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I'm reading some paper where they haven't really defined their notation very well (or I've missed something). You can see the image below.

What does the square bracket and star mean precisely? The context is that a space (with a group acting on it) is partition into $[G_1, G_2]$. Is it some representation or some lift, something to do with cosets or something else?

enter image description here

Paper link is here: https://arxiv.org/abs/2002.11259

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  • $\begingroup$ I've never seen this notation before. Stating the specific paper would be more helpful. It looks like it's just some generic "we're defining a new structure starting with something that has a pre-existing structure, so to distinguish them we do such-and-such". $\endgroup$ Mar 31, 2021 at 20:59
  • $\begingroup$ @zibadawatimmy Thanks I forgot to post the link. I now think it is likely a "lift" or something like that. So the element is on the partitioned space is lifted to the full space via asterisk or something like that. They really didn't define it in the body of the paper. $\endgroup$
    – mathtick
    Mar 31, 2021 at 21:02
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    $\begingroup$ the square brackets denote the unit of a quantity, say if $e_1=5$ m/s then $[e_1]=$m/s; the equations express the obvious fact that if you change units from meters to kilometers and from seconds to hours, that then the unit of velocity becomes km/h; this has very little to do with group theory, it's basic dimensional analysis. $\endgroup$ Mar 31, 2021 at 21:11
  • $\begingroup$ @CarloBeenakker I don't think so. The square brackets are used for that elsewhere in the paper but I don't see how for example [e_1]^* = e_2 if you think e_2 can be units AND a quantity. $\endgroup$
    – mathtick
    Mar 31, 2021 at 21:19
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    $\begingroup$ The notation seems introduced p21-22, having a look makes me quite confused and I don't have the patience to convert this into standard mathematical language (I doubt it says anything deep). (Less important, but "will" in a theorem doesn't sound very serious...) $\endgroup$
    – YCor
    Mar 31, 2021 at 21:34

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