Lowering operators on $K$-$k$-Schur functions and a lowering operator formula for closed $K$-$k$-Schur functions

TL;DR

This paper systematically studies lowering operators on $K$-$k$-Schur functions, deriving a formula for closed functions and providing a combinatorial proof for a conjecture related to $k$-Schur Katalan functions.

Contribution

It introduces a lowering operator formula for closed $K$-$k$-Schur functions and offers a new combinatorial proof for a conjecture on $k$-Schur Katalan functions.

Findings

Derived a lowering operator formula for closed $K$-$k$-Schur functions.

Provided a combinatorial proof for a conjecture on $k$-Schur Katalan functions.

Enhanced understanding of the role of lowering operators in $K$-$k$-Schur functions.

Abstract

This paper gives a systematic study of the lowering operators acting on the $K$-$k$-Schur functions, motivated by the pivotal role played by the operators in the definition and study of Katalan functions. A lowering operator formula for closed $K$-$k$-Schur functions is obtained. As an application, a combinatorial proof is provided to a conjecture on closed $k$-Schur Katalan functions, posed by Blasiak, Morse and Seelinger, and recently proved by Ikeda, Iwao and Naito by a different method.