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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,881
Hobart, Tasmania
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:



Browder - 1 - Proposition 12.25  ... ... PART 1 .png
Browder - 2 - Proposition 12.25  ... ... PART 2 .png


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