Algecom-XXIII
The meeting is planned twice a year, and the next iteration will take place at Loyola University in Chicago on Saturday April 15, 2023. You and your students are invited to join us!.
Algecom Committee:
AlGeCom XXIII Date: Saturday April 15, 2023
Location: Loyola University in Chicago
Cuneo Hall 312, Loyola University in Chicago
Campus map: https://www.luc.edu/media/lucedu/pdfs-campusmaps/lsc.pdf
Parking:
Available in the Main Parking Structure (P1) for a price of $7. For more information read https://www.luc.edu/campustransportation/generalinformation/visitorsandguests/
Local Organizers (questions related to Algecom XXIII):
Rafael González D’León, Aaron Lauve, and Peter Tingley
ALGECOM is partially funded by a grant from the National Science Foundation, which provides limited support for students and postdocs to offset the cost of attendance. Everyone should apply by filling out the registration form below and indicating their interest in financial support by Tuesday March 14, 2023 (and we appreciate early applications). Funding decisions will be made shortly thereafter.
Participants are encouraged to present their work during the poster session. Please indicate your title and abstract in the registration form, or by sending an email to rgonzalezdleon@luc.edu
Tentative Schedule:
(All times are in the Central Time Zone)
9:15-10:00am Coffee, Snacks and Registration
10:00-11:00am Nantel Bergeron (York University)
"Excedance quotient of the Bruhat order, Quasisymmetric Varieties and Temperley-Lieb algebras":
We define a set of permutations $QSV_n$ with the following properties. $QSV_n$ is a basis of the Temperley-Lied Algebra $TL_n(2)$. When we consider the permutations in $QSV_n$ as points $(\sigma_1,\sigma_2,\ldots,\sigma_n)$, then the vanishing ideal $I(QSV_n)$ is such that the ideal of top homogeneous component is $\langle QSym_{n}^{+} \rangle.$
We have several byproducts of our construction. We give a partition of the symmetric group $S_n$ into equivalent classes given by excedence classes $C_\lambda$. The indexes $\lambda$ are non-crossing partition of $\{1,2,\ldots,n\}$. For each $\lambda$, the class $C_\lambda$ is an interval of the (strong) Bruhat order containing a unique element on $QSV_n$ (which is the maximum in the interval) and a unique $321$-avoiding permutation (which is the minimum). We show that in fact any section of the excedence classes produce a basis of $TL_n$. We show that the Bruhat order induce a well-defined order on the quotient $S_n/\!\!\sim$ by excedence classes. This allows us to also introduce a weak order on $S_n\big/\!\!\sim$.
Joint work with Lucas Gagnon.
11:15-12:15am Laura Colmenarejo (NC State University)
"A Murnaghan-Nakayama rule for the quantum Schubert polynomials":
Several \emph{linear algebra problems} are very interesting in the ring of symmetric polynomials. One of them is understanding \emph{combinatorially} how to multiply polynomials from different bases and expand the resulting symmetric polynomial in one of the bases. The classical Murnaghan–Nakayama rule is a formula for the product of a Schur symmetric polynomial and a Newton power sum. It is as fundamental as the Pieri rule, and the resulting formulas from the Murnaghan-Nakayama rule are significantly more compact. The Schubert polynomials are a very interesting generalization of Schur polynomials due to their connection with the cohomology of the flag variety in algebraic geometry. In this talk, I will present a general rule for multiplying a quantum Schubert polynomial by a quantum power sum polynomial, achieved by relating the structure constants to the classical case. We will review the classical and quantum stories and discover the combinatorics behind each version.
This project is joint work with Carolina Benedetti, Nantel Bergeron, Franco Saliola, and Frank Sottile.
12:15am-2:00pm Lunch
2:00-3:00pm Rosa Orellana (Dartmouth College)
"Characters of planar diagram algebras via symmetric functions":
Diagram algebras arise as centralizer algebras of groups or algebras when these act diagonally on $k$-fold tensor space. The construction gives rise to a tower or algebras $A_0 \subset A_1 \subset A_2 \subset \cdots$. Each algebra $A_k$ has a basis indexed by graphs with $2k$ vertices and the product can be described via concatenation of the graphs. A widely studied subclass of diagram algebras are those whose diagrams do not allow crossings of edges, these are called planar. The classical example of a planar algebra is the Temperley-Lieb algebra, $TL_k$, whose dimensions are given by the Catalan numbers.
Benkart and Halverson constructed a new class of planar algebras whose dimension is given by the Motzkin numbers and it is closely related to the Temperley-Lieb algebras. One objective of this talk is to give a construction of a new tower of planar algebras whose dimensions are given by the Riordan numbers. A second objective is to describe the characters of these planar algebras.
Joint work with N. Wallace and M. Zabrocki
3:15-4:15pm Mario Sanchez (Cornell University)
"Geometry of the Chromatic Symmetric Function of Trees":
Stanley's chromatic symmetric function is a generalization of the chromatic polynomial of a graph that encodes coloring information for graphs. One open conjecture is that non-isomorphic trees have different chromatic symmetric functions. In this talk, I will give two geometric interpretations of these functions for trees. The first interprets the chromatic symmetric function of a tree as an element in the Chow ring of the permutahedral variety opening the conjecture to algebraic geometric methods. The second describes this open conjecture in terms of the theory of valuations on generalized permutahedra. These are functions on polytopes which satisfy certain inclusion-exclusion relations with respect to subdivisions. From this perspective, we make progress on the conjecture by constructing new valuations on generalized permutahedra. We will primarily focus on this convex geometric interpretation for this talk.
4:30-5:30pm Poster Session:
Power graphs of finite groups. Amrita Acharyya (University of Toledo)
The Primitive Eulerian polynomial. Jose Bastidas (LACIM - Université du Québec à Montréal)
Invariant Theory for the Free Left-Regular Band and a q-Analogue. Patricia Commins (University of Minnesota)
A partially commutative generalization of the immaculate functions and their duals. Spencer Daugherty (North Carolina State University)
Counting Parking Sequences and Parking Assortments Through Permutations. Kimberly Harry and Jan Kretschmann (University of Wisconsin Milwaukee)
Rowmotion on Rooted Trees. Jamie Kimble (Michigan State University)
Splines on Cayley Graphs of the Symmetric Group. Nathan Lesnevich (Washington University in St Louis)
Hopf monoids and polynomial invariants. Yichen Ma (Cornell University)
Oriented Matroid Circuit Polytopes: Type A. Jodi McWhirter (Washington University in St. Louis)
Bender-Knuth Involutions on Linear Extensions of Posets. Son Nguyen (University of Minnesota)
Binomial Schubert Determinantal Ideals. Ada Stelzer (UIUC)
A Cancellation-Free Antipode Formula for a Hopf Monoid of Partial Orders. John Whelan (Cornell University)
The Isomorphism Problem for cominuscule Schubert Varieties. Weihong Xu (Virginia Tech)
6:00pm Dinner:
Beard and Belly. 6157 N Broadway, Chicago, IL 60660. https://beardandbellychicago.com/
Lodging:
Rooms are available close to campus at the Hampton Inn Chicago North-Loyola Station (1209 W Albion Ave, Chicago, IL 60626) and at the Super 8 by Wyndham (7300 N Sheridan Rd, Chicago, IL 60626). Other recommended hotels in Evanston are the Hilton Orrington/Evanston (1710 Orrington Ave) and the Holiday Inn Chicago North-Evanston. There are of course other hotels in Chicago and Evanston that you can choose from.