Strange Expectations
Abstract
Let gcd(a,b)=1. J. Olsson and D. Stanton proved that the maximum number of boxes in a simultaneous (a,b)-core is (a^2-1)(b^2-1)/24, and that this maximum was achieved by a unique core. P. Johnson combined Ehrhart theory with the polynomial method to prove D. Armstrong's conjecture that the expected number of boxes in a simultaneous (a,b)-core is (a-1)(b-1)(a+b+1)/24. We extend P. Johnson's method to compute the variance to be ab(a-1)(b-1)(a+b)(a+b+1)/1440. By extending the definitions of "simultaneous cores" and "number of boxes" to affine Weyl groups, we give uniform generalizations of all three formulae above to simply-laced affine types. We further explain the appearance of the number 24 using the "strange formula" of H. Freudenthal and H. de Vries.
Cite
@article{arxiv.1508.05293,
title = {Strange Expectations},
author = {Marko Thiel and Nathan Williams},
journal= {arXiv preprint arXiv:1508.05293},
year = {2015}
}
Comments
31 pages, 3 figures