Multiplicity and indiscernibility

The indistinguishability of bosons and fermions has been an essential part of our ideas of quantum mechanics since the 1920s. But what is the mathematical basis for this indistinguishability? An answer was provided in the group representation theory that developed alongside quantum theory and quickly became a major part of its mathematical structure. In the 1930s such a complex and seemingly abstract theory came to be rejected by physicists as the standard functional analysis picture presented by John von Neumann took hold. The purpose of the present account is to show how indistinguishability is explained within representation theory.