Welcome to the Rutgers Symmetric Functions & Probability Theory seminar page.
This seminar runs at 12:10 PM on Tuesdays in Hill 705.
The organizers are Swee Hong Chan , Konstantin Matveev, Siddhartha Sahi , Hong Chen and Jike Liu.
This page is currently maintained by Jike Liu. If you wish to join our mailing list, please click here or email jl4069@scarletmail.rutgers.edu .
For past talks, click here.
Speaker: Swee Hong Chan (Rutgers)
Title: Breaking the infinite barrier in the 1/3-2/3 Conjecture
Abstract: The celebrated 1/3–2/3 conjecture in poset theory asserts that: every finite poset that is not a chain contains two elements x and y such that, in a uniformly random linear extension, the probability that x precedes y lies between 1/3 and 2/3. The previously best general result, due to Brightwell, Felsner, and Trotter in 1995, replaces 1/3 by (5-sqrt{5}/10) approximately 0.2764. A striking obstruction to improving this bound is that it is sharp for infinite posets, suggesting that any further progress must exploit the finiteness of the poset in an essential way. In this talk, we discuss an improvement of this bound by a strictly positive absolute constant. This is the first improvement since 1995 and, in particular, breaks the aforementioned infinite-poset barrier. This is a joint work with Aires, Pak, and Panova.
Speaker: Tuong Le (Princeton)
Title: Skew hives, skew skeps, skew schur log-concavity
Abstract: Skeps are a variant of hives that counts Littlewood–Richardson coefficients. They were recently introduced by Speyer to prove a conjecture of Lam, Postnikov and Pylyavskyy. We introduce skew hives and skew skeps, generalizing hives and skeps, and explain how to use L-log concavity techniques to prove a skew Schur generalization of the Lam–Postnikov–Pylyavskyy conjecture. We discuss implications on Newell-Littlewood numbers, log-concavity of Koike–Terada symmetric functions and shadow skew Schur functions. Finally, we explain the bijection between skew skeps and Nguyen–Nguyen–Woodruff peelable tableaux. This is joint work with Son Nguyen.
Speaker: Tianyi Yu (Université du Québec à Montréal)
Title: Standard Crystals for symmetric Grothendieck Polynomials
Abstract: Schur polynomials form a fundamental basis for symmetric polynomials. Motivated by geometry and representation theory, researchers have expanded various polynomials into the Schur basis positively, including (i) skew Schur polynomials, (ii) products of Schur polynomials, and (iii) Stanley symmetric polynomials. Standard crystals provide an elegant framework that effectively demonstrates these Schur expansions. Symmetric Grothendieck polynomials are non-homogeneous analogues of Schur polynomials, arising from the K-theory of flag varieties. Analogous expansions into symmetric Grothendieck polynomials have gained significant attention over the past decades: Buch established the K-theoretic analogues of (i) and (ii), while Buch, Kresch, Shimozono, Tamvakis, and Yong resolved (iii). In this talk, we present an analogue of standard crystal theory, introducing a powerful new tool for establishing symmetric Grothendieck positivity. This framework not only recovers the three K-theoretic expansions mentioned above but also sheds light on related problems, including a conjecture of Ikeda and Naruse. This work is based on joint work with Eric Marberg and Kam Hung Tong.
Speaker: Jike Liu (Rutgers)
Title: Four-Functions Inequalities, Skeps, and Schur Positivity
Abstract: Schur functions form one of the fundamental bases of the ring of symmetric functions. By comparing coefficients in the Schur basis, one obtains the partial order of Schur positivity, which provides a natural setting for studying analogues of classical correlation and log-concavity inequalities in symmetric function theory. A recent breakthrough in this direction is Speyer’s proof of the Lam–Postnikov–Pylyavskyy conjecture, based on a new combinatorial model for Littlewood–Richardson coefficients called skeps. On the global inequality side, Chan–Chen–Pak–Soskin established a Schur-positive analogue of the Ahlswede–Daykin four-functions theorem and conjectured a Schur-positive version of the Klartag–Lehec four-functions inequality. In this talk, I will explain how scalar four-functions inequalities can be combined with Speyer’s skep model to obtain a general Schur-positive four-functions theorem that can recovers the results above and proves the conjectured Klartag–Lehec inequality. I will also discuss how the result can be applied to K-theoretic symmetric functions, yielding analogous inequalities in the Schur order for sign-normalized Grothendieck polynomials and dual stable Grothendieck functions. This is based on joint work with Swee Hong Chan.