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THE SMASH PRODUCT OF SYMMETRIC FUNCTIONS. EXTENDED ABSTRACT
 

Summary: THE SMASH PRODUCT OF SYMMETRIC FUNCTIONS.
EXTENDED ABSTRACT
MARCELO AGUIAR, WALTER FERRER, AND WALTER MOREIRA
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 endo-
morphisms 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.
R´esum´e. Nous construisons une nouvelle op´eration parmi les repr´esentations du groupe
sym´etrique qui interpole entre les produits interne et externe. Ces derniers sont d´efinis
en termes du produit tensoriel et de l'induction des repr´esentations. D'apr`es Malvenuto
et Reutenauer, nous passons des fonctions sym´etriques aux fonctions sym´etriques non

  

Source: Aguiar, Marcelo - Department of Mathematics, Texas A&M University

 

Collections: Mathematics