This table contains Hilbert bases (HB) and Markov bases (MB) of affine semigroups obtained as images of maximal rank valuations on the coordinate rings of three different classes of
algebraic varieties. The number of elements in the Hilbert and Markov bases are
upper bounds on the number of generators and relations needed to present the
associated coordinate ring. Information about each variety appears below, more
about the techniques used to produce these semigroups can be found in the papers
https://arxiv.org/abs/1403.3990 and https://arxiv.org/abs/1103.2484. All
computations are carried out in 4ti2 http://www.4ti2.de/.
Graphs and Character Varieties
$$SL_2(\mathbb{C})$$ | HB: 3 | MB: 0 |
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$$SL_3(\mathbb{C})$$ | HB: 10 | MB: 2 |
$$SL_4(\mathbb{C})$$ | HB: 89 | MB: 2240 |
$$SL_5(\mathbb{C})$$ | HB: 12955 |
$$SL_2(\mathbb{C})$$ | HB: 3 | MB: 0 |
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$$SL_3(\mathbb{C})$$ | HB: 10 | MB: 2 |
$$SL_4(\mathbb{C})$$ | HB: 44 | MB: 360 |
$$SL_5(\mathbb{C})$$ | HB: 1561 |
$$SL_2(\mathbb{C})$$ | HB: 7 | MB: 1 |
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$$SL_3(\mathbb{C})$$ | HB: 65 | MB: 1029 |
$$SL_2(\mathbb{C})$$ | HB: 7 | MB: 1 |
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$$SL_3(\mathbb{C})$$ | HB: 341 | MB: 49583 |
$$SL_2(\mathbb{C})$$ | HB: 8 | MB: 2 |
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$$SL_3(\mathbb{C})$$ | HB: 328 | MB: 64923 |
$$SL_2(\mathbb{C})$$ | HB: 8 | MB: 2 |
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$$SL_3(\mathbb{C})$$ | HB: 213 | MB: 16563 |
$$SL_2(\mathbb{C})$$ | HB: 8 | MB: 2 |
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$$SL_3(\mathbb{C})$$ | HB: 65 | MB: 1029 |
$$SL_2(\mathbb{C})$$ | HB: 23 | MB: 75 |
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$$SL_2(\mathbb{C})$$ | HB: 22 | MB: 67 |
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$$SL_2(\mathbb{C})$$ | HB: 21 | MB:60 |
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$$SL_2(\mathbb{C})$$ | HB: 20 | MB: 55 |
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$$SL_2(\mathbb{C})$$ | HB: 15 | MB: 20 |
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$$SL_2(\mathbb{C})$$ | HB: 20 | MB: 45 |
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$$SL_2(\mathbb{C})$$ | HB: 20 | MB: 44 |
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$$SL_2(\mathbb{C})$$ | HB: 22 | MB: 63 |
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$$SL_2(\mathbb{C})$$ | HB: 22 | MB: 63 |
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$$SL_2(\mathbb{C})$$ | HB: 18 | MB: 36 |
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$$SL_2(\mathbb{C})$$ | HB: 18 | MB: 36 |
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$$SL_2(\mathbb{C})$$ | HB: 17 | MB: 26 |
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$$SL_2(\mathbb{C})$$ | HB: 22 | MB: 63 |
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$$SL_2(\mathbb{C})$$ | HB: 22 | MB: 63 |
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$$SL_2(\mathbb{C})$$ | HB: 22 | MB: 63 |
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$$SL_2(\mathbb{C})$$ | HB: 22 | MB: 63 |
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$$SL_2(\mathbb{C})$$ | HB: 15 | MB: 24 |
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Trees and Configurations of full principal Flags
$$SL_3(\mathbb{C})$$ | HB: 8 | MB: 1 |
