Weighted $K$-$k$-Schur functions and their application to the $K$-$k$-Schur alternating conjecture

TL;DR

This paper introduces weighted K-k-Schur functions, unifying existing functions and proving the K-k-Schur alternating conjecture for many partitions, advancing understanding of K-theoretic symmetric functions.

Contribution

It defines weighted K-k-Schur functions, extending previous functions, and proves the K-k-Schur alternating conjecture for a broad class of partitions.

Findings

Resolved the K-k-Schur alternating conjecture for all strictly decreasing k-bounded partitions.

Unified and extended K-k-Schur functions within a new weighted framework.

Provided new insights into the combinatorial structure of K-theoretic symmetric functions.

Abstract

We introduce the new concept of weighted $K$-$k$-Schur functions -- a novel family within the broader class of Katalan functions -- that unifies and extends both $K$-$k$-Schur functions and closed $k$-Schur Katalan functions. This new notion exhibits a fundamental alternating property under certain conditions on the indexed $k$-bounded partitions. As a central application, we resolve the $K$-$k$-Schur alternating conjecture -- posed by Blasiak, Morse, and Seelinger in 2022 -- for a wide class of $k$-bounded partitions, including all strictly decreasing $k$-bounded partitions. Our results shed new light on the combinatorial structure of $K$-theoretic symmetric functions.