Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A birational lifting of the Stanley-Thomas word on products of two
  chains   The dynamics of certain combinatorial actions and their liftings to actions
at the piecewise-linear and birational level have been studied lately with an
eye towards questions of periodicity, orbit structure, and invariants. One key
property enjoyed by the rowmotion operator on certain finite partially-ordered
sets is homomesy, where the average value of a statistic is the same for all
orbits. To prove refined versions of homomesy in the product of two chain
posets, 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 the
piecewise-linear, birational, and noncommutative realms. Although the map is no
longer a bijection, so cannot be used to prove periodicity directly, it still
gives enough information to prove the homomesy at the piecewise-linear and
birational levels (a result previously shown by D. Grinberg, S. Hopkins, and S.
Okada). Even at the noncommutative level, the Stanley--Thomas word of a poset
labeling rotates cyclically with the lifting of antichain rowmotion. Along the
way we give some formulas for noncommutative antichain rowmotion that we hope
will be first steps towards proving the conjectured periodicity at this level.

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The categories of this paper are: gr-qc
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: q-alg
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ph
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The categories of this paper are: mtrl-th