## Topics in Combinatorial Representation Theory

### Abstracts

 PDF version   Ming Fang (Chinese Academy of Sciences) Schur functors and dominant dimension Abstract:Schur-Weyl duality has been playing important roles in the study of general linear groups, BGG category O and many other contexts. The main idea is to link two algebras or their module categories by investigating the so called Schur functors. In this talk, we exploit dominant dimension to study the cohomological property of Schur functors and derive some new characterizations of this homological dimension.  Some applications will also be discussed. This is a joint work with Steffen Koenig.   Masao Ishikawa (University of the Ryukyus)     Slide A Pfaffian analogue of the Hankel determinants and the Selberg integrals Abstract: In the previous work with Hiroyuki Tagawa and Jiang Zeng, we established a Pfaffian analogue of the Hankel determinants of q-Catalan numbers. In the proof we used the Pfaffian decomposition and Jackson's formula for the basic hypergeometric seris ${}_{6}\varphi_{5}$. In this talk, we present another proof which uses the de Bruijn formula and reduces it to Askey's q-Selberg's integral formula. We also attack the open problems in the previous paper via de Bruijn's formula.   Masao Ishikawa (University of the Ryukyus)     Lecture NoteHiroyuki Tagawa (Wakayama University) A generalization of the Mehta-Wang determinant and the Askey-Wilson polynomials Abstract:Metha and Wang calculated a determinant involving Gamma function to complete the calculation of the Mellin transform of the probability density of the determinant of a random quaternion self-dual matrix taken from the gaussian symplectic ensemble. Nishizawa established a q-analogue of this determinant. In our previous work we established a Pfaffian identity whose entries involves the moments of the little q-Jacobi orthogonal polynomials. In this talk we evaluate a determinant which generalizes Nishizawa's determinant and our privious Pfaffian. The answer is given by a product multiplied by an Askey-Wilson polynomial. To establish this identity we also present the Krattenthaler-type evaluation of this determinant (i.e., we choose arbitrary n rows and the first n columns, and we establish a formula).   Ryoichi Kase (Osaka University)The number of arrows in a tilting quiver over a path algebra of type A or D Abstract: A tilting quiver is a directed edge graph which has the set of basic tilting modules as the set of vertices, and has the set of mutations as the set of arrows. For a finite dimensional path algebra over a algebraically closed field, the tilting quiver is a finite type precisely when this path algebra is a Dynkin type. In this case the number of vertices in a tilting quiver is given by Fomin and Zelevinski. In this talk we show that the number of arrows in a tilting quiver is independent of the orientation, and give it when a path algebra is type A or D.   Anatol N. Kirillov (Kyoto University)Algebraic and combinatorial properties of Dunkl operators at critical level Abstract:The Dunkl operators have been introduced by C. Dunkl in the middle of 80's of last century, since that time have found numerous applications in Mathematics and Mathematical Physics. There are three kinds of Dunkl operators, namely, rational, trigonometric and elliptic.   In my talk I introduce a certain noncommutative inhomogeneous quadratic algebra and distinguish set of mutually commuting element in it in such a way that three kinds of Dunkl operators at critical level will corresponds to some representations of the quadratic algebra in question. The main purpose of my talk is to explain some algebraic and combinatorial properties of that "universal Dunkl elements" (UDE), as well as to outlook surprising connections of UDE with quantum and classical Schubert calculus, elliptic hypergeometric series, volume of Chan-Robbins polytopes and so on. (partly joint with T. Maeno)   Masaki Mori (University of Tokyo)     SlideRepresentation theory of wreath product in non-integral rank Abstract:Recently P. Deligne created a family of categories, namely "the representation category of the symmetric group $S_t$" for each rank t which is not necessarily a natural number. In this talk we generalize his construction for representation categories of wreath products (of groups, algebras, etc.). This construction gives us a 2-functor which sends an arbitrary tensor category to a new tensor category. By applying it to the representation category of some group, we obtain the representation category of the wreath product of the group for non-integral rank.   Kentaro Nagao (Nagoya University) Matrix counting and the refined topological vertex of the conifold Abstract:The refined topological vertex is a 1-parameter deformation of the Gromov-Witten theory of toric Calabi-Yau 3-folds, which is defined only in terms of physics or combinatorics. It is expected that the refined topological vertex is realized via motivic Donaldson-Thomas theory. In the joint work with Andrew Morrison, Sergey Mozgovoy and Balazs Szendroi, we check the coincidence in the case of the conifold. In this talk, I will explain the combinatorial part (counting the number of matrices which satisfy a certain equation) of the proof.  Norihiro Nakashima (Hokkaido University)A basis for the module of differential operators of order 2 on the braid hyperplane arrangement Abstract:The module of derivations which preserving a defining polynomial of an arrangement is studied in a rich literature. We call it the module of A-derivations. There is a well-known basis for the module of A-derivations when A is the braid arrangement. We generalize the module of A-derivations into the module of high order differential operators. We call it the module of A-differential operators. By using the essential symmetric polynomial, we construct a basis for the module of A-differential operators of order 2 when A is a braid arrangement.  