How to work with the action of the symmetric group in Sage?

Ask SageMath Archive
,
1

I wonder if there is a more natural or intuitive way to think about the action of an element of the symmetric group on the set it permutes. I will illustrate my concern with a simple example:

in: S=SymmetricGroup(3)
in: a=S((1,2))
in: b=S((2,3))
in: a*b
out: (1,3,2)
in: [a(b(i)) for i in range(1,4)]
out: [2, 3, 1]
in: [b(a(i)) for i in range(1,4)]
out: [3, 1, 2]
in: [(a*b)(i) for i in range(1,4)]
out: [3, 1, 2]

I understand that Sage is internally consistent here; my confusion is about how to reconcile this behavior with the usual mathematical notation for the permutation action.

What I have in mind intuitively is the following hypothetical:

in: [(i)(a*b) for i in range(1,4)] == [((i)a)b for i in range(1,4)] 
out: True

where I would denote the right action of an element w on the index i by (i)w.

Maybe I am missing a standard or recommended approach in Sage for thinking about permutation actions in this situation, and I would appreciate any clarification.

A potentially related question.

2

Comments on the original question

3

First, you may be wrong about the “usual mathematical notation” - please check this MO discussion and in particular the answer of Brendan McKay there.

Second, the question you cited does provide a workaround via use of Permutations.

4

See also this discussion at sage-devel: https://groups.google.com/g/sage-deve…

5

Brendan’s explanation helps, and thinking in terms of i^w clarifies the convention. My remaining confusion isn’t about left-to-right composition, but about notation in Sage.

In Sage, w(i) denotes the action i^w, even though it looks like ordinary function application. I find that this notation makes me commit unintentional mistakes when switching viewpoints.

As a workaround, I’ve been explicitly separating the two by defining

def perm_eval(w, i): return w.inverse()(i)

so evaluation behaves the way I intuitively expect from function notation.

Ideally, I’d like to work only with something like Brendan’s notation i^w in Sage and, if possible, block or avoid the w(i) syntax altogether so I can’t make accidental mistakes. I don’t know whether Sage supports this.

6

I agree that i^w might be a better notation for a right action than my hypothetical (i)w; I wasn’t aware that i^w is the more standard notation for this kind of action. And Sage’s notation w(i) for a right action might cause mistakes. Thanks for letting me know.