What are double cosets in group theory?

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Discussion Overview

The discussion revolves around the concept of double cosets in group theory, specifically focusing on their notation and meaning as encountered in a paper. Participants seek clarification on the definitions and implications of these mathematical constructs.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant introduces the concept of double cosets as encountered in a specific paper and requests help in understanding the notation used.
  • Another participant questions whether the inquiry is about the general concept of double cosets or the specific sets mentioned in the paper.
  • A participant expresses confusion regarding the notation involving slashes and backslashes, suggesting that the slash indicates a quotient and inquires about the meaning of the backslash.
  • One participant explains that the backslash distinguishes left cosets from right cosets, indicating that the two slashes serve to denote different operations without deeper implications.
  • A later reply reflects on the realization that the groups involved are subgroups of G, noting that this understanding clarifies the associative properties relevant to the discussion.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the notation and implications of double cosets, with some clarifications provided but no consensus reached on all aspects of the discussion.

Contextual Notes

There is a lack of clarity on the specific definitions and properties of double cosets, as well as the implications of the notation used, which may depend on the context of the groups involved.

Heidi
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Hi Pfs
It is the first time that reas something about "double cosets"
it was in this paper
https://arxiv.org/pdf/0810.2091.pdf
At page 4 i read
∆1\SU(3)/∆1 = ∆\U(3)/∆
Could you help to understand what are these sets (or cosets)?
thanks
 
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Do you know what a coset (double coset) is? In other words are you asking what double cosets are, or are you asking what these specific ones are?
 
I would like to get the meaning of the slashs and anti slashs
in the wikipedia double coset article i found the notations
H\G/K and (H\G)/K and H\(G/K)
i suppose that the slash is for a quotient but what is the antislash for?
 
Heidi said:
i suppose that the slash is for a quotient but what is the antislash for?
If we have double cosets, then we have to distinguish right and left operations, i.e. multiplication from left and right. The two slashes are meant to do this, backslash for the left coset, and slash for the right coset. There is no deeper meaning.
 
Ah, I got confused until I realized the groups by which we fsctor are all subgroups of G. Then also associaivity kicks in and the triple product on both sides makes sense.
 

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