Searcharxiv⌕ Search

arXiv subjects

Victor Ostrik

Publications and source records attributed to Victor Ostrik.

At least 19 recordsLinked to original sources
arXiv2026

On the Drinfeld center of the Verlinde category $\Ver_p$

We provide some information about the Drinfeld center $\Z(\Ver_p)$ of the Verlinde category $\Ver_p$. We compute explicitly the Cartan matrix, bound quiver and cohomology of $\Z(\Ver_p^+)$, and prove that the category $\mathscr{Z}(\Ver_p^+)$ is wild for every $p\ge 7$. In the special case $p=5$, we show that $\Z(\Ver_5^+)$ has exactly $10$ non-isomorphic indecomposable objects, classify them and describe the Green ring of $\Z(\Ver_5^+)$, and compute the semisimplification of $\Z(\Ver_5^+)$.

math.QA↗
arXiv2026

Twisted Deligne products of semisimple tensor categories

We discuss the classification of twisted Deligne products of two semisimple tensor categories $\mathcal C,\mathcal D$, i.e., categorifications of the tensor product of their Grothendieck rings in which the factors are categorified by $\mathcal C$ and $\mathcal D$. In particular, we show that if both factors have no non-trivial gradings, or if one factor has neither non-trivial gradings nor tensor structures on the identity functor, then the only twisted Deligne product is the ordinary one. Using the work arXiv:2405.10207 by Müller, Peña Pollastri and Plavnik, this gives, in principle, a group-theoretical classification of twisted Deligne products and, more generally, exact factorizations of arbitrary fusion categories. In the Appendix we introduce the notion of categorical $n$-cocycles for $n=2,3,4$ and show that they are all pullbacks of group $n$-cocycles from the universal grading group of the underlying based ring. In the case of $4$-cocycles, this answers a question of Johnson-Freyd, Ostrik and Yu from arXiv:2601.09060.

math.CT↗
arXiv2026

Tensor ideals of abelian type and quantum groups

We initiate a study of tensor ideals in linear rigid monoidal categories that are kernels of linear monoidal functors to abelian monoidal categories. We develop general methods and apply them to the category of tilting modules over quantum groups as well as to some representation categories of finite groups. In an appendix on Duflo involutions in monoidal categories, we make a connection between Duflo involutions in the affine Weyl group and tensor ideals for quantum groups, and prove some of Lusztig's conjectures for arbitrary Coxeter groups, at equal parameters, without invoking the boundedness hypothesis.

math.QA↗
arXiv2026

A non-semisimple Witt class

We describe several infinite families of braided finite tensor categories. A simplest example gives a non-degenerate braided tensor category which is not Witt equivalent to a semisimple category.

math.RT↗
arXiv2026

On the structure of Witt groups and minimal extension conjecture

Let $\mathcal{E}=\text{Rep}(G)$ be a Tannakian fusion category. For a braided fusion category $\mathcal{C}$ over $\mathcal{E}$ we give sufficient and necessary conditions that characterize the Witt relation $[\mathcal{C}]=[\mathcal{E}]$. Then we show the Witt group $\mathcal{W}(\mathcal{E})$ is naturally a direct sum of Witt group $\mathcal{W}:=\mathcal{W}(\text{Vec})$ and the group $\text{H}^4(G,\mathbb{K}^\times)$. Consequently, for any non-degenerate fusion category $\mathcal{C}$ over $\mathcal{E}$, there is a positive integer $n$ (e.g. $n=|G|$) such that $\mathcal{C}^{\boxtimes_\mathcal{E}^n}$ admits a minimal extension.

math.CT↗
arXiv2024

Asymptotic properties of tensor powers in symmetric tensor categories

Let G be a group and V a finite dimensional representation of G over an algebraically closed field k of characteristic p>0. Let $d_n(V)$ be the number of indecomposable summands of $V^{\otimes n}$ of nonzero dimension mod p. It is easy to see that there exists a limit $δ(V):=\lim_{n\to \infty}d_n(V)^{1/n}$, which is positive (and $\ge 1$) iff V has an indecomposable summand of nonzero dimension mod p. We show that in this case the number $$ c(V):=\liminf_{n\to \infty} \frac{d_n(V)}{δ(V)^n}\in [0,1] $$ is strictly positive and $$ \log (c(V)^{-1})=O(δ(V)^2), $$ and moreover this holds for any symmetric tensor category over k of moderate growth. Furthermore, we conjecture that in fact $$ \log(c(V)^{-1})=O(δ(V)) $$ (which would be sharp), and prove this for p=2,3; in particular, for p=2 we show that $c(V)\ge 3^{-\frac{4}{3}δ(V)+1}$. The proofs are based on the characteristic p version of Deligne's theorem for symmetric tensor categories obtained in earlier work of the authors. We also conjecture a classification of semisimple symmetric tensor categories of moderate growth which is interesting in its own right and implies the above conjecture for all $p$, and illustrate this conjecture by describing the semisimplification of the modular representation category of a cyclic p-group. Finally, we study the asymptotic behavior of the decomposition of $V^{\otimes n}$ in characteristic zero using Deligne's theorem and the Macdonald-Mehta-Opdam identity.

math.RT↗
arXiv2023

On a necessary condition for unitary categorification of fusion rings

In arXiv:1910.12059 Liu, Palcoux and Wu proved a remarkable necessary condition for a fusion ring to admit a unitary categorification, by constructing invariants of the fusion ring that have to be positive if it is unitarily categorifiable. The main goal of this note is to provide a somewhat more direct proof of this result. In the last subsection we discuss integrality properties of the Liu-Palcoux-Wu invariants.

