Webs and multiwebs for the symplectic group
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915006061215744 |
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| author | Kenyon, Richard Wu, Haihan |
| author_facet | Kenyon, Richard Wu, Haihan |
| contents | We define $2n$-multiwebs on planar graphs and discuss their relation with $\mathrm{Sp}(2n)$-webs. On a planar graph with a symplectic local system we define a matrix whose Pfaffian is the sum of traces of $2n$-multiwebs. As application we generalize Kasteleyn's theorem from dimer covers to $2n$-multiweb covers of planar graphs with $U(n)$ gauge group.
For $\mathrm{Sp}(4)$ we relate Kuperberg's ``tetravalent vertex'' to the determinant, and classify reduced $4$-webs on some simple surfaces: the annulus, torus, and pair of pants.
We likewise define, for $\mathrm{Sp}(2n)$ and $q=1$, a $2n$-valent vertex corresponding to the determinant, and classify reduced $2n$-webs on an annulus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03153 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Webs and multiwebs for the symplectic group Kenyon, Richard Wu, Haihan Mathematical Physics Probability Representation Theory 82B20, 05E10 We define $2n$-multiwebs on planar graphs and discuss their relation with $\mathrm{Sp}(2n)$-webs. On a planar graph with a symplectic local system we define a matrix whose Pfaffian is the sum of traces of $2n$-multiwebs. As application we generalize Kasteleyn's theorem from dimer covers to $2n$-multiweb covers of planar graphs with $U(n)$ gauge group. For $\mathrm{Sp}(4)$ we relate Kuperberg's ``tetravalent vertex'' to the determinant, and classify reduced $4$-webs on some simple surfaces: the annulus, torus, and pair of pants. We likewise define, for $\mathrm{Sp}(2n)$ and $q=1$, a $2n$-valent vertex corresponding to the determinant, and classify reduced $2n$-webs on an annulus. |
| title | Webs and multiwebs for the symplectic group |
| topic | Mathematical Physics Probability Representation Theory 82B20, 05E10 |
| url | https://arxiv.org/abs/2411.03153 |