the orthogonal group. Walled Brauer algebra The walled Brauer algebra is a subalgebra of . Diagrammatically, it consists of diagrams where the only allowed pairings are of the types , , , . This amounts to having a wall that separates from , and requiring that pairings cross the wall while pairings don't. The walled Brauer algebra is generated by . These generators obey the basic relations of that involve them, plus the two relations (In , these two relations follow from the basic relations.) For a natural integer, let be the natural representation of the general linear group . The walled Brauer algebra has a natural action on , which is related by Schur-Weyl duality to the action of . See also Birman–Wenzl algebra, a deformation of the Brauer algebra. References Representation theory Diagram algebras