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The dynamics of certain combinatorial actions and their liftings to actionsat the piecewise-linear and birational level have been studied lately with aneye towards questions of periodicity, orbit structure, and invariants. One keyproperty enjoyed by the rowmotion operator on certain finite partially-orderedsets is homomesy, where the average value of a statistic is the same for allorbits. To prove refined versions of homomesy in the product of two chainposets, J. Propp and the second author used an equivariant bijection discovered(less formally) by R. Stanley and H. Thomas.  We explore the lifting of this "Stanley--Thomas word" to thepiecewise-linear, birational, and noncommutative realms. Although the map is nolonger a bijection, so cannot be used to prove periodicity directly, it stillgives enough information to prove the homomesy at the piecewise-linear andbirational levels (a result previously shown by D. Grinberg, S. Hopkins, and S.Okada). Even at the noncommutative level, the Stanley--Thomas word of a posetlabeling rotates cyclically with the lifting of antichain rowmotion. Along theway we give some formulas for noncommutative antichain rowmotion that we hopewill be first steps towards proving the conjectured periodicity at this level.

A birational lifting of the Stanley-Thomas word on products of two chains