The Schubert Calculus and Eigenvalue Inequalities for Sums of Hermitian Matrices

Last updated: 14 October 2014

Notation, Conventions

In this section we introduce and discuss language and conventions which will be commonly used in later sections.

${Q}_{d,n}$ will denote the set of strictly increasing integer sequences of length $d$ chosen from $\left\{1,2,3,\dots ,n\right\}\text{.}$ Elements of ${Q}_{d,n}$ will normally appear as lower case Roman letters, e.g. $a=\left(2,5,9\right)$ is an element of ${Q}_{3,10}\text{.}$ The ${i}^{\text{th}}$ component of $a$ will be denoted by ${a}_{i},$ i.e. $a=\left({a}_{1} {a}_{2} \dots {a}_{d}\right)\text{.}$ Given $a\in {Q}_{d,n},$ we define three associated sequences $\stackrel{‾}{a},\stackrel{ˆ}{a},$ and $\stackrel{\sim }{a}$ as follows:

 1) $\stackrel{‾}{a}$ (the reverse of $a\text{)}$ is the element of ${Q}_{d,n}$ defined by $a‾i=n+1- ad+1-i$ 2) $\stackrel{ˆ}{a}$ (the complement of $a\text{)}$ is the element of ${Q}_{n-d,n}$ defined by $aˆi=i+ ∑t=1d δat-t(i),$ where $δx(y)= { 0, if y≤x, 1, if y>x.$ One can check that the components of $\stackrel{ˆ}{a}$ consist of elements of $\left\{1,2,3,\dots ,n\right\}$ which are not components of $a\text{.}$ 3) $\stackrel{\sim }{a}$ (the dual of $a\text{)}$ is the element of ${Q}_{n-d,n}$ which is the reverse of the complement of $a$ (or equivalently, the component of the reverse of $a\text{).}$

A formula for ${\stackrel{\sim }{a}}_{i}$ is $a∼i=??i+ ∑t=1d δat-t(i). Handwritten notes here!!$

If $a\in {Q}_{d,n},$ and $u\in {Q}_{k,d},$ we define $a\circ u$ (the composition of $a$ with $u\text{),}$ as the element of ${Q}_{k,d}$ defined by $(a∘u)i=aui.$

We define a partial order $\le$ on ${Q}_{d,n}$ by setting $a\le b$ iff ${a}_{i}\le {b}_{i}$ for $i=1,2,3,\dots ,d\text{.}$

${Q}_{d,n}^{3}$ will denote the set ${Q}_{d,n}×{Q}_{d,n}×{Q}_{d,n},$ i.e. $Qd,n3= { (a,b,c) | a,b and c∈Qd,n } .$

If $f$ is any map from ${Q}_{d,n}$ to ${Q}_{d,n},$ we use the same symbol $f$ to denote the component-wise map from ${Q}_{d,n}^{3}$ to ${Q}_{d,n}^{3},$ i.e. $f\left(a,b,c\right)=\left(f\left(a\right),f\left(b\right),f\left(c\right)\right)\text{.}$ In the same manner, we define the operation $\circ$ on ${Q}_{d,n}^{3}×{Q}_{k,d}^{3}$ by $\left(a,b,c\right)\circ \left(u,v,w\right)=\left(a\circ u,b\circ v,c\circ w\right)\text{;}$ we also set $\left(a,b,c\right)\le \left(a\prime ,b\prime ,c\prime \right)$ iff $a\le a\prime ,$ $b\le b\prime ,$ $c\le c\prime \text{.}$

Finally, if $\left(a,b,c\right)\in {Q}_{d,n}^{3},$ we let $\mathrm{\Sigma }\left(a,b,c\right)$ denote $∑i=1d ai+bi+ci.$

Notes and References

This is an excerpt from Steven Andrew Johnson's 1979 dissertation The Schubert Calculus and Eigenvalue Inequalities for Sums of Hermitian Matrices.