Weak Bruhat interval modules of the $0$-Hecke algebras for genomic Schur functions

Time: 09:00 to  11:00 Ngày 28/10/2022

Venue/Location: Online

Speaker: Semin Yoo (School of Computational Sciences at Korea Institute for Advanced Study)

This talk will be mostly devoted to convey about the story around our work without details. In 2017, Yong and Pechenik introduced the genomic Schur function as a natural deformation of the ordinary Schur function in the context of the K-theory of Grassmannians. They showed that genomic Schur functions consist of a basis of the ring of symmetric functions, although they are not Schur-positive in general. Recently, Pechenik proved that genomic Schur functions are expanded positively in terms of fundamental quasisymmetric functions, implying the possibility of the existence of nice $0$-Hecke modules corresponding to genomic Schur functions under quasisymmetric characteristics.

Since the mid-2010s, there have been considerable attempts to provide a representation-theoretic interpretation of noteworthy quasisymmetric functions by constructing appropriate $0$-Hecke modules. Very recently, Jung, Kim, Lee, and Oh introduced weak Bruhat interval modules to provide a unified method to study those $H_{n}(0)$-modules.

In this talk, I will construct new $0$-Hecke modules whose quasisymmetric characteristics are the homogeneous components of genomic Schur functions and study their structural properties. Furthermore, I will decompose of our modules into weak Bruhat interval modules and find the projective cover of each summand of the decomposition.

This is joint work with Young-Hun Kim.

ZOOM link: https://us02web.zoom.us/j/86050584049?pwd=U2pNNVovdlQ1SDlEb1loVEdZQU9Odz09 (updated) 

Website Seminar: https://sites.google.com/view/ktv-seminar/