The smash product of symmetric functions. Extended abstract
msra(2004)
摘要
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.
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关键词
non- commutative symmetric function,. hopf algebra,smash product,quasi-symmetric function,schur-weyl duality.,symmetric function,descent algebra,symmetric group,tensor product,quantum algebra
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