Chain-order polytopes: toric degenerations, Young tableaux and monomial bases (Q6634446)

The author studies toric varieties associated with marked chain-order polytopes for the Gelfand-Tsetlin poset and shows that they can be realized as an explicit Gröbner degeneration of the flag variety.\N\NThis construction generalizes the earlier work of \textit{B. Sturmfels} [Algorithms in invariant theory. Wien: Springer-Verlag (1993; Zbl 0802.13002)], and \textit{N. Gonciulea} and \textit{V. Lakshmibai} [Transform. Groups 1, No. 3, 215--248 (1996; Zbl 0909.14028)], and \textit{M. Kogan} and \textit{E. Miller} [Adv. Math. 193, No. 1, 1--17 (2005; Zbl 1084.14049)] on the Gelfand-Tsetlin degeneration to the more general setting of marked chain-order polytopes.\N\NThe key idea in the author's approach is the use of pipe dreams to define realizations of toric varieties in Plücker coordinates. Building on this, the author extends two well-known constructions to arbitrary marked chain-order polytopes: standard monomial theories (such as those arising from semistandard Young tableaux) and PBW-monomial bases in irreducible representations, including the FFLV bases. Additionally, the author introduces the notion of semi-infinite pipe dreams, which enables the construction of an infinite family of poset polytopes, each providing a toric degeneration of the semi-infinite Grassmannian.