Partition algebras

where the LGLn(λ) are irreducible GLn(C)-modules and the S λ k are irreducible Sk-modules. The decomposition in (0.1) essentially makes the study of the representations of GLn(C) and the study of representations of the symmetric group Sk two sides of the same coin. The group GLn(C) has interesting subgroups, GLn(C) ⊇ On(C) ⊇ Sn ⊇ Sn−1, and corresponding centralizer algebras, CSk ⊆ CBk(n) ⊆ CAk(n) ⊆ CAk+ 2 (n), which are combinatorially defined in terms of the “multiplication of diagrams” (see Section 1) and which play exactly analogous “Schur-Weyl duality” roles with their corresponding subgroup