Last updated: 14 October 2014
In this section we introduce and discuss language and conventions which will be commonly used in later sections.
will denote the set of strictly increasing integer sequences of length chosen from Elements of will normally appear as lower case Roman letters, e.g. is an element of The component of will be denoted by i.e. Given we define three associated sequences and as follows:
|1)||(the reverse of is the element of defined by|
|2)||(the complement of is the element of defined by where One can check that the components of consist of elements of which are not components of|
|3)||(the dual of is the element of which is the reverse of the complement of (or equivalently, the component of the reverse of|
A formula for is
If and we define (the composition of with as the element of defined by
We define a partial order on by setting iff for
will denote the set i.e.
If is any map from to we use the same symbol to denote the component-wise map from to i.e. In the same manner, we define the operation on by we also set iff
Finally, if we let denote
This is an excerpt from Steven Andrew Johnson's 1979 dissertation The Schubert Calculus and Eigenvalue Inequalities for Sums of Hermitian Matrices.