The smash product of symmetric functions. Extended abstract
  We construct a new operation among representations of the symmetric group that interpolates between the classical internal and external products, which are defined in terms of tensor product and induction of representations. Following Malvenuto and Reutenauer, we pass from symmetric functions to non-commutative symmetric functions and from there to the algebra of permutations in order to relate the internal and external products to the composition and convolution of linear endomorphisms of the tensor algebra. The new product we construct corresponds to the smash product of endomorphisms of the tensor algebra. For symmetric functions, the smash product is given by a construction which combines induction and restriction of representations. For non-commutative symmetric functions, the structure constants of the smash product are given by an explicit combinatorial rule which extends a well-known result of Garsia, Remmel, Reutenauer, and Solomon for the descent algebra. We describe the dual operation among quasi-symmetric functions in terms of alphabets. 
Marcelo Aguiar, Walter Ferrer, and Walter Moreira