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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2011
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/1106.5068 |
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| _version_ | 1866908697836388352 |
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| author | Bremner, Murray R. Bickis, Mikelis G. Soltanifar, Mohsen |
| author_facet | Bremner, Murray R. Bickis, Mikelis G. Soltanifar, Mohsen |
| contents | Cayley's hyperdeterminant is a homogeneous polynomial of degree 4 in the 8 entries of a 2 x 2 x 2 array. It is the simplest (nonconstant) polynomial which is invariant under changes of basis in three directions. We use elementary facts about representations of the 3-dimensional simple Lie algebra sl_2(C) to reduce the problem of finding the invariant polynomials for a 2 x 2 x 2 array to a combinatorial problem on the enumeration of 2 x 2 x 2 arrays with non-negative integer entries. We then apply results from linear algebra to obtain a new proof that Cayley's hyperdeterminant generates all the invariants. In the last section we show how this approach can be applied to general multidimensional arrays. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1106_5068 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | Cayley's hyperdeterminant: a combinatorial approach via representation theory Bremner, Murray R. Bickis, Mikelis G. Soltanifar, Mohsen Representation Theory 15A72 Cayley's hyperdeterminant is a homogeneous polynomial of degree 4 in the 8 entries of a 2 x 2 x 2 array. It is the simplest (nonconstant) polynomial which is invariant under changes of basis in three directions. We use elementary facts about representations of the 3-dimensional simple Lie algebra sl_2(C) to reduce the problem of finding the invariant polynomials for a 2 x 2 x 2 array to a combinatorial problem on the enumeration of 2 x 2 x 2 arrays with non-negative integer entries. We then apply results from linear algebra to obtain a new proof that Cayley's hyperdeterminant generates all the invariants. In the last section we show how this approach can be applied to general multidimensional arrays. |
| title | Cayley's hyperdeterminant: a combinatorial approach via representation theory |
| topic | Representation Theory 15A72 |
| url | https://arxiv.org/abs/1106.5068 |