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arXiv · 1504.02586

Combinatorics of symplectic invariant tensors

Martin Rubey·Bruce W. Westbury
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Abstract

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.

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Martin Rubey, Bruce W. Westbury. 2015-04-10. Combinatorics of symplectic invariant tensors. https://arxiv.org/abs/1504.02586

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