The following is an informal response written by Professor Laurie R. Santos of Yale University.
This is a tricky one, as there's lots of controversy on the specifics related to these questions in the field of primate cognition. On the one hand, there is evidence that some primates (chimpanzees and capuchin monkeys) can use some kinds of symbolic representations in some situations. For example, both chimpanzees and capuchins can be trained to use tokens to represent different numerosities. Once trained, they can do all kinds of smart things with these new symbols-- including adding and subtracting them, and so on. There's also evidence that chimpanzees can understand some spatial symbols, such as using a scale model of a room to figure out where a piece of food in a real room is hidden. I think this use of a scale geometric model is actually the best evidence for symbol use that's really geometric in nature. That said, it's pretty limited and does appear to require at least some training to get going. To my knowledge, there are not real cases of other spontaneous kinds of symbols. There's nothing like representational drawing in primates (lots of very abstract painting, but nothing that would suggest graphical representations) or anything like spontaneous use of marking/pebbles for counting. So I guess the upshot is that primates can use symbols sometimes, but in very restricted, very scaffolded, and often very limited ways.
Here are also two references that might be of use...
chimpanzee use of scale models: http://www.infantcognitiongroup.com/Portals/0/PapersPosters/KuhlmeierEtAl(1999).pdf
capuchin use of symbols: http://www.sciencedaily.com/releases/2008/06/080610212404.htm
(ed -- note that the comments below were written about a previous revision of this answer, which was just a promise to ask Prof. Santos! Since the question is currently closed, Pete's only option was to edit this answer.)