math.QA↗
arXiv2023

Incompressible tensor categories

A symmetric tensor category $\mathcal D$ over an algebraically closed field $k$ is incompressible if every tensor functor out of $\mathcal D$ is an embedding. E.g., the categories $Vec$ and $sVec$ of (super)vector spaces are incompressible. Moreover, by Deligne's theorem, if char$(k)=0$ then any tensor category of moderate growth uniquely fibres over $sVec$, so $Vec$ and $sVec$ are the only incompressible categories in this class. Similarly, in characteristic $p>0$, we have the incompressible Verlinde category $Ver_p$, and any Frobenius exact category of moderate growth uniquely fibres over $Ver_p$. More generally, the Verlinde categories $Ver_{p^n}$, $Ver_{p^n}^+$ are incompressible, and a key conjecture is that every tensor category of moderate growth uniquely fibres over $Ver_{p^\infty}$. This would make the above the only incompressible categories in this class. We prove a part of this conjecture, showing that every tensor category of moderate growth fibres over an incompressible one. So it remains to understand incompressible categories. We say that $\mathcal D$ is subterminal if it every tensor category admits at most one fibre functor to it, and a Bezrukavnikov category if the class of tensor categories that fibre over $\mathcal D$ is closed under quotients. Clearly, a subterminal Bezrukavnikov category is incompressible, and we conjecture the converse. We prove that $Ver_p$ is Bezrukavnikov, generalizing the result of Bezrukavnikov for $Vec$. We also find intrinsic sufficient conditions for incompressibility and subterminality. Namely, $\mathcal D$ is maximally nilpotent if the growth rates of symmetric powers are minimal. We show that a finite maximally nilpotent category is incompressible, and also subterminal if it satisfies an additional geometric reductivity condition. Then we verify these conditions for $Ver_{2^n}$.

math.CT↗
arXiv2023

On the minimal extension and structure of weakly group-theoretical braided fusion categories

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 group-theoretical non-degenerate fusion category $\mathcal{C}$, assume that $\text{FPdim}(\mathcal{C})=nd$ and $(n,d)=1$. If $(\text{FPdim}(X)^2,d)=1$ for all simple objects $X$ of $\mathcal{C}$, then we show that $\mathcal{C}$ contains a non-degenerate fusion subcategory $\mathcal{C}(\mathbb{Z}_d,q)$. In particular, we obtain that integral fusion categories of FP-dimensions $p^md$ such that $\mathcal{C}'\subseteq \text{sVec}$ are nilpotent and group-theoretical, where $p$ is a prime and $(p,d)=1$.

math.QA↗
arXiv2023

Universal construction in monoidal and non-monoidal settings, the Brauer envelope, and pseudocharacters

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.

math.QA↗
arXiv2022

Super invariant theory in positive characteristic

We study invariant theory of the general linear supergroup in positive characteristic. In particular, we determine when the symmetric group algebra acts faithfully on tensor superspace and demonstrate that the symmetric group does not always generate all invariants.

math.RT↗
arXiv2022

Monoidal Abelian Envelopes with a quotient property

We study abelian envelopes for pseudo-tensor categories with the property that every object in the envelope is a quotient of an object in the pseudo-tensor category. We establish an intrinsic criterion on pseudo-tensor categories for the existence of an abelian envelope satisfying this quotient property. This allows us to interpret the extension of scalars and Deligne tensor product of tensor categories as abelian envelopes, and to enlarge the class of tensor categories for which all extensions of scalars and tensor products are known to remain tensor categories. For an affine group scheme G, we show that pseudo-tensor subcategories of RepG have abelian envelopes with the quotient property, and we study many other such examples. This leads us to conjecture that all abelian envelopes satisfy the quotient property.

math.CT↗
arXiv2022

On Frobenius exact symmetric tensor categories

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 Ver_p (i.e., is the representation category of an affine group scheme in Ver_p) 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 Ver_p (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^{\otimes n} of dimension prime to p.

math.RT↗
arXiv2021

New incompressible symmetric tensor categories in positive characteristic

We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $\bf k$. If ${\rm char}({\bf k})=p>0$, we use this method to construct generalizations ${\rm Ver}_{p^n}$, ${\rm Ver}_{p^n}^+$ of the incompressible abelian symmetric tensor categories defined in arXiv:1807.05549 for $p=2$ and by Gelfand-Kazhdan and Georgiev-Mathieu for $n=1$. Namely, ${\rm Ver}_{p^n}$ is the abelian envelope of the quotient of the category of tilting modules for $SL_2(\bf k)$ by the $n$-th Steinberg module, and ${\rm Ver}_{p^n}^+$ is its subcategory generated by $PGL_2(\bf k)$-modules. We show that ${\rm Ver}_{p^n}$ are reductions to characteristic $p$ of Verlinde braided tensor categories in characteristic zero, which explains the notation. We study the structure of these categories in detail, and in particular show that they categorify the real cyclotomic rings $\mathbb{Z}[2\cos(2π/p^n)]$, and that ${\rm Ver}_{p^n}$ embeds into ${\rm Ver}_{p^{n+1}}$. We conjecture that every symmetric tensor category of moderate growth over $\bf k$ admits a fiber functor to the union ${\rm Ver}_{p^\infty}$ of the nested sequence ${\rm Ver}_{p}\subset {\rm Ver}_{p^2}\subset\cdots$. This would provide an analog of Deligne's theorem in characteristic zero and a generalization of the result of arXiv:1503.01492, which shows that this conjecture holds for fusion categories, and then moreover the fiber functor lands in ${\rm Ver}_p$.

math.RT↗