Eigenvalues of Hermitian matrices and the Belkale-Kumar product
| Speaker: | Kevin Purbhoo |
|---|---|
| Affiliation: | University of Waterloo |
| Room: | Mathematics & Computer Building (MC) 5158 |
Abstract:
The story of this talk begins with the very classical Hermitian sum problem, which asks: if the eigenvalues of two Hermitian matrices are known, what can we say about the possible eigenvalues of their sum?
The
answer
is,
nowadays,
quite
a
bit
(though
it
took
over
a
hundred
years
for
us
to
get
there).
We
know
that
the
possible
lists
of
eigenvalues
form
a
convex
polyhedal
cone.
The
integer
points
in
this
cone
have
an
interpretation
in
representation
theory.
The
facets
of
this
cone
are
described
via
the
geometry
of
Schubert
varieties,
which
in
turn
produces
a
combinatorial
description.
Recently,
Belkale
and
Kumar
introduced
a
ring
structure,
which
they
essentially
defined
by
taking
a
complicated
ring
(the
cohomology
ring
of
a
generalized
flag
variety),
and
changing
the
product
so
as
to
ignore
the
complicated
parts.
Amazingly,
this
simplification
led
to
a
number
breakthroughts
in
generalizing
the
Hermitian
sum
picture,
including
descriptions
of
facets
for
related
problems,
and
information
about
higher
codimension
faces.
My
goal
in
this
talk
is
to
give
an
overview
of
the
mathematics
surrounding
the
Hermitian
sum
problem,
and
to
discuss,
in
particular,
the
combinatorics
of
the
Belkale-Kumar
product.
No
knowledge
of
representation
theory,
Schubert
varieties,
or
cohomology
will
be
assumed.
This
is
joint
work
(in
progress)
with
Allen
Knutson.