The smash product of symmetric functions. Extended abstract

TL;DR

This paper introduces a new algebraic operation called the smash product for symmetric functions and related structures, unifying and extending classical internal and external products through combinatorial and representation-theoretic methods.

Contribution

It constructs the smash product for symmetric functions and non-commutative symmetric functions, providing explicit combinatorial rules and connecting it to representation theory and algebra of permutations.

Findings

Defined the smash product as an interpolation between classical products.

Extended combinatorial rules for structure constants of the smash product.

Linked the smash product to representation operations like induction and restriction.

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.