Combinatorial descriptions of the crystal structure on certain PBW bases
Lusztig's theory of PBW bases gives a way to realize the crystal B(∞) for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations, one for each reduced expression for the longest element of the Weyl group. There...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2020-04-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/6377/pdf |
| _version_ | 1827323961003737088 |
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| author | Ben Salisbury Adam Schultze Peter Tingley |
| author_facet | Ben Salisbury Adam Schultze Peter Tingley |
| author_sort | Ben Salisbury |
| collection | DOAJ |
| description | Lusztig's theory of PBW bases gives a way to realize the crystal B(∞) for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different 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. For ADE types, we give conditions on the reduced expression which ensure that the corresponding crystal operators are given by simple combinatorial bracketing rules. We then give at least one reduced expression satisfying our conditions in every type except E8, and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in types A and D. |
| first_indexed | 2024-04-25T02:01:04Z |
| format | Article |
| id | doaj.art-cc699a97ba5d42849a08c6f08f88cdbc |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:01:04Z |
| publishDate | 2020-04-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-cc699a97ba5d42849a08c6f08f88cdbc2024-03-07T14:55:20ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502020-04-01DMTCS Proceedings, 28th...10.46298/dmtcs.63776377Combinatorial descriptions of the crystal structure on certain PBW basesBen Salisbury0https://orcid.org/0000-0002-7013-5301Adam Schultze1Peter Tingley2Department of Mathematics - University of MichiganDepartment of Mathematics [New York CUNY]Department of Mathematics and Statistics [Chicago]Lusztig's theory of PBW bases gives a way to realize the crystal B(∞) for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different 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. For ADE types, we give conditions on the reduced expression which ensure that the corresponding crystal operators are given by simple combinatorial bracketing rules. We then give at least one reduced expression satisfying our conditions in every type except E8, and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in types A and D.https://dmtcs.episciences.org/6377/pdfcombinatorics[math.math-co]mathematics [math]/combinatorics [math.co] |
| spellingShingle | Ben Salisbury Adam Schultze Peter Tingley Combinatorial descriptions of the crystal structure on certain PBW bases Discrete Mathematics & Theoretical Computer Science combinatorics [math.math-co]mathematics [math]/combinatorics [math.co] |
| title | Combinatorial descriptions of the crystal structure on certain PBW bases |
| title_full | Combinatorial descriptions of the crystal structure on certain PBW bases |
| title_fullStr | Combinatorial descriptions of the crystal structure on certain PBW bases |
| title_full_unstemmed | Combinatorial descriptions of the crystal structure on certain PBW bases |
| title_short | Combinatorial descriptions of the crystal structure on certain PBW bases |
| title_sort | combinatorial descriptions of the crystal structure on certain pbw bases |
| topic | combinatorics [math.math-co]mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/6377/pdf |
| work_keys_str_mv | AT bensalisbury combinatorialdescriptionsofthecrystalstructureoncertainpbwbases AT adamschultze combinatorialdescriptionsofthecrystalstructureoncertainpbwbases AT petertingley combinatorialdescriptionsofthecrystalstructureoncertainpbwbases |