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
· · · f
of linear subspaces of V with dim f
= 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
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

= 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)
Flm(F) of all
flags that are transversal to both f and g. The set Uf,g(R)
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
= span{e1,...,ej} and
= 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
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.
Previous Page Next Page