Related papers: Integral objects and Deligne's category Rep(S_t)

We prove the refined Loughran--Smeets conjecture of Loughran--Rome--Sofos for a wide class of varieties arising as products of conic bundles. One interesting feature of our varieties is that the subordinate Brauer group may be arbitrarily…

The purpose of this article is to study the existence of Deligne's tensor product of abelian categories by comparing it with the well-known ten- sor product of finitely cocomplete categories. The main result states that the former exists…

This article is a sequel to hep-th/9411050, q-alg/9412017. In Chapter 1 we associate with every Cartan matrix of finite type and a non-zero complex number $\zeta$ an abelian artinian category $\FS$. We call its objects {\em finite…

For polynomial representations of $GL_n$ of a fixed degree, H. Krause defined a new internal tensor product using the language of strict polynomial functors. We show that over an arbitrary commutative base ring $k$, the Schur functor…

The representation theory of 0-Hecke-Clifford algebras as a degenerate case is not semisimple and also with rich combinatorial meaning. Bergeron et al. have proved that the Grothendieck ring of the category of finitely generated…

We study the quotient of $\mathcal{T}_n = Rep(GL(n|n))$ by the tensor ideal of negligible morphisms. If we consider the full subcategory $\mathcal{T}_n^+$ of $\mathcal{T}_n$ of indecomposable summands in iterated tensor products of…

Let $V=\C^N$ with $N$ odd. We construct a $q$-deformation of $\End_{Sp(N-1)}(V^{\otimes n})$ which contains $\End_{U_q\sl_N}(V^{\otimes n})$. It is a quotient of an abstract two-variable algebra which is defined by adding one more generator…

Starting with a k-linear or DG category admitting a (homotopy) Serre functor, we construct a k-linear or DG 2-category categorifying the Heisenberg algebra of the numerical K-group of the original category. We also define a 2-categorical…

We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also…

We extend the classical Schur-Weyl duality between representations of the groups $SL(n,\C)$ and $\sN$ to the case of $SL(n,\C)$ and the infinite symmetric group $\sinf$. Our construction is based on a "dynamic," or inductive, scheme of…

We prove an analog of Deligne's theorem for finite symmetric tensor categories $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $2$. Namely, we prove that every such category $\mathcal{C}$…

We define an extension of the affine Brauer algebra, the type B/C affine Brauer algebra. This new algebra contains the hyperoctahedral group and it naturally acts on $END_K(X \otimes V^{\otimes k})$ for Orthogonal and Symplectic groups.…

We consider the possibility of semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. Conditions are given for the existence or nonexistence of coherent associative structures for such fusion rules,…

We discuss the category $\cal I$ of level zero integrable representations of loop algebras and their generalizations. The category is not semisimple and so one is interested in its homological properties. We begin by looking at some…

We prove that, for any fields $k$ and $\mathbb{F}$ of characteristic $0$ and any finite group $T$, the category of modules over the shifted Green biset functor $(kR_{\mathbb{F}})_T$ is semisimple.

Let $K$ be a field and $G$ be a group of its automorphisms. If $G$ is precompact then $K$ is a generator of the category of smooth (i.e. with open stabilizers) $K$-semilinear representations of $G$. There are non-semisimple smooth…

We introduce the marked Brauer algebra and the marked Brauer category. These generalize the analogous constructions for the ordinary Brauer algebra to the setting of a homogeneous bilinear form on a $\mathbb{Z}_2$-graded vector space. We…

The representation category $\mathcal{A} = Rep(G,\epsilon)$ of a supergroup scheme $G$ has a largest proper tensor ideal, the ideal $\mathcal{N}$ of negligible morphisms. If we divide $\mathcal{A}$ by $\mathcal{N}$ we get the semisimple…

We investigate objects in symmetric tensor categories that have simultaneously finite symmetric and finite exterior algebra. This forces the characteristic of the base field to be $p>0$, and the maximal degree of non-vanishing symmetric and…

We prove a universal property of Deligne's category $\uRep^{ab}(S_d)$. Along the way, we classify tensor ideals in the category $\uRep(S_d)$.