11,379 research outputs found
Constructing Involutive Tableaux with Guillemin Normal Form
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.Comment: This article co-evolved with "Degeneracy of the Characteristic
Variety," arXiv:1410.6947 and most notation is shared. However, be aware that
the meaning of the indices i,j,k,l and the space Y is not the same between
these article
Extending the Coinvariant Theorems of Chevalley, Shephard--Todd, Mitchell and Springer
We extend in several directions invariant theory results of Chevalley,
Shephard and Todd, Mitchell and Springer. Their results compare the group
algebra for a finite reflection group with its coinvariant algebra, and compare
a group representation with its module of relative coinvariants. Our extensions
apply to arbitrary finite groups in any characteristic.Comment: The applications and Examples in section 4 have been extende
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