Christian Krattenthaler

Classical Group Character Calculus by the C.P.L.M.S. tableaux

We consider identities for classical group characters which feature characters of "nearly rectangular" shape, by which we mean shapes that are rectangular except that one row or column is allowed to be shorter. It turns out that there are quite a few attractive identities, all of which have obvious representation-theoretic meaning. However, the main emphasis of the talk will be on the COMBINATORIAL OBJECTS that are needed to prove these identities, namely certain TABLEAUX by DeConcini, Procesi, Lakshmibai, Musili and Seshadri.

These tableaux (which come from algebraic geometry) have a uniform (though complicated) definition for all classical groups, and are quite different from other tableau descriptions for classical group characters (like King's tableaux, Sundaram's tableaux, or Proctor's tableaux). Particularly in the symplectic case, the definition of the tableaux will be explained in detail and how these tableaux can be effectively used for proving character identities.