Title: Two-colouring hypersurface complements in open Richardson varities
| Speaker: | Kevin Purbhoo |
| Affiliation: | University of Waterloo |
| Room: | MC 5417 |
Abstract:
Given
an
algebraic
hypersurface
$H
\subset
\mathbb{R}^n$,
we
can
always
2-colour
the
components
of
the
complement
$\mathbb{R}^n
\setminus
H$
such
that
adjacent
components
are
of
opposite
colours.
However,
this
property
does
not
necessarily
continue
to
hold
if
we
replace
$\mathbb{R}^n$
by
a
space
with
a
non-trivial
topology
(e.g.
a
torus).
We
wanted
to
know:
does
this
2-colouring
property
hold
for
open
Richardson
varieties
in
the
real
Grassmannian?
It
turns
out,
the
answer
is
yes.
To
prove
this,
we
showed
that
the
coordinate
ring
of
open
Richardson
variety
is
a
unique
factorization
domain
over
any
field,
which
implies
the
result.
Our
proof
uses
a
non-trivial
theorem
of
Knutson-Lam-Speyer
about
positroid
varieties.
This
is
joint
work
with
Jake
Levinson.