Invariant Algebras of Low-Rank Neutrino Matter Flows: Krylov-Plücker Invariants, the 3+1 Toric Ring, and Controlled Deformations
Abstract
Matter changes the eigenvalues and mixing matrix of the neutrino propagation Hamiltonian while entering the flavor-basis Hamiltonian through a low-dimensional additive deformation. We develop a unified algebraic description of the quantities that are insensitive to this deformation. The invariant algebra is formulated as the joint kernel of commuting matter-translation derivations inside the flavor-rephasing invariant ring. For diagonal matter potentials, this gives a general decomposition into unaffected diagonal directions and an off-diagonal cycle algebra, together with an exact dimension formula for an arbitrary number of flavors and independent matter parameters. For matter spurions with common low-rank support, block-Krylov rows generate exterior-power covariants whose maximal minors are matter independent. Their mass-basis form is a Plücker expansion: rank one produces the familiar Vandermonde factorization, whereas higher rank generically produces a sum of independent Plücker terms. The standard 3+1 system provides our principal nontrivial example. Its global off-diagonal algebra is the toric cycle ring of the complete four-vertex flavor graph, generated by edge moduli, triangle cycles, and quadrangle cycles; the commonly used eleven invariants constitute a generic local coordinate system rather than a global polynomial generating set. We further formulate independently varying matter compositions and off-diagonal nonstandard interactions as, respectively, commuting exact flows and controlled deformations of the invariant algebra. This framework separates the existence of exact invariants from the stronger and exceptional property of single-monomial spectral factorization.
Related
- Capacity Overflow, Effective Stochasticity, and Phase-B Invariants: Critical Deficit, Markov Closure, and Invariant Selection under Finite Projection
- Isospin-invariant Skyrme density functional investigation of quadrupole deformations in the isospin triplets of bfA=70, 74 and bf78
- Deciphering Matter Invariants via Renormalization Group Equations for Neutrino Oscillations
- Generating Functions for Casimir Invariants of Simple Lie Algebras
- Radiative Neutrino Mass in a Nonholomorphic T' Modular Invariant Model
BibTeX
@misc{lu2026invariant,
title = {{Invariant Algebras of Low-Rank Neutrino Matter Flows: Krylov-Plücker Invariants, the $3+1$ Toric Ring, and Controlled Deformations}},
author = {Jianlong Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13915},
eprint = {2609.13915},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13915},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Invariant Algebras of Low-Rank Neutrino Matter Flows: Krylov-Plücker Invariants, the 3+1 Toric Ring, and Controlled Deformations
AU - Jianlong Lu
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13915
UR - https://arxiv.org/abs/2609.13915
AB - Matter changes the eigenvalues and mixing matrix of the neutrino propagation Hamiltonian while entering the flavor-basis Hamiltonian through a low-dimensional additive deformation. We develop a unified algebraic description of the quantities that are insensitive to this deformation. The invariant algebra is formulated as the joint kernel of commuting matter-translation derivations inside the flavor-rephasing invariant ring. For diagonal matter potentials, this gives a general decomposition into unaffected diagonal directions and an off-diagonal cycle algebra, together with an exact dimension formula for an arbitrary number of flavors and independent matter parameters. For matter spurions with common low-rank support, block-Krylov rows generate exterior-power covariants whose maximal minors are matter independent. Their mass-basis form is a Plücker expansion: rank one produces the familiar Vandermonde factorization, whereas higher rank generically produces a sum of independent Plücker terms. The standard 3+1 system provides our principal nontrivial example. Its global off-diagonal algebra is the toric cycle ring of the complete four-vertex flavor graph, generated by edge moduli, triangle cycles, and quadrangle cycles; the commonly used eleven invariants constitute a generic local coordinate system rather than a global polynomial generating set. We further formulate independently varying matter compositions and off-diagonal nonstandard interactions as, respectively, commuting exact flows and controlled deformations of the invariant algebra. This framework separates the existence of exact invariants from the stronger and exceptional property of single-monomial spectral factorization.
ER -