S. Assaf - A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs

dmtcs:3626 - Discrete Mathematics & Theoretical Computer Science, January 1, 2008, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) - https://doi.org/10.46298/dmtcs.3626
A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs

Authors: S. Assaf 1

  • 1 Department 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.


Volume: DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)
Section: Proceedings
Published on: January 1, 2008
Imported on: May 10, 2017
Keywords: 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]
Funding:
    Source : OpenAIRE Graph
  • PostDoctoral Research Fellowship; Funder: National Science Foundation; Code: 0703567

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