Spring 2023

Title: Vexillary double Edelman--Greene coefficients are Graham positive

Abstract: Lam, Lee, and Shimozono (LLS) recently introduced backstable double Schubert polynomials to represent classes in the cohomology ring of the infinite flag variety. Using these polynomials, they introduce double Stanley symmetric functions, which expand into double Schur functions with polynomial coefficients called double Edelman--Greene coefficients. They prove that double Edelman--Greene coefficients are Graham positive. For vexillary permutations, we use a bijection of Weigandt to convert this result to a statement for skew flagged double Schur functions, where we give an explicit combinatorial formula for double Edelman--Greene coefficients that is manifestly Graham positive. Our methods extend to the K-theoretic setting, partially affirming a later conjecture of LLS now proven geometrically by Anderson. This is joint work with Zach Hamaker and Tianyi Yu.

Title: Combinatorics of exceptional collections in type A-tilde

Abstract: We will define quivers of type A-tilde, their representations, and exceptional collections of these representations. We will then introduce a combinatorial model of these representations, based on the one constructed by Garver, Igusa, Matherne, and Ostroff for type A, by drawing strands on a copy of the integers. We will see that collections of strands called strand diagrams are in bijection with exceptional collections in type A-tilde. After, we will introduce a different combinatorial model consisting of arcs on an annulus that is also in bijection with exceptional collections in type A-tilde. Using the arc diagrams, we will place exceptional collections into finitely many infinite families, and then using a labeling of the strand diagrams, we will show that for a certain orientation of the quiver, the number of families is counted by a generalization of the Catalan numbers known as the Rothe numbers, or the Rothe-Hagan coefficients of the first type.


Title: Chromatic Symmetric Functions and RSK for (3 + 1)-free Posets

Abstract: In 1995, Stanley introduced the chromatic symmetric function of a graph, a symmetric function analog of the classical chromatic polynomial of a graph. The Stanley-Stembridge e-positivity conjecture is a long-standing conjecture that states that the chromatic symmetric function of a certain class of graphs, called incomparability graphs of (3+1)-free posets, has nonnegative coefficients when expanded in the elementary symmetric function basis. In 1996, Gasharov described Schur expansion of the chromatic symmetric function for this class of graphs in terms of P-tableau, a generalization of a standard Young tableau. An open problem is to find a bijective proof of this expansion for all (3+1)-free posets.

Title: Bargain hunting in a Coxeter group

Abstract: Petersen and Tenner defined the depth statistic for Coxeter group elements which, in the symmetric group, can be described in terms of a cost-minimization problem over the factorizations of a permutation into transpositions. We generalize that cost function to the other classical (finite and affine) Weyl groups, letting the cost of an individual reflection t be the distance between the integers transposed by t in the combinatorial representation of the group (à la Eriksson and Eriksson). Arbitrary group elements then have a well-defined cost, obtained by minimizing the sum of the transposition costs among all factorizations of the element. We show that the cost of arbitrary elements can be computed directly from the elements themselves using a simple, intrinsic formula.  This work is joint with Bridget Tenner.


Title: Schur-like Bases of QSym and NSym: Properties of the Shin Functions

Abstract: A Schur-like basis of the noncommutative symmetric functions is one whose commutative image is the Schur basis of the symmetric functions.  The canonical Schur-like bases of NSym are the immaculate basis, the shin basis, and the Young noncommutative Schur basis, each of which reflects the properties of the Schur functions in interesting and surprisingly different ways. For each of these bases, we will discuss tableaux interpretations, various types of multiplication rules, and dual bases in the quasisymmetric functions. We will then present new results on the shin functions including a creation operator and Jacobi-Trudi rule for certain cases, along with observations about the multiplicative structure.

