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 λ for the symmetric group Σd for which the Tate cohomology Ĥi(Σd,Y λ) does not vanish identically, but vanishes for approximately 1 3d32 consecutive degrees. We conjecture these vanishing ranges are maximal among all Σd-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/