
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 WisconsinStout, Menomonie, WI 547512506, 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 torsionfree exterior differential system or partial differential equation via the CartanKähler 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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