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$$SL_4(\mathbb{C})$$ | HB: 18 | MB: 15 |
$$SL_5(\mathbb{C})$$ | HB: 45 | MB: 234 |
$$SL_6(\mathbb{C})$$ | HB: 166 | MB: 7039 |
$$SL_7(\mathbb{C})$$ | HB: 1369 |
$$SL_8(\mathbb{C})$$ | HB: 39219 |
$$SL_2(\mathbb{C})$$ | HB: 6 | MB: 1 |
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$$SL_3(\mathbb{C})$$ | HB: 22 | MB: 35 |
$$SL_4(\mathbb{C})$$ | HB: 103 | MB: 2194 |
$$SL_5(\mathbb{C})$$ | HB: 1515 |
$$SL_2(\mathbb{C})$$ | HB: 10 | MB: 5 |
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$$SL_3(\mathbb{C})$$ | HB: 52 | MB: 364 |
$$SL_4(\mathbb{C})$$ | HB: 754 |
$$SL_2(\mathbb{C})$$ | HB: 15 | MB: 15 |
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$$SL_3(\mathbb{C})$$ | HB: 114 | MB: 2562 |
$$SL_2(\mathbb{C})$$ | HB: 15 | MB: 15 |
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$$SL_3(\mathbb{C})$$ | HB: 114 | MB: 2530 |
$$SL_2(\mathbb{C})$$ | HB: 21 | MB: 35 |
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$$SL_3(\mathbb{C})$$ | HB: 240 |
$$SL_2(\mathbb{C})$$ | HB: 21 | MB: 35 |
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$$SL_3(\mathbb{C})$$ | HB: 240 |
$$SL_2(\mathbb{C})$$ | HB: 28 | MB: 70 |
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$$SL_3(\mathbb{C})$$ | HB: 494 |
Trees and Configurations of full Flags
$$SL_2(\mathbb{C})$$ | HB: 1 | MB: 0 |
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$$SL_3(\mathbb{C})$$ | HB:2 | MB:0 |
$$SL_4(\mathbb{C})$$ | HB 12 | MB: 28 |
$$SL_5(\mathbb{C})$$ | HB: 22 | MB: 89 |
$$SL_6(\mathbb{C})$$ | HB:4464 |
$$SL_7(\mathbb{C})$$ | HB: 43177 |
$$SL_2(\mathbb{C})$$ | HB: 2 | MB: 0 |
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$$SL_3(\mathbb{C})$$ | HB: 8 | MB: 3 |
$$SL_4(\mathbb{C})$$ | HB: 142 | MB: 7388 |
$$SL_2(\mathbb{C})$$ | HB: 6 | MB: 5 |
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$$SL_3(\mathbb{C})$$ | HB: 67 | MB: 1465 |
$$SL_2(\mathbb{C})$$ | HB: 5 | MB: 1 |
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$$SL_3(\mathbb{C})$$ | HB: 463 |
$$SL_2(\mathbb{C})$$ | HB: 6 | MB: 2 |
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$$SL_3(\mathbb{C})$$ | HB:1061 |
$$SL_2(\mathbb{C})$$ | HB: 36 | MB: 406 |
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$$SL_3(\mathbb{C})$$ | HB: 16646 |
$$SL_2(\mathbb{C})$$ | HB: 36 | MB: 406 |
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Configurations in projective space
$$SL_3(\mathbb{C})$$ | HB: 1 | MB: 0 |
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$$SL_3(\mathbb{C})$$ | HB: 1 | MB: 0 |
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$$SL_4(\mathbb{C})$$ | HB: 1 | MB: 0 |
$$SL_3(\mathbb{C})$$ | HB: 6 | MB: 5 |
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$$SL_4(\mathbb{C})$$ | HB: 1 | MB: 0 |
$$SL_5(\mathbb{C})$$ | HB: 1 | MB: 0 |
$$SL_3(\mathbb{C})$$ | HB: 7 | MB: 2 |
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$$SL_4(\mathbb{C})$$ | HB: 6 | MB: 2 |
$$SL_3(\mathbb{C})$$ | HB: 7 | MB: 2 |
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$$SL_4(\mathbb{C})$$ | HB: 5 | MB: 1 |
$$SL_3(\mathbb{C})$$ | HB: 273 | MB: 32449 |
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$$SL_3(\mathbb{C})$$ | HB: 282 | MB: 34828 |
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$$SL_3(\mathbb{C})$$ | HB: 2327 | MB: |
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[(0,1),(1,2),(2,3),(3,4),(4,5),(5,6)] | |
$$SL_3(\mathbb{C})$$ | HB: 443 | MB: |
---|