Unique rectification in $d$-complete posets: towards the $K$-theory of Kac-Moody flag varieties

TL;DR

This paper establishes the existence of unique rectification targets in a broad class of $d$-complete posets, advancing the combinatorial understanding of $K$-theoretic Schubert calculus in Kac-Moody flag varieties.

Contribution

It unifies previous theories by proving unique rectification targets exist in many $d$-complete posets, leading to conjectural formulas for $K$-theoretic structure constants.

Findings

Existence of unique rectification targets in large classes of $d$-complete posets.

Development of conjectural positive formulas for $K$-theoretic Schubert structure constants.

Progress towards unifying combinatorial models in $K$-theory and Kac-Moody geometry.

Abstract

The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of "unique rectification targets" in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to $K$-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's "$d$-complete posets" to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of $d$-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain $K$-theoretic Schubert structure constants in the Kac-Moody setting.