Cayley tables in a special form and nxn permutation matrices representing a group with n elements

To begin, let me introduce the Kronecker delta function, 𝞭(x,y), where x and y are both elements of some set S.

If two things are the same, the delta function returns a 1. If not, the function returns a zero. 

 

Easy peasy, lemon squeezy.

 

Next, a Cayley table in special form. So far, the top row and left most column were transposes of each other, the same group element in position k of the top row as in position k of the leftmost column.

 

This time, we will line up the group elements with their inverses, so if x is in position k in the top row, position k in the leftmost column will be x⁻¹.

Now, we will do the Kronecker delta of the Cayley table with the identity element (1).

   

This is what the identity matrix should look like, ones along the mail diagonal, zeros everyplace else, which is to say we use the Kronecker delta on i and j for every entry mij in matrix M.

 

Remarkably enough, if we do the Kronecker delta on our Cayley table with any element in S3, we will get the 6x6 matrix that represents that element.

Matrix multiplication agrees with the combination operator of the permutations. More than that, if we want to turn the identity matrix into an even permutation mtarix such as (123) or (132), it will take an even number of row swaps or column swaps. To change the identity into (12), (13) or (23), we will do an odd number of column swaps. 

 

The biography of Leopold Kronecker is remarkable. An excellent student, he was also from a wealthy family and had little interest in teaching, only research at the top level. He was finally persuaded to teach at the age of 39, but students found his lectures hard to follow.

 

His most famous quote is "God made the integers, all the rest is the work of man." He had a visceral hatred of the new concepts of infinity presented by Georg Cantor, and did not like the definition of any irrational real number being defined as the limit of some infinite series of rational numbers. It should be noted that Gauss, born long before Kronecker and Cantor, did do research into infinity and decided not to publish, fearing it would just cause controvesry.

 

The next post will be about invariants in conjugacy classes of permutation matrices.