Multilinear hyperquiver representations
We count singular vector tuples of a system of tensors assigned to the edges of a directed hypergraph. To do so, we study the generalisation of quivers to directed hypergraphs. Assigning vector spaces to the nodes of a hypergraph and multilinear maps to its hyperedges gives a hyperquiver representat...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
| Published: |
Springer
2024
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| Summary: | We count singular vector tuples of a system of tensors assigned to the edges
of a directed hypergraph. To do so, we study the generalisation of quivers to directed
hypergraphs. Assigning vector spaces to the nodes of a hypergraph and multilinear maps to
its hyperedges gives a hyperquiver representation. Hyperquiver representations generalise
quiver representations (where all hyperedges are edges) and tensors (where there is only one
multilinear map). The singular vectors of a hyperquiver representation are a compatible
assignment of vectors to the nodes. We compute the dimension and degree of the variety of
singular vectors of a sufficiently generic hyperquiver representation. Our formula specialises
to known results that count the singular vectors and eigenvectors of a generic tensor. Lastly,
we study a hypergraph generalisation of the inverse tensor eigenvalue problem and solve it
algorithmically. |
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