A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs
For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight $0$. Standard Young tableaux also parametrize the vertices of dual...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
2008-01-01
|
| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/3626/pdf |
| _version_ | 1797270429557063680 |
|---|---|
| author | S. Assaf |
| author_facet | S. Assaf |
| author_sort | S. Assaf |
| collection | DOAJ |
| description | For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight $0$. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explain this connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the $0$-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs. |
| first_indexed | 2024-04-25T02:04:08Z |
| format | Article |
| id | doaj.art-8df2d41e635f49ea9ef5b1e83585a929 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:04:08Z |
| publishDate | 2008-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-8df2d41e635f49ea9ef5b1e83585a9292024-03-07T14:38:06ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502008-01-01DMTCS Proceedings vol. AJ,...Proceedings10.46298/dmtcs.36263626A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphsS. Assaf0Department of Mathematics [Philadelphia]For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight $0$. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explain this connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the $0$-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.https://dmtcs.episciences.org/3626/pdfschur-weyl dualityzero weight spacescrystal graphsdual equivalence graphs[math.math-co] mathematics [math]/combinatorics [math.co][info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | S. Assaf A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs Discrete Mathematics & Theoretical Computer Science schur-weyl duality zero weight spaces crystal graphs dual equivalence graphs [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs |
| title_full | A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs |
| title_fullStr | A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs |
| title_full_unstemmed | A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs |
| title_short | A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs |
| title_sort | combinatorial realization of schur weyl duality via crystal graphs and dual equivalence graphs |
| topic | schur-weyl duality zero weight spaces crystal graphs dual equivalence graphs [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/3626/pdf |
| work_keys_str_mv | AT sassaf acombinatorialrealizationofschurweyldualityviacrystalgraphsanddualequivalencegraphs AT sassaf combinatorialrealizationofschurweyldualityviacrystalgraphsanddualequivalencegraphs |