Summer School
Geometry and Combinatorics in Representation Theory of Lie Algebras Crystals, Path-Model, Quiver Varieties October 4 - 8, 2010 Cologne, Germany |
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Organizers Ghislain Fourier (Cologne) Michael Ehrig (Bonn) | ||

Supported by DFG-Schwerpunkt 1388 "Darstellungstheorie" |

Monday | Tuesday | Wednesday | Thursday | Friday | |
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9:00 - 10:00 | Feigin I | Feigin II | Feigin III | Overview | Gaussent IV |

10:00 - 10:30 | Coffee | Coffee | Coffee | Coffee | Coffee |

10:30 - 11:30 | Gaussent I | Gaussent II | Gaussent III | Feigin IV | Savage IV |

11:30 - 12:00 | M. Lanini | T. Pecher | Savage III (until 12:30) | M. Meng | Shimozono IV (until 12:30) |

12:00 - 14:00 | Lunch | Lunch | Lunch | Lunch | Lunch |

14:00 - 15:00 | Savage I | Savage II | Shimozono III | ||

15:00 - 15:30 | Coffee | Coffee | Coffee | ||

15:30 - 16:30 | Shimozono I | Shimozono II | C. Gutschwager (15:30 - 16:00) C. Ikenmeyer (16:00 - 16:30) | ||

16:30 - 17:00 | W. Gnedin | M. Bennett | S. Markouski |

In our lectures we will focus on the PBW deformation of the Lie theory. This deformation is based on the deformation of the universal enveloping algebra of a nilpotent subalgebra to the symmetric algebra and turns out to be very interesting from its own side and because of various applications to the classical Lie theory. We will discuss representation theoretical, algebro-geometric and combinatorial aspects of the story.

Our lectures will be based on the preprints arXiv:1002.0674 and arXiv:1007.0646.

I Buildings

This first lecture will be devoted to recall the definition of the Bruhat-Tits building associated to a reductive group over a discrete valuation field and the one of its building at infinity.

("Buildings", by Brown, or, "Lectures on Buildings" by Ronan, of course, and more advanced, Bruhat and Tits, "Groupes réductifs sur un corps local I" and II)

II MV-cycles versus LS-paths

Both groups of letters (MV and LS) will be defined and we will introduce LS-galleries to show the connection between them.

(articles of Littelmann : "A Littlewood-Richardson rule for symmetrizable Kac-Moody algebra" or "Pahs and root operators in representation theory"; articles of Mirkovic and Vilonen "Perverse sheaves on affine grassmannians and Langlands duality"; our article with Littelmann "LS galleries, the path model, and MV cycles")

III Saturation

We will give a sketch of the proof of the saturation conjecture given by Kapovich, Leeb and Millson. We will focus on the folding and unfolding of triangles in the BT-building.

(article of Kapovich, Leeb and Millson "Polygons in buildings and their refined side lengths", and also, "A path model for geodesics in Euclidean buildings and its applications to representation theory")

IV Hall-Littlewood polynomials

By going to galleries in the one-skeleton of the BT-building, we will be able to give a generalisation of a type-A-formula of MacDonald to any type.

(article with Littelmann : One-skeleton galleries, Hall-Littlewood polynomials and the path model")

We will give an introduction to the geometric realization of crystal graphs via the quiver varieties of Lusztig and Nakajima. The emphasis is on motivating the constructions through concrete examples. If times permits, we will also discuss the relation between the geometric construction of crystals and combinatorial realizations using Young tableaux.

Main reference: A. Savage, Lectures on geometric realization of crystals, arXiv:1003.5019. These are lecture notes from a similar course given at a summer school at The University of Ottawa in 2009. References to primary sources can be found in the bibliography of these notes.

For a combinatorialist, Schubert calculus involves a nice ring with distinguished basis called the Schubert basis, and the goal is to give obviously positive formulas for the nonnegative integers which give the expansion into the Schubert basis, of the product of basis elements.

These rings come from nice spaces such as projective space, Grassmannians, and flag varieties. The Schubert bases come from paving these spaces by cells, copies of (complex) Euclidean spaces of various dimensions.

Linear algebra will be used to informally describe the cell decompositions and the inclusion relations among the closures of the cells. In this way the Bruhat order on permutations and partitions will arise naturally.

