Entropy of affine permutations and universality of affine atomic lengths

Abstract

We introduce and study the notion of entropy of affine permutations and prove that it coincides with the atomic length associated with the sum of the fundamental weights for a type $A$ affine root system, as defined by the first two authors. We then establish an analogue of the Granville-Ono theorem by showing that any nonnegative integer can be realised as the entropy of an affine permutation or alternatively, as the size of a core multipartition as introduced by the last two authors. Our proof uses an additive combinatorics theorem due to Hall on difference sets of permutations modulo $n$. More generally, we give a polynomial expression of the atomic length associated with any dominant weight in affine type $A$ and investigate the problem of its universality. Beyond type $A$, we are able to prove that the entropy of affine type $C_n$ permutations is universal when $2n+1$ is prime. This is achieved by establishing an analogue of Hall's theorem for the hyperoctahedral group based on Alon's combinatorial Nullstellensatz. We also propose conjectures generalising the results presented in the paper, each supported by computational evidence. Finally, we show that in any affine classical type, the problem of the universality of the atomic length simplifies in large rank when the weight considered is conveniently adjusted.