User talk:Simplifix/Archive Aug2008

Burnside ring looks good

 * (reply to your message)

Thanks for creating it! This article has been needed for a long time, and your version is right on target.

Most of my changes were minor: I tidied up some italics, converted some of the images to text, did some technical things to references, and made a few minor spelling or punctuation changes.

I changed two mathematical things: Burnside ring is the set of formal differences of isomorphism classes of (exactly what you said). I briefly changed examples to a subsection, but you are right, it should be a section. I added one sentence to say that marks were analogous to characters, as in, they made dealing with whatever objects just a matter of dealing with component-wise addition and multiplication of integers.

History of multiplication: I'm not sure. Decomposing tensor products of representations goes back before Dress for sure, but I'm not sure whether this was phrased in terms of the Burnside ring.

The advantage of image: looks the same for all browsers, and all users. The disadvantage of image: looks the same for all browsers and all users, even those with poor eyesight who need larger fonts or text readers. As you mentioned, poor browser support for the unicode text makes converting to text only a partial fix. I changed the &cup; back, since html does a poor job of subscripts for big operators (like \sum and \cup). If you ever need to do both a subscript and a superscript, you are pretty much stuck with the tags. One more disadvantage of images: load time. If a page has a ton of images, like WP:RD/MATH, and you reload it, FF jumps around as the images load, making it hard to keep your place. Also if wikipedia is slow that day, then even the first load is jerky. I try to use text as much as possible, but leave images for the few complicated formulas.

I'll probably change \mapsto back as well (or feel free to).

Ring structure: Right now I'm trying to decide if it is worth indicating more explicitly the ring structure of the examples. I've always found the cohomology ring structure descriptions unsatisfying. For S3, I would say Omega(S3) = Z[x,y]/(x(x-2),y(1+x-y)) = Z, so with Z-basis 1,x,y,xy, but this doesn't seem very satisfying. Here x is 2,0,2,0 and y is 3,1,0,0. JackSchmidt (talk) 17:49, 15 February 2008 (UTC)


 * (reply to message)

I actually did mean the linear representation, but probably my perspective is warped. I only care about the Burnside ring because (linear) permutation representations happen to be interesting when you reduce them mod p, and the Burnside ring provides a tidy way to summarize half of this, and the representation ring nearly the other half. I tend to think of G-sets as Z-linear representations that just happen to have a special form, and of the Burnside ring as a (Z-form of a) subring of the complex representation ring (which also needs an article, hint hint). I actually wanted to emphasize the similarity in that opening sentence, but basically it is just a throwaway sentence to help the reader gain some context, so feel free to fix it up. JackSchmidt (talk) 18:33, 15 February 2008 (UTC)

Representation ring
Hello. Please see my edits to representation ring. In particular, notice this difference:
 * 5-3=2
 * 5 &minus; 3 = 2
 * 5 &minus; 3 = 2
 * 5 &minus; 3 = 2

A minus sign is longer than a stubby little hyphen, and when it represents a binary operation, a space preceeds and follows it. (In TeX, the software takes care of this automatically; in non-TeX notation, you do it by hand.) See Manual of Style (mathematics). Michael Hardy (talk) 21:59, 31 March 2008 (UTC)

Another issue of the kind above is suggested by Affine Grassmannian (manifold): In expressions like g(x), the parentheses should not be italicized. Nor should digits and things like "det", "log", "max", etc. All this matches TeX style. Michael Hardy (talk) 00:07, 24 April 2008 (UTC)