Title: Graph representatives of positroid strata (juggling meets quantum physics)
| Speaker: | Michaela Rombach |
| Affiliation: | University of California |
| Room: | MC 5417 |
Abstract:
In
2013
Knutson,
Lam
and
Speyer
showed
that
the
poset
of
bounded
juggling
patterns
is
isomorphic
to
the
poset
of
positroids.
Bounded
juggling
patterns
are
affine
permutations
that
satisfy
,
for
all
.
Positroids
are
a
class
of
matroids,
introduced
by
Alexander
Postnikov,
that
arise
from
matrices
of
rank
with
real
entries
such
that
all
maximal
minors
are
nonnegative.
Positroids
have
many
nice
properties.
For
example,
they
are
closed
under
restriction,
contraction,
and
duality.
There
is
a
map
from
graphs
to
matroids
that
is
not
onto,
creating
the
class
of
so-called
graphic
matroids,
where
the
independent
sets
of
the
matroid
are
the
forests
in
the
graph.
We
show
that
every
juggling
pattern
and
therefore
positroid
stratum
in
the
Grassmannian
has
a
planar
graph
representative,
using
planar
bicolored
(plabic)
graphs.
These
diagrams
are
at
the
heart
of
a
recent
breakthrough
in
particle
physics
as
they
present
a
new
type
of
Feynman
diagram
that
computes
scattering
amplitudes.
Joint
work
with
Allen
Knutson.