Homepage of Thorsten Heidersdorf

## Privatdozent Dr. Thorsten Heidersdorf

Mathematisches Institut,
Room N1.006
Endenicher Allee 60
53115 Bonn

Office hours: By appointment

Email: thorsten@math.uni-bonn.de

Research group on Algebra and Representation theory

Research interests: Representation theory and tensor categories. In particular:
• Deligne categories (certain families of universal tensor categories)
• Diagram algebras
• Stable representation theory of the symmetric group and some finite groups of Lie type
• Construction and classification of thick ideals and tensor ideals
• Representations of algebraic supergroups, in particular tensor product decompositions, the Duflo-Serganova cohomology functor and character/dimension formulae; applications to mathematical physics
• Homotopical methods
• Representations of quantum groups at roots of unity, their super versions and possible applications to quantum knot invariants.
• Tilting modules for quantum groups and algebraic groups

For general information about representation theory I recommend the Wikipedia entry and the Quanta article The ‘Useless’ Perspective That Transformed Mathematics. I have written an outreach article about one of my favorite projects for the MfO snapshot series: Truncated fusion rules for supergroups.

### Teaching

SS 2019: Algebra 1 ("commutative algebra")

WS 2019: Graduate Seminar on Representation Theory: Hopf algebras and tensor categories. Seminar page

WS 2020/21: Advanced Algebra I: Lie algebras and their representations

SS 2021:
WS 2021/22:
• Graduate Seminar on Advanced Algebra: Nichols algebras and generalizations of universal enveloping algebras
• Coordinator/Assistent Linear Algebra 1
• Oberseminar Representation Theory (joint with Jacob Matherne and Catharina Stroppel)
SS 2022:
WS 2022: Graduate Seminar on Representation Theory: Representations of Lie superalgebras

Current and former students

If you are interested in writing a Bachelor or Master thesis under my supervision, please contact me. Typical topics range from representations of Lie superalgebras and quantum groups to monoidal categories.

### Publications / Preprints / Reports

1. Truncated fusion rules for supergroups, written for the MfO (Oberwolfach) snapshot series, preliminary version

2. Gruson-Serganova character formulas and the Duflo-Serganova cohomology functor. (joint with M. Gorelik), Preprint 2021,
Abstract

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.

3. Semisimplification for algebraic supergroups., Oberwolfach reports 2106a, 2021. (local link)

4. Interpolation and semisimplification of monoidal categories., Habilitation thesis 2020, University of Bonn.

5. Semisimplicity of the DS functor for the orthosymplectic Lie superalgebra. (joint with M. Gorelik), Advances in Mathematics 394, 2022, (arXiv version)
Abstract

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

6. Generalized negligible morphisms and their tensor ideals. (joint with H. Wenzl), Selecta Mathematica volume 28, 2022, (arXiv version)
Abstract

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

7. Monoidal abelian envelopes and a conjecture of Benson--Etingof. (joint with K. Coulembier, I. Entova-Aizenbud), to appear in Algebra & Number Theory, Preprint 2019, (arXiv version)
Abstract

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

8. Homotopy quotients and comodules of supercommutative Hopf algebras. (joint with R.Weissauer), Preprint 2019, 77 pages, (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

9. Semisimplification of representation categories. Oberwolfach Reports 1848, 2018. (local link)

10. On classical tensor categories attached to the irreducible representations of the General Linear Supergroup $GL(n|n)$. (joint with R. Weissauer), Preprint 2018, 97 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.

11. Deligne categories and representations of the infinite symmetric group. (joint with Daniel Barter, Inna Entova-Aizenbud), Advances in Mathematics 346, 2018, 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.

12. On supergroups and their semisimplified representation categories. 28 pages, Algebr. Represent. Theory Vol.22, Issue 4, 2019 (arXiv version)
AbstractThe 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

13. 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.

14. Pieri type rules and $GL(2|2)$ tensor products. (joint with R. Weissauer), Alg. Repr. Theory 24 (2021), 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.

15. Cohomological tensor functors on representations of the General Linear Supergroup. (joint with R. Weissauer), 128 pages, Mem. Am. Math. Soc. Volume 270 Number 1320 (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.

16. 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.