Generalized hook and content numbers identities
For any cell \(x=(i,j)\) in a skew diagram \(\lambda/\eta\) one defines the hook pair \((\text{arm}(x),\text{leg}(x))=(\lambda_i-j,\lambda'_j-i)\) and the content pair \((j-1,i-1)\). Thus, the diagram determines the multisets of hook pairs and content pairs. The author proves various identities between these multisets. In particular, these identities imply new ``hook'' formulas for the dimensions of the irreducible \(\text{GL}(n)\)-representations. In the more general situation of the Jack symmetric functions \(P_{\lambda}^{(\alpha)}\), these identities give new ``\(\alpha\)-hook'' formulas for \(P_{\lambda}^{(\alpha)}(1,\ldots,1)\). An important feature of the paper is that beyond some standard definitions, the main results and their proofs are completely elementary and no previous knowledge of any kind is assumed on the part of the reader. The proofs are inductive. Recently, bijective proofs of the main results of the present paper and of [\textit{A. Regev}, Eur. J. Comb. 21, No. 7, 959-966 (2000; Zbl 0962.05062)] have been given in [\textit{C. Krattenthaler}, Electron. J. Comb. 7, No. 1, Research paper R27 (2000; Zbl 0939.05011)].
scientific article; zbMATH DE number 1538576