Double-partition Quantum Cluster Algebras

TL;DR

This paper introduces a new family of quantum cluster algebras indexed by double partitions, exploring their structure, mutations, and basis elements, expanding the theoretical framework of quantum algebraic structures.

Contribution

It defines and studies a novel class of quantum cluster algebras linked to double partitions and broken lines, with detailed mutation and basis properties.

Findings

Algebras are equal to their upper cluster algebras.

Variables are elements of the dual canonical basis.

Mutations can be grouped into quantum line mutations.

Abstract

A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but subclasses have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties. Equivalently, they are indexed by broken lines $L$. By grouping together neighboring mutations into quantum line mutations we can mutate from the cluster algebra of one broken line to another. Compatible pairs can be written down. The algebras are equal to their upper cluster algebras. The variables of the quantum seeds are given by elements of the dual canonical basis. This is the final version, where some arguments have been expanded and/or improved and several typos corrected. Full bibliographic details: Journal of Algebra (2012), pp. 172-203 DOI information: 10.1016/j.jalgebra.2012.09.015