# Tensors and the Alternation Operator ... Browder, Proposition 12.25 ... ...

#### Peter

##### Well-known member
MHB Site Helper
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 12: Multilinear Algebra ... ...

I need some help in order to fully understand the proof of Proposition 12.2 on pages 277 - 278 ... ...

Proposition 12.2 and its proof read as follows:  In the above proof by Browder (near the end of the proof) we read the following:

" ... ... To see also that $$\displaystyle A( \beta \otimes \alpha ) = 0$$, we observe that $$\displaystyle \beta \otimes \alpha = \ ^{ \sigma }( \alpha \otimes \beta )$$ where $$\displaystyle \sigma$$ is the permutation which sends $$\displaystyle (1, \cdot \cdot \cdot , r+s )$$ to $$\displaystyle (r+1, \cdot \cdot \cdot , r+s, 1 \cdot \cdot \cdot , r )$$ ... ... "

My question ... or more accurately problem is that given $$\displaystyle \sigma$$ as defined by Browder I cannot verify that $$\displaystyle \beta \otimes \alpha = \ ^{ \sigma }( \alpha \otimes \beta )$$ is true ...

My working is as follows:

Let $$\displaystyle \alpha \in \bigwedge^r$$ and let $$\displaystyle \beta \in \bigwedge^s$$ ... ...

Then we have ...

$$\displaystyle \beta \otimes \alpha (v_1, \cdot \cdot \cdot , v_{ r+s } ) = \beta ( v_1, \cdot \cdot \cdot , v_s ) \alpha ( v_{ s+1 }, \cdot \cdot \cdot , v_{ r+s } )$$

and

$$\displaystyle \alpha \otimes \beta ( (v_1, \cdot \cdot \cdot , v_{ r+s } ) = \alpha ( v_1 , \cdot \cdot \cdot , v_r ) \beta ( v_{ r+1 }, \cdot \cdot \cdot , v_{ r+s } )$$

Now consider $$\displaystyle \sigma$$ where ...

... $$\displaystyle \sigma$$ sends $$\displaystyle (1, \cdot \cdot \cdot , r+s )$$ to $$\displaystyle (r+1, \cdot \cdot \cdot , r+s, 1 \cdot \cdot \cdot , r )$$

We have

$$\displaystyle ^{ \sigma }( \alpha \otimes \beta ) (v_1, \cdot \cdot \cdot , v_{ r+s } )$$

$$\displaystyle = \alpha \otimes \beta ( v_{ r+1 }, \cdot \cdot \cdot , v_{ r+s }, v_1, \cdot \cdot \cdot , v_r )$$ ... ... hmm ... this does not appear to be correct ...

BUT ... ... if we consider $$\displaystyle \sigma_1$$ where ...

... $$\displaystyle \sigma_1$$ sends $$\displaystyle (1, \cdot \cdot \cdot , r+s )$$ to $$\displaystyle (s+1, \cdot \cdot \cdot , r+s, 1 \cdot \cdot \cdot , s )$$

then we have

$$\displaystyle ^{ \sigma_1 } ( \alpha \otimes \beta ) (v_1, \cdot \cdot \cdot , v_{ r+s } )$$

$$\displaystyle = ( \alpha \otimes \beta ) ( v_{ s+1 }, \cdot \cdot \cdot , v_{ r+s }, v_1, \cdot \cdot \cdot , v_s )$$

$$\displaystyle = \alpha ( v_{ s+1 }, \cdot \cdot \cdot , v_{ r+s } ) \beta ( v_1, \cdot \cdot \cdot , v_s )$$

$$\displaystyle = \beta \otimes \alpha$$ ...

Given that my working differs from Browder ... I suspect I have made an error ...

Can someone please point out the error(s) in my working ...

Help will be much appreciated ...

Peter