Some time will be spent on explicit realizations of the nice rings and their Schubert bases and especially the combinatorics arising from them.

Although this won't be covered in the lectures, flag varieties can be used to construct the finite-dimensional representations of semisimple Lie groups via the Borel-Weil-Bott construction.

The product of two Schubert classes can be decomposed as a linear combination of Schubert classes. The Littlewood-Richardson coefficients are the coefficients in this linear combination. We will give a lower bound for the number of Schubert classes in this linear combination. Furthermore, we will present an inequality satisfied by the Littlewood-Richardson coefficients and present the equivalence of the Schubert product and skew characters of the symmetric group and skew Schur functions.

Littlewood-Richardson coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group GL(n,C). They have a wide variety of interpretations in combinatorics, representation theory and geometry. Mulmuley and Sohoni pointed out that it is possible to decide the positivity of Littlewood-Richardson coefficients in polynomial time. This follows by combining the saturation property of Littlewood-Richardson coefficients (shown by Knutson and Tao 1999) with the well-known fact that linear optimization is solvable in polynomial time. We design an explicit combinatorial polynomial time algorithm for deciding the positivity of Littlewood-Richardson coefficients. This algorithm is highly adapted to the problem and it is based on ideas from the theory of optimizing flows in networks.

In an ongoing project we try to interpret well-known identities of Kazhdan-Lusztig polynomials on a categorical level. In particular, we translate such identities into the category of Braden-MacPherson sheaves on a moment graph. These sheaves became important in representation theory due to recent work of Fiebig, Juteau, Mautner and Williamson.

Global Weyl modules, for generalized loop algebras, have been defined and studied for any dominant integral weight. We show that the space of morphisms between Global Weyl modules shares some properties with the space of morphisms between Verma modules.

We give an explicit crystal morphism between Nakajima monomials and the subset of those monomials which give a realization of crystal bases for finite dimensional irreducible modules over the quantized enveloping algebra for Lie algebras of type A and C. This morphism provides a connection between arbitrary Nakajima monomials and Nakashima Kashiwara tableaux. Moreover it gives rise to an insertion scheme for Nakajima monomials which coincides with the insertion scheme for tableaux under the above connection.

In the talk we discuss certain results on diagonal locally simple Lie algebras. The main part of these results is presented in the author’s preprint ”Locally simple subalgebras of diagonal Lie algebras”, arxiv.org/abs/1002.1684. Diagonal Lie algebras are defined as direct limits of finite-dimensional Lie algebras under diagonal injective homomorphisms. An explicit description of the isomorphism classes of diagonal locally simple Lie algebras is given in the paper [A. A. Baranov, A. G. Zhilinskii, Diagonal direct limits of simple Lie algebras, Comm. Algebra, 27 (1998), 2749-2766]. In the preprint already mentioned all locally simple Lie subalgebras of any diagonal locally simple Lie algebra are classified up to isomorphism. In the talk we discuss some methods used for this classification.

Many aspects of the representation theory of GL(n) can be described via Standard Young tableaux: E.g. the definition of Schur polynomials or the decomposition of tensor products by means of Littlewood-Richardson coefficients. The aim of this talk is to present a "symplectic" version of tableaux and to sketch their applications to the representation theory of Sp(2n).

In my talk, I am going to explain a new approach to classify the indecomposable Harish-Chandra modules over the Lie group SL(2,R). This problem was posed by I.M Gelfand on the International Congress of Mathematicians in 1970 and studied by many authors including Nazarova and Roiter, Khoroshkin, Bondarenko, Crawley-Boevey, Deng and others. However, the exact combinatorics of the indecomposable Harish-Chandra modules as well as their homological properties remained to be clarified. One can show that this classification problem reduces to a description of representations of a certain quiver with relations. The category of these representations is equivalent to the non-trivial block of the category of perverse sheaves on complex (2 × 2)-matrices stratified by rank, which were studied by Braden and Grinberg in 1999. Our method extends a construction, suggested by Burban and Drozd in 2002. It is based on the technique of derived categories and certain problems of linear algebra called representations of bunches of semi-chains which were solved by Bondarenko in 1988. Our approach leads to a complete description of invariants, projective resolutions and bases of indecomposable Harish-Chandra modules over SL(2,R).