arXiv subjects
Publications and source records attributed to Thorsten Heidersdorf.
We establish the existence of abelian envelopes for interpolation categories of wreath product groups $G\wr S_n$, for a fixed finite group $G$ with the symmetric groups $S_n$, for $n\ge0$. Our approach consists of showing directly via essentially combinatorial methods that certain generalized restriction functors admit adjoints.
We show that any upper finite or essentially finite highest weight category where the standard objects have linear projective resolutions and the costandard objects have linear injective resolutions is Koszul. This extends the result of Agoston, Dlab, and Lukacs to the case of infinite highest weight categories. We apply this result to Khovanov algebras and representations of classical Deligne categories and show that these are Koszul.
We study the semisimplification of the full karoubian subcategory generated by the irreducible finite dimensional representations of the algebraic supergroup $GL(m|n)$ over an algebraically closed field of characteristic zero. This semisimplification is equivalent to the representations of a pro-reductive group $H_{m|n}$. We show that there is a canonical decomposition $H_{m|n} \cong GL(m\!-\! n) \times H_{n|n}$, thereby reducing the determination of $H_{m|n}$ to the equal rank case $m\! =\! n$ which was treated in a previous paper.
A general construction of Knop creates a symmetric monoidal category $\mathcal{T}(\mathcal{A},δ)$ from any regular category $\mathcal{A}$ and a fixed degree function $δ$. A special case of this construction are the Deligne categories $\underline{\operatorname{Rep}}(S_t)$ and $\underline{\operatorname{Rep}}(GL_t(\mathbb{F}_q))$. We discuss when a functor $F:\mathcal{A} \to \mathcal{A}'$ between regular categories induces a symmetric monoidal functor $\mathcal{T}(\mathcal{A},δ) \to \mathcal{T}(\mathcal{A}',δ')$. We then give a criterion when a pair of adjoint functors between two regular categories $\mathcal{A}, \ \mathcal{A}'$ lifts to a pair of adjoint functors between $\mathcal{T}(\mathcal{A},δ)$ and $\mathcal{T}(\mathcal{A}',δ')$.
We develop the theory of projective endofunctors for modules of Khovanov algebras $K$ of type B. In particular we compute the composition factors and the graded layers of the image of a simple module under such a projective functor. We then study variants of such functors for a subquotient $e\tilde{K}e$. Via a comparison of two graded lifts of the Brauer algebra we relate the Khovanov algebra to the Brauer algebra and use this to show that projective functors describe translation functors on representations of the orthosymplectic supergroup $\mathrm{OSp}(r|2n)$. As an application we get a description of the Loewy layers of indecomposable summands in tensor powers of the natural representation of $\mathrm{OSp}(r|2n)$.
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.
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 $G_λ = (H_λ^0)_{der}$ corresponding to the tannakian subcategory in $Rep(H_n)$ generated by an irreducible representation $L(λ)$. 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 $π_0(H_n)$.
We study the Deligne interpolation categories $\underline{\mathrm{Rep}}(GL_{t}(\mathbb{F}_q))$ for $t\in \mathbb{C}$, first introduced by F. Knop. 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 (an 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 $\underline{\mathrm{Rep}}(GL_{t}(\mathbb{F}_q))$ 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 $GL_{\infty}(\mathbb{F}_q)$.
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 prove an elementary geometric description of the thick ideals in quantum type A and propose a similar one in the modular case.
We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. As a main result we prove that the category of finite-dimensional representations of a semisimple simply connected algebraic group is the abelian envelope of the category of tilting modules. Benson and Etingof conjectured that a certain limit of finite symmetric tensor categories is tensor equivalent to the finite dimensional representations of $SL_2$ in characteristic $2$. We use our results on the abelian envelopes to prove this conjecture and its variants for any prime $p$.
We establish an explicit formula for the character of an irreducible finite-dimensional representation of $\mathfrak{gl}(m|n)$. The formula is a finite sum with integer coefficients in terms of a basis $\mathcal{E}_μ$ (Euler characters) of the character ring. We prove a simple formula for the behaviour of the ``superversion'' of $\mathcal{E}_μ$ 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 combinatorial formulas for superdimensions, dimensions and $\mathfrak{g}_0$-decompositions for $\mathfrak{gl}(m|n)$ and $\mathfrak{osp}(m|2n)$.
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 $γ: \mathcal{C} \to Ho \mathcal{C}$ 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 study its homotopy category $Ho \mathcal{T}$.
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 $DS_x(L(λ))$ in terms of the arc diagram attached to $λ$.
We establish a connection between two settings of representation stability for the symmetric groups $S_n$ over $\mathbb{C}$. One is the symmetric monoidal category ${\rm Rep}(S_{\infty})$ of algebraic representations of the infinite symmetric group $S_{\infty} = \bigcup_n S_n$, related to the theory of ${\bf FI}$-modules. The other is the family of rigid symmetric monoidal Deligne categories $\underline{\rm Rep}(S_t)$, $t \in \mathbb{C}$, together with their abelian versions $\underline{\rm Rep}^{ab}(S_t)$, constructed by Comes and Ostrik. We show that for any $t \in \mathbb{C}$ the natural functor ${\rm Rep}(S_{\infty}) \to \underline{\rm Rep}^{ab}(S_t)$ is an exact symmetric faithful monoidal functor, and compute its action on the simple representations of $S_{\infty}$. Considering the highest weight structure on $\underline{\rm Rep}^{ab}(S_t)$, we show that the image of any object of ${\rm Rep}(S_{\infty})$ has a filtration with standard objects in $\underline{\rm Rep}^{ab}(S_t)$. As a by-product of the proof, we give answers to the questions posed by P. Deligne concerning the cohomology of some complexes in the Deligne category $\underline{\rm Rep}(S_t)$, and their specializations at non-negative integers $n$.
We define and study cohomological tensor functors from the category $T_n$ of finite-dimensional representations of the supergroup $Gl(n|n)$ into $T_{n-r}$ for $0 <r \leq n$. In the case $DS: T_n \to T_{n-1}$ we prove a formula $DS(L) = \bigoplus Π^{n_i} L_i$ for the image of an arbitrary irreducible representation. In particular $DS(L)$ is semisimple and multiplicity free. We derive a few applications of this theorem such as the degeneration of certain spectral sequences and a formula for the modified superdimension of an irreducible representation.
We describe the image of the canonical tensor functor from Deligne's interpolating category $Rep(GL_{m-n})$ to $Rep(GL(m|n))$ attached to the standard representation. This implies explicit tensor product decompositions between any two projective modules and any two Kostant modules of $GL(m|n)$, covering the decomposition between any two irreducible $GL(m|1)$-representations. We also obtain character and dimension formulas. For $m>n$ we classify the mixed tensors with non-vanishing superdimension. For $m=n$ we characterize the maximally atypical mixed tensors and show some applications regarding tensor products.
The representation category $\mathcal{A} = Rep(G,ε)$ 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 representation category of a pro-reductive supergroup scheme $G^{red}$. We list some of its properties and determine $G^{red}$ in the case $GL(m|1)$.
We describe indecomposable objects in Deligne's category $\underline{\operatorname{Re}}\!\operatorname{p}(O_δ)$ and explain how to decompose their tensor products. We then classify thick ideals in $\underline{\operatorname{Re}}\!\operatorname{p}(O_δ)$. As an application we classify the indecomposable summands of tensor powers of the standard representation of the orthosymplectic supergroup up to isomorphism.