# 2021-05-12

## 10:28

See Serre’s uniformity conjecture.

## Pavel Etingof, Frobenius exact symmetric tensor categories

Source: Frobenius exact symmetric tensor categories - Pavel Etingof. IAS Geometric/modular representation theory seminar. https://www.youtube.com/watch?v=7L06K7SL5qw

• Looking at modular representations of finite groups.

• irreducible representations: hard, but a lot is known.
• indecomposable objects of a category representations: very hard, very little is known. Hard
• See tensor ideal, Krull-Schmidt theorem : decomposition into indecomposables is essentially unique.

• Can take split Grothendieck ring.

• symmetric tensor category $$\mathsf{C}$$:

• $$k{\hbox{-}}$$linear, so enriched in $${ \mathsf{Vect} }_{/k}$$ for $$k= \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$$ (no assumption on characteristic) So morphisms are vector spaces and composition is bilinear.
• abelian category
• Artinian category : objects have finite length and $$\dim_K \mathsf{C}(X, Y) < \infty$$
• monoidal category : $$(\otimes, \one)$$ satisfying associativity and the pentagon axiom
• symmetric monoidal category : a symmetric braiding $$X\otimes Y \xrightarrow{\tau_{XY}} Y\otimes X$$ such that $$\tau_{YX} \circ \tau_{XY} = \operatorname{id}$$.
• Unsorted/rigid (objects in a category) : existence of duals and morphisms $$X\to X {}^{ \vee }$$ plus rigidity axioms
• Compatibility of additive/multiplicative structures: implied when $$\otimes$$ is bilinear on morphisms.
• $$\mathop{\mathrm{End}}_{\mathsf{C}}(\one) = k$$.
• Example: $${\mathsf{Rep}}_k(G)$$ the category of finite-dimensional representations of $$G$$. Take $$G=1$$ to recover $${ \mathsf{Vect} }_{/k}$$.

• Can replace a group here by an affine group scheme
• Such a category is tannakian if there exists a fiber functor : a symmetric tensor functor $$F: \mathsf{C} \to { \mathsf{Vect} }_{/k}$$.

• Preserves tensor structure (hexagon axiom), preserves braiding, is exact.
• Implies automatically faithful.
• Can take forgetful functor from representations to underlying vector space.
• Called “fiber” because for spaces and local systems, one can take a fiber at a point
• Deligne-Milne show this is unique.
• Can define scheme of tensor automorphisms, $$G = \underline{\mathop{\mathrm{Aut}}}^\otimes(F) \in {\mathsf{Grp}}{\mathsf{Sch}}_{/k}$$.
• For an (additive category rigid symmetric monoidal category $$\mathsf{C}_{/k}$$ with $$\mathop{\mathrm{End}}_{\mathsf{C}}(\one) = k$$, for any $$f\in \mathsf{C}(X, X)$$ we can define its trace $$\operatorname{Tr}(f) \in \mathop{\mathrm{End}}_{\mathsf{C}}(\one)$$:

• Can define a categorical dimension of a category $$\dim X \in K$$ as $$\dim X\coloneqq\operatorname{Tr}(\operatorname{id}_X)$$.
• See theory of semisimplification.
• A morphism $$f\in \mathsf{C}(X, Y)$$ is negligible if for all $$g\in \mathsf{C}(Y, X)$$ we have $$\operatorname{Tr}(f\circ g) = 0$$.
• Negligible morphisms form a tensor ideal of morphisms: stable under composition and tensor product with other morphisms.
• Can form quotient $$\mkern 1.5mu\overline{\mkern-1.5mu\mathsf{C}\mkern-1.5mu}\mkern 1.5mu \mathsf{C} / \mathsf{N}$$: full subcategory where $$\mkern 1.5mu\overline{\mkern-1.5mu\mathsf{C}\mkern-1.5mu}\mkern 1.5mu(X, Y) \coloneqq\mathsf{C}(X, Y) / \mathsf{N}(X, y)$$, i.e. form the vector space quotient of the hom sets.
• Still monoidal.
• Generally nasty, but if trace of any nilpotent endomorphism in $$\mathsf{C}$$ (e.g. when $$\mathsf{C}$$ admits a monoidal functor to an abelian STC), then $$\mkern 1.5mu\overline{\mkern-1.5mu\mathsf{C}\mkern-1.5mu}\mkern 1.5mu$$ is a semisimple category STC, and in particular is abelian and every object is a direct sum of simple objects.
• True if $$\mathsf{C}$$ is abelian: take any nilpotent endomorphism, filter by kernels of powers and take the associated graded. Trace of original equals trace of associated graded, but the latter is zero.
• Can compute trace after pushing through an abelian functor.
• $$\mkern 1.5mu\overline{\mkern-1.5mu\mathsf{C}\mkern-1.5mu}\mkern 1.5mu$$ is the semisimplification of $$\mathsf{C}$$
• simple objects of a category in $$\mkern 1.5mu\overline{\mkern-1.5mu\mathsf{C}\mkern-1.5mu}\mkern 1.5mu$$ are indecomposable objects of a category in $$\mathsf{C}$$ of nonzero dimension.
• This procedure is forcing Schur’s Lemma to be true!
• Run into functors that aren’t exact on either side, e.g. the Frobenius.
• But still additive, and every additive functor on a semisimple category is exact.
• Frobenius functors: take $$X^{\otimes p}$$, allow cyclic permutations $$c$$ with $$c^p = 1$$, get equivariance with respect to $${\mathbb{Z}}/p$$.
• Thus $$X^{\otimes p} \in \mathsf{C} \boxprod {\mathsf{Rep}}_{/k} {\mathbb{Z}}/p$$, the Deligne tensor product.
• Can take $$\operatorname{id}_{\mathsf{C}} \boxprod \ss$$, i.e. semisimplification on the 2nd component, to get an additive monoidal twisted-linear functor $$\mathop{\mathrm{Fr}}: \mathsf{C} \to \mathsf{C} \boxprod \Ver_p$$
• Here $$\Ver_p$$ is a Verlinde category: just denotes $$\mkern 1.5mu\overline{\mkern-1.5mu\mathsf{C}\mkern-1.5mu}\mkern 1.5mu$$ with a modified tensor product formula…?
• Can filtration $$X^{\otimes p}$$ by $$\mathop{\mathrm{Fr}}_i \coloneqq\ker (1-c)^i$$.