Checking alternating tensors

Question

How do I check that $$f(x,y)=x_1y_2-x_2y_1+x_1y_1$$ is an alternating tensor? I did check that f is a tensor, but how do I know if it is alternating by direct computation?

Thanks in advance!

Answer 1

We can write this, with help of determinant like:

$$f(x,y) = {x^1}{y^2} - {x^2}{y^1} + {x^1}{y^1} = \left| {\begin{array}{*{20}{c}} {{x^1}}&{{y^1}}\\ {{x^2}}&{{y^2}} \end{array}} \right| + {x^1}{y^1}$$

Then switching arguments:

$$f(y,x) = \left| {\begin{array}{*{20}{c}} {{y^1}}&{{x^1}}\\ {{y^2}}&{{x^2}} \end{array}} \right| + {x^1}{y^1} = - \left| {\begin{array}{*{20}{c}} {{x^1}}&{{y^1}}\\ {{x^2}}&{{y^2}} \end{array}} \right| + {x^1}{y^1}$$

$$ - f(y,x) = \left| {\begin{array}{*{20}{c}} {{x^1}}&{{y^1}}\\ {{x^2}}&{{y^2}} \end{array}} \right| - {x^1}{y^1}$$

But this shows:

$$f(x,y) \ne - f(y,x)$$

So this is not an alternating tensor.

If we take:

$$f(x,y) = {x^1}{y^2} - {x^2}{y^1} = \left| {\begin{array}{*{20}{c}} {{x^1}}&{{y^1}}\\ {{x^2}}&{{y^2}} \end{array}} \right|$$

$$f(x,y) = \left| {\begin{array}{*{20}{c}} {{x^1}}&{{y^1}}\\ {{x^2}}&{{y^2}} \end{array}} \right| = - \left| {\begin{array}{*{20}{c}} {{y^1}}&{{x^1}}\\ {{y^2}}&{{x^2}} \end{array}} \right| = - f(y,x)$$

that's alternating. The determinant function for any $n \times n$ square-matrix are examples for multilinear-alternating tensors.