[Paper Review] Deligne categories as limits in rank and characteristic

Nate Harman|arXiv (Cornell University)|Jan 13, 2016

Advanced Algebra and GeometryMathematics8 references18 citations

TL;DR

This paper establishes a novel connection between Deligne's categories $\underline{Rep}(GL_t)^{ab}$ and $\underline{Rep}(S_t)^{ab}$ in characteristic zero and modular representation theory of symmetric and general linear groups in large rank and large characteristic. Using ultraproducts via non-principal ultrafilters, it shows that these Deligne categories arise as limits of representations over $\overline{\mathbb{F}}_p$ as $p \to \infty$, enabling transfer of results between complex rank and positive characteristic settings.

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.

Motivation & Objective

• To reinterpret Deligne's abelian tensor categories $\underline{Rep}(GL_t)^{ab}$ and $\underline{Rep}(S_t)^{ab}$ in characteristic zero as limits of modular representation categories over $\overline{\mathbb{F}}_p$.
• To establish a rigorous framework using ultraproducts and ultrafilters to capture asymptotic behavior of representation categories as both rank and characteristic grow.
• To enable transfer of results between modular representation theory in large characteristic and Deligne categories in complex rank.
• To provide a new perspective on the structure of simple algebras and standard modules in Deligne categories via positive characteristic analogues.

Proposed method

• Utilizes non-principal ultrafilters to construct ultraproducts of sequences of representation categories $\mathrm{Rep}(G_n, \overline{\mathbb{F}}_p)$ for $G_n = GL_n$ or $S_n$.
• Defines the ultraproduct category $\widehat{\mathcal{C}}_\mathcal{U}$ as a limit category, with objects and morphisms as germs of sequences under the ultrafilter.
• Restricts the ultraproduct to a manageable subcategory $\mathcal{C}_\mathcal{U}$ generated by regular representations, ensuring equivalence to Deligne’s categories.
• Applies the ultraproduct construction to show that $\underline{Rep}(S_t)^{ab}$ is the limit of $\mathrm{Rep}(S_{p+n}, \overline{\mathbb{F}}_p)$ as $p \to \infty$.
• Uses the equivalence of multiplicity in standard and irreducible modules in Deligne categories to relate positive characteristic and supergroup representation theory.
• Establishes a correspondence between admissible weights in $\mathfrak{gl}(n+m|m)$ and highest weights in $\mathrm{Rep}(GL_{n+p}, \overline{\mathbb{F}}_p)$ via stabilized Kac module multiplicities.

Experimental results

Research questions

• RQ1 How can Deligne’s categories in complex rank be interpreted as limits of modular representation categories in large characteristic?
• RQ2 What is the precise role of ultrafilters in constructing a categorical limit that captures the asymptotic behavior of representation categories?
• RQ3 Can results from modular representation theory in large rank and characteristic be transferred to Deligne categories in complex rank?
• RQ4 How do the structures of standard and irreducible modules in $\mathrm{Rep}(GL_{n+p}, \overline{\mathbb{F}}_p)$ relate to those in supergroup representations $\mathrm{Rep}(GL(n+m|m))$?
• RQ5 What is the significance of stabilized Kac module multiplicities in the context of Deligne’s abelian envelopes?

Key findings

• The abelian envelope $\underline{Rep}(S_t)^{ab}$ for transcendental $t$ is isomorphic to the ultraproduct limit of $\mathrm{Rep}(S_{p+n}, \overline{\mathbb{F}}_p)$ as $p \to \infty$, providing a new realization of Deligne’s category.
• For fixed bipartitions $\lambda, \mu$, the multiplicity $[K(\lambda), L(\mu)]$ in $\mathrm{Rep}(GL(n+m|m))$ stabilizes as $m \to \infty$ and matches the multiplicity $[\Delta(\lambda), L(\mu)]$ in $\mathrm{Rep}(GL_{n+p}, \overline{\mathbb{F}}_p)$ for sufficiently large $p$.
• The ultraproduct construction yields a category equivalent to $\underline{Rep}(GL_n)^{ab}$, showing that this Deligne category arises as a limit of modular representations in large characteristic.
• The theory enables a characterization of simple commutative algebras in $\underline{Rep}(S_t)^{ab}$ by extending Etingof’s classification from $\mathbb{C}$ to algebraically closed fields in large characteristic.
• The correspondence between modular representations and supergroup representations is formalized via stabilized Kac module multiplicities, linking positive characteristic and supergroup representation theory.
• The paper provides a framework to transfer results between Deligne categories and modular representation theory, exemplified by explicit formulas for irreducible modular representations of symmetric groups.