Maki Nakasuji (Kitasato University)     SlideTokuyama-type formula for factorial Schur functions Abstract:Tokuyama fomrula which is regarded as a deformation of the Weyl character formula for $GL_n({\mathbb C})$ shows Schur function times the Weyl denominator is expressed as a sum over Gelfand-Tsetlin patterns. In this talk, as a generalization of this formula, we will give the Tokuyama-type formula for factorial Schur functions which have the spectral parameters and  the "shift" parameters. It shows that the factorial Schur functions times a deformation of the Weyl denominator is expressed as the partition function for a statistical system based on the six vertex model by using the Yang-Baxter equation. Further, we will give the application of this formula.This is joint work with Daniel Bump and Peter J. McNamara.   Katsuyuki Naoi (University of Tokyo)     SlideDemazure modules, Demazure crystals and the X=M conjecture Abstract:One-dimensional configuration sums are some polynomials defined from tensor products of Kirillov-Reshetikhin crystals, and fermionic forms are some polynoimals defined from affine root systems and dominant integral weights of classical Lie algebras. The X=M conjecture asserts that a one-dimensional sum is equal to the corresponding fermionic form. In this talk, we will give a proof of the X=M conjecture for $A_n^{(1)}$ and $D_n^{(1)}$ using Demazure modules and Demazure crystals.   Masato Okado/Nobumasa Sano (Osaka University)KKR type bijection for the exceptional affine algebra $E_6^{(1)}$ Abstract:For the exceptional affine type $E_6^{(1)}$ we establish a statistic-preserving bijection between the highest weight paths consisting of the simplest Kirillov-Reshetikhin crystal and the rigged configurations. The algorithm only uses the structure of the crystal graph, hence could also be applied to other exceptional types.  Takayuki Okuda (University of Tokyo)Smallest complex nilpotent orbits with real points Abstract:For a complex simple Lie algebra $\mathfrak{g}_\mathbb{C}$ and its real form $\mathfrak{g}$, some complex nilpotent orbits in $\mathfrak{g}_\mathbb{C}$ meets $\mathfrak{g}$. In this talk, we show that there uniquely exists an orbit of minimal positive dimension in such nilpotent orbits.We also determine such the orbit for any $(\mathfrak{g}_\mathbb{C},\mathfrak{g})$ by describing the weighted Dynkin diagram of it.  Euiyong Park (Osaka Unisersity)     SlideIrreducible modules over Khovanov-Lauda-Rouquier algebras of finite classical type Abstract:In this talk, we give an explicit construction of irreducible modules over Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^{\lambda}$ of finite classical type. More precisely, for each element $v$ of $B(\infty)$ (resp. $B(\lambda)$), we first construct the induced representation ${\rm Ind} (\nabla(\mathbf{a};1) \boxtimes \cdots \boxtimes \nabla(\mathbf{a};n))$ labeled by the adapted string $\mathbf{a}$ of $v$. We then show that every irreducible $R$-module (resp. $R^{\lambda}$-module) arises as the simple head of ${\rm Ind} (\nabla(\mathbf{a};1) \boxtimes \cdots \boxtimes \nabla(\mathbf{a};n))$. This construction is compatible with the crystal structure on $B(\infty)$ (resp. $B(\lambda)$). This is joint work with Georgia Benkart(University of Wisconsin), Seok-Jin Kang (Seoul National University) and Se-jin Oh (Seoul National University).    Pavlo Pylyavskyy (University of Minnesota)Electrical networks, Lie theory and crystals Abstract:We introduce a new class of "electrical" Lie groups. These Lie groups, or more precisely their nonnegative parts, act on the space of planar electrical networks via combinatorial operations previously studied by Curtis-Ingerman-Morrow. The corresponding electrical Lie algebras are obtained by deforming the Serre relations of a semisimple Lie algebra in a way suggested by the star-triangle transformation of electrical networks. We also define and study a birational transformation acting on cylindrical electrical networks called the electrical R-matrix. The ultradiscretization of this transformation is the combinatorial R-matrix for certain affine crystals.  Yuichiro Tanaka (University of Tokyo)A generalized Cartan decomposition for connected compact Lie groups and its application Abstract:We consider the tensor product representation of irreducible representationsof connected compact Lie groups. J.R.Stembridge classified the multiplicity-free tensor product representations by using a combinatorial method (2003). In this talk, we give a generalization of the Cartan decomposition for non-symmetric cases motivated by the notion of visible actions on complex manifolds, which was introduced by Kobayashi. It leads us to a geometric proof of multiplicity-free property of some tensor product representations.   Shunsuke Tsuchioka (University of Tokyo)Quiver Hecke superalgebras Abstract:We introduce two families of superalgebras $R_n$ and $RC_n$ which are weakly Morita superequivalent each other. The quiver Hecke superalgebra $R_n$ is a generalization of the Khovanov-Lauda-Rouquier algebras. We show that, after suitable specialization and completion, the quiver Hecke-Clifford superalgebra $RC_n$ is isomorphic to the affine Hecke-Clifford superalgebras and its rational degeneration. This is a joint work with Seok-Jin Kang and Masaki Kashiwara.  Yutaka Yoshii (Nara National College of Technology)Weyl modules and principal series modules Abstract:Pillen has proved that the highest weight vectors of "most" Weyl modules  generates principal series modules for finite Chevalley groups. Here we would like to generalize his result.