Realizing large gaps in cohomology for symmetric group modules

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 ${\hat{H}}^i\left({\Sigma }_d,Y^{\lambda }\right)$ 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.