Realizing large gaps in cohomology for symmetric group modules

Hemmer, David

doi:10.2140/ant.2012.6.825

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Realizing large gaps in cohomology for symmetric group modules

David J. Hemmer

Vol. 6 (2012), No. 4, 825–832

DOI: 10.2140/ant.2012.6.825

Abstract

Using results of the author with Cohen and Nakano, we find examples of Young modules $Y^{λ}$ for the symmetric group $Σ_{d}$ for which the Tate cohomology ${\hat{H}}^{i} (Σ_{d}, Y^{λ})$ 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 $Σ_{d}$ -modules with nonvanishing cohomology. The best known upper bound on such vanishing ranges stands at ${(d - 1)}^{2}$ , 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

Mathematical Subject Classification 2010

Primary: 20C30

Milestones

Received: 20 June 2011

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/

Vol. 6, No. 4, 2012