# The Unapologetic Mathematician

## The Special Linear Group (and others)

We’ve got down the notion of the general linear group $\mathrm{GL}(V)$ of a vector space $V$, including the particular case of the matrix group $\mathrm{GL}(n,\mathbb{F})$ of the space $\mathbb{F}^n$. We also have defined the orthogonal group $\mathbb{O}(n,\mathbb{F})$ of $n\times n$ matrices over $\mathbb{F}$ whose transpose and inverse are the same, which is related to the orthogonal group $\mathrm{O}(V,B)$ of orthogonal transformations of the real vector space $V$ preserving a specified bilinear form $B$. Lastly, we’ve defined the group $\mathrm{U}(n)$ of unitary transformations on $\mathbb{C}^n$$n\times n$ complex matrices whose conjugate transpose and inverse are the same.

For all of these matrix groups — which are all subgroups of some appropriate $\mathrm{GL}(n,\mathbb{F})$ — we have a homomorphism to the multiplicative group of $\mathbb{F}$ given by the determinant. We originally defined the determinant on $\mathrm{GL}(n\mathbb{F})$ itself, but we can easily restrict it to any subgroup. We actually know that for unitary and orthogonal transformations the image of this homomorphism must lie in a particular subgroup of $\mathbb{F}^\times$. But in any case, the homomorphism must have a kernel, and this kernel turns out to be important.

In the case of the general linear group $\mathrm{GL}(V)$, the kernel of the determinant homomorphism consists of the automorphisms of $V$ with determinant ${1}$. We call this subgroup of $\mathrm{GL}(V)$ the “special linear group” $\mathrm{SL}(V)$, and transformations in this subgroup are sometimes called “special linear transformations”. Of course, we also have the particular special linear group $\mathrm{SL}(n,\mathbb{F})\subseteq\mathrm{GL}(n,\mathbb{F})$. When we take the kernel of any of the other groups, we prepend the adjective “special” and an $\mathrm{S}$ to the notation. Thus we have the special orthogonal groups $\mathrm{SO}(V,B)$ and $\mathrm{SO}(n,\mathbb{F})$ and the special unitary group $\mathrm{SU}(n)$.

In a sense, all the interesting part of the general linear group is contained in the special linear subgroup. Outside of that, what remains is “just” a scaling. It’s a little more complicated than it seems on the surface, but not much.

September 8, 2009 Posted by | Algebra, Linear Algebra | 6 Comments