The Schubert Calculus and Eigenvalue Inequalities for Sums of Hermitian Matrices
Last updated: 14 October 2014
Horn's Conjecture and the Hersch–Zwahlen Methods
The set introduced by Horn [Hor1962]
is defined recursively as follows:
Horn conjectured that
for all and he showed that for all
but further, that
The following lemma was the key tool used.
(The "pushing" lemma for
and let be three integers such that
Horn proved this lemma using the Cauchy inequalities relating the eigenvalues of a hermitian matrix to those of a principal submatrix. Briefly, one shows that given
which can be "pushed" to
This can be done except for one special case in
which is treated by a separate argument. For the number of special cases becomes unmanageable.
The problem addressed by Horn may be reformulated as follows
There is a one-to-one correspondence between and
If the same correspondence is applied to one obtains
It can be shown that
Horn's conjecture is thus equivalent to the conjecture that
In [HZw1962] J. Hersch and B. P. Zwahlen developed a new technique for establishing membership in
We introduce some notation relevant to their method. Let
denote affine over a field
A is a sequence
of nested subspaces
A tower is said to be of type if and
Let be a
We call a Schubert variety of type
where denotes the field of complex numbers.
Theorem 2.7 shows that Horn's conjecture would be true if
In [Zwa1966], Zwahlen investigated this possibility. He was able to show that
and for all
but could not quite verify that
Zwahlen's methods were purely geometrical, and required no special properties of the group field
In [TTh1974], R. C. Thompson made clear the problem with Zwahlen's methods. Thompson showed that
for all but that in general,
His approach started by establishing the following lemma.
(The "pushing" lemma for
where is an arbitrary field. Let be three integers chosen from
Lemma 2.8 is directly analogous to lemma 2.1, Horn's "pushing" lemma. Using 2.8, Thompson was able to follow Horn's approach and show that all elements of
can be obtained by pushing from elements of
except for one special case. A difficult analysis of this special case showed that
for all if is algebraically closed, but not otherwise.
Identities for and
The main assertion of the present work is that
for all if The proof presented required the machinery of algebraic geometry.
This machinery, specifically the Schubert Calculus, is discussed in Section 3. We close Section 2 with some results which first suggested the plausibility of our assertion.
||Proof of 2.9.
Let be arbitrary
respectively. Then there exist
type respectively, such that
Hence there exists a subspace such that
so there exist
of type respectively such that
there exists subspace such that
In particular, Notice that for
Since are arbitrary
and the proof is complete.
We remark that Theorem 2.9 holds true if is replaced by
is any field.
||Proof of 2.10.
We introduce notation for use in this and subsequent proofs. Let
The proof of 2.10 is by induction on The result is trivial for
Assume the result holds for
Then by definition,
We must show
holds, and so (*) holds.
We conclude this section by noting that the proof of 2.9 yields the following
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.