Get An Introduction to Quasisymmetric Schur Functions: Hopf PDF

Combinatorics

By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

ISBN-10: 1461472997

ISBN-13: 9781461472995

ISBN-10: 1461473004

ISBN-13: 9781461473008

An advent to Quasisymmetric Schur Functions is geared toward researchers and graduate scholars in algebraic combinatorics. The aim of this monograph is twofold. the 1st aim is to supply a reference textual content for the elemental thought of Hopf algebras, specifically the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric capabilities and connections among them. the second one objective is to provide a survey of effects with appreciate to a thrilling new foundation of the Hopf algebra of quasisymmetric capabilities, whose combinatorics is similar to that of the well known 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

36). 9] shows that for NSym the antipode, which they denote ω˜ , is an anti-automorphism. The automorphism of NSym here referred to as ψ is implicitly used at the beginning of [34, Section 4] to give an abbreviated description of some of the transition matrices between bases of NSym, for example, ψ (hα ) = eα . For each of these involutions it is worth noting what are the images of bases of interest. For example, on Sym we have ω (hλ ) = eλ and ω (mλ ) = fλ , where { fλ } is the basis of forgotten symmetric functions.

17. 15 has descent 1 and ascents 2 and 3. 13, we can describe a (w, γ )partition as a map f from w to the positive integers satisfying 1. f (wi ) f (wi+1 ), 2. f (wi ) = f (wi+1 ) implies i is an ascent of (w, γ ) or equivalently, i is a descent of (w, γ ) implies f (wi ) < f (wi+1 ). We now introduce generating functions for P-partitions. 18. Let (P, γ ) be a labelled poset. For any (P, γ )-partition f , denote by x f the monomial x f = ∏ x f (p) . p∈P Then the weight enumerator of (P, γ ) is the formal power series F(P, γ ) defined by F(P, γ ) = ∑ x f , where the sum is over all (P, γ )-partitions f .

Let λ = (λ1 , . . , λk ) be a partition. Then the monomial symmetric function mλ is defined by λ mλ = ∑ xλi11 · · · xikk , where the sum is over all k-tuples (i1 , . . , ik ) of distinct indices that yield distinct monomials. We define m0/ = 1. 5. We have m(2,1) = x21 x12 + x22 x11 + x21 x13 + x23 x11 + x21 x14 + x24 x11 + x22 x13 + x23 x12 + · · · . Moreover, since the mλ are independent we have Symn = span{mλ | λ n}. The basis of monomial symmetric functions is not the only interesting and useful basis.

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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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