After thinking about the problem I added some code to the cart module so that the HTTP headers of the response given to the browser tells him to not cache it because I knew that IE can be quite strange with caching issues and that it might help.
Now that doesn't mean that the problem is solved... Hopefully, it won't.
For the record, I added to the file components/com_hikashop/views/product/tmpl/cart.php the code
if(JRequest::getWord('tmpl','')=='component'){
header( 'Cache-Control: no-store, no-cache, must-revalidate' );
header( 'Cache-Control: post-check=0, pre-check=0', false );
header( 'Pragma: no-cache' );
}
at the top of the file just after the php start tag.