Organizers: Sin-Myung Lee (KIAS, sinmyunglee@kias.re.kr), Jaeseong Oh (HCMC KIAS, jsoh@kias.re.kr)

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Upcoming Talks:

• Brendon Rhoades (University of California, San Diego, USA)

Sep. 5 (Thu), 2024, 10:00am-11:00am (Korea time, UTC+9)

Title: TBA

Abstract: TBA

Past talks:

• Hideya Watanabe (Rikkyo University, Japan)

Jun. 21 (Fri), 2024, 10:00am-11:00am (Korea time, UTC+9)

Title: Combinatorial $K$-matrices arising from affine quantum symmetric pairs of type $A$

Abstract: A combinatorial $R$-matrix is a solution to the combinatorial Yang-Baxter equation, which plays a vital role in discrete dynamical systems such as box-ball systems. It has been known that the theory of crystal bases arising from quantum groups provides us with a framework to construct combinatorial $R$-matrices systematically. In a dynamical system which has a boundary, the role of the combinatorial Yang-Baxter equation is replaced by the combinatorial reflection equation, whose solutions are called combinatorial $K$-matrices. In this talk, based on a joint work with Hiroto Kusano and Masato Okado, I will explain how to construct combinatorial $K$-matrices systematically by means of the theory of crystal bases arising from affine quantum symmetric pairs of type $A$.

• Theo Douvropoulos (Brandeis University, USA)

May. 24 (Fri), 2024, 10:00am-11:00am (Korea time, UTC+9)

Title: Parking functions on the cluster complex and Cyclic Sieving Phenomena

Abstract: The associahedron K_n and its dual, the type-A cluster complex Y(S_n), with their many realizations and generalizations are central objects in algebraic combinatorics. Their faces are labeled by subdivisions of an (n+2)-gon, they have Catalan-many vertices/facets, and their f-vectors are given by the Kirkman numbers.

In past joint work with Josuat-Verges we gave a refinement of the f-vector of Y(n) that keeps track of a cycle type associated to each polygonal subdivision, and is expressed in terms of structure coefficients of Haiman's parking space of diagonal coinvariants. We will present in this talk two research works built on these structures.

In the first work, joint with Josuat-Verges, we give a topological, equivariant version of the combinatorial f-to-h reciprocity for Y(n). We label each polygonal subdivision with a set partition and get as a result a simplicial complex that carries a parking space representation in its top homology. This presents the result of the recent preprint arxiv:2402.03052.

In the second still in progress work, joint with Josuat-Verges and Sommers, we give q-versions of these refined f-vectors (obtained naturally via the parking space interpretation) and prove cyclic sieving phenomena (CSPs) for the natural cyclic rotation of the polygon. In particular, we show that the q-Kirkman numbers satisfy a new, weighted type of CSP. We present further a version that keeps track of dihedral symmetries via the qt-Schroder numbers, utilizing the full grading of the parking space.

Most of these results make sense for all real reflection groups but we will focus our presentation to the symmetric group case. We will end with open questions in each approach or possible common extensions.

• Oleksandr Tsymbaliuk (Purdue University, USA)

May. 16 (Thu), 2024, 7:30pm-8:30pm (Korea time, UTC+9)  (Note the unusual day and time!)

Title: Lyndon words and fused currents in shuffle algebra 

Abstract: Classical q-shuffle algebras provide combinatorial models for the positive half U_q(n) of a finite quantum group. We define a loop version of this construction, yielding a combinatorial model for the positive half U_q(Ln) of a quantum loop group. In particular, we construct a PBW basis of U_q(Ln) indexed by standard Lyndon words, generalizing the work of Lalonde-Ram, Leclerc and Rosso in the U_q(n) case. We also connect this to Enriquez' degeneration A of the elliptic algebras of Feigin-Odesskii, proving a conjecture that describes the image of the embedding U_q(Ln) -> A in terms of pole and wheel conditions. The talk shall conclude with the shuffle interpretations of fused currents proposed by Ding-Khoroshkin. This is based on joint works with Andrei Negut.

• Arun Ram  (University of Melbourne, Australia)

Apr. 26 (Fri), 2024, 11:30am-12:30pm (Korea time, UTC+9)

Title: A new definition of integral form Macdonald polynomials

Abstract:  Mellit explained the Modified Macdonald polynomials can be viewed as a generating function for the number of points in parabolic affine Springer fibers. In this talk we explain that the plethystic transformation that relates integral form Macdonald polynomials and modified Macdonald polynomials has an interpretation in terms of traces on the Hecke algebra. This leads to an interpretation of the integral form Macdonald polynomials as generating functions for the number of points Lusztig varieties.

• Joshua Wen (Northeastern University, USA)

Apr. 12 (Fri), 2024, 10:00am-11:00am (Korea time, UTC+9)

Title: Wreath Macdonald polynomials

Abstract: Defined by Haiman, wreath Macdonald polynomials are generalizations of the well-known Macdonald polynomials to wreath products of cyclic groups with symmetric groups. For a fixed cylic group Z/rZ, these can be viewed as partially-symmetric polynomials, where there are r families of symmetric variables. Many results for the usual Macdonald polynomials should have analogues in the wreath setting: e.g. Macdonald operators, bispectral duality, evaluation formulas, and norm formulas. Precise conjectures for these analogues can be tricky to write down and even more difficult to prove. A guiding principle is that various quantum algebraic methods in the classical Macdonald theory should have generalizations in the wreath setting. I will present work, joint with Daniel Orr and Mark Shimozono, that studies these polynomials via the rank r quantum toroidal algebra.

• Tianyi Yu (University of California, San Diego, USA)

Apr. 5 (Fri), 2024, 2:00pm-3:00pm (Korea time, UTC+9)

Title: Analogue of Fomin-Stanley algebra on bumpless pipedreams 

Abstract: Schubert polynomials are distinguished representatives of Schubert cells in the cohomology of the flag variety. Pipedreams (PD) and bumpless pipedreams (BPD) are two combinatorial models of Schubert polynomials. There are many classical perspectives to view PDs: Fomin and Stanley represented each PD as an element in the NilCoexter algebra; Lenart and Sottile converted each PD into a labeled chain in the Bruhat order. In this talk, we unravel the BPD analogues of both viewpoints. One application of our results is a simple bijection between PDs and BPDs via Lenart’s growth diagram.

• Huanchen Bao (National University of Singapore, Singapore)

Mar. 15 (Fri), 2024, 3:00pm-4:00pm (Korea time, UTC+9)

Title: Acyclic matchings on Bruhat intervals

Abstract: Discrete Morse theory, developed by Forman, is an efficient tool to determine the homotopy type of a regular CW complex. The theory has been reformulated by Chari in purely combinatorial terms of acyclic matchings on the face poset. In this talk, I will discuss explicit constructions of such acyclic matchings on Bruhat intervals using reflection orders. As an application, we show the totally nonnegative Springer fibres are conctractible, verifying a conjecture of Lusztig. This is based on joint work with Xuhua He.