New PDF release: An Introduction to Quasisymmetric Schur Functions: Hopf

By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

ISBN-10: 1461472997

ISBN-13: 9781461472995

ISBN-10: 1461473004

ISBN-13: 9781461473008

An creation to Quasisymmetric Schur Functions is aimed toward researchers and graduate scholars in algebraic combinatorics. The objective of this monograph is twofold. the 1st objective is to supply a reference textual content for the fundamental thought of Hopf algebras, particularly the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric features and connections among them. the second one target is to provide a survey of effects with appreciate to an exhilarating new foundation of the Hopf algebra of quasisymmetric features, whose combinatorics is comparable to that of the popular Schur functions.

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Additional resources for An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux

Example text

Given a monomial xαi11 · · · xikk , we say it has degree n if (α1 , . . , αk ) n. Furthermore we say a formal power series has finite degree if each monomial has degree at most m for some nonnegative integer m, and is homogeneous of degree n if each monomial has degree n. 1. A symmetric function is a formal power series f ∈ Q[[x1 , x2 , . ]] such that 1. The degree of f is finite. α 2. For every composition (α1 , . . , αk ), all monomials xαi11 · · · xikk in f with distinct indices i1 , . . , ik have the same coefficient.

Therefore if we index the rows and columns of these matrices using a linear extension of refinement we can deduce NSymn = span{eα | α n} = span{hα | α n} = span{rα | α n}. 1 Products and coproducts With each of the bases introduced the product of two such functions is not hard to describe. 22) and rα rβ = rα ·β + rα β. 23) For example, h(1,4,1,2) h(3,1,1) = h(1,4,1,2,3,1,1) and r(1,4,1,2)r(3,1,1) = r(1,4,1,2,3,1,1) + r(1,4,1,5,1,1). Meanwhile, the coproduct is easy to describe for the elementary and complete homogeneous noncommutative symmetric functions n Δ (en ) = ∑ ei ⊗ en−i i=0 n Δ (hn ) = ∑ hi ⊗ hn−i.

21, then adding |α | to each label of the second chain. 18, it is clear that 42 3 Hopf algebras F(u, γ )F(v, δ ) = F(u + v, γ + δ ), where γ + δ is the labelling of the disjoint union u + v that maps ui → γ (ui ) and v j → δ (v j ). Thus Fα Fβ = F(u, γ )F(v, δ ) = F(u + v, γ + δ ) = ∑ F(w, γ + δ ) ∑ Fα (w) , w∈L (u+v) = w∈L (u+v) where α (w) |w| satisfies set(α (w)) = D(w, γ + δ ). 13). We have just seen that the product of quasisymmetric functions corresponds to operations on labelled posets. We shall now see that the same is true for coproduct and the antipode.

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An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux by Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

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