Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras
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. [\textit{P. H. Dondi} and \textit{P. D. Jarvis}, J. Phys. A 14, 547--563 (1981; Zbl 0449.17002), \textit{A. B. Balantekin} and \textit{I. 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)].
scientific article; zbMATH DE number 4000172