Title: Following Pipe Dreams
| Speaker: | Nantel Bergeron |
|
Affiliation: | York University |
| Room: | MC 5501 |
Abstract: Together with Sara Billey in 1993, we have introduced combinatorial objects to encode monomials occurring in Schubert polynomials. At the time, we called them RC-diagrams as they encode some positioned reduced expressions for a given permutations. They were renamed Pipe Dreams by Knutson and Miller as their pictorial representation evoke the Pipe Dream game from the same era.
As
it
turn
out,
these
combinatorial
objects
are
interesting
in
many
con-
texts.
In
our
original
work
they
were
envisioned
as
a
generalization
of
Young
tableaux.
Later
they
were
used
by
Knutson
and
Miller
to
encode
the
toric
degeneration
of
Schubert
varieties.
This
naturally
led
them
to
introduce
some
simplicial
complexes
of
subwords
that
are
conjectured
to
be
polytopal
in
many
cases.
The
faces
of
the
polytope
would
be
indexed
by
certain
pipedreams.
Very
recently,
with
Pilaud
and
Ceballos,
we
have
introduced
new
algebraic
operations
on
pipe
dreams.
By
considering
subalgebras,
we
exhibit
very
interesting
families
of
pipe
dreams
that
are
enumerated
by
Catalan
number
Crn+1
for
a
x
r.
A
well
we
have
a
family
that
is
in
bijection
with
certain
lattice
paths
studied
by
Bousquet-melou
and
Mishna.