This paper characterizes the first cohomology group H(G,M) where M is a Banach space (with norm || ||M) that is also a left CG module such that the elements of G act onM as continuous C-linear transformations. We study this group for G an infinite, finitely generated group. Of particular interest are the implications of the vanishing of the group H(G,M). The first result is that H(G,CG) imbeds ...

The non-relativistic quark model has been a useful tool in the study of hadrons. Baryons and mesons are described by quantum mechanical wavefunctions for nonrelativistic constituent quarks. The lowest lying baryons, the 81/2 and 103/2, are three quark states with wavefunctions which are completely antisymmetric in color, and completely symmetric in position and spin-flavor. The properties of ba...

We present a program that allows for the computation of tensor products of irreducible representations of Lie algebras A−G based on the explicit construction of weight states. This straightforward approach (which is slower and more memoryconsumptive than the standard methods to just calculate dimensions of the tensor product decomposition) produces Clebsch-Gordan coefficients that are of intere...

After careful introduction and discussion of the concepts involved, procedures are developed to compute Racah and Clebsch-Gordan coefficients for general r-fold tensor products of the U(N) groups. In the process, the multiplicity of a given irreducible representation (irrep) in the direct sum basis is computed, and generalized Casimir operators are introduced to uniquely label the multiple irre...

We study the tensor category of modules over a semisimple bialgebra H under the assumption that irreducible H -modules of the same dimension >1 are isomorphic. We consider properties of Clebsch–Gordan coefficients showing multiplicities of occurrences of each irreducible H -module in a tensor product of irreducible ones. It is shown that, in general, these coefficients cannot have small values....

We prove an inequality for the Kostka Foulkes polynomials Kλ,μ(q). As a corollary, we obtain a nontrivial lower bound for the Kostka numbers and a new proof of the Berenstein Zelevinsky weight-multiplicity-one-criterium. The concept of Young tableau plays an important role in the representation theory of the symmetric and general linear groups. Based on the pioneering fundamental works of G. Fr...

We study a class of representations called “calibrated representations” of the degenerate double affine Hecke algebra and those of the rational Cherednik algebra of type GLn. We give a realization of calibrated irreducible modules as spaces of coinvariants constructed from integrable modules over the affine Lie algebra ĝl m . Moreover, we give a character formula of these irreducible modules in...

Stanley has studied a symmetric function generalization X G of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of X G in terms of elementary symmetric functions has nonnegative coeecients if G is a clawfree incomparabil-ity graph. Here we present a new approach that combines Gasharov's work on the conjecture with E~...

In [LLT1], Lascoux, Leclerc and Thibon give some factorisation formulas for Hall-Littlewood functions when the parameter q is specialized at roots of unity. They also give formulas in terms of cyclic characters of the symmetric group. In this article, we give a generalization of these specializations for different versions of the Macdonald polynomials and we obtain similar formulas in terms of ...