correction to the last problem

[Vavilov Nikolai]

 

Carissimi,

 

after the class, we continued to discuss how to correct the last problem with Pasha and Stefano,

and Pasha immediately came up with an observation, that elements of order 8 split into TWO

conjugacy classes of orders 6, namely

 

1\pm i, 1\pm j, 1\pm k

 

and

 

-1\pm i, -1\pm j, -1\pm k

 

It is indeed hard to imagine how +1 could be conjugate to -1 (the trace of the first guy in the

faithful 2-dimensional complex representation is +2, whereas of the second one is -2 --- so,

Pasha, these are the FAITHFUL 2-dimensional COMPLEX representations, rather than the

2-dimensional real one with the image S_3, the last two lines of the table).

In other words, the correct list of classes is

 

1A, 2A, 3A, 4A, 4B, 6A, 8A, 8B,

 

of orders

 

1, 1, 8, 6, 12, 8, 6, 6

 

of which -3A=6A, -4A=4A, -4B=4B and -8A=-8B

 

THUS, in the initial printed text one should correct ONE item, 4C to 8B.

 

In other words, all preimages of transpositions are conjugate (we should have jumped to this

conclusion at the moment, when we noticed that i+j is conjugate to i-j).

But preimages of elements of order 4 fall into TWO conjugacy classes, 1+i and -1+i.

 

Now, I hope, there are no more contradictions, and one can complete the character table.

 

Sorry for that! I don't know, what made me denote the class 4C knowing that its elements go

to elements of order 4 in S_4 and that multiplication by -1 does not change the order!

 

Since the order of the elements of the class does not play any role here (both classes

8A and 8B are real), of course the initial problem could be completed to the correct character

character table even with this mistake in the statement. It's only an attempt to correct the

mistake, that lead to a contradiction, and definitive correction. This is a quantum phenomenon,

that only upon a FIRST try a quantum system chooses its state, after which at the SECOND

try you get a WRONG answer, so it's only the THIRD try that works.

 

You know, Japanese samurai used to say, that ALL details should be triple-checked:

"Even a roasted chicken should be tied".

 

N.V.