Symmetric function

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of 
  
    
      
        n
      
    
    {\displaystyle n}
  
 variables is symmetric if its value is the same no matter the order of its <a href="/facts/Argument_of_a_function/QkuPc28I">arguments</a>. For example, a function 
  
    
      
        f
        
          (
          
            
              x
              
                1
              
            
            ,
            
              x
              
                2
              
            
          
          )
        
      
    
    {\displaystyle f\left(x_{1},x_{2}\right)}
  
 of two arguments is a symmetric function if and only if 
  
    
      
        f
        
          (
          
            
              x
              
                1
              
            
            ,
            
              x
              
                2
              
            
          
          )
        
        =
        f
        
          (
          
            
              x
              
                2
              
            
            ,
            
              x
              
                1
              
            
          
          )
        
      
    
    {\displaystyle f\left(x_{1},x_{2}\right)=f\left(x_{2},x_{1}\right)}
  
 for all 
  
    
      
        
          x
          
            1
          
        
      
    
    {\displaystyle x_{1}}
  
 and 
  
    
      
        
          x
          
            2
          
        
      
    
    {\displaystyle x_{2}}
  
 such that 
  
    
      
        
          (
          
            
              x
              
                1
              
            
            ,
            
              x
              
                2
              
            
          
          )
        
      
    
    {\displaystyle \left(x_{1},x_{2}\right)}
  
 and 
  
    
      
        
          (
          
            
              x
              
                2
              
            
            ,
            
              x
              
                1
              
            
          
          )
        
      
    
    {\displaystyle \left(x_{2},x_{1}\right)}
  
 are in the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> of 
  
    
      
        f
        .
      
    
    {\displaystyle f.}
  
 The most commonly encountered symmetric functions are <a href="/facts/Polynomial_function/Lzak8VVx">polynomial functions</a>, which are given by the <a href="/facts/Symmetric_polynomial/GNhguVcn">symmetric polynomials</a>.
</p><p>A related notion is <a href="/facts/Alternating_polynomial/uzUEDCFQ">alternating polynomials</a>, which change sign under an interchange of variables. Aside from polynomial functions, <a href="/facts/Symmetric_tensor/CWKBfgnv">tensors</a> that act as functions of several vectors can be symmetric, and in fact the space of symmetric 
  
    
      
        k
      
    
    {\displaystyle k}
  
-tensors on a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
  
    
      
        V
      
    
    {\displaystyle V}
  
 is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to the space of <a href="/facts/Homogeneous_polynomials/WUHL7Kct">homogeneous polynomials</a> of degree 
  
    
      
        k
      
    
    {\displaystyle k}
  
 on 
  
    
      
        V
        .
      
    
    {\displaystyle V.}
  
 Symmetric functions should not be confused with <a href="/facts/Even_and_odd_functions/tGHWJqAa">even and odd functions</a>, which have a different sort of symmetry.
</p>

Symmetric function open-in-new

Symmetric function