Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential rol... more Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. We calculate the canonical distinguished involutions in the affine Weyl groups of rank ≤ 7. We also prove some partial results relating canonical distinguished involutions and Dynkin's diagrams of the nilpotent orbits in the Langlands dual group.
Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential rol... more Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In this note we calculate the canonical distinguished involutions in the affine Weyl groups of rank ≤ 7. We also prove some partial results relating canonical distinguished involutions and Dynkin's diagrams of the nilpotent orbits in the Langlands dual group.
We develop a theory of Frobenius functors for symmetric tensor categories (STC) C over a field k ... more We develop a theory of Frobenius functors for symmetric tensor categories (STC) C over a field k of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F : C → C ⊠ Verp, where Verp is the Verlinde category (the semisimplification of Rep k (Z/p)); a similar construction of the underlying additive functor appeared independently in [Co]. This generalizes the usual Frobenius twist functor in modular representation theory and also the one defined in [O], where it is used to show that if C is finite and semisimple then it admits a fiber functor to Verp. The main new feature is that when C is not semisimple, F need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor C → Verp. We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of F , and use it to show that for categories with finitely many simple objects F does not increase the Frobenius-Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which F is exact, and define the canonical maximal Frobenius exact subcategory Cex inside any STC C with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius-Perron dimension is preserved by F. One of our main results is that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to Verp. This is the strongest currently available characteristic p version of Deligne's theorem (stating that a STC of moderate growth in characteristic zero is the representation category of a supergroup). We also show that a sufficiently large power of F lands in Cex. Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category (namely, one having a fiber functor into the category of representations of the triangular Hopf algebra k[d]/d 2 with d primitive and R-matrix R = 1 ⊗ 1 + d ⊗ d), and show that a STC with Chevalley property is (almost) Frobenius exact. Finally, as a by-product, we resolve Question 2.15 of [EG1].
A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraic... more A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic p version of this theorem. Namely we show that a pre-Tannakian category over an algebraically closed field of characteristic p > 0 admits a fiber functor into the Verlinde category Verp (i.e., is the representation category of an affine group scheme in Verp) if and only if it has moderate growth and is Frobenius exact. This implies that Frobenius exact pre-Tannakian categories of moderate growth admit a well-behaved notion of Frobenius-Perron dimension. It follows that any semisimple pre-Tannakian category of moderate growth has a fiber functor to Verp (so in particular Deligne's theorem holds on the nose for semisimple pre-Tannakian categories in characteristics 2, 3). This settles a conjecture of the third author from 2015. In particular, this result applies to semisimplifications of categories of modular representations of finite groups (or, more generally, affine group schemes), which gives new applications to classical modular representation theory. For example, it allows us to characterize, for a modular representation V , the possible growth rates of the number of indecomposable summands in V ⊗n of dimension prime to p. To Vera Serganova on her 60th birthday with admiration. Date: September 2, 2022. 1 7.1. The subcategory C *. 19 7.2. The enhanced Frobenius functor on C * 20 7.3. Graded categories 20 7.4. Concluding the proof 22 7.5. Algebraically closed fields 22 8. Applications 23 8.1. Frobenius-Perron dimensions in pre-Tannakian categories of moderate growth 23 8.2. Classification of semisimple pre-Tannakian categories of moderate growth in positive characteristic 27 8.3. Growth rates in modular representation theory 28 8.4. Benson's conjecture 30 8.5. Characteristic p > 2 31 8.6. p-adic dimensions 33 Appendix A. A faithfulness result for symmetric groups 34 References 35
We show that any slightly degenerate weakly group-theoretical fusion category admits a minimal no... more We show that any slightly degenerate weakly group-theoretical fusion category admits a minimal non-degenerate extension. Let d be a positive square-free integer, given a weakly grouptheoretical non-degenerate fusion category C, assume that FPdim(C) = nd and (n, d) = 1. If (FPdim(X) 2 , d) = 1 for all simple objects X of C, then we show that C contains a non-degenerate fusion subcategory C(Z d , q). In particular, we obtain that integral fusion categories of Frobenius-Perron dimensions p m d such that C ′ ⊆ sVec are nilpotent and group-theoretical, where p is a prime and (p, d) = 1.
We prove a universal property of Deligne's category Rep ab (S d). Along the way, we classify tens... more We prove a universal property of Deligne's category Rep ab (S d). Along the way, we classify tensor ideals in the category Rep(S d).
Communications in Mathematical Physics, Jan 18, 2014
We give a new way to derive branching rules for the conformal embedding (ŝl n) m ⊕ (ŝl m) n ⊂ (ŝl... more We give a new way to derive branching rules for the conformal embedding (ŝl n) m ⊕ (ŝl m) n ⊂ (ŝl nm) 1. In addition, we show that the category C(ŝl n) 0 m of degree zero integrable highest weight (ŝl n) m-representations is braided equivalent to C(ŝl m) 0 n with the reversed braiding.
We propose a conjectural extension to positive characteristic case of a well known Deligne's theo... more We propose a conjectural extension to positive characteristic case of a well known Deligne's theorem on the existence of super fiber functors. We prove our conjecture in the special case of semisimple categories with finitely many isomorphism classes of simple objects.
We classify semisimple module categories over the tensor category of representations of quantum S... more We classify semisimple module categories over the tensor category of representations of quantum SL(2).
We study generalized Deligne categories and related tensor envelopes for the universal two-dimens... more We study generalized Deligne categories and related tensor envelopes for the universal two-dimensional cobordism theories described by rational functions, recently defined by Sazdanovic and one of the authors.
We prove a general result which implies that the global and Frobenius-Perron dimensions of a fusi... more We prove a general result which implies that the global and Frobenius-Perron dimensions of a fusion category generate Galois invariant ideals in the ring of algebraic integers.
We prove a general result which implies that the global and Frobenius-Perron dimensions of a fusi... more We prove a general result which implies that the global and Frobenius-Perron dimensions of a fusion category generate Galois invariant ideals in the ring of algebraic integers.
We give a characterization of Drinfeld centers of fusion categories as nondegenerate braided fusi... more We give a characterization of Drinfeld centers of fusion categories as nondegenerate braided fusion categories containing a Lagrangian algebra. Further we study the quotient of the monoid of non-degenerate braided fusion categories modulo the submonoid of the Drinfeld centers and show that its formal properties are similar to those of the classical Witt group.
This paper clarifies basic definitions in the universal construction of topological theories and ... more This paper clarifies basic definitions in the universal construction of topological theories and monoidal categories. The definition of the universal construction is given for various types of monoidal categories, including rigid and symmetric. It is also explained how to set up the universal construction for non-monoidal categories. The second part of the paper explains how to associate a rigid symmetric monoidal category to a small category, a sort of the Brauer envelope of a category. The universal construction for the Brauer envelopes generalizes some earlier work of the first two authors on automata, power series and topological theories. Finally, the theory of pseudocharacters (or pseudo-representations), which is an essential tool in modern number theory, is interpreted via one-dimensional topological theories and TQFTs with defects. The notion of a pseudocharacter is studied for Brauer categories and the lifting property to characters of semisimple representations is established in characteristic 0 for Brauer categories with at most countably many objects. The paper contains a brief discussion of pseudo-holonomies, which are functions from loops in a manifold to real numbers similar to traces of the holonomies along loops of a connection on a vector bundle on the manifold. It concludes with a classification of pseudocharacters (pseudo-TQFTs) and their generating functions for the category of oriented two-dimensional cobordisms in the characteristic 0 case.
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