Strange Expectations

TL;DR

This paper extends known formulas for the maximum, expected value, and variance of boxes in simultaneous (a,b)-cores, generalizes these to affine Weyl groups, and explores the underlying algebraic structures and formulas.

Contribution

It introduces a unified approach to compute maximum, expectation, and variance for simultaneous cores and generalizes these results to affine Weyl groups.

Findings

Maximum number of boxes in a simultaneous (a,b)-core is (a^2-1)(b^2-1)/24.

Expected number of boxes in a simultaneous (a,b)-core is (a-1)(b-1)(a+b+1)/24.

Variance of boxes in a simultaneous (a,b)-core is ab(a-1)(b-1)(a+b)(a+b+1)/1440.

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.