Standard Young tableaux are fillings of the diagram for a partition of an integer k with the numbers {1,2, …, k}. They are used to encode dimensions and basis elements of representations of the symmetric group. I will explain how these tableaux arise from restriction of irreducible representations in the symmetric group algebra and then show how this generalizes to a family of algebras called “diagram algebras” (which includes, for instance, the Brauer and Temperley-Lieb algebras). The dimensions of the irreducible representations for diagram algebras can be encoded with subsets of set valued tableaux. All of this is done with the goal of resolving some of the most difficult and well-known problems in combinatorial representation theory (e.g. Kronecker, plethysm and restriction).

This is joint work with Laura Colmenarejo, Rosa Orellana, Franco Saliola and Anne Schilling