User:Dratman

Ralph Dratman

This article currently hews to the convention that permutations are to be applied from right to left. This follows the f(g(x)) paradigm, where the parentheses specify the order of operations. Personally I prefer to apply permutations from left to right. Nevertheless, I can see the appeal of the other approach, even if I don't personally like it. But I do insist that a difference in conventions ought not to impede the reader's path through the material, and I hope on that we might all agree. (If not to help readers, why would any of us spend time on this godforsaken---sorry, on this obscure talk page?)

Now when the competition between two opposite conventions in notation becomes ideological, I think the reader's study can be impeded. I suggest that the following, quoted from the page as it is at the time I write this, constitutes an ideological roadblock:

Huh? What is the reader supposed to make of the idea that the given equation is "somewhat unfortunate"? Here, I believe, we see the victory of ideology over utility. What the text thinks is "unfortunate" is that the conventional order of matrix multiplication, namely left to right, betrays the principle of applying permutations from right to left!

As a newcomer to group theory, I attempt to work out some examples by hand, and some by Mathematica. As it happens, Mathematica uses the left-to-right convention. This, in turn, fits with the mechanical method of composing permutations by moving around the sequence to be permuted "by address #2" rather than "by value".

To illustrate this tedious dichotomy, suppose I set out to rearrange the sequence (s,n,a,r,k) by the permutation (3,5,1,2,4). Note that I begin with a non-numeric sequence. Had I started with a numeric sequence, one step below would have been saved. But had I started with the particular sequence (1,2,3,4,5), all the results would be the same, I think. This is not simple.

Implicitly, the permutation written as (3,5,1,2,4) refers to

However, the above can still be interpreted at least two ways.

Using the "BY ADDRESS #1 --- WHICH IS LESS COMMON, I THINK --- " convention, I interpret the permutation to mean "send the contents of position 1 to position 3; send the contents of position 2 to position 5; send the contents of position 3 to position 1; send the contents of position 4 to position 2, and finally (with no actual choice remaining), send the contents of position 5 to position 4. The result of BY ADDRESS #1 is (a,r,s,k,n).

Using the "BY ADDRESS #2" convention --- WHICH IS MORE COMMON, I THINK ---, I interpret the permutation to mean "send the contents of position 3 to position 1; send the contents of position 5 to position 2; send the contents of position 1 to position 3; send the contents of position 2 to position 4, and finally (with no actual choice remaining), send the contents of position 4 to position 5. The result of BY ADDRESS #2 is (a,k,s,n,r).

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