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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.18071 |
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| _version_ | 1866915755475337216 |
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| author | Knill, Oliver |
| author_facet | Knill, Oliver |
| contents | The connection Laplacian L and the Dirac matrix D are both n x n matrices defined from a given finite simplicial complex G with n sets. In both cases, there is interlacing of the eigenvalues for subcomplexes. This gives general upper bounds of the eigenvalues both for L and D in terms of inclusion or intersection degrees. We conjecture that L always dominates both D and the inverse of L in a weak Loewner sense. In a second part we look at dynamical systems (G,T), where T is a simplicial map on G. Both L and D generalize to dynamical versions of L and D. The modified L is still unimodular with an explicit Green function inverse and modified Dirac part still comes from an exterior derivative d. We also review the Lefschetz fixed point theorem for a simplicial map T on a simplicial complex G which implies the Brouwer fixed point theorem: any simplicial map on a contractible finite abstract simplicial complex G has a fixed simplex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18071 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Remarks about Connection and Dirac matrices Knill, Oliver Combinatorics Discrete Mathematics 05C50 15Bxx The connection Laplacian L and the Dirac matrix D are both n x n matrices defined from a given finite simplicial complex G with n sets. In both cases, there is interlacing of the eigenvalues for subcomplexes. This gives general upper bounds of the eigenvalues both for L and D in terms of inclusion or intersection degrees. We conjecture that L always dominates both D and the inverse of L in a weak Loewner sense. In a second part we look at dynamical systems (G,T), where T is a simplicial map on G. Both L and D generalize to dynamical versions of L and D. The modified L is still unimodular with an explicit Green function inverse and modified Dirac part still comes from an exterior derivative d. We also review the Lefschetz fixed point theorem for a simplicial map T on a simplicial complex G which implies the Brouwer fixed point theorem: any simplicial map on a contractible finite abstract simplicial complex G has a fixed simplex. |
| title | Remarks about Connection and Dirac matrices |
| topic | Combinatorics Discrete Mathematics 05C50 15Bxx |
| url | https://arxiv.org/abs/2601.18071 |