Plücker Relations Finally
I’ve always been frustrated and confused by the statement of the theorem defining plucker relations. There are too many letters! In addition to the m and n which give the size of a matrix, there are also p, q, s, t, and then a,b,c which are indexed by the previous six letters. I’m currently reading this expository paper by Bruns and Conca – http://arxiv.org/pdf/math/0302058v3 . Unfortunately these notational woes still exist. I think i’ve just figured out the point again and am writing it down as a note to myself:
The basic idea is that you want to look at maximal minors of an m by n matrix with m < n. Then you basically pick some number of columns (more than m, less than n) - call these “shifty columns” and then some other columns split into two sets – left columns and right columns.
What we’ll do next is write down a standard product of two maximal minors
(Minor 1)(Minor 2)
Put all the “left columns” on the far left, and all the “right columns” on the far right.
(Left guys, Stuff)(Stuff, Right Guys)
The idea is that we take the shifty columns and throw them into the “stuff” portion of the previous line. There are just two rules:
1. Split the shifty columns so that both sets of parentheses describe a maximal minor (but then again it wouldn’t make sense otherwise)
2. Make sure that the “Stuff” in each parenthesis is written in increasing order.
Then the theorem says that if we take the sum over all ways of splitting the stuff up (multiplied by appropriate signs of permutations) we get 0. Or something like that. This isn’t specific at all, but it’s specific to get the idea across which is likely all that you really need to read a paper that uses this, say to prove the Straightening Law.
Notes: A few days later I tried working out some examples, and there’s one more key point: The number of shifty columns needs to be more than the size of each minor. In other words, you can’t say that (13|24)=(12|34). You have to permute at least 3 guys in this case!
