  We define two refinements of the skew length statistic on simultaneous core
partitions. The first one relies on hook lengths and is used to prove a refined
version of the theorem stating that the skew length is invariant under
conjugation of the core. The second one is equivalent to a generalisation of
Shi tableaux to the rational level of Catalan combinatorics. These rational Shi
tableaux encode dominant $p$-stable elements in the affine symmetric group. We
prove that the rational Shi tableau is injective, that is, each dominant
$p$-stable affine permutation is determined uniquely by its Shi tableau.
Moreover, we provide a uniform generalisation of rational Shi tableaux to Weyl
groups, and conjecture injectivity in the general case.
