Approximate Representations and Approximate Homomorphisms
Speaker: | Cris Moore |
---|---|
Affiliation: | University of New Mexico and the Santa Fe Institute |
Room: | Mathematics & Computer Building (MC) 5158 |
Abstract:
Approximate algebraic structures play a defining role in arithmetic combinatorics. I will discuss approximate representations of finite groups: functions from G to the unitary group U(d) that act like homomorphisms within some error. We bound the expected error in terms of d/d_min, where d_min is the dimension of the smallest nontrivial genuine representation of G. As an application, we bound the extent to which a function f from G to another finite group H can be an approximate homomorphism. We show that if H's representations are significantly smaller than G's, no such f can be "much more homomorphic" than a random function.
We
interpret
these
results
as
showing
that
if
G
is
quasirandom,
that
is,
if
d_min
is
large,
then
G
cannot
be
embedded
in
a
small
number
of
dimensions,
or
in
a
less-quasirandom
group,
without
significant
distortion
of
G's
multiplicative
structure.
We
also
prove
that
our
bounds
are
tight
by
showing
that
minors
of
genuine
representations
and
their
polar
decompositions
are
essentially
optimal
approximate
representations.
This
is
joint
work
with
Alex
Russell
of
the
University
of
Connecticut.