Abstract
Using cocommutativity of the Hopf algebra of symmetric functions, certain skew Schur functions are proved to be equal. Some of these skew Schur function identities are new.
Highlights
The Schur symmetric functions are a nice basis for symmetric functions
It will be helpful to think of the Hopf algebra of symmetric functions as a quotient of a more naive Hopf algebra defined in terms of shapes but without such identities
The main result is that β ◦ γ ∼ β∗ ◦ γ (Theorem 4.12) provided the ends of the composition do not have any extraneous ribbons
Summary
The Schur symmetric functions are a nice basis for symmetric functions. Skew Schur functions are very nice symmetric functions, but contain more mysteries. This paper uses very different techniques to find equalities between skew Schur functions. The problem of skew Schur identities can be seen as a special case of a question of Schur positivity. The special kind of identities considered above, equalities between two skew Schur functions, forms the equality case of the question of when the difference between two skew Schur functions is Schur positive. Both Schur positivity and Schur positivity of differences are questions which have had much study, see for example [6, 7, 8, 9] and the references therein
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