Ultraproduct and oligomorphic-group constructions of interpolation categories for finite classical groups agree, and the categories depend only on a parameter t, with an additional parity label in the orthogonal case.
Deligne categories as limits in rank and characteristic
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
We give new interpretations of the Deligne categories $\underline{Rep}(GL_t)$ and $\underline{Rep}(S_t)$ (and their abelian envelopes) over $\mathbb{C}$ in terms of modular representations of general linear and symmetric groups of large rank in large characteristic. In particular we make sense of the sentence "$\underline{Rep}(S_n)$ is the limit of $Rep(S_{p+n})$ over $\bar{\mathbb{F}}_p$ as $p$ goes to infinity". We then give examples of how to pass results between these different settings.
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math.RT 1years
2025 1verdicts
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Classical interpolation categories
Ultraproduct and oligomorphic-group constructions of interpolation categories for finite classical groups agree, and the categories depend only on a parameter t, with an additional parity label in the orthogonal case.