﻿Order Relation on the Permutation Symbols in the Ehresmann Subvariety Class Associated to the Distinguished Monomials of Flag Manifolds

Vol.3 No.7(2013), Article ID:38122,6 pages DOI:10.4236/apm.2013.37083

Order Relation on the Permutation Symbols in the Ehresmann Subvariety Class Associated to the Distinguished Monomials of Flag Manifolds

Received June 2, 2013; revised July 8, 2013; accepted August 5, 2013

Keywords: Natural Asset; Financial Value; Neural Network

ABSTRACT

In this paper, we use the theory of lexicographical and graded lexicographical orders to compare two distinguished monomials through their codes of invariants and study the effect of this comparison on their respective defining permutation symbols in the Ehresmann subvariety classes.

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

andthe 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 manifoldthat 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

andthen 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.

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