# On the isotypic decomposition of cohomology modules of symmetric semi-algebraic sets: polynomial bounds on multiplicities

## Saugata Basu, Purdue University

## April 7, 2015

We consider symmetric (as well as multi-symmetric)
real algebraic varieties and semi-algebraic sets, as well
as symmetric complex varieties in affine and projective spaces,
defined by polynomials of fixed degrees.
We give polynomial (in the dimension of the ambient space) bounds on
the number of irreducible representations of the symmetric group which
acts on
these sets, as well as their multiplicities,
appearing in the isotypic decomposition of their cohomology modules
with
coefficients in a field of characteristic $0$.
We also give some applications of our methods in proving lower bounds
on the
degrees of defining polynomials of certain symmetric semi-algebraic
sets,
as well as improved bounds on the Betti numbers of the images under
projections
of (not necessarily symmetric) bounded real algebraic sets.

Finally, we conjecture that the multiplicities
of the irreducible representations of the symmetric group
in the cohomology modules of symmetric semi-algebraic sets defined by
polynomials of fixed degrees
are computable with polynomial complexity, which would imply
that the Betti numbers of such sets are also computable with
polynomial complexity. This is in contrast with general semi-algebraic
sets,
for which this problem is provably hard ($\#\mathbf{P}$-hard).

(Joint work with Cordian Riener).