Stanley's classification conjecture for indecomposable upho modular lattices

Let PP be an upho modular lattice, meaning that PP is of finite type and its principal order filter at every element is isomorphic to PP. Suppose that PP is indecomposable and contains a complemented interval of rank 33. Stanley's classification conjecture. Then PP is isomorphic to the poset of submodules of RdR^d of finite colength, ordered by reverse inclusion, for some local principal ideal domain RR with a finite residue field. This conjecture proposes that, apart from decomposable cases, these module posets account for upho modular lattices containing a complemented rank-33 interval; the paper records the conjecture but does not discuss its proof or status.

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Primary source

Yibo Gao, Joshua Guo, Karthik Seetharaman and Ilaria Seidel, “The Rank-Generating Functions of Upho Posets”, arXiv:2011.01916 (2020).

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