Séminaire de Physique Théorique

All unitary representations of su(p,qIm)

by Prof. Dmytro VOLIN (Trinity College Dublin & IHES)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


35, route de Chartres, F-91440 Bures-sur-Yvette (France)

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.