Talk:Unitary group

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[edit] Other fields

Could someone add a definition of a unitary group over a finite field? These are pretty important in group theory however I don't have the background to do this. TooMuchMath 02:42, 14 April 2006 (UTC)

I think someone has done this to an extent, but it is not clear if the contributions were correct. From the standpoint of finite groups, you do not definite the unitary group for just one field, but for a degree two field extension F < K with some field automorphism f of order two. Then for every matrix A in GL(n,K), you define A^* to be the matrix formed from A by first taking the transpose and then applying f to each entry. (A^*)^* = A, and the matrices satisfying AA^* = 1 form a group called U(n,K/F). When K is finite, then F and K uniquely define each other and every F has a K (but not every K has an F); notation varies about whether the group is called U(n,q) or U(n,q^2) where q is the size of F and q^2 is the size of K.
At any rate, back to the current revision. I don't think U(n,C) has a well-defined meaning. For A in U(n,K/F), 1 = det(AA^*) = det(A)*det(A)^f, so Norm_{K/F}(det(A))=1. When K=F, that means f=1, and det(A)^2=1, so U(n,C/C) is at best O(n,C), but really should be undefined since [C:C]=1, not 2.
One method to motivate U(n,C)=GL(n,C) would be to say that U is the twisted group of Lie type with Dynkin diagram of type A, and over an algebraically closed field there are no possible twists, so instead ofa twisted version of GL(n,C), you just get GL(n,C). Both GL(n,C) and O(n,C) as answers seem incredibly fishy. JackSchmidt (talk) 06:41, 14 December 2007 (UTC)
My modification of the section "Other Fields" was in the following vein: there is an algebraic group over R, call it G, whose real points G(R) are U(n) as defined in the lead of this article (namely as the real Lie group consisting of elements of GL(n,C) such that AA*=I). In the notation of your second paragraph above, G=U(n,C/R). This is the standard way to consider the unitary group. I'll discuss its C-valued points below.
There are of course many many unitary groups. I've personally mainly only dealt with unitary groups related to a totally complex extension of a totally real field, though just looking some stuff up now here's what seems to be going on (see page 103 of Milne's notes Algebraic Groups and Arithmetic Groups available here [1] for example). Milne says take a field k and let K be a degree 2 separable k-algebra, then there's a unique k-automorphism a\mapsto\overline{a} of K such that a=\overline{a} if, and only if, a\in k. This allows one to define an involution on the set of nxn matrices Mn(K) that sends A to A^*:=\overline{A}^t. The standard n-variable unitary group over k with respect to K/k would then be the algebraic group over k whose points over a k-algebra R would be
U(n,K/k)(R):=\{A\in GL(n,K\otimes_kR):AA^*=I\}
(where for a\otimes r\in K\otimes_kR we define \overline{a\otimes r}=\overline{a}\otimes r). More generally though, one could take an arbitrary invertible Hermitian matrix (where here this means Φ such that Φ=Φ*) in Mn(K) (or to avoid picking a basis, pick an n-dimensional K-vector space and a non-degenerate hermitian form Ψ on V) and define the unitary group in n-variables over k with respect to K/k and Φ (or Ψ) as the algebraic group over k whose points over a k-algebra R are
U(n,K/k,\Phi)(R):=\{A\in GL(n,K\otimes_kR):A\Phi A^*=\Phi\}
(or if you've chose the Ψ route, as the elements of GL(VR) that are Ψ-invariant). For example, taking C over R and Φ the diagonal (m+n)x(m+n) matrix with m +1's and n -1's, this would give the group over R whose R points are the real Lie group generally denoted U(m,n).
As for the fact that U(n)(C)=U(n,C/R)(C)=GL(n,C), in general U(n,K/k)(K)=GL(n,K). Indeed, first note that K\otimes_kK\cong K\oplus K as k-algebras by sending x\otimes y\mbox{ to }(xy,\overline{x} y), then \overline{x\otimes y}=\overline{x}\otimes y\mbox{ is sent to }(\overline{x}y,xy), so you can see that for (x,y) in K\oplus K, \overline{(x,y)}=(y,x). Then GL(n,K\otimes_kK)=GL(n,K\oplus K)=GL(n,K)\times GL(n,K). Tracing the definition of everything, and element (A,B) in GL(n,K)xGL(n,K) is in U(n,K/k)(K) if ABt=I and BAt=I, which are both the same equation and are satisfied if, and only if, B=(At)-1. So U(n,K/k)(K)={(A,(At)-1):A is in GL(n,K)} and this is clearly just GL(n,K). I've said quite a bit, so I'll leave it there for now. RobHar (talk) 09:19, 14 December 2007 (UTC)
