Constructing Involutive Tableaux with Guillemin Normal Form

TL;DR

This paper provides a detailed analysis of Guillemin normal form, introducing an explicit quadratic condition that characterizes involutive tableaux, thereby advancing the classification of involutive exterior differential systems.

Contribution

It introduces a new quadratic condition that enhances Guillemin normal form and offers a systematic way to classify involutive systems.

Findings

Derived an explicit quadratic involutivity condition.

Enhanced understanding of Guillemin normal form.

Characterized involutive tableaux explicitly.

Abstract

Involutivity is the algebraic property that guarantees solutions to an analytic and torsion-free exterior differential system or partial differential equation via the Cartan-K\"ahler theorem. Guillemin normal form establishes that the prolonged symbol of an involutive system admits a commutativity property on certain subspaces of the prolonged tableau. This article examines Guillemin normal form in detail, aiming at a more systematic approach to classifying involutive systems. The main result is an explicit quadratic condition for involutivity of the type suggested but not completed in Chapter IV, \S 5 of the book Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, and Griffiths. This condition enhances Guillemin normal form and characterizes involutive tableaux.