Lars Thorge Jensen
I am a graduate student of Geordie Williamson at the
Max Planck Institute for Mathematics in Bonn.
Research Interests
My research interests focus on (Geometric) Representation Theory, Categorification and Homological Algebra.
Currently, I am trying to transfer as much as possible from Kazhdan-Lusztig cell theory in characteristic 0 to positive characteristic.
My project is described in more detail in my research proposal, which I wrote
while applying for postdoctoral positions in November 2016.
Topics which I am interested in and which I am trying to gain a deeper understanding of are:
- the Hecke category and its algebraic and geometric incarnations (Soergel bimodules, perverse and parity sheaves),
- Representation theory of reductive algebraic groups (in positive characteristic),
- (Affine) flag manifolds and the geometry of Schubert varieties,
- the affine Grassmannian and the geometric Satake equivalence,
- Kazhdan-Lusztig cell theory in characteristic 0,
- the 2-braid group and Rouquier complexes,
- Hodge theory.
Publications and Preprints
- The p-canonical Basis of Hecke algebras
Joint with Geordie Williamson.
To appear in Perspectives on Categorification, AMS Contemporary Math.
We describe a positive characteristic analogue of the
Kazhdan-Lusztig basis of the Hecke algebra of a crystallographic
Coxeter system and investigate some of its properties. Using
Soergel calculus we describe an algorithm to calculate this basis.
We outline some known or expected applications in modular representation theory.
We conclude by giving several examples.
- The 2-braid group and Garside normal form
To appear in MathZeitschrift.
We investigate the relation between the Garside normal form for positive braids and the 2-braid group defined by Rouquier.
Inspired by work of Brav and Thomas we show that the Garside normal form is encoded in the action of the 2-braid group on a
certain categorified left cell module. This allows us to deduce the faithfulness of the 2-braid group in finite type. We also
give a new proof of Paris' theorem that the canonical map from the generalized braid monoid to its braid group is injective in
arbitrary type.
Talks
- Mai 2017: The 2-braid group and Garside normal form, Séminaire d'algèbre and géométrie,
Laboratoire de Mathématiques Nicolas Oresme, Université de Caen.
- October 2016: The p-canonical basis of Hecke algebras and p-cells, Seminar on Representation
Theory, RIMS, Kyoto, Japan.
- June 2016: Blocks of Schur algebras and algebraic Groups, Max Planck Institute for Mathematics, Bonn.
- May 2016: The p-canonical basis of Hecke algebras, Oberseminar: Algebra, Zahlentheorie und
algebraische Geometrie, Uni Freiburg.
- November 2015: The Hopfological algebra of Khovanov and Qi, Max Planck Institute for Mathematics, Bonn.
- November 2015: The 2-braid group and Garside normal form, Séminaire Quantique, Institut de
Recherche Mathématique Avancée, Université de Strasbourg.
- August 2015: Perverse Sheaves on the Flag Variety and Category O (Part 2), BIREP Summer
School on Koszul Duality, Bad Driburg.
- April 2015: Structure Theory of Reductive Groups, Classification of Simple Modules,
Seminar on the Representation Theory of Reductive Groups in Positive Characteristic,
Max Planck Institute for Mathematics, Bonn.
- October 2014: Series of Survey Talks on Kazhdan-Lusztig Cell Theory, Max Planck Institute for
Mathematics, Bonn.
Here are some slides from one of the talks.
- June 2014: Representations of Quantum Groups for semi-simple Lie Algebras, Summer
School on Quiver Hecke Algebras, Cargèse, Corsica, France.
Other Documents
Link Collection
Homepages of some other graduate students:
Other useful links: