Weyl Character Research Articles

Reduced determinantal forms for characters of the classical Lie groups

The reduced determinantal forms for the character of all the irreducible representations of the classical Lie groups are derived from Weyl's character formulae. The derivations use a lemma due to Frobenius and the results are expressed in terms of characters of representations specified by hook diagrams. The results are appropriate to all irreducible representations whether specified by ordinary or composite Young diagrams, and whether tensor or spinor in nature.

On the Weyl character formula forSU(n)

In this Note, we draw a straight line between the representation theory of SU(3) and the SU(3)-classification schemes in particle physics. Our approach is based on that of Weyl, but we have in mind the versions which appear, “in modern dress,” in Adams and Bott. Our formulation brings an important part of particle physics into line with two contemporary accounts of compact Lie groups.

Conditions on a Compact Connected Lie Group Which Insure a "Weyl Character Formula"

Conditions on a Compact Connected Lie Group Which Insure a "Weyl Character Formula"

Conditions on a compact connected Lie group which insure a “Weyl character formula”

A theorem showing the equivalence of three conditions on a compact connected Lie group is proved. Among the corollaries is an extended “Weyl character formula” as originally stated by Bott.

Weyl's character formula for algebraic groups

Weyl's character formula for algebraic groups

A branching law for the symplectic groups

Branching laws for the irreducible tensor representations of the general linear and orthogonal groups are well-known. Furthermore, these laws have a simple form [1]. In the case of the symplectic groups, however, the branching law becomes more complicated and is expressed in terms of a determinant. We derive this result here bybrute force applied to the Weyl character formulas, though it could also have been obtained from a more sophisticated treatment of representation theory contained in some unpublished work of Kostant. The Branching law* Let Vn be an ^-dimensional vector space over the complex field. The symplectic group in n dimensions, Sp(n/2), is the set of all linear transformations ae &(V n), under which a nondegenerate skew-symmetric bilinear form on Vn x V* is invariant, [3]. If is the bilinear form on Vn x Vn and ae ξ?(V n), then

Reduction of Representations of SUmn with respect to the Subgroup SUm ⊗ SUn

With a view to application to theories such as the Wigner supermultiplet theory of nuclear structure and its recent extension into the domain of particle physics, the reduction of representations of the special unitary group SUmn into representations of its SUm ⊗ SUn subgroup is considered. Proof is given of the result, that, in the case of a representation of SUmn of plurality type pmn, this reduction yields only those irreducible representations of SUm ⊗ SUn, whose SUm and SUn parts have plurality types pm and pn equal to pmn modulo m and modulo n, respectively. A practical method of obtaining reductions is described and a tabulation of results in the case of SU6 and SU2 ⊗ SU3 is made. The method depends on the use of a very concise version of the Weyl character formula for unitary groups which does not seem to have been used elsewhere.

Divided differences and the Weyl character formula in equivariant K-theory

Let $X$ be a topological space and $G$ a compact connected Lie group acting on $X$. Atiyah proved that the $G$-equivariant K-group of $X$ is a direct summand of the $T$-equivariant K-group of $X$, where $T$ is a maximal torus of $G$. We show that this direct summand is equal to the subgroup of $K_T^*(X)$ annihilated by certain divided difference operators. If $X$ consists of a single point, this assertion amounts to the Weyl character formula. We also give sufficient conditions on $X$ for $K_G^*(X)$ to be isomorphic to the subgroup of Weyl invariants of $K_T^*(X)$.