Talk:Double coset

Mistake

Hi,

I think the definition of multiplication in Hecke algebra is acually wrong here. Fish11223344 (talk) 17:05, 1 March 2016 (UTC)[reply]

Can you be more specific, and more importantly, supply a reference. Thanks. Bill Cherowitzo (talk) 18:16, 1 March 2016 (UTC)[reply]

The discussion in "Products in the free abelian group on the set of double cosets" contained errors. I have removed part of the section; someone can fix it and add it back. The issues surround the formula [HxK][KyL] = sum [HabL] where a,b run over right coset reps of H in HxK and left coset reps of L in KyL. Let's just take H=K=L. First: the formula gives [H][H] = [H]. But the description via multiplication of left/right invariant functions in the group algebra gives [H][H] = #H [H]. So right off the bat there is a normalization issue. Working over a field this wouldn't be a problem (just re-normalize) but over Z this doesn't work. So over Z the algebra being described by that formula is really not the same as the algebra obtained by multiplication in the group algebra.

But in any case the given formula can't be correct. For example consider G = S_3 and H = cyclic of order 2. Then there are two double cosets, H and G-H. The forumla [HxK][KyL] = sum [HabL] gives [H]*[H] = [H] and [H]*[G-H] = 2*[G-H]. But this is not associative (consider [H]*[H]*[G-H] bracketed two ways). The issue is that the indicator function of the cosets Ha and bL don't convolve to give the indicator function of HabL, but rather #H * #L / #(HabL) times this, and this factor has to be taken into account in the formula. 24 July 2020.

Another mistake?

"If H is normal, then H \ G is a group, and the right action of K on this group factors through the right action of H \ HK. It follows that H \ G / K = HK \ G. "

If H is the trivial group you get G / K = K \ G, which is not true if K is not normal. I think something is wrong here. — Preceding unsigned comment added by 176.231.159.63 (talk) 08:49, 22 April 2021 (UTC)[reply]

It looks to me like left versus right got mixed up; we should have H \ G / K = G / HK if H is normal. I've changed the article. Ozob (talk) 03:55, 26 April 2021 (UTC)[reply]

Thank you. Now I don't I understand the explanation: "the right action of K on this group factors through the right action of H \ HK". It is quite obvious that HgK=gHK if H is normal.

Well, I think it's a matter of phrasing. The article is observing that H \ G / K can be computed as (H \ G) / K, and that this is also (H \ G) / (H \ HK) = G / HK. You are observing that the set of double cosets HgK is the same as the set of cosets gHK. These probably amount to the same thing if you fill in every little detail. If you think you can clarify the article, please go ahead! Ozob (talk) 17:38, 30 April 2021 (UTC)[reply]