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.

In representation theory

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 $L_{\nu }(V)$ for a Young diagram $\nu$ . So for some nonnegative multiplicities $a_{\lambda ,\mu ,\nu }$ there is an isomorphism

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 $a_{\lambda ,\mu ,\nu }$ .^[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)).

References

^ Littlewood (1936, p. 52, 1944, p. 329)
^ Weyman, Jerzy (2003). Cohomology of Vector Bundles and Syzygies. Cambridge University Press. doi:10.1017/CBO9780511546556. ISBN 9780511546556.

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: 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

[1] Littlewood (1936, p. 52, 1944, p. 329)

[weyman-2] Weyman, Jerzy (2003). Cohomology of Vector Bundles and Syzygies. Cambridge University Press. doi:10.1017/CBO9780511546556. ISBN 9780511546556.

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