MAAGC 2018 ABSTRACTS

Sylvie Corteel (CNRS, LIAFA, Université Denis Diderot - Paris 7, Miller Institute at UC Berkeley):

Title: Lecture Hall Tableaux

Abstract: The lecture hall partitions were introduced by Bousquet-Mélou and Eriksson in 1997 by showing that they are the inversion vectors of elements of the parabolic quotient $\tilde{C}_n/C_n$. Since 1997, a lot of beautiful combinatorial techniques were developed to study these objects and their generalisations. These use basic hypergeometric series, geometric combinatorics, real rooted polynomials... Some of those results can be found in the survey paper by C. D. Savage "The Mathematics of lecture hall partitions". Here we take a different approach and show that these objects are also multivariate moments of the Little $q$-Jacobi polynomials. The multivariate moments were introduced by Williams
and me in the context of asymmetric exclusion processes. The benefit of this new approach is that we define a tableau analogue of lecture hall partitions and we show that their generating function is a beautifulproduct. This uses a mix of orthogonal polynomials techniques, non intersecting lattice paths and $q$-Selberg integral. This is joint work with Jang Soo Kim (SKKU).

The talk will not require any prerequisite on the subjects.

Allen Knutson (Cornell): 

Title: Schubert calculus, scattering amplitudes, and cotangent bundles

Abstract: I'll first recall the AJS/Billey and puzzle formulae for Schubert classes and Schubert calculus, and explain how both can be interpreted as "scattering amplitudes". Then I'll explain how this leads to a quick and easy proof of old and new puzzle rules for Grassmannian, 2-step, and 3-step Schubert calculus. This approach, based on representations of quantized affine algebras, suggests that there should be a richer problem involving the quantum parameter. I'll explain the Maulik-Okounkov classes in K-theory of the cotangent bundle, how puzzles are actually "about" the multiplication of closely related classes to those, and what they can tell us about 4-step Schubert calculus. This is joint work with Paul Zinn-Justin.

Franco Saliola (UQAM):

Title: A Murnaghan-Nakayama rule for quantum cohomology of the flag manifold

Abstract: I will present a general rule for multiplying a quantum Schubert polynomial by a quantum power sum polynomial. This is achieved by relating the structure constants for the multiplication of a quantum Schubert polynomial by a hook quantum Schur polynomial with the structure constants for the (classical) multiplication of a Schubert polynomial by a hook Schur polynomial. This is joint work with C Benedetti, N Bergeron, L Colmenarejo and F Sottile. 

Monica Vazirani (UC Davis): 

Title: A Schur-Weyl-like construction of the rectangular DAHA representation

Abstract: Young diagrams and tableaux are everywhere in type A representation theory. In particular, it is well-known that bases of irreducible S_n representations are indexed by standard tableaux. There are affine (infinite-dimensional) analogues of this statement as well. It was by analyzing a basis of a particular space of invariants that is indexed by standard periodic tableaux that we were able to identify that space as the "rectangular" representation of the double affine Hecke algebra (DAHA) in type A.

More precisely, Jordan constructed a Schur-Weyl-like functor from quantum D-modules on special linear groups to representations of the DAHA, building on the work of Calaque-Enriquez-Etingof, Lyubashenko-Majid, and Arakawa-Suzuki. When we input the module of quantum functions into the functor, the output is L(k^N), the irreducible DAHA representation indexed by an N by k rectangle. For the specified parameters, L(k^N) is Y-semisimple, i.e. one can diagonalize the Dunkl operators. We give an explicit combinatorial description of this module via its Y-weight basis in terms of periodic tableaux of rectangular shape. This is joint work with David Jordan.