I think your definition should work well in all of the contexts I care about. The only minor problem is the full notation would be quite long: U(n,K/k,Φ)(L) is defined for all fields k, all separable 2 dimensional k-algebras K (which defines the automorphism f and the matrix operator *), all elements Φ of GL(n,K) with Φ=Φ*, and all k-algebras L. (Here algebra is associative with 1, I think).
U(n,K/k,\Phi)(L) = \{ A \in GL(n,K \otimes_k L) : A\cdot (\Phi \otimes 1) \cdot A^* = \Phi \otimes 1 \}
For applications to finite simple groups, Φ is required to be non-degenerate and K is required to be a field (that is, K=k × k and K=k[x]/(x2) are not allowed), and then all such groups U(n, K/k, Φ) for a fixed finite field k are isomorphic and depend only on the order q of k, or on the order q^2 of K, and so they can be denoted either U(n, q) or U(n, q2). The independence on Φ is interesting and often discussed in books on classical groups, so it seems quite reasonable to mention Φ even in the finite field case.
Presumably for your applications it would be interesting to mention U(m+n, C/R, I_m ⊕ -I_n)(R) = U(m,n).
Does it sound reasonable to include the "full" definition and notation, and then a nice list of examples? I am fine with writing up the finite field case, but I'd want you to handle the case where Φ actually mattered. JackSchmidt (talk) 18:13, 14 December 2007 (UTC)
I read over Milne's notes (I had been reading some books on linear algebraic groups and those sufficed yesterday). His notes are quite clear and answered my silly question: Surely there are only two types of separable k-algebras of dimension 2, one a field extension (possibly many isoclasses) and one k x k (unique), and only one inseparable algebra, k[x]/(x^2) (also unique; has a unique auto of degree 2 in char not 2, but lots of other autos). It seems silly to use such language when the only interesting example is a degree two field extension. Also to make sure, there is no need to consider K=k x k for your calculation of U(n,C/R,1)(C), right? I only ask because the calculation for U(n,(C x C)/C,1)(C)=GL(n,C) is fairly similar.
Would it better then to require K to be a field extension of degree 2? I think including Φ is a good idea. Me personally, I don't see much interesting about evaluating the group scheme thingy on other k-algebras than k, but in some sense the U(n,K/k,1)(K)=GL(n,K) is interesting.
In other words, I'm leaning more towards the "standard" definition of a unitary space of a k-vector space and a K/k sesqui-linear form. Grove's Classical Groups text has such a point of view. JackSchmidt (talk) 06:15, 15 December 2007 (UTC)
I think it is definitely important to be able to view U(n) as an algebraic group and be able to evaluate it on k-algebras other than k. For example, the recent proof of the Sato-Tate conjecture, as well as the attacks on the main conjecture of Iwasawa theory for GL(2) all use in an essential way these unitary groups and the theory of their automorphic representations, Shimura varieties, and associated Galois representations. They are important in the study of the Langlands program as one of the examples of reductive algebraic groups that give rise to Shimura varieties (unlike the the groups GL(n) themselves).
One could avoid saying degree two separable algebra and just consider degree two extensions, but I find it aesthetically better to do the more general thing and mention that really there's only the two cases (degree two extensions and kxk). I think it's nice that one can obtain GL(n) from this construction as well. This is similar to the classification of central simple algebras of dimension 4: there are all the various division algebras (quaternion algebras) and then there's the 2x2 matrices (and this is related to the current situation as one can construct a unitary group as the units in a central simple algebra).
The hard part will be to do this in a way that will be readable and understandable. Perhaps one could start with just the usual U(n) as a matrix group (over R), and at first just mention that there are generalizations, first U(m,n) as a matrix group (over R), and then the more general unitary groups over arbitrary fields, giving examples and maybe some isomorphisms that must exist for small unitary groups over small finite fields (that you probably know about). (I'd also like something on the unitary similitude group that is mentioned in the lead).
