| Date |
Speaker |
Title |
| May 12 | Andrew Gitlin (UC Berkeley) | A vertex model for LLT polynomials and k-tilings of the Aztec diamond
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| | We describe a Yang-Baxter integrable colored vertex model, from which we construct a class of partition functions that equal the LLT polynomials of Lascoux, Leclerc, and Thibon. Using the vertex model formalism, we can prove many properties of these polynomials. We also use the vertex model to study k-tilings (k-tuples of domino tilings) of the Aztec diamond of rank m, where we assign a weight to each k-tiling depending on the number of vertical dominos and the number of "interactions" between the tilings. We compute the generating polynomials of the k-tilings, and prove some combinatorial results about k-tilings in certain limits of the interaction strength.
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| Feb 3 | Victor Wang (UBC) | P-partition power sums
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The Hopf algebra of symmetric functions is spanned by several important bases, including by power sum symmetric functions, which encode the class values of the characters of the symmetric group under the Frobenius characteristic map. We introduce in this talk the basis of combinatorial power sums, naturally refining the power sum symmetric functions, for the larger Hopf algebra of quasisymmetric functions. Our construction is motivated by the theory of (weighted) P-partitions, the combinatorics of which will allow us to describe formulas for products, coproducts and classical quasisymmetric involutions, as well as give combinatorial rules for the expansion into the monomial and fundamental bases of quasisymmetric functions. Joint work with Farid Aliniaeifard and Stephanie van Willigenburg.
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| May 26 | no talk | Combinatorial and Algebraic Enumeration conference at Waterloo
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| June 2 (12pm) | Per Alexandersson (KTH) | Cyclic sieving with focus on open problems |
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The cyclic sieving phenomenon (CSP) connects a cyclic group action on a family of combinatorial objects with some q-analog of that set. We discuss some recent results and open problems for standard and semistandard tableaux, as well as some other families of combinatorial objects.
Several open problems with various levels of difficulty will be presented.
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| June 9 | Zachary Hamaker (University of Florida) | Virtual characters of permutation statistics (in person) |
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Functions of permutations are studied in a wide variety of fields including probability, statistics and theoretical computer science. I will introduce a method for studying such functions using representation theory and symmetric functions. As a consequence, one can extract detailed information about asymptotic behavior of many permutation statistics with respect to non-uniform measures that are invariant under conjugation. The key new tool is a combinatorial formula called the path Murnaghan-Nakayama rule that gives the Schur expansion of a novel basis of the ring of symmetric functions. This is joint work with Brendon Rhoades. |
| June 16 | Christian Gaetz (Harvard) | 1-skeleton posets of Bruhat interval polytopes (in person) |
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Bruhat interval polytopes are a well-studied class of
generalized permutohedra which arise as moment map images of
various toric varieties and totally positive spaces in the flag
variety. I will describe work in progress in which I study the 1-
skeleta of these polytopes, viewed as posets interpolating between
weak order and Bruhat order. In many cases these posets are
lattices and the polytopes, despite not being simple, have
interesting h-vectors.
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| June 23 |
| social hour
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| June 30 | Thomas McConville (Kennesaw State University) | Determinantal formulas with major indices |
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Krattenthaler and Thibon discovered a beautiful formula for the determinant of the matrix indexed by permutations whose entries are q^maj( u*v^{-1} ), where “maj” is the major index. Previous proofs of this identity have applied the theory of nonsymmetric functions or the representation theory of the Tits algebra to determine the eigenvalues of the matrix. I will present a new, more elementary proof of the determinantal formula. Then I will explain how we used this method to prove several conjectures by Krattenthaler for variations of the major index over signed permutations and colored permutations. This is based on joint work with Donald Robertson and Clifford Smyth. |
| July 7 | Emily Gunawan (University of Oklahoma) | Box-ball systems, RSK, and Motzkin paths |
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A box-ball system (BBS) is a discrete dynamical system whose dynamics come from the balls jumping according to certain rules. A permutation on n objects gives a BBS state by assigning its one-line notation to n consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau called the soliton decomposition of the box-ball system. The shape of the soliton decomposition is called the BBS partition. An exciting discovery (made in 2019 by Lewis, Lyu, Pylyavskyy, and Sen) is that the BBS partition and its conjugate record permutation statistics similar to the classical Greene’s theorem statistics.
The well-known Robinson—Schensted algorithm is a bijection from permutations w to pairs of standard tableaux P(w), Q(w) of the same shape. We will discuss a few new results which relates BBS to these P and Q tableaux:
(1) The soliton decomposition of a permutation w is a standard tableau if and only if it is equal to P(w).
(2) The Q tableau of a permutation completely determines the dynamics of the corresponding box-ball system.
(3) We present a bijection between Motzkin paths and a class of involutions whose soliton decompositions are standard.
This talk is based on joint work with B. Drucker, E. Garcia, A. Rumbolt, R. L. Silver (arxiv.org/abs/2112.03780); M. Cofie, O. Fugikawa, M. Stewart, D. Zeng (SUMRY 2021); and S. Hong, M. Li, R. Okonogi-Neth, M. Sapronov, D. Stevanovich, and H. Weingord (SUMRY 2022).
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| July 14 | Kevin Purbhoo (U Waterloo) | An identity in the group algebra of the symmetric group |
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Come with me on a magical journey into the mysterious world of inverse Wronskians. |
| July 21 | no talk | FPSAC 2022 |
| July 28 | Jinyoung Park (Stanford) | Thresholds |
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Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its "expectation-threshold," which is a natural (and often easy to calculate) lower bound on the threshold. In this talk, I will present recent progress on this topic. Based on joint work with Huy Tuan Pham.
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