Alternating dual Pieri rule conjecture and $k$-branching conjecture of closed $k$-Schur Katalan functions

TL;DR

This paper proves two conjectures related to closed $k$-Schur Katalan functions, specifically the alternating dual Pieri rule and the $k$-branching conjecture, for large enough $k$ and strictly decreasing partitions.

Contribution

It provides the first positive proofs of the alternating dual Pieri rule for large $k$ and strictly decreasing partitions, and the $k$-branching conjecture for strictly decreasing partitions.

Findings

Proves the alternating dual Pieri rule for large $k$ and strictly decreasing partitions.

Establishes the $k$-branching conjecture for strictly decreasing partitions.

Advances understanding of $k$-Schur Katalan functions and related combinatorial conjectures.

Abstract

For closed $k$-Schur Katalan functions $\fg{\lambda}{k}$ with $k$ a positive integer and $\lambda$ a $k$-bounded partition, Blasiak, Morse and Seelinger proposed the alternating dual Pieri rule conjecture and the $k$-branching conjecture. In the present paper, we positively prove the first one for large enough $k$ and for strictly decreasing partitions $\lambda$ respectively, as well as the second one for strictly decreasing partitions $\lambda$.