J. W. B. Hughes

Character formulas for irreducible modules of the Lie superalgebras sl(m/n)

Kac distinguished between typical and atypical finite‐dimensional irreducible representations of the Lie superalgebras sl(m/n) and provided an explicit character formula appropriate to all the typical representations. Here, the range of validity of some character formulas for atypical representations that have been proposed are discussed. Several of them are of the Kac–Weyl type, but then it is proved that all formulas of this type fail to correctly give the character of one particular atypical representation of sl(3/4). Having ruled out, therefore, all such formulas, a completely new extension of the Kac–Weyl character formula is proposed. The validity of this formula in the case of all covariant tensor irreducible representations is proved, and some evidence in support of the conjecture that it covers all irreducible representations of sl(m/n) is presented.

Representations of Osp(2, 1) and the metaplectic representation

Shift operator techniques are used to treat the irreducible representations of the superalgebra Osp(2, 1). Apart from obtaining the well known gradestar dispin representations which arise when the even part is the compact SU(2) algebra, the case when the star conditions on the even part are those satisfied by the noncompact SU(1, 1) algebra is also treated. In this case no gradestar representations arise, and the star representations are found to consist of the direct sum of two discrete series representations of SU(1, 1). One of these representations can be realized in terms of functions of a single complex variable, and turns out to be a simple example of a metaplectic representation.

On the composition factors of Kac modules for the Lie superalgebras sl(m/n)

In the classification of finite‐dimensional modules of Lie superalgebras, Kac distinguished between typical and atypical modules. Kac introduced an induced module, the so‐called Kac module V(Λ) with highest weight Λ, which was shown to be simple if Λ is a typical highest weight. If Λ is an atypical highest weight, the Kac module is indecomposable and the simple module V(Λ) can be identified with a quotient module of V(Λ). In the present paper the problem of determining the composition factors of the Kac modules for the Lie superalgebra sl(m/n) is considered. An algorithm is given to determine all these composition factors, and conversely, an algorithm is given to determine all the Kac modules containing a given simple module as a composition factor. The two algorithms are presented in the form of conjectures, and illustrated by means of detailed examples. Strong evidence in support of the conjectures is provided. The combinatorial way in which the two algorithms are intertwined is both surprising and interesting, and is a convincing argument in favor of the solution to the composition factor problem presented here.

SU(2)×SU(2) shift operators and representations of SO(5)

SU(2)×SU(2) shift operators analogous to the SU(2) shift operators developed and used by the author for the classification and analysis of representations of Lie algebras in an SU(2) or SO(3) basis are obtained for the SU(2)×SU(2) Lie algebra in the case where one has an additional set of operators forming an irreducible four‐dimensional tensor representation of SU(2)×SU(2). The shift operators obtained are used to treat the representations of SO(5) in an SU(2)×SU(2) basis.

Nonscalar extension of shift operator techniques for SU (3) in an O(3) basis. I. Theory

A set of relations is set up which connect quadratic products of the shift operators Okl (k = 0,1,2), which are nonscalar with respect to the O(3) subgroup of SU (3). The usefulness of these relations is illustrated by the calculation of the eigenvalues of the scalar shift operator O0l for various irreducible representations ( p,q) of SU (3).

Nonscalar extension of shift operator techniques for SU (3) in an O(3) basis. II. Applications

In a preceding paper relations have been derived which connect nonscalar quadratic shift operator products. Here, the extreme usefulness of these relations is demonstrated by the example of the O0l ‐eigenvalue calculation for the cases l = p−i (i = 0,1,2,3, and 4), where ( p,q) is any SU (3) representation. For the first time a case of threefold l‐degeneracy is completely solved in a pure analytical way.

Unimodal polynomials associated with Lie algebras and superalgebras

It is well known that the theory of Lie algebras can be used to prove the unimodality of certain polynomials. Recently, it was shown that Lie superalgebras also give rise to unimodal polynomials. In this paper, we present a unified approach to the question. We do not discuss the technical details of Lie algebras and superalgebras, but give several old and new examples of unimodal polynomials associated with these structures.

A pair of commuting scalars for G(2) ⊇ SU(2)×SU(2)

G(2) ⊇ SU(2)×SU(2) is a two‐missing‐labels problem, and therefore in order to give a complete and orthogonal specification of states of irreducible representations of G(2) in an SU(2)×SU(2) basis, one needs to find a pair of commuting Hermitian operators which are scalar with respect to the SU(2)×SU(2) subalgebra. A theorem due to Peccia and Sharp states that there are, apart from the Lie algebra invariants, twice as many functionally independent scalars as missing labels. Here two commuting SU(2)×SU(2) scalars are obtained, both of sixth order in the G(2) basis elements. They are in fact combinations of five scalars of different tensorial types, indicating that the functionally independent ones are in general insufficient to provide the lowest‐order commuting scalars. An expression for the sixth‐order invariant of G(2) is also obtained.

Irreducible representations of the central extension of Sl(2) ΛT2

Using shift operator techniques a classification is given of the irreducible star representations of the central extension algebra C(Sl(2)ΛT2). It is found to possess two generic series of such representations, together with an isolated representation which is just the metaplectic representation of Sl(2). This is the only representation it possesses in common with the superalgebra Osp(2, 1).

SO(4) shift operators and representations of SO(5)

Some new SO(4) shift operators are constructed and used to analyse the irreducible representations of SO(5) in an SO(4) basis. These techniques can be applied to other groups, such as G(2), which possesses an SO(4) subgroup.