In a Borel subalgebra U(B) of the sl2 loop algebra, we introduce a highest weight vector Ψ. We call such a representation of U(B) that is generated by Ψ highest weight. We define a generalization of the Drinfeld polynomial for a finitedimensional highest weight representation of U(B). We show that every finitedimensional highest weight representation of the Borel subalgebra is irreducible if th...

A basis for each finite-dimensional irreducible representation of the symplectic Lie algebra sp(2n) is constructed. The basis vectors are expressed in terms of the Mickelsson lowering operators. Explicit formulas for the matrix elements of generators of sp(2n) in this basis are given. The basis is natural from the viewpoint of the representation theory of the Yangians. The key role in the const...

Let [Formula: see text] be a commutative Noetherian ring. For finitely generated text]-module text], Northcott introduced the reducibility index of which is number submodules appearing in an irredundant irreducible decomposition submodule text]. On other hand, for Artinian Macdonald proved that sum-irreducible representation does not depend on choice representation. This called sum-reducibility...

We prove that for any partition (λ1, . . . , λd2) of size dm there exists k ≥ 1 such that the tensor square of the irreducible representation of the symmetric group Skdm with respect to the rectangular partition (km, . . . , km) contains the irreducible representation corresponding to the stretched partition (kλ1, . . . , kλd2). We also prove a related approximate version of this statement in w...

The linear 2-dim irreducible representations of the dihedral groups (Dn) are interpreted as classical linear operators of geometrical optics. It is shown that the 2-dim irreducible representation of D4 is simply the refractive group described by Campbell [Optom. Vision Sci. 74, 381 (1997)]. The dihedral Fourier-inverse mechanism is introduced and shown to provide a systematic connection between...

Consider the general linear group GLM over the complex field. The irreducible rational representations of the group GLM can be labeled by the pairs of partitions μ = (μ1, μ2 , . . .) and μ̃ = (μ̃1, μ̃2 , . . .) such that the total number of non-zero parts of μ and μ̃ does not exceed M . Let Vμμ̃ be the irreducible representation corresponding to such a pair. Regard the direct product GLN ×GLM as a s...

In the representation theory of nite groups it is useful to know which ordinary irreducible representations remain irreducible modulo a prime p. For the symmetric groups S n this amounts to determining which Specht modules are irreducible over a eld of characteristic p. Throughout this note we work in characteristic 2, and in this case we classify the irreducible Specht modules, thereby verifyi...

A unified and systematic scheme for construction of differential operator realization of any irreducible representation of sl(n) is developed. The q-analogue of this unified scheme is used to construct q-difference operator realization of any irreducible representation of Uq(sl(n)). Explicit results for Uq(sl(2)), Uq(sl(3)) and Uq(sl(n)) are given. Research partialy supported by Sharif Universi...