Hessenberg patch ideals, geometric vertex decomposition, and Grobner bases

Abstract: Hessenberg varieties are subvarieties of the flag variety $Flags(\mathbb{C}^n)$, the study of which have rich interactions with symplectic geometry, representation theory, and equivariant topology, among other research areas, with particular recent attention arising from its connections to the famous Stanley-Stembridge conjecture in combinatorics. The special case of regular nilpotent Hessenberg varieties has been much studied, and in this talk I will describe some work in progress analyzing the local defining ideals of these varieties. In particular, using some techniques relating liason theory, geometric vertex decomposition, and the theory of Grobner bases (following work of Klein and Rajchgot), we are able to show that, for the coordinate patch corresponding to the longest word $w_0$, the local defining ideal for any indecomposable Hessenberg variety is geometrically vertex decomposable, and we find an explicit Grobner basis for a certain monomial order. This is a report on joint work in progress with Sergio Da Silva.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

Geometry, Physics, and Representation Theory Seminar

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