# The Wedge Product ... Tu, Section 3.7 ... ...

#### Peter

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I am reading Loring W.Tu's book: "An Introduction to Manifolds" (Second Edition) ...

I need help in order to fully understand Tu's section on the wedge product (Section 3.7 ... ) ... ...

The start of Section 3.7 reads as follows:

In the above text from Tu we read the following:

" ... ... for every permutation $$\displaystyle \sigma \in S_{ k + l }$$, there are $$\displaystyle k!$$ permutations $$\displaystyle \tau$$ in $$\displaystyle S_k$$ that permute the first $$\displaystyle k$$ arguments $$\displaystyle v_{ \sigma (1) }, \cdot \cdot \cdot , v_{ \sigma (k) }$$ and leave the arguments of $$\displaystyle g$$ alone; for all $$\displaystyle \tau$$ in $$\displaystyle S_k$$, the resulting permutations $$\displaystyle \sigma \tau$$ in $$\displaystyle S_{ k + l }$$ contribute the same term to the sum, ... ... "

Since I did not completely understand the above quoted statement I developed an example with $$\displaystyle f \in A_2 (V)$$ and $$\displaystyle g \in A_3 (V)$$ ... ... so we have

$$\displaystyle f \wedge g ( v_1 , \cdot \cdot \cdot , v_5)$$

$$\displaystyle = \frac{1}{2!} \frac{1}{3!} \sum_{ \sigma \in S_5 } f ( v_{ \sigma (1) }, v_{ \sigma (2) } ) g( v_{ \sigma (3) }, v_{ \sigma (4) }, v_{ \sigma (5) } )$$

Now ... translating Tu's quoted statement above into the terms of the example we have ... ...

" ... for every permutation $$\displaystyle \sigma \in S_{ 2 + 3 }$$ there are $$\displaystyle 2! = 2$$ permutations $$\displaystyle \tau$$ in $$\displaystyle S_2$$ that permute the first $$\displaystyle 2$$ arguments $$\displaystyle v_{ \sigma (1) }, v_{ \sigma (2) }$$ and leave the arguments of $$\displaystyle g$$ alone ... ... "

Now, following the above quoted text ... consider a specific permutation $$\displaystyle \sigma$$ ... say

$$\displaystyle \sigma_1 = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 1 \end{bmatrix}$$

Now there are $$\displaystyle 2!= 2$$ permutations $$\displaystyle \tau$$ in $$\displaystyle S_2$$ that permute the first $$\displaystyle k = 2$$ arguments $$\displaystyle ( v_{ \sigma (1) }, v_{ \sigma (2) } ) = ( v_2, v_3 )$$ ...

[Note ... one of the $$\displaystyle k!$$ permutations is essentially $$\displaystyle \sigma_1$$ itself ... ]

These permutations may be represented by (is this correct?)

... in $$\displaystyle S_2$$ ...

$$\displaystyle \tau_1 = \begin{bmatrix} 2 & 3 \\ 2 & 3 \end{bmatrix}$$

$$\displaystyle \tau_2 = \begin{bmatrix} 2 & 3 \\ 3 & 2 \end{bmatrix}$$

and in $$\displaystyle S_{ 2 + 3 }$$ ...

$$\displaystyle \tau_1 = \begin{bmatrix} 2 & 3 & 4 & 5 & 1 \\ 2 & 3 & 4 & 5 & 1 \end{bmatrix}$$

$$\displaystyle \tau_2 = \begin{bmatrix} 2 & 3 & 4 & 5 & 1 \\ 3 & 2 & 4 & 5 & 1 \end{bmatrix}$$

Now the resulting permutations $$\displaystyle \sigma \tau$$ are supposed to contribute the same term to the sum ...

... ... $$\displaystyle \sigma_1 \tau_1 = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 1 \end{bmatrix} \begin{bmatrix} 2 & 3 & 4 & 5 & 1 \\ 2 & 3 & 4 & 5 & 1 \end{bmatrix}$$

$$\displaystyle = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 1 \end{bmatrix}$$

and

$$\displaystyle \sigma_1 \tau_2 = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 1 \end{bmatrix} \begin{bmatrix} 2 & 3 & 4 & 5 & 1 \\ 3 & 2 & 4 & 5 & 1 \end{bmatrix}$$

$$\displaystyle = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 4 & 3 & 5 & 1 \end{bmatrix}$$

Now the two permutations are not identical ... but ... maybe the difference is accounted for in the $$\displaystyle ( \text{ sgn } )$$ function somehow ... but how ... ?

Can someone please clarify the above ...

Peter

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