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In algebra, '''plethysm''' is an operation on [[ring of symmetric functions|symmetric functions]] introduced by [[Dudley E. Littlewood]],<ref>{{harvs|txt|last=Littlewood|year1=1936|loc1=p. 52|year2=1944|loc2=p. 329|author-link=Dudley Ernest Littlewood}}</ref> who denoted it by {''λ''}&nbsp;⊗&nbsp;{''μ''}. The word "plethysm" for this operation (after the Greek word πληθυσμός meaning "multiplication") was introduced later by {{harvs|txt|last=Littlewood|year1=1950|loc1=p. 289|year2=1950b|loc2=p.274}}, who said that the name was suggested by M. L. Clark.
If symmetric functions are identified with operations in [[lambda ring]]s, then plethysm corresponds to composition of operations.
==In representation theory==
Let ''V'' be a [[vector space]] over the [[complex number]]s, considered as a [[group representation|representation]] of the [[general linear group]] GL(''V''). Each [[Young diagram]] λ corresponds to a [[Schur functor]] ''L''<sub>λ</sub>(-) on the category of GL(''V'')-representations. Given two Young diagrams λ and μ, consider the decomposition of ''L''<sub>λ</sub>(L<sub>μ</sub>(''V'')) into a [[direct sum]] of [[irreducible representation]]s of the group. By the [[representation theory]] of the general linear group we know that each summand is isomorphic to <math>L_\nu(V)</math> for a Young diagram <math>\nu</math>. So for some nonnegative multiplicities <math>a_{\lambda,\mu,\nu}</math> there is an isomorphism
:<math>L_\lambda(L_\mu(V)) = \bigoplus_{\nu} L_\nu(V)^{\oplus a_{\lambda, \mu, \nu}}.</math>
The '''problem of (outer) plethysm''' is to find an expression for the multiplicities <math>a_{\lambda, \mu, \nu}</math>.<ref name="weyman">{{cite book| first=Jerzy | last=Weyman | title = Cohomology of Vector Bundles and Syzygies | year = 2003 | publisher = Cambridge University Press | isbn = 9780511546556 | doi = 10.1017/CBO9780511546556}}</ref>
This formulation is closely related to the classical question. The [[Character theory|character]] of the GL(''V'')-representation L<sub>λ</sub>(''V'') is a symmetric function in dim(''V'') variables, known as the [[Schur polynomial]] ''s''<sub>λ</sub> 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''<sub>λ</sub>}⊗{''s''<sub>μ</sub>} . The second piece of data is precisely the character of ''L''<sub>λ</sub>(''L''<sub>μ</sub>(''V'')).
==References==
{{reflist}}
*{{citation|zbl=0013.14602
|last=Littlewood|first= D. E.
|title=Polynomial concomitants and invariant matrices
|journal= [[J. London Math. Soc.]] |volume=11|issue=1|pages= 49–55 |year=1936
|doi=10.1112/jlms/s1-11.1.49}}
*{{citation|mr=0010594
|last=Littlewood|first= D. E.
|title=Invariant theory, tensors and group characters
|journal=[[Philosophical Transactions of the Royal Society A]] |volume= 239|year=1944|issue=807 |pages= 305–365
|jstor=91389|doi=10.1098/rsta.1944.0001|doi-access=free}}
*{{Citation | last1=Littlewood | first1=Dudley E. | title=The theory of group characters and matrix representations of groups | url=https://www.ams.org/bookstore?fn=20&arg1=alggeom&item=CHEL-357-H | publisher=AMS Chelsea Publishing, Providence, RI | isbn=978-0-8218-4067-2 | mr=0002127 | year=1950}}
*{{citation|mr=0045079
|last=Littlewood|first= D. E.
|title=A University Algebra|publisher= William Heinemann, Ltd.|place= Melbourne, London, Toronto|year= 1950b|url=https://books.google.com/books?id=pu3uAAAAMAAJ&q=Clark}}
[[Category:Symmetric functions]]