Mercedes Rosas

Combinatoria Algebraica en Sevilla

Miembros:

• Emmanuel Briand

• Mercedes Rosas

Experimental mathematics:

• Emmanuel Briand mantains a Database of Piecewise Quasipolynomial Functions.

  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.

• Adrian Lillo’s SageMath notebook Weary drivers

  the record-preserving bijection and the various statistics appearing in our work on record codes and parking functions.

• Emmanuel Briand’s SageMath notebook the144symmetries

  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.

• Emmanuel Briand’s SageMath notebook On the growth of the Kronecker coefficients: Companion Worksheet

  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.

Estudiantes:

• Adrian Lillo.

  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 Ocampo

  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 Calvo Sánchez

  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 Puerto

  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

Antiguos Estudiantes y visitantes:

• Laura Colmenarejo

  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 Trandafir

  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 Gutiérrez Cáceres

  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 Esquivias.

  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.

Conferences and Workshops:

• Algebraic Combinatorics in Sevilla

  Virtual Seminaire

• 91 Séminaire Lotharingien de Combinatoire

  Salobreña, March 17 - 20, 2024.
  Invited Speakers: Petter Brändén and Ira Gessel.

• School and Workshop “Mathematical fundations of quantum information”

  Sevilla, Spain, 23-27 november 2009.

• Diagonally symmetric polynomials and applications

  Castro-Urdiales, October 15 - 19, 2007

Funding:

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