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 ![](https://www.scirp.org/html/6-5300485\a808371a-7e72-4006-a890-9b0ab563bef9.jpg)
(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
![](https://www.scirp.org/html/6-5300485\9309d842-aa0e-497c-b329-6740bf29fddb.jpg)
be a fixed flag. An Ehresmann symbol is a matrix
(4)
where
are the integers such that
![](https://www.scirp.org/html/6-5300485\fc315d63-1fbe-4ca8-af4d-ca8c45129f7b.jpg)
.
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 ![](https://www.scirp.org/html/6-5300485\52ff729d-6832-4b37-a958-c651a79646c7.jpg)
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)
![](https://www.scirp.org/html/6-5300485\c0f97def-dd4d-4cfb-b45f-e62df0f281de.jpg)
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,
![](https://www.scirp.org/html/6-5300485\b3f90733-3612-4f5e-948e-df376b804d2e.jpg)
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 ![](https://www.scirp.org/html/6-5300485\0c545180-bf61-4dc4-a3db-3d40a8250ac5.jpg)
where
, that is,
![](https://www.scirp.org/html/6-5300485\ef05ceaa-0b22-4dd1-b472-e55258bdcf4c.jpg)
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, ![](https://www.scirp.org/html/6-5300485\cc5bf46b-e5b5-41e8-982b-13398262dcfb.jpg)
The collection of distinguished monomials is denoted by ![](https://www.scirp.org/html/6-5300485\df0bb3d3-27b5-436d-8841-3418721b70ec.jpg)
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 ![](https://www.scirp.org/html/6-5300485\d6cb0303-0ffc-4c2f-8aac-a4e503e2699e.jpg)
![](https://www.scirp.org/html/6-5300485\89f8af25-bd32-4a31-8bbb-31ef9b22cbf5.jpg)
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
![](https://www.scirp.org/html/6-5300485\84d1b518-3dab-4057-beda-9c51e66cdb8b.jpg)
if ![](https://www.scirp.org/html/6-5300485\64d1cfe5-f2e8-4bc1-a0eb-cccf14edd1b9.jpg)
![](https://www.scirp.org/html/6-5300485\224e3848-635c-46a6-8001-39e6dd8a4a0b.jpg)
Definition 6 (Graded lexicographic Order). Let
, the collection of
-tuple exponents of distinguished monomials. We say
![](https://www.scirp.org/html/6-5300485\b9d03980-edd4-4538-ad48-7897d656d2a1.jpg)
if
![](https://www.scirp.org/html/6-5300485\c2a6a164-957b-4009-8404-49c8194bc37f.jpg)
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
![](https://www.scirp.org/html/6-5300485\d62c58b4-6108-4b08-98a2-b0417d490573.jpg)
of dimension
. We say
![](https://www.scirp.org/html/6-5300485\44572ff3-54ac-4a79-a776-35d0321128b3.jpg)
if
![](https://www.scirp.org/html/6-5300485\8fc7d8ab-a8b3-4190-80c9-3bebae7965d9.jpg)
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
![](https://www.scirp.org/html/6-5300485\d8fd02fb-695b-4a94-86fb-ec2ed14c390c.jpg)
where
![](https://www.scirp.org/html/6-5300485\9ac79b3c-cc7d-416b-82a7-9cf15e9434db.jpg)
Remark 3. Equivalence of permutation symbols is an equivalence relation. Each of the partitions is called the Ehresmann cycle class and denoted by
![](https://www.scirp.org/html/6-5300485\ecce7c51-8aa0-47ad-b360-7d611a21623f.jpg)
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
![](https://www.scirp.org/html/6-5300485\eb0a9588-eb3a-4e6f-9bac-a86fa03c381b.jpg)
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 ![](https://www.scirp.org/html/6-5300485\3075dfed-a459-4ecc-a55b-42ceb0c8e684.jpg)
of any cycle
in the class ![](https://www.scirp.org/html/6-5300485\3c396d9f-349a-408c-8099-89bbf2f30817.jpg)
given by
which coincides with the degree of the distinguished monomial of the cycle. The ![](https://www.scirp.org/html/6-5300485\8626abb9-3b30-4d3c-9100-d9a8377a3f08.jpg)
is precisely the dimension of the flag manifold
that is,
and hence
![](https://www.scirp.org/html/6-5300485\c36e555e-1238-4bd7-b3d6-8e682a829cd2.jpg)
Theorem 2. Let
![](https://www.scirp.org/html/6-5300485\6ece30b2-bca8-4a7e-ad83-6b0316ca0352.jpg)
be the Ehresmann cycle class of dimension
in the cohomology of
, and let
![](https://www.scirp.org/html/6-5300485\90db1381-940f-4aa4-95de-473476c8016d.jpg)
be the disjoint union of such classes. Let
![](https://www.scirp.org/html/6-5300485\15ba79e1-119f-4197-bb71-c2e6b10f06c1.jpg)
be the graded monoid of distinguished monomials of degrees
in the cohomology ring of the flag manifold
. Then there is a natural bijection
![](https://www.scirp.org/html/6-5300485\88e11337-31ea-4e59-86fa-3f0f9408cc96.jpg)
between
and
.
