Vol. 6, No. 4, 2012

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Realizing large gaps in cohomology for symmetric group modules

David J. Hemmer

Vol. 6 (2012), No. 4, 825–832
Abstract

Using results of the author with Cohen and Nakano, we find examples of Young modules ${Y}^{\lambda }$ for the symmetric group ${\Sigma }_{d}$ for which the Tate cohomology ${\stackrel{̂}{H}}^{i}\left({\Sigma }_{d},{Y}^{\lambda }\right)$ does not vanish identically, but vanishes for approximately $\frac{1}{3}{d}^{3∕2}$ consecutive degrees. We conjecture these vanishing ranges are maximal among all ${\Sigma }_{d}$-modules with nonvanishing cohomology. The best known upper bound on such vanishing ranges stands at ${\left(d-1\right)}^{2}$, due to work of Benson, Carlson and Robinson. Particularly striking, and perhaps counterintuitive, is that these Young modules have maximum possible complexity.

Keywords
symmetric group, cohomology, Young module
Primary: 20C30
Milestones
Accepted: 13 August 2011
Published: 25 July 2012
Authors
 David J. Hemmer Department of Mathematics University at Buffalo, SUNY College of Arts and Sciences The State University of New York Buffalo, NY 14260-2900 United States http://math.buffalo.edu/~dhemmer/