Geometry, Combinatorics, and Integrable Systems  

  Autumn 2017

Time: Thursdays 3-4pm
Location: MA 317


August 31  
Thurs, 3pm 
Max Glick 
The limit point of the pentagram map
September 7  
Thurs, 3pm 
September 14  
Thurs, 3pm 
September 21  
Thurs, 3pm 
Yuancheng Xie 
Generalized hypergeometric functions on the Grassmannian and integrable systems of hydrodynamic type
September 28  
Thurs, 3pm 
Ed Richmond 
(Oklahoma State)  
Pattern avoidance and fiber bundle structures on Schubert varieties
October 5  
Thurs, 3pm 
Tair Akhmejanov  
Affine Growth Diagrams
October 12  
Thurs, 3pm 
(fall break) 
October 19  
Thurs, 3pm 
MW 154
Cristian Lenart 
Combinatorics of Lusztig's t-analogue of weight multiplicity
October 26  
Thurs, 3pm 
November 2  
Thurs, 3pm 
November 9  
Thurs, 3pm 
November 16  
Thurs, 3pm 
November 30  
Thurs, 3pm 


(Glick): The pentagram map is a discrete dynamical system defined on the space of polygons in the plane. In the first paper on the subject, R. Schwartz proved that the pentagram map produces from each convex polygon a sequence of successively smaller polygons that converges exponentially to a point. We investigate the limit point itself, giving an explicit description of its Cartesian coordinates as roots of certain degree three polynomials.

(Xie): In this talk I will describe a connection between generalized hypergeometric functions on the Grassmannian and integrable systems of hydrodynamic type. This talk consists of the following four parts: (1) Symmetries of hydrodynamic-type systems in Riemann invariant form; (2) Generalized hypergeometric functions defined on the Grassmannian and their confluences; (3) Integrable hydrodynamic-type systems and confluences of Lauricella-type hypergeometric functions; (4) Some preliminary results on generalizations to the Grassman Gr(r,n) with r > 2. A special case of the compatibility conditions of hydrodynamic-type systems in Riemann invariant form leads to Euler-Poisson-Darboux(EPD) system. This system admits Lauricella type hypergeometric functions as solutions, which are special cases of Aomoto-Gel'fand hypergeometric functions defined on the Grassman Gr(r, n) with r=2. In this way each Lauricella type hypergeometric functions gives rise to a hierarchy of integrable systems of hydrodynamic type in Riemann invariant form. The confluences of the classical Gauss hypergeometric functions can be generalized to Aomoto-Gel'fand hypergeometric functions. The confluences of a generalized Lauricella type hypergeometric function also produce hierarchies of integrable systems of hydrodynamic type which are not necessarily in Riemann invariant form anymore. In this talk I will describe these constructions and connections and also some preliminary results on generalizations to the Grassman Gr(r,n) with r>2.

(Richmond): In this talk, I will discuss joint work with Tim Alland where we give a pattern avoidance criteria for determining when the projection map from the flag variety to a Grassmannian induces a fiber bundle structure on a Schubert variety. To do this, we introduce the notion of split pattern avoidance and show that a Schubert variety has such a fiber bundle structure if and only if the corresponding permutation avoids the split patterns 3|12 and 23|1. Continuing, we also characterize when a Schubert variety is an iterated fiber bundle of Grassmannian Schubert varieties in terms of usual pattern avoidance.

(Akhmejanov): We introduce a new type of growth diagram, arising from the geometry of the affine Grassmannian for GLm. These affine growth diagrams are in bijection with the clambda many components of the polygon space Poly(lambda) for lambda a sequence of minuscule weights and clambda the Littlewood--Richardson coefficient. Unlike Fomin growth diagrams, they are infinite periodic on a staircase shape, and each vertex is labeled by a dominant weight of GLm. Letting $m$ go to infinity, a dominant weight can be viewed as a pair of partitions, and we recover the RSK correspondence and Fomin growth diagrams within affine growth diagrams. The main combinatorial tool used in the proofs is the n-hive of Knutson-Tao-Woodward. The local growth rule previously appeared in van Leeuwen's work on Littelmann paths, so our results give a geometric interpretation of this combinatorial rule.

(Lenart): Lusztig defined the Kostka-Foulkes polynomial Kλ,μ(t) as a t-analogue of the multiplicity of a weight μ in the irreducible representation of highest weight λ of a semisimple Lie algebra. This polynomial has remarkable properties, such as being an affine Kazhdan-Lusztig polynomial. Finding combinatorial formulas for Kλ,μ(t) beyond type An has been a long-standing problem. In joint work with Cédric Lecouvey, we give the first such formula, for Kλ,0(t) in type Cn, using combinatorics of Kashiwara's crystal graphs; the special case μ=0 is, in fact, the most complex one. Related aspects and applications will be discussed. I will also mention the so-called atomic decomposition of Kostka-Foulkes polynomials, as well as its relevance to the geometric construction of representations given by the Satake correspondence. The talk will be largely self-contained.

Past Seminars

Spring 2017