# Plethysm

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In algebra, **plethysm** is an operation on [symmetric functions](/source/Ring_of_symmetric_functions) introduced by [Dudley E. Littlewood](/source/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 txt, who said that the name was suggested by M. L. Clark.

If symmetric functions are identified with operations in [lambda rings](/source/Lambda_ring), then plethysm corresponds to composition of operations.

## In representation theory

Let *V* be a [vector space](/source/Vector_space) over the [complex numbers](/source/Complex_number), considered as a [representation](/source/Group_representation) of the [general linear group](/source/General_linear_group) GL(*V*). Each [Young diagram](/source/Young_diagram) λ corresponds to a [Schur functor](/source/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](/source/Direct_sum) of [irreducible representations](/source/Irreducible_representation) of the group. By the [representation theory](/source/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](/source/Character_theory) of the GL(*V*)-representation *L*λ(*V*) is a symmetric function in dim(*V*) variables, known as the [Schur polynomial](/source/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

1. txt

1. Weyman, Jerzy (2003). *Cohomology of Vector Bundles and Syzygies*. Cambridge University Press. [doi:10.1017/CBO9780511546556](https://doi.org/10.1017/CBO9780511546556). ISBN 9780511546556.

- Littlewood, D. E. (1936), "Polynomial concomitants and invariant matrices", *[J. London Math. Soc.](/source/J._London_Math._Soc.)*. **11** (1): 49–55, [doi:10.1112/jlms/s1-11.1.49](https://doi.org/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](/source/Philosophical_Transactions_of_the_Royal_Society_A)*. **239** (807): 305–365, [doi:10.1098/rsta.1944.0001](https://doi.org/10.1098/rsta.1944.0001). [JSTOR 91389](https://www.jstor.org/stable/91389). MR 0010594
- Littlewood, Dudley E. (1950), [*The theory of group characters and matrix representations of groups*](https://www.ams.org/bookstore?fn=20&arg1=alggeom&item=CHEL-357-H), AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4067-2. MR 0002127
- Littlewood, D. E. (1950b), ["A University Algebra"](https://books.google.com/books?id=pu3uAAAAMAAJ&q=Clark), Melbourne, London, Toronto: William Heinemann, Ltd., MR 0045079

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