Quantum Chevalley Formula

SageMath implementation of the small quantum cohomology algebra of a partial flag variety G/P, based on the Quantum Chevalley Formula of Fulton–Woodward (2004).

Requirements

SageMath

Usage

sage: load("quantum-chevalley.sage")
sage: QH = QuantumCohomologyPartialFlagVariety("A3", (1,))

The constructor takes a Dynkin type and a tuple of simple root indices specifying the parabolic subgroup P. To get the maximal parabolic P_k, pass a singleton (k,).

Quantum multiplication by a divisor:

sage: QH.quantum_chevalley(1, QH.module.monomial(QH.schubert_basis[3]))
q1*B[1]

Matrix of quantum multiplication:

sage: QH.quantum_chevalley_matrix(1)
[ 0  1  0  0]
[ 0  0  1  0]
[ 0  0  0  1]
[q1  0  0  0]

Quantum spectrum (eigenvalues of multiplication by the anticanonical class):

sage: QH.quantum_spectrum()
[4, -4, -4*I, 4*I]

Reference

W. Fulton, C. Woodward, On the quantum product of Schubert classes, J. Algebraic Geom. 13 (2004), 641–661.

Authors

• Pieter Belmans
• Maxim Smirnov