Proofs and generalizations of a homomesy conjecture of Propp and Roby

Let GG be a group acting on a set XX of combinatorial objects, with finite orbits, and consider a statistic ξ:X→Cξ:X→C. Propp and Roby defined the triple (X,G,ξ)(X,G,ξ) to be homomesic   if for any orbits O1,O2O1,O2, the average value of the statistic ξξ is the same, that is 1|O1|∑x∈O1ξ(x)=1|O2|∑y∈O2ξ(y). In 2013 Propp and Roby conjectured the following instance of homomesy. Let SSY Tk(m×n) denote the set of semistandard Young tableaux of shape m×nm×n with entries bounded by kk. Let SS be any set of boxes in the m×nm×n rectangle fixed under 180° rotation. For T∈SSY Tk(m×n), define σS(T)σS(T) to be the sum of the entries of TT in the boxes of SS. Let 〈P〉〈P〉 be a cyclic group of order kk where PP acts on SSY Tk(m×n) by promotion. Then (SSY Tk(m×n),〈P〉,σS) is homomesic.We prove this conjecture, as well as a generalization to cominuscule posets. We also discuss analogous questions for tableaux with strictly increasing rows and columns under the K-promotion of Thomas and Yong, and prove limited results in that direction.