The quantum k-Bruhat order

TL;DR

The paper advances the study of the quantum $k$-Bruhat order by modeling maximal chains as sequences of transpositions encoded in a free quantum monoid $\mathcal{F}_n^{\mathbf{q}}$ acting on the $q$-extended symmetric group $S_n[\mathbf{q}]$. It develops a robust framework of equivalence-preserving transformations (Flattening, Cyclic Shift, Horizontal Flip, Vertical Flip) to compare operator compositions and then derives a comprehensive catalog of low-degree equivalences (degrees 2–4) including both zero and non-zero cases, plus two infinite families for arbitrary degree. The results culminate in a conjectured complete set of defining relations for $\mathcal{F}_n^{\mathbf{q}}$, which would yield a quantum analogue of the classical monoid $\mathcal{M}_n$ and shed light on the multiplicative structure of quantum Schubert polynomials via the quantum Monk's rule. If proven, these equivalences would provide a combinatorial handle on quantum structure constants and open new pathways to understanding quantum cohomology in a Schubert-theoretic context.

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.

Figures (15)

• Figure 1: (a) 2-Bruhat order on $S_3$ and (b) Grassmannian Bruhat order on $S_3$
• Figure 2: Isomoprhic intervals in (a) $\le_2$, (b) $\le_3$, and (c) $\preceq$
• Figure 3: Quantum 2-Bruhat order on $S_3[\mathbf{q}]$
• Figure 4: Diagrammatic presentation of $\mathbf{v}_{14}\mathbf{v}_{62}\mathbf{v}_{16}\in\mathcal{F}^{\mathbf{q}}_6$
• Figure 5: Operator

Theorems & Definitions (94)

• Theorem 1.1: Monk's rule Macdonald
• Theorem 1.2: Quantum Monk's rule Quantum
• Proposition 3.1: BS1
• Theorem 3.2: BS1
• Definition 3.3
• Theorem 3.4: BS2
• Theorem 3.5
• proof
• Remark 4.1
• Proposition 4.2: qkB1