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

Symplectic determinant laws and invariant theory