### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 053, 14 pages      arXiv:1410.7593      https://doi.org/10.3842/SIGMA.2015.053

### Constructing Involutive Tableaux with Guillemin Normal Form

Abraham D. Smith
Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, Menomonie, WI 54751-2506, USA

Received December 15, 2014, in final form July 01, 2015; Published online July 09, 2015

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-Kähler 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, § 5 of the book Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, and Griffiths. This condition enhances Guillemin normal form and characterizes involutive tableaux.

Key words: involutivity; tableau; symbol; exterior differential systems.

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