This talk will be mostly devoted to convey the story around our work without details.  

In 2017, Yong and Pechenik [3] 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 [2] proved that genomic Schur functions are expanded positively in terms of fundamental quasisymmetric functions, implying the possibility of the existence of nice -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 -Hecke modules. 

Very recently, Jung, Kim, Lee, and Oh [1] introduced weak Bruhat interval modules to provide a unified method to study those -modules.

In this talk, I will construct new -Hecke modules whose quasisymmetric characteristics are the homogeneous components of genomic Schur functions and study their structural properties. Furthermore, I will decompose 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.

[1] W.-S. Jung, Y.-H. Kim, S.-Y. Lee, and Y.-T. Oh, Weak Bruhat interval modules of the 0-Hecke algebra, Math. Z., 301:3755–3786, (2022).

[2] O. Pechenik, The genomic Schur function is fundamental-positive, Ann. Comb., 24(1):95–108, (2020).

[3] O. Pechenik and A. Yong, Genomic tableaux, J. Algebraic Combin, 45(3):649–685, (2017).