Order Relation on the Permutation Symbols in the Ehresmann Subvariety Class Associated to the Distinguished Monomials of Flag Manifolds ()
1. Introduction
A flag 
is a nested system
(1)
, 
of subspaces of P(V), the projective space of an (n + 1)-dimensional vector space V over
, the field of complex numbers. The set of all such flags is called flag manifold and will be denoted by
. The general linear group 
acts transitively on
. Let E be a fixed reference flag in
. The isotropic group of 
is a Borel subgroup 
so that
(2)
Its dimension is
. The flag manifold F(n + 1)
is the disjoint union of 
-orbits indexed by elements of symmetric group 
(3)
The major interest in this direction has been on the cohomology of these manifolds, where by cohomology, we mean in a general sense; singular and equivariant, K-theory and equivariant K-theory. For each of these theories, there are two descriptions of cohomology. One is in terms of Ehresmann classes, which are cohomology associated to the Ehresmann subvarieties of 
given in terms of permutation symbols. There is one Ehresmann class for each permutation symbol [1]. The Ehresmann classes form a basis for the cohomology over its ground ring and the other is in terms of generators and relations called the Borel-Hirzebruch basis elements [2].
Definition 1. Let
be a fixed flag. An Ehresmann symbol is a matrix
(4)
where 
are the integers such that
.
Following Monk [3], the 
row of this symbol is to be interpreted as a Schubert condition 
on the element 
of
. The matrix represents a subvariety of 
consisting of all the flags F satisfying the conditions:
(5)
Definition 2. The variety of 
is said to be irreducible(and the corresponding symbol is called an irreducible symbol) if for every 
, there exists 
such that 
The set of all such irreducible varieties is called the Ehresmann base.
Remark 1. Writing a matrix for each irreducible symbol is unwieldy and Monk [3] suggested representing the matrix by a permutation 
of 
where 
is the new element in the 
row and 
is the missing integer. Conversely every permutation of 
determines an irreducible symbol and hence the number of elements in the Ehresmann base is
.
It has been proved that the dimension of the subvariety represented by the matrix when irreducible is
(6)
2. Distinguished Monomials
It is well known in [4-6] that the flag manifold 
comes equipped with a flag of tautological vector bundles 
and associated sequence of line bundles
,
. The 
possess natural hermitian structures induced from the standard hermitian metric 
on 
-dimensional vector space 
over
. For
, we denote by
, the 
-dimensional Chern form on 
of the hermitian line bundle 
[7-9]. In other words, they represent the Chern classes 
in the cohomology of
. The only nontrivial Chern class is the first Chern class, which is an element of the second cohomology group of the manifold [10]. The cohomology ring 
is therefore, generated by the Chern classes
.
There is indeed a correspondence between the permutation symbols and the
, viz,
and it is interesting to note that any permutation symbol can be identified uniquely with certain product of these generators. These specialized products are called the distinguished monomials.
Definition 3. Let 
be any cycle of the Ehresmann subvariety class of dimension 
in the cohomology of the flag manifold
, then the product 
is the distinguished monomial of 
where
, that is,
Example 1. The distinguished monomial of the cycle 
in the Ehresmann cycle class of dimension 2 of the cohomology of F(4) is given by
.
Definition 4. The degree 
of the distinguished monomial 
is given by
, the index of the cycle
, that is, 
The collection of distinguished monomials is denoted by 
3. Main Results
We now compare any two distinguished monomials and study the effect of this comparison on their respective defining cycles via the code of invariants
, the collection of 
-tuple exponents of distinguished monomials. In order to this, we impose ordering on these monomials. In practice, we shall assume the following relation on the generators 
Several orderings can be defined on set of monomials but due to the characterization of
, it seems lexicographic order and graded lexicographic order are most appropriate.
Definition 5 (Lexicographic Order). Let
and
the collection of 
-tuple exponents of distinguished monomials. 
if in the vector difference
, the left-most nonzero entry is positive. We shall write
if 
Definition 6 (Graded lexicographic Order). Let
, the collection of 
-tuple exponents of distinguished monomials. We say
if
or 
and
.
The distinguished monomial ordering relation on 
on the code of invariants
, the set of 
-tuple of collection of monomials is well-ordered. By the distinguished monomial ordering relation on 
in this context, we mean graded lexicographic order on 
and denote it by
.
Definition 7. Let 
and 
be any two cycles in the Ehresmann cycle class
of dimension
. We say
if
Remark 2. In general, the ordering extends over the the Ehresmann base
. In other words, the ordering still holds even if the cycles are not equivalent.
