The authors provide here the definitive account of the generalization to the general linear Lie superalgebra \(\mathfrak{pl}(V)=\mathfrak{pl}(k,\ell)\) of Schur’s classical work on the irreducible representations of the general Lie algebra \(\mathfrak{gl}(V)=\mathfrak{gl}(k)\). In particular, Schur’s proof that the permutation action of \(S_ n\) and the diagonal action of \(\mathfrak{gl}(V)\) on \(\text{End}(V^{\otimes n})\) centralise one another is extended to the case of \(V=T\oplus U\), with \(\dim T=k\) and \(\dim U=\ell\). The action of \(S_ n\) restricted to \(T^{\otimes n}\) is as before but involves a sign factor on restriction to \(U^{\otimes n}\). This action centralises that of \(\mathfrak{pl}(V)\) and vice versa on \(\mathrm{End}(V^{\otimes n})\). It is proved that the corresponding irreducible representations may be specified by a set \(\Gamma(k,\ell;n)\) of partitions \(\lambda =(\lambda_ 1,\lambda_ 2,\dots)\) of \(n\) for which \(\lambda_ j\leq \ell\) for \(j\geq k+1\) so that the corresponding Young diagrams lie inside a hook shaped region of arm-height \(k\) and leg-width \(\ell.\)
The proof of this key result involves a generalization of the Robinson-Schensted-Knuth construction and certain new semistandard tableaux in two sets of variables. The characters of irreducible representations of \(\mathfrak{pl}(V)\) are given explicitly, along with branching rules, dimension formulae and a factorization theorem appropriate to the case \(\lambda_ k\geq \ell\).
The work is self-contained but overlaps to some extent independent developments published elsewhere. [P. H. Dondi andP. D. Jarvis, J. Phys. A 14, 547–563 (1981;Zbl 0449.17002),A. B. Balantekin andI. Bars, J. Math. Phys. 22, 1149–1162 (1981;Zbl 0469.22017); 22, 1810–1818 (1981;Zbl 0547.22014), the reviewer, Lect. Notes Phys. 180, 41–47 (1983;Zbl 0529.17005), and Ars. Comb. 16A, 269–287 (1983;Zbl 0547.22016)].

Reviewer: R. C. King (Southampton)

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