Double Quantum Schubert Cells and Quantum Mutations

TL;DR

This paper constructs explicit quantum seeds related to Schubert cells in quantum flag manifolds, introduces mutation operations, and demonstrates their applications in prime ideals, cluster algebras, and minors.

Contribution

It provides a new explicit construction of quantum seeds and mutation operations for Schubert cells, enabling simplified proofs and deeper understanding of quantum flag varieties.

Findings

Quantum seeds are explicitly constructed for pairs of minimal coset representatives.

Mutations relate different quantum seeds, forming a mutation graph.

Applications include results on prime ideals, cluster algebras, and minors.

Abstract

Let ${\mathfrak p}\subset {\mathfrak g}$ be a parabolic subalgebra of s simple finite dimensional Lie algebra over ${\mathbb C}$. To each pair $w^{\mathfrak a}\leq w^{\mathfrak c}$ of minimal left coset representatives in the quotient space $W_p\backslash W$ we construct explicitly a quantum seed ${\mathcal Q}_q({\mathfrak a},{\mathfrak c})$. We define Schubert creation and annihilation mutations and show that our seeds are related by such mutations. We also introduce more elaborate seeds to accommodate our mutations. The quantized Schubert Cell decomposition of the quantized generalized flag manifold can be viewed as the result of such mutations having their origins in the pair $({\mathfrak a},{\mathfrak c})= ({\mathfrak e},{\mathfrak p})$, where the empty string ${\mathfrak e}$ corresponds to the neutral element. This makes it possible to give simple proofs by induction. We exemplify this in three directions: Prime ideals, upper cluster algebras, and the diagonal of a quantized minor.