## 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:

 500000 Symmetric functions and generalizations 5e+10 Combinatorial aspects of representation theory
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