Symmetrization

In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables.Conversely, anti-symmetrization converts any function in n variables into an antisymmetric function. In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables.Conversely, anti-symmetrization converts any function in n variables into an antisymmetric function. Let S {displaystyle S} be a set and A {displaystyle A} an abelian group. Given a map α : S × S → A {displaystyle alpha colon S imes S o A} , α {displaystyle alpha } is termed a symmetric map if α ( s , t ) = α ( t , s ) {displaystyle alpha (s,t)=alpha (t,s)} for all s , t ∈ S {displaystyle s,tin S} . The symmetrization of a map α : S × S → A {displaystyle alpha colon S imes S o A} is the map ( x , y ) ↦ α ( x , y ) + α ( y , x ) {displaystyle (x,y)mapsto alpha (x,y)+alpha (y,x)} . Similarly, the anti-symmetrization or skew-symmetrization of a map α : S × S → A {displaystyle alpha colon S imes S o A} is the map ( x , y ) ↦ α ( x , y ) − α ( y , x ) {displaystyle (x,y)mapsto alpha (x,y)-alpha (y,x)} . The sum of the symmetrization and the anti-symmetrization of a map α is 2α.Thus, away from 2, meaning if 2 is invertible, such as for the real numbers, one can divide by 2 and express every function as a sum of a symmetric function and an anti-symmetric function. The symmetrization of a symmetric map is its double, while the symmetrization of an alternating map is zero; similarly, the anti-symmetrization of a symmetric map is zero, while the anti-symmetrization of an anti-symmetric map is its double. The symmetrization and anti-symmetrization of a bilinear map are bilinear; thus away from 2, every bilinear form is a sum of a symmetric form and a skew-symmetric form, and there is no difference between a symmetric form and a quadratic form. At 2, not every form can be decomposed into a symmetric form and a skew-symmetric form – for instance, over the integers, the associated symmetric form (over the rationals) may take half-integer values, while over Z / 2 Z , {displaystyle mathbf {Z} /2mathbf {Z} ,} a function is skew-symmetric if and only if it is symmetric (as 1 = −1). This leads to the notion of ε-quadratic forms and ε-symmetric forms.