How to permute quantum systems continuously?

I will show how to use basic facts from representation theory to derive a unitary version of Cayley's theorem (it allows embedding any finite group in a continuous subgroup of the unitary group). When applied to the symmetric group, this can be used to permute quantum systems in a continuous fashion. I will illustrate how this works for a small number of systems and conclude with some interesting open questions. My talk is loosely based on arXiv:1508.00860. For background on representation theory see http://www.cs.umd.edu/~amchilds/teaching/w13/l06.pdf.