# Notation for subgroup of $\mathrm{GL}(n)$ consisting of matrices with determinant +1 or -1?

$\mathrm{SL}(n)$ is the standard notation for the subgroup of matrices with determinant $1$, but what about the subgroup of all matrices with determinant $+1$ or $-1$? Is there a standard notation for this in the literature?

I am considering using $\mathrm{SL}_{\pm}(n)$ to denote this group, but if there is already a standard notation in the literature I would prefer to use that instead.

• With coefficients in what? Mar 22, 2018 at 0:23
• @Bernard $\mathbb{R}$ in my case, though $\mathbb{C}$ is fine too. In the latter case we would want matrices $g$ with $|\det g|=1$.
– ಠ_ಠ
Mar 22, 2018 at 1:30
• Does "$\ker | \det(\bullet) |$" work for you?
– lhf
Mar 22, 2018 at 2:06
• @lhf Yeah I guess if there isn't a standard notation I might as well use that instead.
– ಠ_ಠ
Mar 22, 2018 at 21:31

As a group theorist I can't recall seeing such a notation and so I would suggest that it is sufficiently uncommon that you can use your own notation without offense. I might suggest it be a superscript since many conventions that include the field will place the dimension or the field in the subscript position, e.g. $SL^{\pm}_n(K)$ and $SL^{\pm}_K(n)$ would both fit with common patterns.
That said, I wouldn't invest much in promoting the notation as it seems it would open pointless debate. The notation schemes that seem to be here to stay revolve around a connection to a geometric condition, a Lie condition, or are necessary to give names to associated simple groups. So while $\det(x)=\pm 1$ is natural and motivate $SL^{\pm}(n)$, someone might argue it natural to include other roots of unity or whatever subgroup $U$ of scalars. But something like $SL^U(n)$ is starting to get silly. Some group theorist like notation like $2.SL(n)$ to indicate a degree 2 extension (a double cover) but that leaves it as ambiguous "which" extension is implied. Still, a common notation you would find in print. You could also write $\mathbb{Z}_2\ltimes SL(n)$ but that starts to get verbose and suffers the same ambiguity as $2.SL(n)$.
• I'm pretty sure I already saw $SL_\pm$ or $SL^\pm$. Such a notation is in any case very intuitive. The semidirect notation awkwardly assumes to artificially fix a semidirect decomposition.
• I agree the $\pm$ notation is intuitive, and my point was not to encourage the semidirect product vocabulary, but rather to suggest that any notation is bound to have alternatives that strike others as better. E.g. the semidirect notation gets at relevant group structure where as the $\pm$ just says what is in the group. That might be a worthwhile addition to the notation in some context. Mar 26, 2018 at 17:43