"Representation Theory in Geometry,
Topology, and combinatorics", 28 October-7 November, 2013,
The University of Melbourne, Australia
My research addresses the
interplay between representation theory, algebraic combinatorics, and
I'm currently working on three main projects:
Fock Spaces, Hecke Algebras
and Representations of Quantum groups at a root of unity
In their paper, Leclerc and Thibon
gave a combinatorial description of the Fock Space for the general
linear group and showed that the canonical basis they introduced here coincides with a
Kazhdan-Lusztig basis for certain parabolic Hecke modules. The goal of this ongoing
project with Arun Ram and Paul Sobaje is to generalise Leclerc-Thibon
construction to any complex reductive algebraic group. By Lusztig
conjecture on quantum groups at a root of unity, this will provide us
with a combinatorial model to compute multiplicities of the simples in
the Weyl modules (in the root-of-1-case).
O at a Positive Level and
Applications to Critical Level Representation Theory
we can consider its Bruhat graph.
Fiebig proved in his paper,
indecomposable projective objects of (a deformed version of) category O
at a positive level by looking at indecomposable Braden-MacPherson
sheaves on the Bruhat graph. In an ongoing joint project with
Peter Fiebig, we consider Braden-MacPherson sheaves on the stable
moment graph of g (that I have
defined in this paper) and we investigate the
corresponding modules in the affine category O at a positive level.
Moment Graph Combinatorics for Semi-infinite Schubert Varieties
Braden and MacPherson proved
in this paper that it is
possible to compute the stalks of the local intersection cohomology of
the Schubert varieties by looking at finite intervals of their
moment graphs and performing a combinatorial algorithm. Any
affine Kac-Moody algebra comes with its stable moment graph (that I
have defined in this paper
) and performing the
Braden-MacPherson algorithm on finite intervals of this graph allows us
to compute the stalks of the local intersection cohomology of
semi-infinite Schubert varieties. I would like to understand the
geometric reason of this phenomenon.
Papers and Preprints
On the stable moment graph of an
affine Kac-Moody algebra,
preprint 2012, arXiv:1210.3218,
Trans. Amer. Math. Soc.
In this paper, I introduce the stable moment
graph of an affine Kac-Moody algebra g: a certain oriented graph with
set of vertices in bijection with the alcoves in the fundamental
chamber and with edges labeled by coroots of g. The study of
indecomposable Braden-MacPherson sheaves on finite intervals (deep
enough in the fundamental chamber) of the stable moment graph leads to
a categorical analogue of a stabilisation property for affine
Kazhdan-Lusztig polynomials proven by Lusztig here.
the notion of push-forward functor in the
category of sheaves on a moment graph and prove that it is right
adjoint to the pullback functor introduced in my previous paper.
via sheaves on
preprint 2012, arXiv:1208.1492, to
in Pac. J. Math.
this work, I generalise Fiebig's definition of the category of special
modules of a Coxeter group (see his paper) to the parabolic setting
and show that this provides a weak categorification of a parabolic
Hecke Module. I define here left translation functors, that turn out to
be a very important tool in my paper on the stable moment graph.
In the last section, I briefly discuss the relation of parabolic
special modules with non-critical singular blocks of (an equivariant
version of) category O for symmetrisable Kac-Moody algebras.
polynomials, DMTCS Proceedings, 24th International
Conference on Formal Power Series and Algebraic Combinatorics (FPSAC
This is an
extended abstract for the FPSAC conference in 2012, that took place in
Nagoya, Japan. It is a survey of the main results of my dissertation,
focusing on the combinatorial side of the story. In particular, I
recall some properties of Braden-MacPherson sheaves, proven here and here, that provide a categorical
lifting of properties of Kazhdan-Lusztig polynomials. I also discuss
the definition of category of k-moment graphs.
2011, arXiv:1103.2282, J. of
Alg. 370 (2012), 152-170.
this paper, I introduce several strategies to interpret in terms of
Braden-MacPherson sheaves certain elementary properties of
Kazhdan-Lusztig polynomials. I define the pullback functor in the
category of sheaves on a moment graph and prove that the pullback of an
isomorphism maps indecomposable Braden-MacPherson sheaves to
indecomposable Braden-MacPherson sheaves. This fact provides a trick
that we use also in the proof of the main theorem of the paper on the stable moment graph.
Since the arguments we provide work in any characteristic (under
certain technical assumptions), by a theorem of Fiebig and
Williamson proven here, the result of this paper
tell us that the stalks of indecomposable parity sheaves in positive
characteristic behave very similarly to the ones of intersection
cohomology complexes in characteristic zero, even in cases in which
are not perverse!