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

TL;DR

The paper addresses two conjectures on closed $k$-Schur Katalan functions $\tilde{\mathfrak{g}}_{\lambda}^{(k)}$: an alternating dual Pieri rule and a $k$-branching property. It develops a new Mirror Lemma II and a detailed lowering-operator calculus to express $L_z\tilde{\mathfrak{g}}_{\lambda}^{(k)}$ as a linear combination of lower-degree functions, enabling precise signs and positivity analysis. The authors prove the alternating dual Pieri rule for sufficiently large $k$ and for strictly decreasing partitions, and prove the $k$-branching conjecture for strictly decreasing partitions, by establishing shift-invariance and controlled expansions with $G_{1^m}^{\perp}$. Together, these results provide a robust positivity framework and deepen the link between $k$-Schur $K$-theoretic objects and branching phenomena in the affine/quantum Schubert setting, with potential influence on quantum $K$-theory and affine Grassmannian combinatorics. The approach combines root-ideal Katalan calculus, Mirror Lemmas, and a structured lowering-operator analysis to achieve new structural insights for $\Lambda_{(k)}$ and its embedding into $\Lambda_{(k+1)}$.

Abstract

For closed $k$-Schur Katalan functions $\fgλ{k}$ with $k$ a positive integer and $λ$ 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 $λ$ respectively, as well as the second one for strictly decreasing partitions $λ$.

Theorems & Definitions (35)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Remark 2.2
• Example 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6