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 |
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Springer
2024
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| _version_ | 1824458624402980864 |
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| author | Muller, T Nanda, V Seigal, A |
| author_facet | Muller, T Nanda, V Seigal, A |
| author_sort | Muller, T |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2025-02-19T04:28:51Z |
| format | Journal article |
| id | oxford-uuid:b8d0c36d-5b73-42b7-a848-2263ce200cc9 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2025-02-19T04:28:51Z |
| publishDate | 2024 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:b8d0c36d-5b73-42b7-a848-2263ce200cc92024-12-09T12:03:50ZMultilinear hyperquiver representationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b8d0c36d-5b73-42b7-a848-2263ce200cc9EnglishSymplectic ElementsSpringer2024Muller, TNanda, VSeigal, AWe 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. |
| spellingShingle | Muller, T Nanda, V Seigal, A Multilinear hyperquiver representations |
| title | Multilinear hyperquiver representations |
| title_full | Multilinear hyperquiver representations |
| title_fullStr | Multilinear hyperquiver representations |
| title_full_unstemmed | Multilinear hyperquiver representations |
| title_short | Multilinear hyperquiver representations |
| title_sort | multilinear hyperquiver representations |
| work_keys_str_mv | AT mullert multilinearhyperquiverrepresentations AT nandav multilinearhyperquiverrepresentations AT seigala multilinearhyperquiverrepresentations |