Two different types of Deligne categories have been defined to interpolate the finite dimensional complex representations of the hyperoctahedral group. The first one, initially defined by Knop and then further studied by Likeng and Savage, uses a categorical analogue of the permutation representation as a tensor generator. The second one, due to Flake and Maassen, is tensor generated by a categorical analogue of the reflection representation. We construct a symmetric monoidal functor between the two and show that it is an equivalence of symmetric monoidal categories.
Mathematics Subject Classification: 17B10, 18D10, 20F55
We study the Deligne categories $Rep(GL_t(\mathbb{F}_q))$ for $t\in \mathbb{C}$. These categories interpolate the categories of finite dimensional complex representations of the finite general linear group $GL_n(\mathbb{F}_q)$. We describe the morphism spaces in this category via generators and relations.
We show that the generating object of this category (analogue of the representation $\mathbb{C}\mathbb{F}_q^n$ of $GL_n(\mathbb{F}_q)$) carries the structure of a Frobenius algebra with a compatible $\mathbb{F}_q$-linear structure; we call such objects $\mathbb{F}_q$-linear Frobenius spaces, and show that $Rep(GL_t(\mathbb{F}_q))$ is the universal symmetric monoidal category generated by such an $\mathbb{F}_q$-linear Frobenius space of categorical dimension $t$.
In the second part of the paper, we prove a similar universal property for a category of representations of $GL_{\infty}(\mathbb{F}_q)$.
Mathematics Subject Classification: 05E05, 18D10, 20C30.
We establish an explicit formula for the supercharacter of an irreducible representation of $\mathfrak{gl}(m|n)$. The formula is a finite sum with positive integer coefficients in terms of a basis $\mathcal{E}_{\mu}$ (Euler characters) of the super character ring. We prove a simple formula for the behaviour of $\mathcal{E}_{\mu}$ in the $\mathfrak{gl}(m|n)$ and $\mathfrak{osp}(m|2n)$-case under the map $ds$ on the supercharacter ring induced by the Duflo-Serganova cohomology functor $DS$. As an application we get a combinatorial formula for the superdimension of an irreducible $\mathfrak{osp}(m|2n)$ representation.
Mathematics Subject Classification: 17B10, 17B20, 17B55, 18D10.
We prove that the Duflo-Serganova functor $DS_x$ attached to an odd nilpotent element $x$ of $\mathfrak{osp}(m|2n)$ is semisimple, i.e. sends a semisimple representation $M$ of $\mathfrak{osp}(m|2n)$ to a semisimple representation of $\mathfrak{osp}(m-2k|2n-2k)$ where $k$ is the rank of $x$. We prove a closed formula for $DS_x(L(\lambda))$ in terms of the arc diagram attached to $\lambda$.
Mathematics Subject Classification: 17B10, 17B20, 17B55, 18D10
We introduce a generalization of the notion of a negligible morphism and study the associated tensor ideals and thick ideals. These ideals are defined by considering deformations of a given monoidal category $\mathcal{C}$ over a local ring $R$. If the maximal ideal of $R$ is generated by a single element, we show that any thick ideal of $\mathcal{C}$ admits an explicitely given modified trace function. As examples we consider various Deligne categories and the categories of tilting modules for a quantum group at a root of unity and for a semisimple, simply connected algebraic group in prime characteristic. We propose an elementary geometric description of the thick ideals in quantum and modular type A.
Mathematics Subject Classification: 18D10, 20G05
We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. Benson and Etingof conjectured that a certain limit of finite symmetric tensor categories is tensor equivalent to the finite dimensional representations of SL2 in characteristic 2. We use our results on the abelian envelopes to prove this conjecture.
Mathematics Subject Classification: 18D10, 20G05
We study induced model structures on Frobenius categories. In particular we consider the case where $\mathcal{C}$ is the category of comodules of a supercommutative Hopf algebra $A$ over a field $k$. Given a graded Hopf algebra quotient $A \to B$ satisfying some finiteness conditions, the Frobenius tensor category $\mathcal{D}$ of graded $B$-comodules with its stable model structure induces a monoidal model structure on $\mathcal{C}$. We consider the corresponding homotopy quotient $\gamma: \mathcal{C} \to Ho \mathcal{C}$ and the induced quotient $\mathcal{T} \to Ho \mathcal{T}$ for the tensor category $\mathcal{T}$ of finite dimensional $A$-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in $Ho \mathcal{T}$. We apply these results in the $Rep (GL(m|n))$-case and and study its homotopy category $Ho \mathcal{T}$.
Mathematics Subject Classification: 16T15, 17B10, 18D10, 18E40, 18G55, 20G05, 55U35
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 irreducible representations up to parity shifts, its quotient is a semisimple tannakian category $Rep(H_n)$ where $H_n$ is a pro-reductive algebraic group. We determine the connected derived subgroup $G_n \subset H_n$ and the groups $G_{\lambda} = (H_{\lambda})_{der}^0$ corresponding to the tannakian subcategory in $Rep(H_n)$ generated by an irreducible representation $L(\lambda)$. This gives structural information about the tensor category $Rep(GL(n|n))$, including the decomposition law of a tensor product of irreducible representations up to summands of superdimension zero. Some results are conditional on a hypothesis on $2$-torsion in $\pi_0(H_n)$.
Mathematics Subject Classification: 17B10, 17B20, 17B55, 18D10, 20G05.