Lemma 1. If 
and 
are any two irreducible symbols of the Ehresmann subvarieties in the the flag manifold
, then 
is equivalent to 
if and only if
where
Remark 3. Equivalence of permutation symbols is an equivalence relation. Each of the partitions is called the Ehresmann cycle class and denoted by
where 
is the dimension of the class and hence the flag manifold 
is given by the disjoint union:
(7)
Theorem 1. Let 
be an Ehresmann cycle class of the flag manifold
. Let 
be the subcollection of the distinguished monomials of degree 
of 
in the cohomology ring of the manifold
. Then the dimension of of the class
is expressed in terms of the degree of the monomials, that is
Proof. The dimension of any Ehresmann cycle class in the flag manifold 
has been proved by Ehresmann[3] and given by
(8)
where
. Extending the summation to accommodate 
automatically puts 
which makes equation 8 still stable. In this case, 
turns out to be index 
of any cycle 
in the class 
given by 
which coincides with the degree of the distinguished monomial of the cycle. The 
is precisely the dimension of the flag manifold
that is, 
and hence
Theorem 2. Let
be the Ehresmann cycle class of dimension 
in the cohomology of
, and let
be the disjoint union of such classes. Let
be the graded monoid of distinguished monomials of degrees 
in the cohomology ring of the flag manifold
. Then there is a natural bijection
between 
and
.
Proof
We define a map
by
(9)
where 
is a subcollection of
, that is,
Let
and
,
,
.
Suppose that
which implies that
where
From the Theorem 1,
and hence
which implies that
Therefore, 
is well defined.
Suppose that
in other words 
and therefore,
and hence 
is injective.
For any subcollection 
in
. By definition,
implies that 
is the dimesion of the Ehresmann class
in 
such that
.
Theorem 3. If the distinguished monomials of two cycles 
and 
in the the Ehresmann base 
are equal then the two cycles coincide.
Proof
In other words, the theorem says no two distinct cycles share the same distinguished monomial. Suppose that 
and 
are not equivalent in the sense of Lemma 2, this leads to the fact that
and hence different distinguished monomials. Now suppose they are equivalent, this implies that
.
Consider the set 
consisting of
and
.
is a subcollection of 
being the set of 
-tuple exponents of distinguished monomials. Since 
is well ordered, 
has a least element and therefore, the distinguished monomials defined by the two 
-tuple exponents differ.
Corollary 1. If 
is a cycle in the Ehresmann cycle class
of dimension
. Then 
has at most one distinguished monomial.
Proof
Suppose 
is identified with
and
then the
and
.
By the definition of 
, the subset 
consisting of
and 
is singleton in 
and hence 
and 
coincide.
Using the definitions 5 and 6, we shall define ordering on the cycles of the Eheresmaan cycle class
of dimension 
and give some of its intrinsic properties in relation to the corresponding subcollection 
of distinguished monomials of degree
, where 
is given by
(10)
Definition 8. Let 
and 
be any two cycles in the Ehresmann cycle class
of dimension
. We say
if
Remark 4. In general, the ordering extends over the Ehresmann base
. In other words, the ordering still holds even if the cycles are not equivalent.
Definition 9. Let
and 
be Ehresmann cycle classes of dimension 
and 
respectively, We say that
if for all cycles 
and 
in
and 
respectively, 
Given any two subcollections 
and 
of distinguished monomials of degrees 
and 
respectively
if for all distinguished monomials 
and 
in 
and 
respectively,
.
Remark 5. The ordering on Ehresmann classes is characterized by dimensions while that of the subcollections of distinguished monomials is given by degrees.
Theorem 4. Let 
and 
be any two cycles in the Ehresmann cycle class
of dimension
, with distinguished monomials 
and 
respectively then
if and only if
Proof
Suppose that
from 2.1, 
and 
are given by
and
respectively,then there is
, 
in the two 
-tuple exponents
such that 
and for all 
coincides with
, if they exist. Therefore, in the the vector difference
the leftmost nonzero entry is negative and and hence
and the results follows easily. On the other hand suppose
this implies that
there is
, 
such that for all
, 
vanish, if 
exist and
(11)
Let the set 
be the natural descending order of the cycles
and
.
Then 
is negative and
Since 
are the
elements of the sets
and 
respectively and therefore
.
Corollary 2. Let
and 
be Ehresmann cycle classes of dimension 
and 
respectively, and let their corresponding subcollections of their distinguished monomials be 
and 
of distinguished monomials of degrees 
and 
respectively,then
if and only if
.
Corollary 3. Let
be Ehresmann cycles classes in the flag manifold 
of dimensions 
respectively such that
, and Let 
their corresponding subcollections of distinguished monomials of degrees 
respectively,such that 
then the relation
induces the relation 
vice versa.
4. Acknowledgements
The first author would like to acknowledge the support provided by Education Trust Fund(Nigeria), University of Ibadan and University of New Mexico. Albuquerque, USA.