The shape of a random affine Weyl group element, and random core partitions

Details The shape of a random affine Weyl group element, and random core partitions The shape of a random affine Weyl group element, and random core partitions

2011-02-22

arxiv.org/abs/1102.4405v1

Mathematics

Probability

12 pages

Scientific paper

Let W be a finite Weyl group and \tW be the corresponding affine Weyl group. We show that a large element in \tW, randomly generated by (reduced) multiplication by simple generators, almost surely has one of |W|-specific shapes. Equivalently, a reduced random walk in the regions of the affine Coxeter arrangement asymptotically approaches one of |W|-many directions. The coordinates of this direction, together with the probabilities of each direction can be calculated via a Markov chain on W. Our results, applied to type A_{n-1}, show that a large random n-core has a limiting shape which is a piecewise-linear graph, in a similar sense to Vershik and Kerov's work on the shape of a random partition.

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