336 research outputs found
Following Schubert varieties under Feigin's degeneration of the flag variety
We describe the effect of Feigin's flat degeneration of the type
flag variety on its Schubert varieties. In particular, we study when they stay
irreducible and in several cases we are able to encode reducibility of the
degenerations in terms of symmetric group combinatorics. As a side result, we
obtain an identification of some Schubert varieties with Richardson varieties
in higher rank partial flag varieties.Comment: Comments are welcome! 22 pages + Appendix (5 pages), 2 figure
Tropical totally positive cluster varieties
We study the relation between the integer tropical points of a cluster
variety (satisfying the full Fock-Goncharov conjecture) and the totally
positive part of the tropicalization of an ideal presenting the corresponding
cluster algebra. Suppose we are given a presentation of the cluster algebra by
a Khovanskii basis for a collection of -vector valuations associated
with several seeds related by mutations. In presence of a full rank fully
extended exchange matrix we construct the rays of a subfan of the totally
positive part of the tropicalization of the ideal that coincides
combinatorially with the subgraph of the exchange graph of the cluster algebra
corresponding to the collection of seeds. Moreover, geometric information about
Gross-Hacking-Keel-Kontsevich's toric degenerations associated with seeds gets
identified with the Gr\"obner toric degenerations obtained from maximal cones
in the tropicalization. As application we prove a conjecture about the relation
between Rietsch-Williams' valuations for Grassmannians arising from plabic
graphs \cite{RW17} to Kaveh-Manon's work on valuations from the tropicalization
of an ideal \cite{KM16}. In a second application we give a partial answer to
the question if the Feigin-Fourier-Littelmann-Vinberg degeneration of the full
flag variety in type is isomorphic to a degeneration obtained from
the cluster structure.Comment: Comments are very welcom
The degree of the Hilbert-Poincar\'e polynomial of PBW-graded modules
In this note, we study the Hilbert-Poincar\'e polynomials for the PBW-graded
of simple modules for a simple complex Lie algebra. The computation of their
degree can be reduced to modules of fundamental highest weight. We provide
these degrees explicitly.Comment: 7 pages, updated references, improved exposition, journal versio
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