Quantum Frobenius Canonical Basis

---
name: quantum-frobenius-canonical-basis
description: "Quantum Frobenius morphism compatibility analysis with canonical bases. Covers imaginary vectors, tight monomial cones, and canonical basis incompatibility for finite-dimensional simple Lie algebras. Triggers: quantum Frobenius, canonical basis, imaginary vectors, tight monomial cone, Lie algebra, Leclerc vectors, quantum group, representation theory."
category: quantum-math
---

# Quantum Frobenius Morphism and Canonical Basis Compatibility

## Core Result

The quantum Frobenius morphism and its splitting are NOT fully compatible with the canonical basis for any finite-dimensional simple Lie algebra when rank is sufficiently large.

## Key Connections

1. **Incompatibility location**: Same place where Leclerc found his imaginary vectors
2. **Monomial cone issue**: Monomials in the tight monomial cone which do NOT belong to the canonical basis
3. **Rank dependence**: Incompatibility emerges at sufficiently high rank

## Applications

- Analyzing canonical basis stability under quantum Frobenius morphism
- Understanding tight monomial cone structure
- Identifying imaginary vectors in quantum group representations