Proof
We define a map
![](https://www.scirp.org/html/6-5300485\5a788306-a711-4e08-a930-6097e6b27fee.jpg)
by
(9)
where
is a subcollection of
, that is,
![](https://www.scirp.org/html/6-5300485\1829f0db-b483-4680-b55f-c92a3ad29f06.jpg)
Let
![](https://www.scirp.org/html/6-5300485\e5ffada0-a72d-42ed-98b6-05031a59c03d.jpg)
and
,
,
.
Suppose that
![](https://www.scirp.org/html/6-5300485\63d1be67-f8a7-47a8-b15d-0ad6b34b1c57.jpg)
which implies that
![](https://www.scirp.org/html/6-5300485\b1ae399a-c687-48cf-b6ff-f3b70f2abaf4.jpg)
where
![](https://www.scirp.org/html/6-5300485\783e70c0-fa90-4e83-be8d-5d7ca7fc9be9.jpg)
![](https://www.scirp.org/html/6-5300485\077af7e6-9773-47a2-b72c-a034808a6b5b.jpg)
From the Theorem 1,
![](https://www.scirp.org/html/6-5300485\10c74175-5452-4f9d-82db-fbf706bcc0ae.jpg)
and hence
![](https://www.scirp.org/html/6-5300485\1a5c6ad5-3d92-46ae-aa7f-40a45bd92964.jpg)
which implies that
![](https://www.scirp.org/html/6-5300485\aad2e0a5-0f42-4a42-bf58-f42ba018ac46.jpg)
Therefore,
is well defined.
Suppose that
![](https://www.scirp.org/html/6-5300485\d1a04136-3cac-486f-a998-23932729cab8.jpg)
in other words ![](https://www.scirp.org/html/6-5300485\3cafd5ac-38a5-495c-85bf-757fd43af5f4.jpg)
![](https://www.scirp.org/html/6-5300485\fcf1097e-baba-4e49-a1a1-953af989be3e.jpg)
![](https://www.scirp.org/html/6-5300485\571d7398-3337-4f82-a5e6-08b9ac98ef2e.jpg)
![](https://www.scirp.org/html/6-5300485\139b8f45-7f88-4bef-8242-04a21ca6a2da.jpg)
and therefore,
![](https://www.scirp.org/html/6-5300485\b81844fd-78fb-480d-a6a8-8356e4f4afdb.jpg)
and hence
is injective.
For any subcollection
in
. By definition,
implies that
is the dimesion of the Ehresmann class
in ![](https://www.scirp.org/html/6-5300485\aa2466e7-e196-4db2-8084-319fa2929a6e.jpg)
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
![](https://www.scirp.org/html/6-5300485\b7730e79-ce08-449d-b134-9f9d46409b45.jpg)
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
![](https://www.scirp.org/html/6-5300485\b7625b7e-88d1-415d-8c0e-98db5992e8ae.jpg)
of dimension
. Then
has at most one distinguished monomial.
Proof
Suppose
is identified with
and
then the
![](https://www.scirp.org/html/6-5300485\e65014b3-c1a7-4c64-93c7-5a76d8d5c861.jpg)
and
.
