What happens to the number of inversions in a permutation group if we reflect the matrix across its antidiagonal?

(I suppose that Leibniz formula for determinants should be used or maybe showing by induction?)

Thank you in advance.

Printable View

- Nov 14th 2010, 12:53 AMzadirNumber of inversions
What happens to the number of inversions in a permutation group if we reflect the matrix across its antidiagonal?

(I suppose that Leibniz formula for determinants should be used or maybe showing by induction?)

Thank you in advance. - Nov 14th 2010, 07:39 AMDrexel28
- Nov 14th 2010, 08:21 AMzadir
For instance for n=2

$\displaystyle \[

\left[ {\begin{array}{cc}

a_{11} & a_{12} \\

a_{21} & a_{22} \\

\end{array} } \right]

\]$

So the number of inversions in the two groups:

in $\displaystyle a_{11}a_{12}$ it is 0

in $\displaystyle a_{12}a_{21}$ it is 1

(That's why the value of its matrix is $\displaystyle a_{11}a_{12}-a_{12}a_{21}$)

If we reflect it across its antidagonal:

$\displaystyle \[

\left[ {\begin{array}{cc}

a_{22} & a_{12} \\

a_{21} & a_{11} \\

\end{array} } \right]

\]$

And the number of inversions are the same. For n=3 it is a little more complicated. So I don't really know what happens with the number of inversions if n>2. I would be glad if you could help me. - Nov 14th 2010, 09:20 AMDrexel28
I know what you meant, sort of. This is pretty much what I thought. You're taking the idea of an inversion in a permutation $\displaystyle \sigma\in S_n$ to be two values $\displaystyle x,y\in[n]$ such that $\displaystyle x<y$ but $\displaystyle \sigma(x)>\sigma(y)$. But could you explain this a little more? Combinatorially inversions can be seen as connecting the values of your matrix with lines and an inversion is an incidence of intersecting lines.

So, could you give a precise definition and not an example? It's clear if you interpret combinatorially (as above) that the number of inversions is invariant under such a flip. - Nov 14th 2010, 11:08 AMzadir
There are some definitions:

Permutation Inversion -- from Wolfram MathWorld

Leibniz formula for determinants - Wikipedia, the free encyclopedia

Thank you that you sacrifice time for me. - Nov 14th 2010, 11:15 AMDrexel28
- Nov 14th 2010, 12:04 PMzadir
Sorry, I see. The determinant of a matrix can be definied by permutations. So the inversion refers to permutation, that refers to determinant and that to the matrix.

Or the last step can be left out, we can say that we reflect the determinant across its minor diagonal.

Here are some links:

A Determinant's relation to permutations

Determinants

And thanks once again.