We study the category $\mathcal{ A}$ of smooth semilinear representations of the infinite symmetric group over the field of rational functions in infinitely many variables. We establish a number of results about the structure of $\mathcal{ A}$, e.g., classification of injective objects, finiteness of injective dimension, computation of the Grothendieck group, and so on. We also prove that $\mathcal{ A}$ is (essentially) equivalent to a simpler linear algebraic category $\mathcal{ B}$, which makes many properties of $\mathcal{ A}$ transparent.
Categories: math.RT math.AC