Universal property of Rep(GL(n, k)) as a free 2-rig on a dimension-n object

Establish that for any field k of characteristic zero and integer n ≥ 1, the 2-rig Rep(GL(n, k)) is the free 2-rig on an object x of dimension n, meaning that the nth exterior power An(x) is invertible with respect to the tensor product, i.e., verify that Rep(GL(n, k)) satisfies the universal property characterizing the free 2-rig on such an object.

Background

The paper proves that for the affine monoid M(n, k) of n×n matrices over a field k of characteristic zero, the representation 2-rig Rep(M(n, k)) is the free 2-rig on an object of subdimension n (i.e., An+1(x) = 0). This settles a conjecture from earlier work and is achieved via Tannaka reconstruction and a theory of quotient 2-rigs and 2-ideals.

Building on this, the authors conjecture an analogous universal property for the general linear group GL(n, k): that its representation 2-rig Rep(GL(n, k)) is free on an object of (bosonic) dimension n, where “dimension n” means the nth exterior power is invertible in the symmetric monoidal structure. Proving this would extend the established monoid result to a fundamental reductive group.