title)£ºË«iÁ¿×ÓȺU^j£¨n£©ºÍU^i£¨n£©£¨The twin i-quantum groups U^j(n) and U^i(n)£©

»ã±¨ÈË (Speaker)£ºJie Du½ÌÊÚ£¨°Ä´óÀûÑÇÐÂÄÏÍþ¶ûÊ¿´óѧ£©

»ã±¨¹¦·ò (Time)£º2025Äê3ÔÂ25ÈÕ (Öܶþ) 16:00

»ã±¨µØÖ· (Place)£ºÐ£±¾²¿GJ303

Ô¼ÇëÈË(Inviter)£ºÕźìÁ«½ÌÊÚ

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»ã±¨ÌáÒª£º When I. Schur used representations of the symmetric groups S_r to determine polynomial representations of the complex general linear group GL_n(C), certain finite-dimensional algebras, known as Schur algebras, played a bridging role between the two. The well-known Schur duality summarizes the relation between the representations of GL_n(C) and S_r. Over almost a hundred years, this duality has profoundly influenced representation theory and has evolved in various forms such as the Schur-Weyl duality, Schur-Weyl-Brauer duality, Schur-Weyl-Sergeev duality, and so on. In this talk, I will discuss a latest development, which I call the Schur-Weyl-Hecke duality, by Huanchen Bao and Weiqiang Wang. Based on joint work with Yadi Wu, I will focus on the investigation of the i-quantum groups U^j(n) and U^i(n) and their associated q-Schur algebras S^j(n, r) and S^i(n, r) of types B and C, respectively. This includes short (element) multiplication formulas, long (element) multiplication formulas, and triangular relations in S^j(n, r) and S^i(n, r). We will also give realisations of Beilinson¨CLusztig¨CMacPherson type for both U^j(n) and U^i(n) and discuss their Lusztig forms. This allows us to link representations of U^j(n) and U^i(n) with those of finite orthogonal and symplectic groups.