Hodge dual operators and model algebras for rational representations of the general linear group

Abstract In this paper, we construct a family of algebras each of whose members is a multiplicity free sum of irreducible rational representations of the general linear group GL n ( C ) . We then use the properties of a generalized version of the Hodge dual operator to determine an explicit basis for each of these algebras, and by restriction, we obtain an explicit basis for each of the irreducible rational representations of GL n ( C ) . Our results cleanly extends classical algebro-combinatorial results on polynomial representations of GL n ( C ) to rational representations.