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