RAQ2014 ) TEL Fax

Transcription

1 RAQ TEL Fax hiroyuki@sci.u-toyama.ac.jp 5/12( ) RAQ2014 ) *. * (1, 2, 3 ) ( (2, 3, 4 ), (2, 3, 4 ), (1, 2, 3 ). 3 *. * ). : 1

2 p.s. 2

3 5 4 RAQ :00-17:00 ( ) Littlewood Richardson 17:15-18:15 ( ) Cluster Variables on Double Bruhat Cells and Monomial Realizations of Crystal Bases 9:30-10:30 ( ) 11:00-12:00 K I 13:15-14:15 Andrei Negut (Columbia & RIMS) Quantum algebras, shuffle algebras and Hilbert schemes 14:30-15:30 Schrödinger representations from the viewpoint of monoidal categories 16:00-17:00 ( ) Hochschild cohomology of q-schur algebras 17:15-18:15 ( ) Rational shuffle conjecture and affine Springer fibers of type A 3

4 9:30-10:30 sl(2, 1)ˆ BGG resolution 11:00-12:00 K II 13:45-14:45 ( ) K Schur P Decomposition tableaux 15:15-16:15 A construction of irreducible representations of the quantized function algebra C[SL n ] v 16:45-17:45 9:30-10:30 ( ) Relaxed Verma 11:00-12:00 Demazure subcrystals of crystal bases of level-zero extremal weight modules over quantum affine algebras hiroyuki@sci.u-toyama.ac.jp, oshima@aitech.ac.jp 4

5 1 2 sl(2, 1)ˆ BGG resolution sl(2, 1)ˆ BGG resolution, Bowcock-Taormina (1997), Semikhatov-Taormina ( 2001), Serganova (2010)., Semikhatov- Taormina,. 3 ( ) Cluster Variables on Double Bruhat Cells and Monomial Realizations of Crystal Bases For semi simple simply connected algebraic group G and elements u, v of its Weyl group W, it is known that the coordinate ring C[G u,v ] of the double Bruhat cell G u,v is isomorphic to an upper cluster algebra Ā(i) C and the generalized minors (k; i) are the cluster variables of C[G u,v ] ([A.Berenstein, S,Fomin, A, Zelevinsky, 2005]). On the other hand, it is known that a Zariski open set of G u,v is biregular isomorphic to H C l(u)+l(v) 0 ([S. Fomin, and A. Zelevinsky, 1998]). Here, H is a maximal torus of G. In this talk, for G = SL r+1 (C), we consider (k; i) as the functions on H C l(u)+l(v). Then they become polynomials with coefficient 1. And we can express each monomial in those polynomials in terms of monomial realization of crystal. 5

6 4 Schrödinger representations from the viewpoint of monoidal categories The Drinfel d double D(A) of a finite-dimensional Hopf algebra A is a Hopf algebraic counterpart of the monoidal center construction. Majid introduced an important representation of D(A), which he called the Schrödinger representation. We study this representation from the viewpoint of the theory of monoidal categories. One of our main results is as follows: If two finitedimensional Hopf algebras A and B over a field k are monoidally Morita equivalent, i.e., there exists an equivalence F : A Mod B Mod of k-linear monoidal categories, then the equivalence D(A) Mod D(B) Mod induced by F preserves the Schrödinger representation. Here, A Mod for an algebra A means the category of left A-modules. As an application, we construct a family of invariants of finitedimensional Hopf algebras under the monoidal Morita equivalence. This family is parameterized by braids. The invariant associated to a braid b is, roughly speaking, defined by coloring the closure of b by the Schrödinger representation. We investigate what algebraic properties this family have and, in particular, show that the invariant associated to a certain braid closely relates to the number of irreducible representations. ( ) 5 K 6 ( ) K Schur P Decomposition tableaux 7 ( ) Rational shuffle conjecture and affine Springer fibers of type A Haglund, Haiman, Loehr, Remmel, and Ulyanov conjectured a formula for bigraded Frobenius series of the diagonal coinvariant rings in terms of combinatorics of parking functions. In this talk, we give an explanation of the combinatorial side of their conjecture or its generalization, called rational shuffle conjecture, by the geometry of affine Springer fibers of type A. 6

7 8 ( ) Kontsevich-Soibelman Donaldson-Thomas 9 ( ) A construction of irreducible representations of the quantized function algebra C[SL n ] v The quantized function algebra C[SL n ] v is a q-analogue of the coordinate ring of the special linear group SL n (C). The (unitarizable) irreducible representations of C[SL n ] v were constructed and classified by Soibelman et al. In this talk, I will give a new explicit construction of irreducible representations of C[SL n ] v and explain the relation between Soibelman s construction and ours. Our construction is direct in the sense that the calculation results deduced from the defining relations of C[SL n ] v are the main tools. 10 Andrei Negut (Columbia & RIMS) Quantum algebras, shuffle algebras and Hilbert schemes : This talk will be a brief survey on an important quantum algebra A which goes by many names: doubly deformed W 1+infinity algebra, elliptic Hall algebra, stable limit of DAHA, quantum toroidal gl 1 etc. We will focus on the presentation of A as a double shuffle algebra, and show how this helps obtain certain explicit formulas in two very important representations: the bosonic Fock space and the K-theory of the Hilbert scheme 7

8 11 Demazure subcrystals of crystal bases of level-zero extremal weight modules over quantum affine algebras : We give a characterization of the crystal bases of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra. This characterization is given in terms of the initial directions of semi-infinite Lakshmibai-Seshadri paths (SiLS paths), and is established under a suitably normalized isomorphism between the crystal basis of the level-zero extremal weight module and the crystal of SiLS paths. This talk is based on a joint work with Satoshi Naito. 12 ( ) : Littlewood Richardson : Young λ, µ, ν shape λ, weight ν/µ Littlewood Richardson Littlewoood Richardson c ν λµ c ν λµ = cν µλ Littlewood Richardson RAQ 2013 semistandard setvalued λ-good tableau 13 ( ) Hochschild cohomology of q-schur algebras C q-schur algebra S. S S n Hochschild cohomology group HH n (S) := Ext n Se(S, S) (n 0). S e S enveloping algebra S C S op. Hochschild cohomology group. S Hochschild cohomology ring, HH (S) := n 0 HH n (S).,. 8

9 14 ( ) Relaxed Verma GKO A (1) 1 Lie Verma N = 2 Verma Feigin-Semikhatov-Tipunin N = 2 Verma A (1) 1 Lie relaxed Verma Verma ( ) hiroyuki@sci.u-toyama.ac.jp Tel: