The Special Linear Group (and others)

We’ve got down the notion of the general linear group of a vector space , including the particular case of the matrix group of the space . We also have defined the orthogonal group of matrices over whose transpose and inverse are the same, which is related to the orthogonal group of orthogonal transformations of the real vector space preserving a specified bilinear form . Lastly, we’ve defined the group of unitary transformations on — complex matrices whose conjugate transpose and inverse are the same.

For all of these matrix groups — which are all subgroups of some appropriate — we have a homomorphism to the multiplicative group of given by the determinant. We originally defined the determinant on 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 . 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 , the kernel of the determinant homomorphism consists of the automorphisms of with determinant . We call this subgroup of the “special linear group” , and transformations in this subgroup are sometimes called “special linear transformations”. Of course, we also have the particular special linear group . When we take the kernel of any of the other groups, we prepend the adjective “special” and an to the notation. Thus we have the special orthogonal groups and and the special unitary group .

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.