Albert-Ludwigs University Freiburg

Group of algebra and representation theory

Nicolas Libedinsky

I am a Von Humboldt postdoc at the University of Freiburg in the Algebra and Representation Theory group working with  Wolfgang Soergel. Before coming to Freiburg I was a PhD student at the University Paris 7, under the supervision of Raphaël Rouquier. My CV is available here.

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.
  

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.

• New bases of some Hecke algebras via Soergel bimodules , Advances in Math. 228 (2011) 1043-1067.
  

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.

• Presentation of right-angled Soergel categories by generators and relations , J. Pure Appl. Algebra 214 (2010), no. 12, 2265-2278.
  

I find a presentation of Soergel category (as a tensor category) by generators and relations in the right-angled Coxeter group case.

• Équivalences entre conjectures de Soergel , Journal of Algebra, 320 (2008) 2695-2705.
  

I prove that in Soergel's conjecture it is equivalent to use the "easy" geometric representation or the "difficult" reflection faithful representation used before.

• Sur la catégorie des bimodules de Soergel , Journal of Algebra 320 (2008) 2675-2694.
  

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)
  

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.

• My Bachelor thesis, under the supervision of Jorge Soto-Andrade (Universidad de Chile)
  

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).

• My master 1 thesis, under the supervision of Marc Rosso (Ecole Normale Superieure)
  

This is a gentle introduction (in french) to the simplest, non trivial representation theory of quantum groups, namely, quantum sl_2.

Other activities

• Criticas euro mundo global
  

• MOHEIO
  

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