Dublin Institute for Advanced Studies
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Winter Symposium - Abstracts
"Branching rules for Specht modules"
In characteristic 0, the classical branching rule for the symmetric group S_n is a combinatorial formula that describes the modules that occur in the restriction of a Specht module to S_{n-1}. When the characteristic is finite and odd, it is possible to give an analogue of the branching rule that describes the indecomposable summands in the restriction of a Specht module to S_{n-1}. This rule generalizes to the Specht modules of the General Linear group, if the characteristic is not 2 or 3, and also implies that certain S_n decomposition numbers are non zero.John Murray (NUI - Maynooth)
Combinatorial Physics, Normal Order and Model Feynman Graphs
The general normal ordering problem for boson strings is a combinatorial problem. In this talk we restrict ourselves to single-mode boson monomials. This problem leads to elegant generalisations of well-known combinatorial numbers, such as Bell and Stirling numbers. We explicitly give the generating functions for some classes of these numbers. Finally we show that a graphical representation of these combinatorial numbers leads to sets of model field theories, for which the graphs may be interpreted as Feynman diagrams corresponding to the bosons of the theory. The generating functions are the generators of the classes of Feynman diagrams.Allan Solomon (Open Univ.)
Algorithms for computing in matrix groups
This talk deals with algorithms for matrix groups. Currently this is one of the most active areas of Computational Group Theory, which is a domain of algebra comprising the design and implementation of algorithms for groups. In the talk the following new algorithms for computing in nilpotent matrix groups will be presented.(i) testing nilpotence of matrix groups
(ii) primitivity/irreducibility testing of nilpotent matrix groups
(iii) constructing irreducible modules and nonrefinable systems of imprimitivity for nilpotent matrix groups. Additionally, algorithms constructing computer databases of nilpotent matrix groups will be considered.
Alla Detinko (NUIG + Belarus)
Continual Lie algebras and solvable models
First, we give an introduction to the invariant group-theoretical (Lie-algebraic) approach to exactly solvable models (due to Leznov and Saveliev). Toda-type examples of two-dimensional solvable field theories in classical and quantum regions associated to finite and infinite-dimensional Lie algebras will be discussed. Algebraic constructions (vertex operators) related to group-theoretical soliton solutions to the above mentioned models will be explained.Then we recall the notion of Saveliev-Vershik continual Lie algebras, present main examples, and comment on non-commutative generalization. Dynamical systems constructed on the basis of continual Lie algebras will be discussed.
Finally, we comment on recent progress in the theory of non-commutative counterparts to two-dimensional exactly solvable models.
Alexander Zuevsky (Max Planck Institut fuer Mathematik)