Classification of all unitary representations of su(p,q|m) algebra with non-zero p,q,m should have been achieved a while ago, given the current level of the representation theory development. However, to our surprise, the literature on the subject contains some incomplete or incorrect statements, save the well-understood su(2,2|N) case. We therefore decided to address the question from scratch and were able to get a complete and concise description of the unitary dual for generic su(p,q|m).
In the current talk:
- The classification statement is presented in full generality, we also mention all the other real forms of gl(p+q|m,C).
- Shortening conditions naturally arise from considering of all possible choices of the Kac-Dynkin-Vogan diagram at once.
- Schwinger oscillators are used to prove unitarity, with a novel option to work with non-integer weights by representing the oscillator algebra in a generalisation of the Fock module.
- A generalisation of Young diagrams inscribed into a T-hook [almost] bijectively labels the unitary dual. This opens interesting opportunities for new combinatorial identities.