Buildings and combinatorial representation theory
March 26 to March 30, 2007
American Institute of Mathematics,
Palo Alto, California
and Monica Vazirani
This workshop will bring together researchers representing
different perspectives in combinatorial representation theory:
combinatorial, metric, and algebro-geometric.
It has emerged from recent works of Littelmann-Gaussent,
Kapovich-Leeb-Millson, Haines, and others, that Bruhat--Tits buildings
play an essential, not yet well-understood role in combinatorial
representation theory by providing a geometric realization to
existing combinatorial models and linking them to the algebro-geometric
tools of representation theory.
In particular the workshop goals include examining and comparing the
different approaches to the saturation theorem, with an emphasis on
the role of buildings, to get more precise answers (in all types)
and improve the proofs, and possibly also make a sensible Horn conjecture
in other types.
We further aim to understand the different combinatorial models involved
(such as Knutson-Tao honeycombs, MV polytopes, Littelmann path models,
canonical bases), provide a dictionary between them, and lay the groundwork
to enable researchers to apply these tools toward a host of related problems.
Material from the workshop
A list of participants.
The workshop schedule.
A report on the workshop activities.
- Arun Ram, Introduction
to Buildings and Combinatorial Representation Theory:
Kapovich, Overview of connections between buildings and
representation theory, and open problems:
- Arkady Berenstein,
Polytopal models and tropical geometry: pdf
- Jenia Tevelev, Tropical
Geometry and Affine Buildings: pdf
- Stephane Gaussent, LS
Galleries and MV Cycles: pdf
- Joel Kamnitzer,
Mirkovic-Vilonen cycles and polytopes: pdf
- David Nadler, Langlands: pdf
- Allen Knutson, Honeycombs:
Kapovich, Saturation: pdf
- Cristian Lenart,
Models for crystals: pdf
- Daniele Alessandrini.
of group representations.
- Arkady Berenstein
Zelevinsky. Tensor product multiplicities,
canonical bases and totally positive varieties. Inventiones
Mathematicae, 143 (1): 77 – 128, 2001.
- Anders S. Buch.
The saturation conjecture (after A. Knutson and T.
- Joel Kamnitzer.
Mirkovic Vilonen cycles and polytopes.
Kapovich. Generalized triangle
inequalities and their applications.
- Allen Knutson and
The honeycomb model of GL(n) tensor products I:
proof of the saturation conjecture. Journal of the American
Mathematical Society, 12 (4): 1055 – 1090, 1999.
Géométrique de l'involution de Schützenberger et
Applications. PhD thesis, l'Université Claude Bernard -
Lyon 1, 2006.
- Arun Ram.
Alcove walks, Hecke algebras, spherical functions,
crystals and column strict tableaux. Pure and Applied Mathematics
Quarterly, 2: 135 – 183, 2006.
- Guy Rousseau.
Euclidean buildings. Lecture notes from
Summer School 2004: Non-positively curved geometries, discrete groups
Other related information
- Notes on
buildings, Chevalley groups, the flag variety, the affine flag variety,
affine Hecke algebras, loop groups, central extensions, and GL(n)
- Software package for the LS-gallery/alcove
path model (Cristian
For your entertainment (or challenge).
Papers arising from the workshop: