**
Jacob
Campbell,
Department
of
Pure
Mathematics,
University
of
Waterloo**

**
"Characters
of
the
infinite
symmetric
group
and
random
matrices"**

A
famous
theorem
of
Thoma
from
1964
parameterizes
the
extremal
characters
of
the
infinite
symmetric
group
$S_{\infty}$
in
terms
of
certain
scalar
sequences.
An
extremal
character
of
$S_{\infty}$
gives
a
good
example
of
a
trace
on
its
group
algebra,
and
allows
one
to
consider
permutations
as
noncommutative
random
variables.
On
the
other
hand,
in
the
last
few
decades
the
so-called
Jucys-Murphy
elements
of
the
group
algebra,
which
are
related
to
the
star-transpositions
$(1,n)$,
have
played
a
central
role
in
the
representation
theory
of
symmetric
groups.

In
this
talk,
I
will
present
joint
work
with
C.
Koestler
and
A.
Nica
(arXiv:2203.01763)
where
we
prove
a
central
limit-type
theorem
for
the
star-transpositions,
with
respect
to
the
trace
coming
from
certain
Thoma
sequences.
For
the
Thoma
sequence
(0,0,...),
i.e.
in
the
regular
representation,
the
limit
is
semicircular,
as
already
found
by
Biane
in
1995.
More
recently,
this
was
generalized
by
Koestler-Nica
in
2021
for
a
sequence
(1/d,...,1/d,0,0,...);
in
this
case,
the
limit
law
is
the
average
eigenvalue
distribution
of
a
certain
well-known
random
matrix
called
the
d
x
d
traceless
GUE.

We
extend
the
latter
result
to
look
at
all
finite
Thoma
sequences
$(w_1,...,w_d,0,0,...)$
and
land
on
something
unexpected:
the
limit
law
comes
from
a
GUE-like
random
matrix
whose
off-diagonal
entries
live
in
noncommutative
algebras
with
canonical
commutation
relations
(CCR).
I
will
explain
how
Thoma
characters
and
star-transpositions
naturally
point
towards
the
combinatorics
of
GUEs,
through
the
probabilistic
notion
of
exchangeability
and
the
so-called
"genus
expansion"
for
the
GUE,
and
how
the
off-diagonal
noncommutativity
is
introduced
by
varying
the
Thoma
parameter.

This seminar will be held jointly online and in person:

- Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09
- Room: MC 5501