The quantum k-Bruhat order

TL;DR

This paper explores the structure of the quantum k-Bruhat order, identifying chain representations and operator equivalences, to better understand quantum Schubert polynomials and their multiplicative properties.

Contribution

It introduces a free monoid action on a q-extension of the symmetric group, identifying operator equivalences and conjecturing their completeness, advancing the understanding of quantum Bruhat structures.

Findings

Identification of chain structures via transpositions

Discovery of a large family of operator equivalences

Implications for quantum Schubert polynomial multiplication

Abstract

In this paper, we extend the study of the quantum $k$-Bruhat order initiated in the work of Benedetti, Bergeron, Colmenarejo, Saliola, and Sottile concerning the quantum Murnaghan-Nakayama rule. Specifically, identifying maximal chains in intervals of the quantum $k$-Bruhat order with sequences of transpositions, we investigate a naturally associated free monoid $F_n^{\mathbf{q}}$ with an action on a $q$-extension of $S_n$, denoted $S_n[\mathbf{q}]$, which encodes the chain structure of the quantum $k$-Bruhat order. Aside from numerous structural results, our main contribution is an identification of a large family of equivalences satisfied by the elements of $F_n^{\mathbf{q}}$ as operators on $S_n[\mathbf{q}]$. In fact, we conjecture that our list of equivalences is complete. As a consequence of the quantum Monk's rule, a complete understanding of such equivalences can be used to gain information about the multiplicative structure of quantum Schubert polynomials.