Abstract

The complex representation theory of the symmetric group $\mathfrak{S}_n$ is described by the partitions of $n$, in particular, to each partition $\lambda$ one can asosciate an irreducible complex representation. Over a field of characteristic $p$, the irreducible representations can be indexed by the partitions of $p$ that are $p$-regular, that is, for which no part repeats $p$ times or more. The regular representation of $\mathfrak{S}_n$ gives a natural probability measure on the set of partitions of $n$, the Plancherel measure. A spectacular result of Kerov–Vershik and Logan-Shepp (1977) gives an asymptotic limit shape for large partitions taken under the Plancherel measure. In this talk, we will see what does this result become for large $p$-regular partitions. Namely, there is still a limit shape, which is given by the shaking of the Kerov-Vershik-Logan-Shepp curve.