Plethysm

A group theoretic operation which is useful in the study of complex atomic spectra. A plethysm takes a set of functions of a given symmetry type and forms from them symmetrized products of a given degree and other symmetry type . A plethysm

(1)

satisfies the rules

(2)

(3)

(4)

$(A+B) tensor {lambda}=sumGamma_(munulambda)(A tensor {mu})(B tensor {nu}),$

(5)

where is the coefficient of in ,

$(A-B) tensor {lambda}=sum(-1)^rGamma_(munulambda)(A tensor {mu})(B tensor {nu^~}),$

(6)

where ${nu^~}$ is the partition of conjugate to , and

$(AB) tensor {lambda}=sumg_(munulambda)(A tensor {mu})(B tensor {nu}),$

(7)

where is the coefficient of in the inner product (Wybourne 1970).

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References

Littlewood, D. E. "Polynomial Concomitants and Invariant Matrices." J. London Math. Soc. 11, 49-55, 1936.Wybourne, B. G. "The Plethysm of

-Functions" and "Plethysm and Restricted Groups." Chs. 6-7 in Symmetry Principles and Atomic Spectroscopy. New York: Wiley, pp. 49-68, 1970.

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Cite this as:

Weisstein, Eric W. "Plethysm." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Plethysm.html

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