Talk:Fundamental pair of periods

Inconsistent typesetting of omega

This page is currently very inconsistent with its omegas which is visually jarring to me. We have things like $\omega _{1}$ , ω₁, and ω₁... My suggestion is to just use $\omega _{1}$ , etc. Thoughts? Kclisp (talk) 23:35, 25 March 2023 (UTC)[reply]

The appearance isn’t quite so inconsistent in some skins/browsers due to font choice, but I switched it to all LaTeX. Does that solve the issue? –jacobolus (t) 20:22, 2 April 2023 (UTC)[reply]

Ah, I see. Yes, it's much better now, thanks!

Kclisp (talk) 00:43, 3 April 2023 (UTC)[reply]

uniqueness?

If $(\omega _{1},\omega _{2})$ defines a lattice, does not $(\omega _{2},\omega _{1})$ define the same lattice? So why specify that it is an ordered pair? —Tamfang (talk) 19:30, 2 April 2023 (UTC)[reply]

If

(\omega _{1},\omega _{2})

defines a lattice, then any pair

(a\omega _{1}+b\omega _{2},c\omega _{1}+d\omega _{2})

defines the same lattice for any matrix

[a,b;c,d]

in the modular group, as described in the section Fundamental pair of periods § Modular symmetry. That certainly includes the example of

a=d=0,\ b=c=1.

Those two different fundamental pairs would define the same lattice. –jacobolus (t) 19:50, 2 April 2023 (UTC)[reply]

A friendly note: this doesn't answer @Tamfang's main question: why is it ordered? Kclisp (talk) 01:22, 3 April 2023 (UTC)[reply]

As a test of Cunningham's Law, I removed that term. —Tamfang (talk) 22:45, 20 June 2023 (UTC)[reply]

An unordered set is clearly enough to uniquely specify the lattice, but if you want to write lattice elements as collections of coefficients, you had better have some kind of indexing of the pair of periods that you can match up with the indexing scheme on your coefficients. As far as I can tell there is no benefit whatsoever to talking about an unordered set in this context, so I reverted your change. –jacobolus (t) 01:44, 21 June 2023 (UTC)[reply]

If you want to change this, perhaps you could do a literature review of the most widely cited sources mentioning complex lattices, and see how they typically set up their definition. –jacobolus (t) 01:46, 21 June 2023 (UTC)[reply]

I did some quick skimming of about a dozen sources that popped up in Google scholar. Every source I can find defines a "pair of periods" to be a pair (i.e. 2-tuple, ordered list)

(\omega _{1},\omega _{2}).

I can't find a single source which uses an unordered set. –jacobolus (t) 00:21, 22 June 2023 (UTC)[reply]