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The plethysm \(s_\lambda[s_\mu]\) at hook and near-hook shapes. (English) Zbl 1033.05096

Electron. J. Comb. 11, No. 1, Research paper R11, 26 p. (2004); printed version J. Comb. 11, No. 1 (2004).
Summary: We completely characterize the appearance of Schur functions corresponding to partitions of the form \(\nu = (1^a, b)\) (hook shapes) in the Schur function expansion of the plethysm of two Schur functions, \[ s_\lambda[s_\mu] = \sum_{\nu} a_{\lambda, \mu, \nu} s_\nu. \] Specifically, we show that no Schur functions corresponding to hook shapes occur unless \(\lambda\) and \(\mu\) are both hook shapes and give a new proof of a result of Carbonara, Remmel and Yang that a single hook shape occurs in the expansion of the plethysm \(s_{(1^c, d)}[s_{(1^a, b)}]\). We also consider the problem of adding a row or column so that \(\nu\) is of the form \((1^a,b,c)\) or \((1^a, 2^b, c)\). This proves considerably more difficult than the hook case and we discuss these difficulties while deriving explicit formulas for a special case.

MSC:

05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
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