2 1. PRELIMINARIES

called the short Pl¨ ucker relation, which will be of special interest to us.

The Pl¨ ucker embedding shows that the Grassmannian Gk(m) is a smooth alge-

braic projective variety. Its homogeneous coordinate ring C[Gk(m)] is the quotient

of the ring of polynomials in Pl¨ ucker coordinates by the ideal of homogeneous poly-

nomials vanishing on Gk(m). The latter ideal is generated by Pl¨ ucker relations.

1.1.2. Grassmannians Gk(m) are particular examples of (partial) flag vari-

eties. More generally, the (partial) flag variety Fl(V, d) associated to the data

d = (d1,...,dn) with 1 ≤ d1 · · · dn ≤ m = dim V is defined as the set of all

flags f = ({0} ⊂ f

1

⊂ · · · ⊂ f

n)

of linear subspaces of V with dim f

j

= dj ; it can

be naturally embedded into the product of Grassmannians Gd1 (m) ×· · ·× Gdn (m).

The image of this embedding is closed, thus Fl(V, d) is a projective variety. If

d = (1, 2,...,m − 1), then Fl(V, d) is called the complete flag variety and denoted

Fl(V ).

Let us choose a basis (e1,...,em) in V and a standard flag

f0(d) = {{0} ⊂ span{e1,...,ed1 } ⊂ · · · ⊂ span{e1,...,edn }}

in Fl(V, d). The group SL(V ) acts transitively on Fl(V, d). We can use the chosen

basis to identify V with

Fm.

The stabilizer of the point f0(d) under this action is

a parabolic subgroup P ⊂ SLm(F) of elements of the form

⎛

⎜

⎜

⎜

⎜

⎝

A1 · · ·

0 A2

...

.

.

.

.

.

.

... ...

0 · · · 0 An

⎞

⎟

⎟

⎟

⎟

⎠

,

where Ai are di × di invertible matrices. Thus, as a homogeneous space, Fl(V, d)

∼

=

SLm(F)/P. In particular, the complete flag variety is isomorphic to

Flm(F) := SLm(F)/B+,

where B+ is the subgroup of invertible upper triangular matrices over F.

Two complete flags f and g in Flm(F) are called transversal if

dim(f

i

∩

gj)

= max(i + j − m, 0)

for any i, j ∈ [1,m]. When two flags are transversal, we often say that they are in

general position.

Fix two transversal flags f and g and consider the set Uf,g(F)

m

⊂ Flm(F) of all

flags that are transversal to both f and g. The set Uf,g(R)

m

serves as one of our

main motivational examples and will be studied in great detail in the next chapter.

For now, note that, since SLm(F) acts transitively on the set of pairs of transverse

flags, we may assume without loss of generality that

gj

= span{e1,...,ej} and

f

i

= span{em−i+1,...,em}. Then any flag h transversal to f can be naturally

identified with a unipotent upper triangular matrix H whose first i rows span the

subspace

hi.

In order for h to be also transversal to g, the matrix H must satisfy

an additional condition: for every 1 ≤ k ≤ m, the minor of H formed by the first

k rows and the last k columns must be non-zero.