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Electrical Networks, Lagrangian Grassmannians, and Symplectic Groupsстатья
- Авторы: Талалаев Д.В., Казаков Антон Александрович, Бычков Борис Сергеевич, Горбунов Василий Геннадьевич
- Журнал: Moscow Mathematical Journal
- Том: 23
- Номер: 2
- Год издания: 2023
- Первая страница: 133
- Последняя страница: 167
- Аннотация: We refine the result of T. Lam \cite{L} on embedding the space En of electrical networks on a planar graph with n boundary points into the totally non-negative Grassmannian Gr≥0(n−1,2n) by proving first that the image lands in Gr(n−1,V)⊂Gr(n−1,2n) where V⊂R2n is a certain subspace of dimension 2n−2. The role of this reduction in the dimension of the ambient space is crucial for us. We show next that the image lands in fact inside the Lagrangian Grassmannian LG(n−1,V)⊂Gr(n−1,V). As it is well known LG(n−1) can be identified with Gr(n−1,2n−2)∩PL where L⊂⋀n−1R2n−2 is a subspace of dimension equal to the Catalan number Cn, moreover it is the space of the fundamental representation of the symplectic group Sp(2n−2) which corresponds to the last vertex of the Dynkin diagram. We show further that the linear relations cutting the image of En out of Gr(n−1,2n) found in \cite{L} define that space L. This connects the combinatorial description of En discovered in \cite{L} and representation theory of the symplectic group.
- Добавил в систему: Талалаев Дмитрий Валерьевич