Report
Electrical Networks, Lagrangian Grassmannians and Symplectic Groups
| العنوان: | Electrical Networks, Lagrangian Grassmannians and Symplectic Groups |
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| المؤلفون: | Bychkov, Boris, Gorbounov, Vassily, Kazakov, Anton, Talalaev, Dmitry |
| سنة النشر: | 2021 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, Mathematics - Representation Theory, 14M15, 82B20, 05E10 |
| الوصف: | We refine the result of T. Lam \cite{L} on embedding the space $E_n$ of electrical networks on a planar graph with $n$ boundary points into the totally non-negative Grassmannian $\mathrm{Gr}_{\geq 0}(n-1,2n)$ by proving first that the image lands in $\mathrm{Gr}(n-1,V)\subset \mathrm{Gr}(n-1,2n)$ where $V\subset \mathbb{R}^{2n}$ 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 $\mathrm{LG}(n-1,V)\subset \mathrm{Gr}(n-1,V)$. As it is well known $\mathrm{LG}(n-1)$ can be identified with $\mathrm{Gr}(n-1,2n-2)\cap \mathbb{P} L$ where $L\subset \bigwedge^{n-1}\mathbb R^{2n-2}$ is a subspace of dimension equal to the Catalan number $C_n$, 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 $E_n$ out of $\mathrm{Gr}(n-1,2n)$ found in \cite{L} define that space $L$. This connects the combinatorial description of $E_n$ discovered in \cite{L} and representation theory of the symplectic group. Comment: Journal version, minor corrections |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.17323/1609-4514-2023-23-2-133-167 |
| URL الوصول: | http://arxiv.org/abs/2109.13952 |
| رقم الانضمام: | edsarx.2109.13952 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.17323/1609-4514-2023-23-2-133-167 |
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