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Algebraic Combinatorics in Europe
(ACE)
(September 1, 2002 - August 31, 2005)
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Work Programme
Algebraic Combinatorics concerns itself with the study of combinatorial problems arising from other branches of mathematics and, on the other hand, with the application and use of techniques coming from other parts of mathematics to combinatorial problems. Interactions with other fields, such as Coxeter Group Theory, Group Representation Theory, Lie Algebras and their Representation Theory, Commutative Algebra, Number Theory, Algebraic Geometry, Topology, and Invariant Theory, and other sciences, such as Statistical Physics and Computer Science are intrinsic in this research.The main areas of investigation in our project are (but will not be restricted to):
| A. | Coxeter Groups and Associated Polynomials |
| B. | Geometric Combinatorics |
| C. | Enumeration |
Numbers in square brackets [1,..] indicate the nodes participating in that part of the programme:
| [1] Bordeaux | [2] Jerusalem | [3] Linköping | [4] Lyon |
| [5] Marburg | [6] Marne-la-Vallée | [7] Roma | [8] Wien |
A. Coxeter Groups and Associated Polynomials [2, 3, 4, 6, 7, 8]
We are interested in the study of algebras and polynomials associated to Coxeter groups, such as canonical bases of Temperley-Lieb algebras associated to Coxeter groups, Kazhdan-Lusztig and R-polynomials of a Coxeter group, Hall-Littlewood and Macdonald polynomials associated to all root systems, and related representation theory of Lie groups. Our research will focus (among others) on properties of these canonical bases, explicit formulae for Kazhdan-Lusztig and R-polynomials, the non-negativity conjecture for Kazhdan-Lusztig and R-polynomials, the combinatorial invariance conjecture for Kazhdan-Lusztig and R-polynomials, extensions of Hall-Littlewood polynomials and Macdonald polynomials into several directions, in particular non-commutative and quasi-symmetric versions, further study of q-analogues of products of GL(n) representations, defined in terms of ribbon tableaux, etc.B. Geometric Combinatorics [2, 3, 5, 6, 7]
We will on the one hand focus on improving combinatorial methods for determining homology in combinatorial situations, in particular, to examine how to find torsion combinatorially. Among others, this will mean to further develop and refine Discrete Morse Theory. Another line of research will deal with the Charney-Davis conjecture, a discrete analogue of the famous Hopf conjecture on closed Riemannian manifold of nonpositive sectional curvature. Finally, we will further study and develop the powerful method of Algebraic Shifting. Our first goal in this direction is to understand the behaviour of discrete Morse functions under algebraic shifting. The same research strand will also help to develop tools in order to attack the remaining open problems on f-vectors of simplicial complexes, notably the general g-conjecture.C. Enumeration [1, 3, 4, 5, 6, 7, 8]
The objects that we are interested in in particular (but not exclusively) are: plane partitions, tilings, alternating sign matrices and related objects (such as configurations in the six vertex model and configurations in the fully packed loop model), vicious walkers, maps, trees, lattice walks, polyominoes and animals, permutations, elements of Coxeter groups, etc. The aim is to further develop known methods and to invent new ones, in particular in order to understand the many mysteries about alternating sign matrices and related objects, the mysterious relations between them and various families of plane partitions, to solve problems posed by statistical mechanics, to be able to "explain," and in many cases: make rigorous, findings by mathematical physicists on combinatorial models in statistical mechanics, to understand better the combinatorics of the symmetric group and of other Coxeter groups, among many others.
Improving Human Potential and the Socio-Economic Knowledge Base
Contract : HPRN-CT-2001-00272.