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

$$\{\mu\}\otimes\{\nu\}=\sum\{\lambda\}$$

satisfies the rules

$$A\otimes(BC)=(A\otimes B)(A\otimes C)=A\otimes BA\otimes C,$$

$$A\otimes(B\pm C)=A\otimes B\pm A\otimes C$$

$$(A\otimes B)\otimes C=A\otimes(B\otimes C)$$

$$(A+B)\otimes\{\lambda\}=\sum \Gamma_{\mu\nu\lambda}(A\otimes\{\mu\})(B\otimes\{\nu\}),$$

where $\Gamma_{\mu\nu\lambda}$ is the coefficient of $\{\lambda\}$ in $\{\mu\}\{\nu\}$,

$$(A-B)\otimes\{\lambda\}=\sum (-1)^r\Gamma_{\mu\nu\lambda}(A\otimes \{\mu\})(B\otimes\{\tilde\nu\}),$$

where $\{\tilde\nu\}$ is the partition of $r$ conjugate to $\{\nu\}$, and

$$(AB)\otimes\{\lambda\}=\sum g_{\mu\nu\lambda}(A\otimes\{\mu\})(B\otimes\{\nu\}),$$

where $g_{\mu\nu\lambda}$ is the coefficient of $\{\lambda\}$ in the inner product $\{\mu\}\circ\{\nu\}$ (Wybourne 1970).

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.