By the definition of
, the subset
consisting of
and ![](https://www.scirp.org/html/6-5300485\49ba1332-3f0e-40da-9764-3f387c5f225e.jpg)
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
![](https://www.scirp.org/html/6-5300485\9b3790b8-1adf-4791-b8ae-eb1299f04c66.jpg)
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
![](https://www.scirp.org/html/6-5300485\ae6ce00f-5461-4ad8-a3d8-6c3a0cb6b0ff.jpg)
of dimension
. We say
![](https://www.scirp.org/html/6-5300485\f273c343-f417-4457-bcd5-4d3b1aca30f1.jpg)
if
![](https://www.scirp.org/html/6-5300485\8a41c4e6-c3ea-4f57-9e64-30102075f553.jpg)
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 ![](https://www.scirp.org/html/6-5300485\57c9d295-2209-412d-939f-df9efb3a2e22.jpg)
be Ehresmann cycle classes of dimension
and
respectively, We say that
![](https://www.scirp.org/html/6-5300485\42a8309d-5dee-4e17-a306-fb6663166504.jpg)
if for all cycles
and
in
and ![](https://www.scirp.org/html/6-5300485\a07333e8-a7b1-494b-bcf7-5d5fd67f1f72.jpg)
respectively, ![](https://www.scirp.org/html/6-5300485\2de15f86-870b-4df4-b6dc-31c73ebe103c.jpg)
Given any two subcollections
and
of distinguished monomials of degrees
and
respectively
if for all distinguished monomials ![](https://www.scirp.org/html/6-5300485\ab88f38f-50cc-4e4d-a49e-975dddec9d52.jpg)
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
![](https://www.scirp.org/html/6-5300485\8747ab7c-1db6-4fa9-9d02-917874558dcd.jpg)
of dimension
, with distinguished monomials
and
respectively then
![](https://www.scirp.org/html/6-5300485\3f99282b-a387-4c9c-afb9-c5348eabde41.jpg)
if and only if
![](https://www.scirp.org/html/6-5300485\8febdce1-27b7-4090-9bc5-ff7da4214791.jpg)
Proof
Suppose that
from 2.1,
and
are given by
![](https://www.scirp.org/html/6-5300485\c712afa1-0478-4a51-a3f3-a493aa20c66a.jpg)
and
![](https://www.scirp.org/html/6-5300485\b7414d58-b268-475a-b7a3-4fd1762e15ce.jpg)
respectively,then there is
,
in the two
-tuple exponents
![](https://www.scirp.org/html/6-5300485\89165856-1313-44fb-a04f-631de4c177de.jpg)
![](https://www.scirp.org/html/6-5300485\d7c552c0-369d-426f-a896-75d2e0563756.jpg)
such that
and for all
coincides with
, if they exist. Therefore, in the the vector difference
![](https://www.scirp.org/html/6-5300485\ab2a0d0f-e139-4cab-bdec-5e99231fffcb.jpg)
the leftmost nonzero entry is negative and and hence
![](https://www.scirp.org/html/6-5300485\657d85c1-4673-460a-8155-a2613268cbc5.jpg)
and the results follows easily. On the other hand suppose
![](https://www.scirp.org/html/6-5300485\7c2cbc0b-40c1-4012-a638-fa1155f05cee.jpg)
this implies that
![](https://www.scirp.org/html/6-5300485\cf954eb2-dfe6-48a0-8536-bfd5015d90e9.jpg)
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
![](https://www.scirp.org/html/6-5300485\84b943f3-4bf5-4cb3-b9a1-48fca1791d13.jpg)
elements of the sets
and ![](https://www.scirp.org/html/6-5300485\4ff97455-e42d-4e00-903c-789931137b6a.jpg)
respectively and therefore
.
Corollary 2. Let
and ![](https://www.scirp.org/html/6-5300485\1e7c4603-3004-4101-a195-9a84f7ec09ea.jpg)
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
![](https://www.scirp.org/html/6-5300485\2eb7f49c-bf1d-4a1d-b712-7a94d28ad612.jpg)
if and only if
.
Corollary 3. Let
![](https://www.scirp.org/html/6-5300485\43dad864-d4c0-4069-9ace-b7a0feb82058.jpg)
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
![](https://www.scirp.org/html/6-5300485\50c8b86f-c2be-42e1-958e-2a2839f3f0bb.jpg)
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.