Combinatorics of symplectic invariant tensors

TL;DR

This paper provides a combinatorial proof of the second fundamental theorem for symplectic group invariants, explicitly linking invariant tensors to noncrossing matchings and revealing new combinatorial structures.

Contribution

It introduces a transparent, combinatorial proof of the second fundamental theorem for Sp(2n) invariants, connecting invariant tensors to noncrossing perfect matchings.

Findings

Explicit combinatorial description of invariants

Connection to (n+1)-noncrossing perfect matchings

Application to cyclic sieving phenomenon

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.