
A quasipolynomial is a function on integer points given by several polynomials: one for each coset of some full-rank sublattice. The piecewise quasipolynomial functions considered here have as domains of quasipolynomiality the maximal cells of a fan subdividing a cone, and are zero outside this cone. This database provides a full description of some small multiplicity functions from representation theory that are piecewise quasipolynomial in this sense. This includes or will include families of Littlewood-Richardson coefficients, Kronecker coefficients, Plethysm coefficients and alike.
the record-preserving bijection and the various statistics appearing in our work on record codes and parking functions.
We perform explicit calculations with sagemath to show that the Littlewood-Richardson coefficients associated to the representation theory of SL3 afford a group of 144 linear symmetries, which is more than the 12.
This notebook implements some of the calculations presented in the preprint On the growth of the Kronecker coefficients (Emmanuel Briand, Amarpreet Rattan, Mercedes Rosas; 2016 Our calculations also show that this group of symmetries acts transitively on the set of chambers of the corresponding chamber complex. As a consequence, the data of the group and of only one chamber (with the formula for the LR on this chamber) completely describe these LR coefficients.
Building on our decomposition of a Cayley tree via records, closely related to the blob encoding of Kreweras, Moszkowski, and Picciotto, we define parking functions anew and construct a record-preserving bijection with Cayley trees. This leads to an equidistribution of six statistics on both structures. More recently, we have been looking at the record generating functions for trees and forests. A Maria de Maetzu grant funds his PhD research.
Aaron is working towards his doctorate degree with a FPU fellowship. At the moment, Aaron is studying the hook stability phenomena present for the much-studied Kronecker coefficients of the symmetric and the linear general group, and its connection to FI-modules.
Luis wrote his senior and his master thesis on a higher dimensional version of the Combinatorial Laplacian defined in the framework of algebraic topology. Luis’s work is based on results of Kalai, Bernardi and Klivans. Luis is currently writing up the results of his investigation.
Pablo wrote his senior thesis focusing in some bijections related to tree records. His results appear in our preprint The genesis sequence, tree records and endofunctions Next year, he will be starting is Master Leiden as part of ALGANT
Laura finished her PhD thesis titled “Stability in the combinatorics of representation theory” in 2016. Laura, a rising star in the area of Algebraic Combinatorics, currently hold a tenure track position at North Carolina State Univesity.
Stefan was a frecuent visitor to the group during his PhD at Simon Fraser University in Vacouver. During this time, he delved into some ideas regarding the study of Kronecker coefficients, which partially stemmed from our group’s discussions. Additionally, he discovered a new symmetry exhibited by the Littlewood-Richardson coefficients. Afterwards, Stefan came back to Sevilla for postdoctoral position with the quantum information group of Adan Cabello (Sevilla). During this second period, Stefan collaborated with us on questions related to records of trees and forests.
Álvaro’s undergraduate senior thesis yielded two publications on the study of the plethysm of symmetric functions. Following his time in Sevilla, Álvaro authored a Master’s thesis at the University of Bonn under the supervision of Catharina Stroppel and Jacob Matherne. Presently, Álvaro is pursuing his Ph.D. under the guidance of Mark Wildon at Bristol University.
Luis’s and Adrian Lillo senior’s thesis were the result of a collaboration with Álvaro Gutiérrez Cáceres, and the senior members of the group. We provided a combinatorial proof of the Graham and Pollak formula for the determinant of the distance matrix of a tree, utilizing the Gessel-Viennot involution. Nowdays, Luis is working towards his PhD in Sevilla, this time focusing in number theory, and under the direction of Antonio Rojas and Alberto Castaño.
Virtual Seminaire
Salobreña, March 17 - 20, 2024.
Invited Speakers: Petter Brändén and Ira Gessel.
Sevilla, Spain, 23-27 november 2009.
Castro-Urdiales, October 15 - 19, 2007
We are funded by the following research projects::
CoMADmRT: Computational Methods in Algebra, D-modules and Representation Theory (PID2020-117843GB-I00)
FQM-333: Computational Algebra in NonCommutative Rings and Applications