One reason I hadn't tried writing this up before was a lack of references which I still have. Milne does a nice job, but does not address the more general Φ situation, and the only references I've found so far only address this situation when k is a local or global field, and though everything is probably fine with what I said above, I kinda wanted to get an all-encompassing reference before including it in the article. Sorry I'm pretty busy right now, so I won't be able to do much, but I definitely think this page needs a lot more than it currently has, and this is definitely the first step. I'll keep trying to find references (unless you already have one) RobHar (talk) 08:05, 15 December 2007 (UTC)
I have a soft spot for the trivial element of the Brauer group, and apparently group schemes are quite popular, so we might as well include all of it. To make it more digestible though, how about presenting it in stages:
  1. AA*=1 over C (main article)
  2. AΦA*=Φ over C (U(m,n) physics types would like the current article and this addition)
  3. AΦA*=Φ over any dim2 sep K/k, especially K a quadratic extension field of k
  4. Finite field interlude (all Φ are equivalent, only one quadratic extension of any finite field, so U(n,K/k,Φ)=U(n,q)).
  5. AΦA*=Φ is just a set of polynomial equations, so this is an algebraic group (not group scheme)
  6. Polynomial equations make sense over any k-algebra, so this is a group scheme
I guess 1 is already done, 2 I think you know about, 3 either of us could write (I have a reference handy), 4 I can do, 5 and 6 could be combined if you like. For 6, if K,F are algebraically independent extensions of k, then U(n,K/k,Φ)(F) is U(n,KF/F,Φ)(F) which is somewhat pretty. In particular, if k is the rationals, and K is any imaginary quadratic extension and F is the reals, then U(n) is just the R points of U(n,K/k,Φ). When they are not algebraically independent it just looks silly to me, but probably the point is that the theory is uniform, not that the examples are interesting.
I think the huge advantage of this way is that it is easy to write a small part, find sources for it, etc. without compromising either the special case or the general case. JackSchmidt (talk) 23:50, 15 December 2007 (UTC)
Hi Rob & Jack,
Thanks for the detailed discussion and references!
I've taken a stab at writing it up, largely following Jack's outline (which outlines the development of the above discussion). My write-up could surely use some work and references, but hopefully it's a good start.
Nbarth (talk) 00:00, 19 December 2007 (UTC)
Looks good to me. I was painfully precise about which field where in the finite field section (hopefully correct too, it's late). I added SU, PU, and PSU, and included the details on perfection and simplicity. I need to add orders, and check if I know the Sylow and conjugacy class structure. There are also some exception isomorphisms, but I think they are incredibly generic (and weren't obviously visible when I was transcribing Grove into here). I think it is really easy like PSU(2,q^2)=SL(2,q) or so, and the isomorphism is natural for all commutative rings with involution, not just the finite field case. I have an article lying around the office with conjugacy class information, but I can't remember if it was legible.
Thanks for being bold! I have a dream for writing up the finite simple groups on wikipedia in a sane, comprehensive, and consistent way, and it really helps to have skeleton (or even partially fleshy zombie) articles around to improve. JackSchmidt (talk) 04:30, 19 December 2007 (UTC)

[edit] Order of conjugation

I've changed the order of conjugation to A * A = I, rather than AA * = I.

Why, and why do I care?

Because this agrees with how matrices representing forms transform: if Φ is the matrix for the form Ψ, then the matrix for the form Ψ(Av,Aw) is A * ΦA.

The analogous statement for bilinear forms is AtΦA, and this also speaks in favor of making Hermitian forms antilinear in the first variable, rather than the second, as otherwise the matrix for the transformed form would be A^t\Phi \overline A, not A^*\Phi \overline A.

Now, this doesn't make a difference for the definition of the set of classical unitary matrices, as both equations are equivalent to A * = A − 1, but this does make a difference for the action on forms and for the set of unitary matrices for other forms.

Nbarth (talk) 00:08, 19 December 2007 (UTC)