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

Tags: #projects/notes/seminars #lie-theory #higher-algebra/category-theory #higher-algebra/monoidal Refs: tensor category

  • 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.
    • \({ \operatorname{End} }_{\mathsf{C}}(\one) = k\).
  • 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 rigid symmetric monoidal category \(\mathsf{C}_{/k}\) with \({ \operatorname{End} }_{\mathsf{C}}(\one) = k\), for any \(f\in \mathsf{C}(X, X)\) we can define its trace \(\operatorname{Tr}(f) \in { \operatorname{End} }_{\mathsf{C}}(\one)\):

Link to Diagram

  • 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 \({\mathbf{Z}}/p\).
    • Thus \(X^{\otimes p} \in \mathsf{C} \boxprod {\mathsf{Rep}}_{/k} {\mathbf{Z}}/p\), the Deligne tensor product.
    • Can take \(\operatorname{id}_{\mathsf{C}} \boxprod {\mathrm{ss}}\), 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 \(\mathop{\mathrm{Fr}}_i \coloneqq\ker (1-c)^i\).
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