The Robinson―Schensted Correspondence and $A_2$-webs
The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ forms bases for morphism...
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Discrete Mathematics & Theoretical Computer Science
2013-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/2349/pdf |
| _version_ | 1827324006657687552 |
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| author | Matthew Housley Heather M. Russell Julianna Tymoczko |
| author_facet | Matthew Housley Heather M. Russell Julianna Tymoczko |
| author_sort | Matthew Housley |
| collection | DOAJ |
| description | The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ forms bases for morphism spaces. A great deal of recent interest has focused on the combinatorics of invariant webs for tensors powers of $V^+$, the standard representation of the quantum group. In particular, the invariant webs for the 3$n$th tensor power of $V^+$ correspond bijectively to $[n,n,n]$ standard Young tableaux. Kuperberg originally defined this map in terms of a graphical algorithm, and subsequent papers of Khovanov–Kuperberg (1999) and Tymoczko (2012) introduce algorithms for computing the inverse. The main result of this paper is a redefinition of Kuperberg's map through the representation theory of the symmetric group. In the classical limit, the space of invariant webs carries a symmetric group action. We use this structure in conjunction with Vogan's generalized tau-invariant and Kazhdan–Lusztig theory to show that Kuperberg's map is a direct analogue of the Robinson–Schensted correspondence. |
| first_indexed | 2024-04-25T02:01:50Z |
| format | Article |
| id | doaj.art-6e38352011b141d38c2cba4258a2c65b |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:01:50Z |
| publishDate | 2013-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
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| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-6e38352011b141d38c2cba4258a2c65b2024-03-07T14:52:36ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502013-01-01DMTCS Proceedings vol. AS,...Proceedings10.46298/dmtcs.23492349The Robinson―Schensted Correspondence and $A_2$-websMatthew Housley0Heather M. Russell1Julianna Tymoczko2Brigham Young UniversityUniversity of Southern CaliforniaSmith College [Northampton]The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ forms bases for morphism spaces. A great deal of recent interest has focused on the combinatorics of invariant webs for tensors powers of $V^+$, the standard representation of the quantum group. In particular, the invariant webs for the 3$n$th tensor power of $V^+$ correspond bijectively to $[n,n,n]$ standard Young tableaux. Kuperberg originally defined this map in terms of a graphical algorithm, and subsequent papers of Khovanov–Kuperberg (1999) and Tymoczko (2012) introduce algorithms for computing the inverse. The main result of this paper is a redefinition of Kuperberg's map through the representation theory of the symmetric group. In the classical limit, the space of invariant webs carries a symmetric group action. We use this structure in conjunction with Vogan's generalized tau-invariant and Kazhdan–Lusztig theory to show that Kuperberg's map is a direct analogue of the Robinson–Schensted correspondence.https://dmtcs.episciences.org/2349/pdfrobinson―schenstedweb basiskazhdan―lusztig theoryyoung tableau[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Matthew Housley Heather M. Russell Julianna Tymoczko The Robinson―Schensted Correspondence and $A_2$-webs Discrete Mathematics & Theoretical Computer Science robinson―schensted web basis kazhdan―lusztig theory young tableau [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | The Robinson―Schensted Correspondence and $A_2$-webs |
| title_full | The Robinson―Schensted Correspondence and $A_2$-webs |
| title_fullStr | The Robinson―Schensted Correspondence and $A_2$-webs |
| title_full_unstemmed | The Robinson―Schensted Correspondence and $A_2$-webs |
| title_short | The Robinson―Schensted Correspondence and $A_2$-webs |
| title_sort | robinson schensted correspondence and a 2 webs |
| topic | robinson―schensted web basis kazhdan―lusztig theory young tableau [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2349/pdf |
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