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Modules for the cyclic group of order p
The indecomposable modules for $Z/p$ over a field of characteristic p are just the Jordan blocks $J_n$ with eigenvalue $1$ and length $1 le n le p$. Tensor products of these modules are easy to describe. Back in the 1970s, Almkvist and Fossum studied exterior and symmetric powers of $J_n$ and found the surprising formula $Lambda^d(J_{n+d}) cong S^d(J_{n+1})$ for $n+dle p$. They did not succeed in evaluating other Schur functors on the modules $J_n$, and Fossum stated this as an open problem. In this talk I describe how to calculate the effect of a Schur functor for any partition of $d<p$ on $J_n$. The recipe generalises the formulas of Almkvist and Fossum, and involves a generalisation of the Gaussian polynomials and a hook formula. Exactly the same recipe describes the effect of any Schur functor on any irreducible representation of SL(2,C) in characteristic zero.
- Speaker
- Dave Benson (University of Aberdeen)