Young—Capelli symmetrizers in superalgebras^†

Abstract

Let Super_n[U [unk] V] be the nth homogeneous subspace of the supersymmetric algebra of U [unk] V, where U and V are Z₂-graded vector spaces over a field K of characteristic zero. The actions of the general linear Lie superalgebras pl(U) and pl(V) span two finite-dimensional K-subalgebras B and [unk] of End_K(Super_n[U [unk] V]) that are the centralizers of each other. Young—Capelli symmetrizers and Young—Capelli *-symmetrizers give rise to K-linear bases of B and [unk] containing orthogonal systems of idempotents; thus they yield complete decompositions of B and [unk] into minimal left and right ideals, respectively.

• polarization operators
• symmetrized Young bitableaux
• Lie superalgebras
• representation theory
• Capelli's identities