The Grothendieck--Teichmüller group and the stable symplectic category

We continue our study of the stable symplectic category that was begun by the first author. In this document we consider a canonical representation of the stable symplectic category and study the Motivic Galois group of monoidal automorphisms of this representation. In particular, we observe that this Galois group contains a natural subgroup isomorphic to the abelian quotient of the Grothendieck--Teichmüller group. We also study other related algebraic invariants like the rational Waldhausen K-theory of the E-infinity ring spectrum of coefficients Ω, of the stable sympelctic category, and its relation to the symplectomorphism group of an object.