Martina
Lanini

Research Fellow

Department Mathematics and Statistics

The University of Melbourne

Parkville VIC 3010

Australia

martina.lanini@unimelb.edu.au

New Mini workshop and Conference on "Representation Theory in Geometry, Topology, and combinatorics", 28 October-7 November, 2013, The University of Melbourne, Australia

Research

My research addresses the interplay between representation theory, algebraic combinatorics, and algebraic geometry.

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).

Periodic
Structures
in
the
Affine
Category
O at a Positive Level and
Applications to Critical Level Representation Theory

Given a symmetrisable Kac-Moody algebra g, we can consider its Bruhat graph. Fiebig proved in his paper, that it is possible to describe 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, to appear in 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. Along the way, I introduce 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.

Categorification of a parabolic Hecke module via sheaves on moment graphs, preprint 2012, arXiv:1208.1492, to appear in Pac. J. Math.

In 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.

Moment graphs and Kazhdan-Lusztig polynomials, DMTCS Proceedings, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), 491-502.

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.

Kazhdan-Lusztig combinatorics in the moment graph setting, preprint 2011, arXiv:1103.2282, J. of Alg. 370 (2012), 152-170.

In 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 they are not perverse!

Research Fellow

Department Mathematics and Statistics

The University of Melbourne

Parkville VIC 3010

Australia

martina.lanini@unimelb.edu.au

New Mini workshop and Conference on "Representation Theory in Geometry, Topology, and combinatorics", 28 October-7 November, 2013, The University of Melbourne, Australia

Research

My research addresses the interplay between representation theory, algebraic combinatorics, and algebraic geometry.

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).

Given a symmetrisable Kac-Moody algebra g, we can consider its Bruhat graph. Fiebig proved in his paper, that it is possible to describe 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, to appear in 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. Along the way, I introduce 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.

Categorification of a parabolic Hecke module via sheaves on moment graphs, preprint 2012, arXiv:1208.1492, to appear in Pac. J. Math.

In 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.

Moment graphs and Kazhdan-Lusztig polynomials, DMTCS Proceedings, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), 491-502.

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.

Kazhdan-Lusztig combinatorics in the moment graph setting, preprint 2011, arXiv:1103.2282, J. of Alg. 370 (2012), 152-170.

In 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 they are not perverse!