Martin Rubey ; Bruce W. Westbury - Combinatorics of symplectic invariant tensors

dmtcs:2508 - Discrete Mathematics & Theoretical Computer Science, January 1, 2015, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) - https://doi.org/10.46298/dmtcs.2508
Combinatorics of symplectic invariant tensorsArticle

Authors: Martin Rubey 1; Bruce W. Westbury 2

  • 1 Fakultät für Mathematik und Geoinformation [Wien]
  • 2 Department of Mathematics, University of Warwick

An important problem from invariant theory is to describe the subspace of a tensor power of a representation invariant under the action of the group. According to Weyl's classic, the first main (later: 'fundamental') theorem of invariant theory states that all invariants are expressible in terms of a finite number among them, whereas a second main theorem determines the relations between those basic invariants.Here we present a transparent, combinatorial proof of a second fundamental theorem for the defining representation of the symplectic group $Sp(2n)$. Our formulation is completely explicit and provides a very precise link to $(n+1)$-noncrossing perfect matchings, going beyond a dimension count. As a corollary, we obtain an instance of the cyclic sieving phenomenon.


Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Section: Proceedings
Published on: January 1, 2015
Imported on: November 21, 2016
Keywords: invariant tensors,cyclic sieving phenomenon,matchings,classical groups,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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