Combinatorial descriptions of the crystal structure on certain PBW bases
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2016
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| _version_ | 1866910939041759232 |
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| author | Salisbury, Ben Schultze, Adam Tingley, Peter |
| author_facet | Salisbury, Ben Schultze, Adam Tingley, Peter |
| contents | Using the theory of PBW bases, one can realize the crystal $B(\infty)$ for any semisimple Lie algebra over $\mathbf{C}$ using Kostant partitions as the underlying set. In fact there are many such realizations, one for each reduced expression for the longest element of the Weyl group. There is an algorithm to calculate the actions of the crystal operators, but it can be quite complicated. Here we show that, for certain reduced expressions, the crystal operators can also be described by a much simpler bracketing rule. We give conditions describing these reduced expressions, and show that there is at least one example in every type except possibly $E_8$, $F_4$ and $G_2$. We then discuss some examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1606_01978 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Combinatorial descriptions of the crystal structure on certain PBW bases Salisbury, Ben Schultze, Adam Tingley, Peter Quantum Algebra Combinatorics Representation Theory Using the theory of PBW bases, one can realize the crystal $B(\infty)$ for any semisimple Lie algebra over $\mathbf{C}$ using Kostant partitions as the underlying set. In fact there are many such realizations, one for each reduced expression for the longest element of the Weyl group. There is an algorithm to calculate the actions of the crystal operators, but it can be quite complicated. Here we show that, for certain reduced expressions, the crystal operators can also be described by a much simpler bracketing rule. We give conditions describing these reduced expressions, and show that there is at least one example in every type except possibly $E_8$, $F_4$ and $G_2$. We then discuss some examples. |
| title | Combinatorial descriptions of the crystal structure on certain PBW bases |
| topic | Quantum Algebra Combinatorics Representation Theory |
| url | https://arxiv.org/abs/1606.01978 |