Ring of symmetric functions: Difference between revisions

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== Symmetric polynomials ==

{{ main article | Symmetric polynomial }}

The study of symmetric functions is based on that of symmetric polynomials. In a [[polynomial ring]] in some finite set of indeterminates, a polynomial is called ''symmetric'' if it stays the same whenever the indeterminates are permuted in any way. More formally, there is an [[group action|action]] by [[ring homomorphism|ring automorphism]]s of the [[symmetric group]] ''S_n'' on the polynomial ring in ''n'' indeterminates, where a permutation acts on a polynomial by simultaneously substituting each of the indeterminates for another according to the permutation used. The [[Invariant (mathematics)#Unchanged under group action|invariants]] for this action form the subring of symmetric polynomials. If the indeterminates are ''X''₁,…,''X''_''n'', then examples of such symmetric polynomials are

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==See also==

* The ring of symmetric functions is the [[Exp ring]] of the integers.

* [[Newton's identities]]

*The ring of symmetric functions is the [[Exp ring]] of the integers.

* [[Quasisymmetric function]]

*[[Newton's identities]]

*[[Quasisymmetric function]]

==References==

{{Reflist}}

==Further reading==

<div class="references">

* Macdonald, I. G. ''Symmetric functions and Hall polynomials.'' Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 1979. viii+180 pp.&nbsp;ISBN 0-19-853530-9 {{MathSciNet|id=84g:05003}}

<references/>

* Macdonald, I. G. ''Symmetric functions and Hall polynomials.'' Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp.&nbsp;ISBN 0-19-853489-2 {{MathSciNet|id=96h:05207}}

</div>

* [[Richard P. Stanley|Stanley, Richard P.]] ''Enumerative Combinatorics'', Vol. 2, Cambridge University Press, 1999. ISBN 0-521-56069-1 (hardback) ISBN 0-521-78987-7 (paperback).

*Macdonald, I. G. ''Symmetric functions and Hall polynomials.'' Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 1979. viii+180 pp.&nbsp;ISBN 0-19-853530-9 {{MathSciNet|id=84g:05003}}

*Macdonald, I. G. ''Symmetric functions and Hall polynomials.'' Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp.&nbsp;ISBN 0-19-853489-2 {{MathSciNet|id=96h:05207}}

*[[Richard P. Stanley|Stanley, Richard P.]] ''Enumerative Combinatorics'', Vol. 2, Cambridge University Press, 1999. ISBN 0-521-56069-1 (hardback) ISBN 0-521-78987-7 (paperback).

[[Category:Polynomials]]

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[[Category:Permutations]]

[[Category:Types of functions]]

[[eo:Simetria funkcio]]

[[it:Funzione simmetrica]]

[[fi:Symmetrinen funktio]]