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Интеллектуальная Система Тематического Исследования НАукометрических данных |
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Electrical networks, Lagrangian Grassmannians and symplectic groupsстатья
- Авторы: Bychkov Boris, Gorbounov Vassily, Kazakov Anton, Talalaev Dmitry
- Журнал: ArXiv e-prints
- Год издания: 2021
- Аннотация: We refine the result of T. Lam on embedding the space En of electrical networks on a planar graph with n boundary points into the totally nonnegative Grassmannian Gr≥0(n−1,2n) by proving first that the image lands in Gr(n−1,V)⊂Gr(n−1,V) 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 L is the space of the fundamental representation of the 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 by T.Lam define that space L. This connects the combinatorial description of En discovered by T.Lam and representation theory of the symplectic group.
- Добавил в систему: Казаков Антон Александрович