By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg
An creation to Quasisymmetric Schur Functions is aimed toward researchers and graduate scholars in algebraic combinatorics. The target of this monograph is twofold. the 1st objective 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 aim is to provide a survey of effects with recognize 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 info for An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux
We are now ready to define a Hopf algebra. 4. Let (H , m, u, Δ , ε ) be a bialgebra. Then H is a Hopf algebra if there is a linear map S : H → H such that m ◦ (S ⊗ id) ◦ Δ = u ◦ ε = m ◦ (id ⊗ S) ◦ Δ . Thus, in Sweedler notation, S satisfies ∑ S(h1)h2 = ε (h)1 = ∑ h1S(h2) for all h ∈ H , where 1 is the identity element of H . The map S is called the antipode of H . A subset I ⊆ H is a Hopf ideal if it is both an ideal and coideal, and S(I ) ⊆ I . A map f : H → H between Hopf algebras is a Hopf morphism if it is both an algebra and coalgebra morphism, and f ◦ SH = SH ◦ f where SH and SH are respectively the antipodes of H and H .
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 .
Meanwhile, if we insert 3, then we have 7 7 7 ←3 = 5 6 6 6 7 2 4 5 5 5 6 1 3 4 6 2 4 4 1 3 3 6 where the bold cells again indicate the insertion path. Similarly we have Schensted insertion for reverse tableaux , which inserts a positive integer k1 into a semistandard or standard reverse tableau Tˇ and is denoted by Tˇ ← k1 . 1. If k1 is less than or equal to the last entry in row 1, place it at the end of the row, else 2. find the leftmost entry in that row strictly smaller than k1 , say k2 , then 3.
An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux by Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg