Symmetric function

Given any function ${\displaystyle f}$ in ${\displaystyle n}$ variables with values in an abelian group, a symmetric function can be constructed by summing values of ${\displaystyle f}$ over all permutations of the arguments. Similarly, an anti-symmetric function can be constructed by summing over even permutations and subtracting the sum over odd permutations. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions ${\displaystyle f.}$ The only general case where ${\displaystyle f}$ can be recovered if both its symmetrization and antisymmetrization are known is when ${\displaystyle n=2}$ and the abelian group admits a division by 2 (inverse of doubling); then ${\displaystyle f}$ is equal to half the sum of its symmetrization and its antisymmetrization.

• Consider the real function
  $${\displaystyle f(x_{1},x_{2},x_{3})=(x-x_{1})(x-x_{2})(x-x_{3}).}$$

  

  By definition, a symmetric function with ${\displaystyle n}$ variables has the property that

  $${\displaystyle f(x_{1},x_{2},\ldots ,x_{n})=f(x_{2},x_{1},\ldots ,x_{n})=f(x_{3},x_{1},\ldots ,x_{n},x_{n-1}),\quad {\text{ etc.}}}$$

  

  In general, the function remains the same for every permutation of its variables. This means that, in this case,

  $${\displaystyle (x-x_{1})(x-x_{2})(x-x_{3})=(x-x_{2})(x-x_{1})(x-x_{3})=(x-x_{3})(x-x_{1})(x-x_{2})}$$

  

  and so on, for all permutations of ${\displaystyle x_{1},x_{2},x_{3}.}$

• Consider the function
  $${\displaystyle f(x,y)=x^{2}+y^{2}-r^{2}.}$$

  

  If ${\displaystyle x}$ and ${\displaystyle y}$ are interchanged the function becomes

  $${\displaystyle f(y,x)=y^{2}+x^{2}-r^{2},}$$

  

  which yields exactly the same results as the original ${\displaystyle f(x,y).}$

• Consider now the function
  $${\displaystyle f(x,y)=ax^{2}+by^{2}-r^{2}.}$$

  

  If ${\displaystyle x}$ and ${\displaystyle y}$ are interchanged, the function becomes

  $${\displaystyle f(y,x)=ay^{2}+bx^{2}-r^{2}.}$$

  

  This function is not the same as the original if ${\displaystyle a\neq b,}$ which makes it non-symmetric.

• Alternating polynomial
• Elementary symmetric polynomial – homogeneous symmetric polynomial in which each possible monomial occurs exactly once with coefficient 1Pages displaying wikidata descriptions as a fallback
• Even and odd functions – Mathematical functions with specific symmetries
• Quasisymmetric function
• Ring of symmetric functions
• Symmetrization – process that converts any function in n variables to a symmetric function in n variablesPages displaying wikidata descriptions as a fallback
• Vandermonde polynomial – determinant of Vandermonde matrixPages displaying wikidata descriptions as a fallback

• F. N. David, M. G. Kendall & D. E. Barton (1966) Symmetric Function and Allied Tables, Cambridge University Press.
• Joseph P. S. Kung, Gian-Carlo Rota, & Catherine H. Yan (2009) Combinatorics: The Rota Way, §5.1 Symmetric functions, pp 222–5, Cambridge University Press, ISBN 978-0-521-73794-4.