The Schubert Calculus and Eigenvalue Inequalities for Sums of Hermitian Matrices

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $\{1,2,3,\dots ,n\}\text{.}$ Elements of $Q_{d,n}$ will normally appear as lower case Roman letters, e.g. $a=(2,5,9)$ is an element of $Q_{3,10}\text{.}$ The $i^{\text{th}}$ component of $a$ will be denoted by $a_i,$ i.e. $a=\left(a_1\hspace{0.17em}{a}_2\hspace{0.17em}\dots \hspace{0.17em}{a}_d\right)\text{.}$ Given $a\in Q_{d,n},$ we define three associated sequences $\bar{a},\hat{a},$ and $\stackrel{\sim }{a}$ as follows:

1)	$\bar{a}$ (the reverse of $a\text{)}$ is the element of $Q_{d,n}$ defined by $${\bar{a}}_i=n+1-a_{d+1-i}$$
2)	$\hat{a}$ (the complement of $a\text{)}$ is the element of $Q_{n-d,n}$ defined by $${\hat{a}}_i=i+\sum _{t=1}^d{\delta }_{a_t-t}\left(i\right),$$ where $${\delta }_x\left(y\right)=\{\begin{array}{ll}0,& \text{if}\hspace{0.17em}y\le x,\\ 1,& \text{if}\hspace{0.17em}y>x\text{.}\end{array}$$ One can check that the components of $\hat{a}$ consist of elements of $\{1,2,3,\dots ,n\}$ 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 $${\stackrel{\sim }{a}}_i=\text{??}i+\sum _{t=1}^d{\delta }_{a_t-t}\left(i\right)\text{.}\text{Handwritten notes here!!}$$

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 $${(a\circ u)}_i=a_{u_i}\text{.}$$

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 $Q_{d,n}\times Q_{d,n}\times Q_{d,n},$ i.e. $$Q_{d,n}^3=\left\{(a,b,c)\hspace{0.17em}\right|\hspace{0.17em}a,b\hspace{0.17em}\text{and}\hspace{0.17em}c\in Q_{d,n}\}\text{.}$$

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. $f(a,b,c)=(f\left(a\right),f\left(b\right),f\left(c\right))\text{.}$ In the same manner, we define the operation $\circ$ on $Q_{d,n}^3\times Q_{k,d}^3$ by $(a,b,c)\circ (u,v,w)=(a\circ u,b\circ v,c\circ w)\text{;}$ we also set $(a,b,c)\le (a\prime ,b\prime ,c\prime )$ iff $a\le a\prime ,$ $b\le b\prime ,$ $c\le c\prime \text{.}$

Finally, if $(a,b,c)\in Q_{d,n}^3,$ we let $\mathrm{\Sigma }(a,b,c)$ denote $$\sum _{i=1}^d{a}_i+b_i+c_i\text{.}$$

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.

page history