Realizability and nonrealizability of Dickson algebras as cohomology rings

Authors:
Larry Smith and R. M. Switzer

Journal:
Proc. Amer. Math. Soc. **89** (1983), 303-313

MSC:
Primary 55R40; Secondary 55R35

DOI:
https://doi.org/10.1090/S0002-9939-1983-0712642-4

MathSciNet review:
712642

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Abstract: Fix a prime and let be an -dimensional vector space over . The general linear group of acts on the polynomial ring on . The ring of invariants has been computed by Dickson, and we denote it by . If we grade by assigning the elements of the degree 2, then becomes a graded polynomial algebra on generators of degrees . The Steenrod algebra acts on in a unique way compatible with the unstability condition and the Cartan formula. The action commutes with the Steenrod algebra action, and so inherits the structure of an unstable polynomial algebra over the Steenrod algebra. In this note we determine explicit formulas for the action of the Steenrod algebra on the polynomial generators of . As a consequence we are able to decide exactly which Dickson algebras can be cohomology rings.

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DOI:
https://doi.org/10.1090/S0002-9939-1983-0712642-4

Article copyright:
© Copyright 1983
American Mathematical Society