Atypical Representations Research Articles

Structure of atypical representations of the Lie superalgebras sl(m/n)

In contrast to Lie algebras, Lie superalgebras contain typical and atypical irreducible representations. Character formulae for typical representations of classical Lie superalgebras have been known for a long time, but the structure of atypical representations remained unsolved. The author constructs the character formula for atypical representations of the special linear Lie superalgebras sl(m/n).

The generalized atypical supertableaux of the orthosymplectic groups OSP(2‖2p)

The classification and the interpretation of the Young supertableaux of the orthosymplectic group OSP(2‖2p) are given. A particular emphasis is made on the generalized atypical supertableaux associated to nonfully reducible atypical representations.

Representations of the superalgebra F4and Young supertableaux

The authors present a method for constructing typical as well as atypical finite-dimensional representations of the superalgebra F4. For such a purpose, Young supertableaux are introduced.

Modification rules for orthosymplectic superalgebras

Modification rules are presented for finite dimensional graded tensor representations of OSp(M/N). With the representations specified by Young supertableaux these rules relate non-standard supertableaux to standard supertableaux via the removal of continuous boundary hooks. All typical tensor representations are treated together with atypical representations which satisfy up to two atypically conditions.

Modification rules for orthosymplectic superalgebras. II. Atypical representations

For pt.I see ibid., vol.19, no.3 p.321-7 (1986). Modification rules are presented for all finite-dimensional, atypical representations of OSp(M/N). With the representations specified by Young supertableaux, these rules relate non-standard supertableaux to standard supertableaux via the removal of continuous boundary hooks.

Unitary massless representations of conformal superalgebras

Massless unitary irreducible representations of the conformal superalgebras SU(2, 2/ N) are shown to be atypical representations. The existence of such representations appears restricted to N⩽4 if the spin condition s⩽2 is imposed on the states.

Application of generating function techniques to Lie superalgebras

The construction of generating functions for multiplicities of irreducible representations of Lie superalgebras from generating functions for characters is examined, and applied in order to obtain polynomial tensors and branching rules. Problems arising because of the existence of typical and atypical representations are discussed in detail. The techniques are applied to osp(1,2), spl(1,2), osp(3,2), and osp(4,2).

Representations of orthosymplectic superalgebras. II. Young diagrams and weight space techniques

For pt.I see ibid., vol.16, p.473 (1983). Finite-dimensional graded tensor representations of OSp(M/N) are enumerated via standard Young diagrams, and their corresponding highest weight and Kac-Dynkin labels are given. A uniform set of conditions on the diagram shape, necessary and sufficient for atypicality, are presented. Weight space techniques are used to provide a complete analysis of an atypical representations for the low-rank cases OSp(2/2), OSp(3/2) and OSp(4/2).

Mixed supertableaux of the superunitary groups. I. SU(n‖1)

Young supertableaux of the graded unitary groups SU(n‖1) are classified and interpreted in terms of representations of the group. A distinction between typical and atypical representations arises naturally, and we propose a new type of generalized atypical supertableaux for non-fully-reducible representations.

Irreducible representations of the [formula omitted] Lie superalgebras

Using Young tableaux, we describe the typical and atypical representations of the SU( 1 N ) Lie superalgebras and present physical applications. A closed formula for the superdimension is given.

Irreducible representations of SU(m/n)

Typical and atypical representations of SU(m/n) are worked out in an explicit and pedestrian way. Examples are given.