antisymmetric mapping

Let $U$ and $V$ be a vector spaces over a field $K$. A bilinear mapping $B:U\times U\to V$ is said to be antisymmetric if

$$B(u,u)=0$$ (1)

for all $u\in U$.

If $B$ is antisymmetric, then the polarization of the anti-symmetry relation gives the condition:

$$B(u,v)+B(v,u)=0$$ (2)

for all $u,v\in U$. If the characteristic of $K$ is not 2, then the two conditions are equivalent.

A multlinear mapping $M:U^k\to V$ is said to be totally antisymmetric, or simply antisymmetric, if for every $u_1,\mathrm{\dots },u_k\in U$ such that

$$u_{i+1}=u_i$$

for some $i=1,\mathrm{\dots },k-1$ we have

$$M(u_1,\mathrm{\dots },u_k)=0.$$

Proposition 1

Let $M\mathrm{:}{U}^k\mathrm{\to }V$ be a totally antisymmetric, multlinear mapping, and let $\pi$ be a permutation of $\mathrm{\{}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}k\mathrm{\}}$. Then, for every $u_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{u}_k\mathrm{\in }U$ we have

$$M(u_{{\pi }_1},\mathrm{\dots },u_{{\pi }_k})=\mathrm{sgn}(\pi )M(u_1,\mathrm{\dots },u_k),$$

where $\mathrm{sgn}\mathit{}\mathrm{(}\pi \mathrm{)}\mathrm{=}\mathrm{\pm }\mathrm{1}$ according to the parity of $\pi$.

Proof. Let $u_1,\mathrm{\dots },u_k\in U$ be given. multlinearity and anti-symmetry imply that

$0$ $=M(u_1+u_2,u_1+u_2,u_3,\mathrm{\dots },u_k)$

$=M(u_1,u_2,u_3,\mathrm{\dots },u_k)+M(u_2,u_1,u_3,\mathrm{\dots },u_k)$

Hence, the proposition is valid for $\pi =(12)$ (see cycle notation). Similarly, one can show that the proposition holds for all transpositions

$$\pi =(i,i+1),i=1,\mathrm{\dots },k-1.$$

However, such transpositions generate the group of permutations, and hence the proposition holds in full generality.

Note.

The determinant is an excellent example of a totally antisymmetric, multlinear mapping.

Title	antisymmetric mapping
Canonical name	AntisymmetricMapping
Date of creation	2013-03-22 12:34:39
Last modified on	2013-03-22 12:34:39
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	10
Author	rmilson (146)
Entry type	Definition
Classification	msc 15A69
Classification	msc 15A63
Synonym	skew-symmetric
Synonym	anti-symmetric
Synonym	antisymmetric
Synonym	skew-symmetric mapping
Related topic	SkewSymmetricMatrix
Related topic	SymmetricBilinearForm
Related topic	ExteriorAlgebra