Title: Type-BC analogs of codominant permutations and unit interval orders

Abstract: Permutations $w$ in $S_n$ for which the (type-A) Schubert variety $\Omega_w$ is smooth are characterized by avoidance of the patterns 3412 and 4231.  The smaller family of codominant permutations, those avoiding the pattern 312, seems to explain a lot about character evaluations at Kazhdan-Lusztig basis elements $C'_w(q)$ of the (type-A) Hecke algebra. In particular, for every Hecke algebra character $\chi$, and every 3412-, 4231-avoiding permutation $w$, there exists a codominant permutation $v$ such that $\chi(C'_w(q)) = \chi(C'_v(q)$. Moreover, these character evaluations can be computed by playing simple games with unit interval orders $P = P(v)$ corresponding to the codominant permutations.  We generalize these facts to the hyperoctahedral group $B_n$ using signed pattern avoidance and an appropriate analog of unit interval orders.

Title: Quantum cellular automata on fusion spin chains

Abstract: Quantum cellular automata (QCA) are models of discrete-time unitary dynamics of quantum spin systems. They can be characterized algebraically as certain automorphisms of the associative algebra generated by local observables of a spin system. We will give a gentle introduction to this topic, and explain some of our recent contributions to the problem of classification of short range entangled phases of QCA for fusion spin chains.

Title: Posets for cluster variables in cluster algebras from surfaces

Abstract: This talk will introduce cluster algebras, with an emphasis on their combinatorics, and describe a recent joint result with Vincent Pilaud and Sibylle Schroll. At the heart of a cluster algebra is a complicated, branching recursion that defines cluster variables (certain rational functions organized into finite sets called clusters). The recursion looks bizarre at first glance (and at subsequent glances), but has surprisingly close connections to various areas of mathematics. For a given cluster algebra, one typically needs a combinatorial model before even formulating the question of how to solve the recurrence. One class of cluster algebras is modeled by surfaces with a finite set of distinguished "marked points", where the cluster variables are indexed by tagged arcs (curves connecting the marked points) and the clusters are collections of arcs that cut the surface into triangles. In this model, solving the recursion means finding a formula for the cluster variable associated to a given tagged arc.  Earlier solutions hint at the importance of the Fundamental Theorem of Finite Distributive Lattices (FTFDL).  We take the FTFDL as a guiding principle to give a new formula for cluster variables, with simpler combinatorics and with simpler proofs that leverage the hyperbolic geometry of the surfaces.  The talk will assume no prior knowledge of cluster algebras, marked surfaces, or distributive lattices.


Title: Lorentzian polynomials in combinatorics and representation theory


Abstract: Lorentzian polynomials were developed independently and simultaneously by Brändén--Huh and by Anari--Liu--Oveis Gharan--Vinzant in 2018.  They have recently played a key role in solving long-standing open problems related to log-concavity in a variety of areas such as combinatorics, knot theory, and computer science.  


The purpose of this talk will be to introduce the class of Lorentzian polynomials and then to share several families of polynomials, from different parts of mathematics, that are either known or expected to be Lorentzian. Two classes I will focus on are the (normalizations of) Schur polynomials and the chromatic symmetric functions of certain graphs appearing in the study of Hessenberg varieties. If time permits, I may discuss stronger conjectures involving stability. The talk will be based on joint work with June Huh, Karola Mészáros, Alejandro Morales, Jesse Selover, and Avery St. Dizier.



Abstract: We describe an algorithm to compute Whitney stratifications of real algebraic varieties. The basic idea is to first stratify the    complexified version of the given real variety using conormal techniques, and then to show that the resulting stratifications admit a description using only real polynomials. This method also extends to stratification problems involving certain basic semialgebraic sets as well as certain algebraic maps. One of the map stratification algorithms described also yields a new method for solving the real root classification problem. This is joint work with Vidit Nanda (Oxford).

Title: Finite-dimensional algebras and the shard intersection order

Abstract: Reading’s shard intersection order is a lattice structure on a finite Coxeter group which is weaker than the weak order. This structure also appears in the representation theory of finite-dimensional algebras as the inclusion order on certain subcategories of modules. Algebraically, the cover relations of this lattice are traversed by applying a “reduction operation” associated to a chosen module. In this talk, we explain the combinatorial analog of this reduction operation in type A. No prior knowledge of finite-dimensional algebras or the shard intersection order will be assumed.