A generalisation of bar-core partitions

Yates, Dean

Algebraic Combinatorics, Tome 5 (2022) no. 4, pp. 667-698

When $p$ and $q$ are coprime odd integers no less than 3, Olsson proved that the $q$-bar-core of a $p$-bar-core is again a $p$-bar-core. We establish a generalisation of this theorem: that the $p$-bar-weight of the $q$-bar-core of a bar partition $\lambda$ is at most the $p$-bar-weight of $\lambda$. We go on to study the set of bar partitions for which equality holds and show that it is a union of orbits for an action of a Coxeter group of type ${\tilde{C}}_{(p-1)/2}\times {\tilde{C}}_{(q-1)/2}$. We also provide an algorithm for constructing a bar partition in this set with a given $p$-bar-core and $q$-bar-core.

Classification : 05E10, 05A17, 20C30
Keywords: representation theory, symmetric group, partitions, projective, bar-core

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