Symplectic determinant laws and invariant theory

We introduce the notion of $\mathit{\text{symplectic}}\mathit{\text{determinant}}\mathit{\text{laws}}$ by analogy with Chenevier’s definition of determinant laws. Symplectic determinant laws are a way to define pseudorepresentations for symplectic representations of algebras with involution over arbitrary $\mathbb{Z}[\frac{1}{2}]$-algebras. We prove that this notion satisfies the properties expected from a good theory of pseudorepresentations, and we compare it to Lafforgue’s ${\text{Sp}}_{2d}$-pseudocharacters. In the process, we compute generators of the invariant algebras $A[M_d^m{]}^G$ and $A[G^m{]}^G$ over an arbitrary commutative ring $A$ when $G\in \{{\text{Sp}}_d,{\mathrm{O}}_d,{\text{GSp}}_d,{\text{GO}}_d\}$, generalizing results of Zubkov.