The semi-linear representation theory of the infinite symmetric group

Nagpal, Rohit; Snowden, Andrew

doi:10.1090/ert/679

The semi-linear representation theory of the infinite symmetric group
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by Rohit Nagpal and Andrew Snowden;

Represent. Theory 28 (2024), 533-551

DOI: https://doi.org/10.1090/ert/679

Published electronically: November 14, 2024

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Abstract:

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.

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