- Dr. Nicolas Libedinsky
-
Mathematisches Institut
Albert-Ludwigs-Universitat Freiburg
Eckerstrase 1
79104 Freiburg im Breisgau
GermanyTel:+49-761-203-5568
Email: nlibedinsky@gmail.com
Research interests
-
Soergel's category of bimodules, mostly in their relation with Lusztig's conjecture in algebraic groups,
with the positivity of Kazhdan-Lusztig polynomials
and to Khovanov homological invariant of links.
- 2-braid groups as defined by R. Rouquier. In particular I would like to understand the interaction between its algebraic and topologic properties, how the Garside structure in braid groups can be "seen" in the 2-braid groups context and Rouquier's conjecture saying that 2-braid groups categorify braid groups for every Coxeter system.
Papers and preprints
- Standard objects in 2-braid groups,
joint with Geordie Williamson,
preprint.
- New bases of some Hecke algebras via Soergel bimodules ,
Advances in Math.
228 (2011) 1043-1067.
- Presentation of right-angled Soergel categories
by generators and relations ,
J. Pure Appl. Algebra
214 (2010), no. 12, 2265-2278.
- Équivalences entre conjectures de Soergel ,
Journal of Algebra, 320 (2008) 2695-2705.
- Sur la catégorie des bimodules de Soergel ,
Journal of Algebra 320
(2008) 2675-2694.
For any Coxeter system, we establish the existence of analogues of standard and costandard objects in 2-braid groups, thus proving a conjecture of Rouquier. This should lead to important information about the "Garside structure" of 2-braid groups and hopefully this will lead to prove that 2-braid groups categorify braid groups for any Coxeter system, as was also conjectured by Rouquier.
This is a first attempt to find explicitely Soergel indecomposable bimodules for extra-large Coxeter systems. I believe that similar bases as the ones described in this paper should exist for every Coxeter system. I expect that a generalization of this methods will give a proof of Soergel's (and Kazhdan-Lusztig positivity) conjecture for large Coxeter systems.
I find a presentation of Soergel category (as a tensor category) by generators and relations in the right-angled Coxeter group case.
I prove that in Soergel's conjecture it is equivalent to use the "easy" geometric representation or the "difficult" reflection faithful representation used before.
I find explicitly all morphisms in Soergel's category and between projective objects in the principal block of BGG category O.
Unpublished texts
- My Ph.D thesis, under the supervision of Raphaël Rouquier (Paris 7)
- My Bachelor thesis, under the supervision of Jorge Soto-Andrade (Universidad de Chile)
- My master 1 thesis, under the supervision of Marc Rosso (Ecole Normale Superieure)
Chapter 1 is a essentially a version of this paper (Soergel bimodules explained by Soergel) with explanations of the obscure points. Sections 2.4 and 2.5 are original and are not included in any other paper. I give a different proof of the fact that Rouquier complexes satisfy the braid relations.
I think that this is at least partially original. I found in a geometric way some intertwining operators that give the irreducible representations of the symmetric group (not very well written).
This is a gentle introduction (in french) to the simplest, non trivial representation theory of quantum groups, namely, quantum sl_2.