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

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.

Qd,n will denote the set of strictly increasing integer sequences of length d chosen from {1,2,3,,n}. Elements of Qd,n will normally appear as lower case Roman letters, e.g. a=(2,5,9) is an element of Q3,10. The ith component of a will be denoted by ai, i.e. a=(a1a2ad). Given aQd,n, we define three associated sequences a,aˆ, and a as follows:

1) a (the reverse of a) is the element of Qd,n defined by ai=n+1- ad+1-i
2) aˆ (the complement of a) is the element of Qn-d,n defined by aˆi=i+ t=1d δat-t(i), where δx(y)= { 0, ifyx, 1, ify>x. One can check that the components of aˆ consist of elements of {1,2,3,,n} which are not components of a.
3) a (the dual of a) is the element of Qn-d,n which is the reverse of the complement of a (or equivalently, the component of the reverse of a).

A formula for ai is ai=??i+ t=1d δat-t(i). Handwritten notes here!!

If aQd,n, and uQk,d, we define au (the composition of a with u), as the element of Qk,d defined by (au)i=aui.

We define a partial order on Qd,n by setting ab iff aibi for i=1,2,3,,d.

Qd,n3 will denote the set Qd,n×Qd,n×Qd,n, i.e. Qd,n3= { (a,b,c)| a,bandcQd,n } .

If f is any map from Qd,n to Qd,n, we use the same symbol f to denote the component-wise map from Qd,n3 to Qd,n3, i.e. f(a,b,c)=(f(a),f(b),f(c)). In the same manner, we define the operation on Qd,n3×Qk,d3 by (a,b,c)(u,v,w)=(au,bv,cw); we also set (a,b,c)(a,b,c) iff aa, bb, cc.

Finally, if (a,b,c)Qd,n3, we let Σ(a,b,c) 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.

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