Algebraic and Enumerative Combinatorics Seminar - Spencer Daugherty

Thursday, November 9, 2023 2:00 pm - 2:00 pm EST (GMT -05:00)

Title: Extended Schur Functions and Bases Related by Involutions

Speaker: Spencer Daugherty
Affiliation: North Carolina State University
Location: MC 6029

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:00 pm.

Abstract: The extended Schur basis and the shin basis generalize the Schur functions to the dual algebras of the quasisymmetric functions and the noncommutative symmetric functions. We define a creation operator and a Jacobi-Trudi rule for certain shin functions and show that a similar matrix determinant expression does not exist for every shin function. We also define and study skew extended Schur functions which connect to skew Schur functions and to the multiplicative structure of the shin functions. Then, we introduce two new pairs of dual bases that result from applying certain involutions to the extended Schur and shin functions. These bases are defined combinatorially by variations on shin-tableaux much like the row-strict extended Schur functions. We will also discuss colored generalizations of the shin and extended Schur functions.