Thorsten Heidersdorf
Max Planck Institut fuer Mathematik, Room 307
Vivatsgasse 5
53111 Bonn
Email: thorsten@math.uni-bonn.de
Research interests: Monoidal structures in representation theory. In particular: Deligne categories (certain families of universal tensor categories), representations of algebraic supergroups, monoidal structures for representations of quantum groups, their super versions and possible applications to quantum knot invariants.
Teaching
SS 2019: Algebra 1 ("commutative algebra")Publications / Preprints / Reports
- Homotopy quotients and comodules of supercommutative Hopf algebras. (joint with R.Weissauer), Preprint 2019, (arXiv version)
Abstract
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 - Semisimplification of representation categories. Oberwolfach Reports 1848, 2018. (local link)
- On classical tensor categories attached to the irreducible representations of the General Linear Supergroup $GL(n|n)$. (joint with R. Weissauer), Preprint 2018, 78 pages, (arXiv version)
Abstract
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. - Deligne categories and representations of the infinite symmetric group. (joint with Daniel Barter, Inna Entova-Aizenbud), to appear in Advances in Mathematics, 31 pages, (arXiv version)
Abstract
We establish a connection between two settings of representation stability for the symmetric groups $S_n$ over $\C$. One is the symmetric monoidal category $\Rep(S_{\infty})$ 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 $\underline{Rep}(S_t)$, $t \in \C$, together with their abelian versions $\underline{Rep}^{ab}(S_t)$, constructed by Comes and Ostrik. We show that for any $t \in \C$ the natural functor $\Rep(S_{\infty}) \to \underline{Rep}^{ab}(S_t)$ is an exact symmetric faithful monoidal functor, and compute its action on the simple representations of $S_{\infty}$. Considering the highest weight structure on $\underline{Rep}^{ab}(S_t)$, we show that the image of any object of $Rep(S_{\infty})$ has a filtration with standard objects in $\underline{Rep}^{ab}(S_t)$. 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 $\underline{Rep}(S_t)$, and their specializations at non-negative integers $n$.
Mathematics Subject Classification: 05E05, 18D10, 20C30. - On supergroups and their semisimplified representation categories. 28pages, to appear in: Algebr. Represent. Theory
(arXiv version)
Abstract
The representation category $\mathcal{A} = Rep(G,\epsilon)$ of a supergroup scheme $G$ has a largest proper tensor ideal, the ideal $\calN$ of negligible morphisms. If we divide $\mathcal{A}$ by $\calN$ 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 $GL(m|1)$.
Mathematics Subject Classification: 17B10, 18D10 - Thick Ideals in Deligne's category $\underline{Rep}(O_\delta)$. (joint with Jonny Comes), J. Algebra {480} Pages 237-265 (2017).
(arXiv version)
Abstract
We describe indecomposable objects in Deligne's category $\underline{Rep}(O_\delta)$ and explain how to decompose their tensor products. We then classify thick ideals in $\underline{Rep}(O_\delta)$. As an application we classify the indecomposable summands of tensor powers of the standard representation of the orthosymplectic supergroup up to isomorphism.
Mathematics Subject Classification: 17B10, 18D10. - Pieri type rules and $GL(2|2)$ tensor products. (joint with R. Weissauer), Preprint 2015, 24 pages,
(arXiv version)
Abstract
We derive a closed formula for the tensor product of a family of mixed tensors using Deligne's interpolating category $\underline{Rep}(GL_{0})$. We use this formula to compute the tensor product of a family of irreducible $GL(n|n)$-representations. This includes the tensor product of any two maximal atypical irreducible representations of $GL(2|2)$.
Mathematics Subject Classification: 17B10, 17B20. - Cohomological tensor functors on representations of the General Linear Supergroup.
(joint with R. Weissauer), 128 pages, to appear in Mem. Am. Math. Soc.
(arXiv version)
Abstract
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 $DS(L) = \bigoplus \Pi^{n_i} L_i$ 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.
Mathematics Subject Classification: 17B10, 17B20, 17B55, 18D10, 20G05. - Mixed tensors of the General Linear Supergroup.
J.Algebra {491} Pages 402-446
(2017).
(arXiv version)
Abstract
We describe the image of the canonical tensor functor from Deligne's interpolating category $\underline{Rep}(GL_{m-n})$ 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.
Mathematics Subject Classification: 17B10, 17B20, 18D10.