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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2402.13978 |
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| _version_ | 1866913925094703104 |
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| author | Gaetz, Christian Pechenik, Oliver Pfannerer, Stephan Striker, Jessica Swanson, Joshua P. |
| author_facet | Gaetz, Christian Pechenik, Oliver Pfannerer, Stephan Striker, Jessica Swanson, Joshua P. |
| contents | Webs give a diagrammatic calculus for spaces of $U_q(\mathfrak{sl}_r)$-tensor invariants, but intrinsic characterizations of web bases are only known in certain cases. Recently, we introduced hourglass plabic graphs to give the first such $U_q(\mathfrak{sl}_4)$-web bases. Separately, Fraser introduced a web basis for Plücker degree two representations of arbitrary $U_q(\mathfrak{sl}_r)$. Here, we show that Fraser's basis agrees with that predicted by the hourglass plabic graph framework and give an intrinsic characterization of the resulting webs. A further compelling feature with many applications is that our bases exhibit rotation-invariance. Together with the results of our earlier paper, this implies that hourglass plabic graphs give a uniform description of all known rotation-invariant $U_q(\mathfrak{sl}_r)$-web bases. Moreover, this provides a single combinatorial model simultaneously generalizing the Tamari lattice, the alternating sign matrix lattice, and the lattice of plane partitions. As a part of our argument, we develop properties of square faces in arbitrary hourglass plabic graphs, a key step in our program towards general $U_q(\mathfrak{sl}_r)$-web bases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13978 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Web bases in degree two from hourglass plabic graphs Gaetz, Christian Pechenik, Oliver Pfannerer, Stephan Striker, Jessica Swanson, Joshua P. Combinatorics Quantum Algebra Representation Theory Webs give a diagrammatic calculus for spaces of $U_q(\mathfrak{sl}_r)$-tensor invariants, but intrinsic characterizations of web bases are only known in certain cases. Recently, we introduced hourglass plabic graphs to give the first such $U_q(\mathfrak{sl}_4)$-web bases. Separately, Fraser introduced a web basis for Plücker degree two representations of arbitrary $U_q(\mathfrak{sl}_r)$. Here, we show that Fraser's basis agrees with that predicted by the hourglass plabic graph framework and give an intrinsic characterization of the resulting webs. A further compelling feature with many applications is that our bases exhibit rotation-invariance. Together with the results of our earlier paper, this implies that hourglass plabic graphs give a uniform description of all known rotation-invariant $U_q(\mathfrak{sl}_r)$-web bases. Moreover, this provides a single combinatorial model simultaneously generalizing the Tamari lattice, the alternating sign matrix lattice, and the lattice of plane partitions. As a part of our argument, we develop properties of square faces in arbitrary hourglass plabic graphs, a key step in our program towards general $U_q(\mathfrak{sl}_r)$-web bases. |
| title | Web bases in degree two from hourglass plabic graphs |
| topic | Combinatorics Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2402.13978 |