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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 {mu} and forms from them symmetrized products of a given degree r and other symmetry type {nu}. A plethysm

 {mu} tensor {nu}=sum{lambda}
(1)

satisfies the rules

 A tensor (BC)=(A tensor B)(A tensor C)=A tensor BA tensor C,
(2)
 A tensor (B+/-C)=A tensor B+/-A tensor C
(3)
 (A tensor B) tensor C=A tensor (B tensor C)
(4)
 (A+B) tensor {lambda}=sumGamma_(munulambda)(A tensor {mu})(B tensor {nu}),
(5)

where Gamma_(munulambda) is the coefficient of {lambda} in {mu}{nu},

 (A-B) tensor {lambda}=sum(-1)^rGamma_(munulambda)(A tensor {mu})(B tensor {nu^~}),
(6)

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

 (AB) tensor {lambda}=sumg_(munulambda)(A tensor {mu})(B tensor {nu}),
(7)

where g_(munulambda) is the coefficient of {lambda} in the inner product {mu} degrees{nu} (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 S-Functions" and "Plethysm and Restricted Groups." Chs. 6-7 in Symmetry Principles and Atomic Spectroscopy. New York: Wiley, pp. 49-68, 1970.

Referenced on Wolfram|Alpha

Plethysm

Cite this as:

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

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