CHAPTER 1
Preliminaries
In this chapter we collect necessary terms and notation that will be used
throughout the book.
1.1. Flag manifolds, Grassmannians, Pl¨ ucker coordinates and Pl¨ucker
relations
1.1.1. Recall that the Grassmannian Gk(m) is the set of all k-dimensional sub-
spaces in an m-dimensional vector space V over a field F. Any k-dimensional sub-
space W of V can be described by a choice of k independent vectors w1,...,wk W ,
and, vice versa, any choice of k independent vectors w1,...,wk V determines a
k-dimensional subspace W = span{w1,...,wk}. Fix a basis in V . Any k indepen-
dent vectors in V give rise to a k × m matrix of rank k whose rows are coordinates
of these vectors. Hence any element X of the Grassmannian can be represented
(non-uniquely) as a k × m matrix
¯
X of rank k. Two k × m matrices represent the
same element of the Grassmannian Gk(m) if one of them can be obtained from the
other one by the left multiplication with a nondegenerate k × k matrix.
Let I be a k-element subset of [1,m] = {1,...,m}. The Pl¨ ucker coordinate xI
is a function on the set of k × m matrices which is equal to the value of the minor
formed by the columns of the matrix indexed by the elements of I. Note that for
any k×m matrix M and any nondegenerate k×k matrix A, the Pl¨ ucker coordinate
xI (AM) is equal to det A · xI (M). The Pl¨ ucker embedding is an embedding of the
Grassmannian into the projective space of dimension
(
m
k
)
−1 that sends X Gk(m)
to a point with homogeneous coordinates
x1,2,...,k(
¯)
X : · · · : xi1,...,ik (
¯)
X : · · · : xm−k+1,m−k+2,...,m(
¯).
X
Pl¨ ucker coordinates are subject to quadratic constraints called Pl¨ ucker rela-
tions. Fix two k-element subsets of the set [1,m], I = {i1,...,ik} and J =
{j1,...,jk}. In what follows, we will sometimes use notation
(1.1) I(iα l) = {i1,...,iα−1,l,iα+1,...,ik}
for α [1,k] and l [1,m].
Then for every I, J and α Pl¨ ucker coordinates satisfy a quadratic equation
(1.2) xI xJ =
k
β=1
xI(iα→jβ
)
xJ(jβ
→iα)
.
If I, J are of the form I = {I , i, j},J = {I , p, q}, where I is a (k 2)-subset
of [1,m] and i, j, p, q are distinct indices not contained in I , then (1.2) reduces to
a 3-term relation
(1.3) xI ijxI
pq
= xI pjxI
iq
+ xI qjxI
pi
1
http://dx.doi.org/10.1090/surv/167/01
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