(Texas A&M University, 2008-10-10) Moreira Rodriguez, Rivera Walter

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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 Heisenberg product
of endomorphisms of the tensor algebra. For symmetric functions, the Heisenberg
product is given by a construction which combines induction and restriction of representations.
For non-commutative symmetric functions, the structure constants of
the Heisenberg 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.