Plethysm

In algebra, plethysm is an operation on symmetric functions introduced by Dudley E. Littlewood,^[1] who denoted it by {λ} ⊗ {μ}. The word "plethysm" for this operation (after the Greek word πληθυσμός meaning "multiplication") was introduced later by Littlewood (1950, p. 289, 1950b, p.274), who said that the name was suggested by M. L. Clark.

If symmetric functions are identified with operations in lambda rings, then plethysm corresponds to composition of operations.

Let V be a vector space over the complex numbers, considered as a representation of the general linear group GL(V). Each Young diagram λ corresponds to a Schur functor L_λ(-) on the category of GL(V)-representations. Given two Young diagrams λ and μ, consider the decomposition of L_λ(L_μ(V)) into a direct sum of irreducible representations of the group. By the representation theory of the general linear group we know that each summand is isomorphic to ${\displaystyle L_{\nu }(V)}$ for a Young diagram ${\displaystyle \nu }$. So for some nonnegative multiplicities ${\displaystyle a_{\lambda ,\mu ,\nu }}$ there is an isomorphism

${\displaystyle L_{\lambda }(L_{\mu }(V))=\bigoplus _{\nu }L_{\nu }(V)^{\oplus a_{\lambda ,\mu ,\nu }}.}$

The problem of (outer) plethysm is to find an expression for the multiplicities ${\displaystyle a_{\lambda ,\mu ,\nu }}$.^[2]

This formulation is closely related to the classical question. The character of the GL(V)-representation L_λ(V) is a symmetric function in dim(V) variables, known as the Schur polynomial s_λ corresponding to the Young diagram λ. Schur polynomials form a basis in the space of symmetric functions. Hence to understand the plethysm of two symmetric functions it would be enough to know their expressions in that basis and an expression for a plethysm of two arbitrary Schur polynomials {s_λ}⊗{s_μ} . The second piece of data is precisely the character of L_λ(L_μ(V)).

• Littlewood, D. E. (1936), "Polynomial concomitants and invariant matrices", J. London Math. Soc., 11 (1): 49–55, doi:10.1112/jlms/s1-11.1.49, Zbl 0013.14602
• Littlewood, D. E. (1944), "Invariant theory, tensors and group characters", Philosophical Transactions of the Royal Society A, 239 (807): 305–365, doi:10.1098/rsta.1944.0001, JSTOR 91389, MR 0010594
• Littlewood, Dudley E. (1950), The theory of group characters and matrix representations of groups, AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4067-2, MR 0002127
• Littlewood, D. E. (1950b), A University Algebra, Melbourne, London, Toronto: William Heinemann, Ltd., MR 0045079