NTNU Mini Lectures on Lie Theory and Combinatorics Chih-Whi Chen Ph.D, NTU Schur-Weyl duality approach to the irreducible character problem over the general linear Lie (super)algebras

2016/09/05-2016/09/09 14:00-16:00 (Math Building, Room 210) Abstract. The Lie superalgebras are generalizations of Lie algebras and are an important tool for physicists in the study of supersymmetries. In particular, their representation theory, have enjoyed a recent surge of considerable interest. This is mainly due to recent works that made connection between Lie superalgebras and other branches in classical Lie theory. But it is also due to the significant progresses made recently, especially, in the irreducible character problem for fnite-dimensional simple Lie superalgebras since Kac’s pioneering works [Kac78] on typical representations of classical Lie superalgebras. Since then the irreducible characters of classical Lie superalgebras in the finite-dimensional modules, and even for modules in the BGG category O have been worked out in [Ser96, Ser98, Br1, GS10, CLW11, CLW15, BW] . This lecture will give a basic overview of the duality approach to solving the problem of irreducible character over the general linear Lie (super)algebras. In the first part, we will review the (classical) Schur-Weyl type dualities. Some basic notions of super duality developed by professors Shun-Jen Cheng, Ngau Lam, Weiqiang Wang and R.B. Zhang will be covered in the second part. We mainly focus on the general linear Lie superalgebras gl(m|n). It will be scheduled in page 2. ∗ Background. There are no prerequisites for being able to understand this lecture. However, the following experiences will be helpful in understanding well both motivations and connections with other fields: (1) Basic knowledge for simple Lie algebras and the representation theory including Cartan subalgebras, root systems, dominant ordering on dual of Cartan subaglebras, Weyl complete reducibility theorem, classification of simple Lie algebras, Linkage principle for gl in the BGG category O, and Kostant’s u-cohomology. (2) Representation theory of symmetric groups including Specht modules, permutation modules, Kostka numbers, Young’s rule, Schur functions, Frobenius characteristic map, Frobenius’ construction of character tables, Jacobi-Trudi formula, and the Littlewood-Richardson principle. In this lecture, the ground field is always C, and (super)modules are always semisimple over standard Cartan subaglaebras. Organizer: Sen-Peng Eu([email protected]), Department of Math, NTNU 1

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1. General linear Lie algebras gl(m) We begin with basic notions in the representation theory of finite-dimensional modules over general linear Lie algebras gl(m). In particular, all finite-dimensional representations can be realized as submodules of tensor products of ”fundamental” irreducibles. A standard approach to irreducible character problem in representation theory will be introduced as well. Follow this approach, the first section will end in the famous Weyl character formula. 2. Schur-Weyl duality for gl(m) In this section, we formulate the Schur-Weyl for gl(m) as a double centralizer property. Some basic notions coming from the representation theory of symmetry groups will be reviewed. As an application of Schur-Weyl duality, we reprove/obtain a combinatorial description of irreducible character formulas of finite-dimensional modules over gl(m). 3. Schur-Sergeev duality for gl(m|n) We now turn to Lie superalgebras. First, the category of finite-dimensional gl(m|n)modules will be described. Our first attempt is the so-called Kac-Weyl (typical) character formula. Then we formulate the Schur-Sergeev theorem as a double centralizer propery gl(m|n) y (Cm|n )⊗d x Sd . As has been mentioned, some combinatorial notions in this version are expected. As an application, we obtain character formulas for polynomial modules, i.e. irreducible submodules of tensor power of natural representations over gl(m|n). 4. Super duality In this section, we introduce the famous BGG category. We formulate super dualities as equivalences between two parabolic BGG categories O and O for gl(m + ∞) and gl(m|∞). We will see that super duality establishes a link of representation theories between Lie algebras and Lie superalgebras. As the first example, Schur-Weyl-Sergeev duality is reproved by using the super duality. Furthermore, we will be able to solve for all finite-dimensional irreducible characters over gl(m|n) via super duality in this section. Finally, we introduce the co-called parabolic Kazhdan-Lusztig-Brundan conjecture for gl(m|n) which gives irreducible characters of infinite-dimensional modules over gl(m|n). Again, it proved by using super duality. 5. Beyond type A(Optional) The final sction is optional. In this section, we introduce Schur-Weyl-Brauer type dualities for other Lie superalgebras, e.g., osp(m|2n), q(n), p(n). Super dualities and its application for Lie superalgebras of type b, c, d will be covered in this section.

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References [Br1]

J. Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n), J. Amer. Math. Soc. 16 (2003), 185–231. [BW] H. Bao and W. Wang, A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs, arXiv:1310.0103. [CLW11] S.-J. Cheng, N. Lam, W. Wang, Super duality and irreducible characters of ortho-symplectic Lie superalgebras, Invent. Math. 183 (2011), 189–224. [CLW15] S.-J. Cheng, N. Lam, and W. Wang, Brundan-Kazhdan-Lusztig conjecture for general linear Lie superalgebras, Duke Math. J. 110 (2015), 617–695. [CW] S.-J. Cheng and W. Wang, Dualities and Representations of Lie Superalgebras. Graduate Studies in Mathematics 144. American Mathematical Society, Providence, RI, 2012. [GS10] C. Gruson and V. Serganova, Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras, Proc. Lond. Math. Soc. 101 (2010), 852–892. [Kac78] V. G. Kac, Lie superalgebras, Adv. Math. 26 (1977), 8–96. [Ser96] V. Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra gl(m|n), Selecta Math. (N.S.) 2 (1996), 607–651. [Ser98] V. Serganova, Characters of irreducible representations of simple Lie superalgebras, Doc. Math., Extra Volume ICM II (1998), 583–593.