Ronald C. King

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.

Standard Young tableaux and weight multiplicities of the classical Lie groups

By examining the branching rules for all irreducible representations of the classical groups U(k), SU(k), SO(2k+1), Sp(2k) and SO(2k) on restriction to U(1)*U(1)*U(1), standard Young tableaux are specified for each of these groups. It is shown that these tableaux determine the corresponding characters of the irreducible representations. The rules for constructing these tableaux are derived and in this way the determination of weight multiplicities is reduced to a simple combinatorial exercise. General formula for such weight multiplicities are given encompassing the most difficult case: namely that of SO(2k). Illustrative examples are provided, including some yielding the explicit k-dependence of weight multiplicities.

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.

New branching rules induced by plethysm

We derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor T? of Young symmetry type ? where ? is an integer partition. The branchings and fixing a vector vi, a symmetric tensor gij = gji and an antisymmetric tensor fij = ?fji, respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function s? ? {?} by the basic M series of complete symmetric functions and the L = M?1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains ?-generalized Newell?Littlewood formulae and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for and , showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine and in some instances non-reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2-cocycles. The algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters. Modification rules are beyond the scope of the present paper, but are briefly discussed.

Symplectic Shifted Tableaux and Deformations of Weyl's Denominator Formula for sp (2 n )

A determinantal expansion due to Okada is used to derive both a deformation of Weyls denominator formula for the Lie algebra sp(2n) of the symplectic group and a further generalisation involving a product of the deformed denominator with a deformation of flagged characters of sp(2n). In each case the relevant expansion is expressed in terms of certain shifted sp(2n)-standard tableaux. It is then re-expressed, first in terms of monotone patterns and then in terms of alternating sign matrices.

Representations and traces of the Hecke algebras Hn(q) of type An−1

The notion of the connectivity class of minimal words in the algebra Hn(q) is introduced and a method of explicitly constructing irreducible representation matrices is described and implemented. Guided by these results, the connection between the Ocneanu trace on Hn(q) and Schur functions is exploited to derive a very simple prescription for calculating the irreducible characters of Hn(q). They appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to Schur functions. Their evaluation involves the use of the Littlewood–Richardson rule, which is proved to apply to Hn(q) just as it does to Sn. Both representation matrices and characters are tabulated.

Construction of orthogonal group modules using tableaux

The program of constructing irreducible modules of the classical groups via Young tableaux is extended to include the orthogonal groups O(n) and SO(n). As in Weyls pioneering work on this topic, each irreducible O(n)-module labelled by a partition λ of l is constructed as a submodule of V ⊗l. The construction is based on new sets of O(n) and SO(n) standard Young tableaux, closely related to those introduced by Proctor. The reduction of an arbitrary O(n) Young tableau to standard form is accomplished through the use of column antisymmetrisation and Garnir relations, as for GL(n), together with a trace removal process, analogous to that introduced by Bercle for Sp(n). For SO(n) it is necessary to define the associate of a Young tableau to effect the final step of the reduction process. The standardisation procedure is algorithmic and allows matrix representations of the Lie algebras so(n) to be constructed explicitly over the field of rational numbers. It is proved that the matrix elements are rational num...

The mixed two-qubit system and the structure of its ring of local invariants

The local invariants of a mixed two-qubit system are discussed. These invariants are polynomials in the elements of the corresponding density matrix. They are counted by means of group-theoretic branching rules which relate this problem to one arising in spin–isospin nuclear shell models. The corresponding Molien series and a refinement in the form of a four-parameter generating function are determined. A graphical approach is then used to construct explicitly a fundamental set of 21 invariants. Relations between them are found in the form of syzygies. By using these, the structure of the ring of local invariants is determined, and complete sets of primary and secondary invariants are identified: there are 10 of the former and 15 of the latter.

Plethysms, replicated Schur functions and series, with applications to vertex operators

Specializations of Schur functions are exploited to define and evaluate the Schur functions sλ[αX] and plethysms sλ[αsν(X))] for any α—integer, real or complex. Plethysms are then used to define pairs of mutually inverse infinite series of Schur functions, Mπ and Lπ , specified by arbitrary partitions π. These are used in turn to define and provide generating functions for formal characters, s(π) λ , of certain groups Hπ , thereby extending known results for orthogonal and symplectic group characters. Each of these formal characters is then given a vertex operator realization, first in terms of the seriesM = M(0) and various L⊥σ dual to Lσ , and then more explicitly in the exponential form. Finally the replicated form of such vertex operators are written down. The characters of the orthogonal and symplectic groups have been found by Schur [34] and Weyl [35] respectively. The method used is transcendental, and depends on integration over the group manifold. These characters, however, may be obtained by purely algebraic methods, . . . . This algebraic method would seem to offer a better prospect of successful application to other restricted groups than the method of group integration. Littlewood D E 1944 Phil. Trans. R. Soc. London, Ser. A 239 (809) 392 PACS numbers: 02.10.−v, 02.10.De, 02.20.−a, 02.20.Hj Mathematics Subject Classification: 05E05, 17B69, 11E57, 16W30, 20E22, 33D52, 43A40 1751-8113/

Composite Young diagrams, supercharacters of U(M/N) and modification rules

The supercharacter of U(M/N) associated with an arbitrary composite Young diagram is defined. The distinction is made between standard and non-standard supercharacters. A modification rule is presented which may be used to express any non-standard supercharacter in terms of standard supercharacters. In the case N=0 the rule reduces to the rule already known to be appropriate to U(M).