Download An Introduction to Quasisymmetric Schur Functions: Hopf by Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg PDF

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 objective of this monograph is twofold. the 1st objective is to supply a reference textual content for the elemental conception of Hopf algebras, particularly the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric features and connections among them. the second one target is to offer a survey of effects with recognize to a thrilling new foundation of the Hopf algebra of quasisymmetric capabilities, whose combinatorics is comparable 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

W9 → 1, 2, 9, 3, 8, 4, 5, 6, 7, hence D(w, γ ) = {3, 5}. Thus the fundamental basis of QSym consists of generating functions for Ppartitions. Observe that the Schur basis of Sym can be viewed as generating functions for semistandard Young tableaux. In fact, P-partitions are a generalization of SSYTs: An SSYT of shape λ is just a P-partition of the labelled poset (Pλ , γλ ) defined as follows. Let Pλ be the poset whose elements are the pairs (i, j) of row and column coordinates for cells in the Young diagram of λ , with order defined by (i, j) (k, l) if i k and j l.

2 The Hopf algebra of symmetric functions Δ (mλ ) = ∑ m μ ⊗ mν λ = μ ·ν 31 n Δ (en ) = ∑ ei ⊗ en−i i=0 Δ (sλ ) = n Δ (hn ) = ∑ hi ⊗ hn−i i=0 ∑ sλ /μ ⊗ sμ = ∑ ∑ cνλ μ sν ⊗ sμ . 19. For the monomial symmetric function m(2,2,1) we have Δ (m(2,2,1) ) =m(2,2,1) ⊗ 1 + m(2,1) ⊗ m(2) + m(2,2) ⊗ m(1) +m(2) ⊗ m(2,1) + m(1) ⊗ m(2,2) + 1 ⊗ m(2,2,1) while for n = 3 we have Δ (e3 ) = 1 ⊗ e3 + e1 ⊗ e2 + e2 ⊗ e1 + e3 ⊗ 1 Δ (h3 ) = 1 ⊗ h3 + h1 ⊗ h2 + h2 ⊗ h1 + h3 ⊗ 1 and the coproduct of the Schur function s(2,1) is Δ (s(2,1) ) = s(2,1) ⊗1+s(2,1)/(1) ⊗s(1) +s(2,1)/(2) ⊗s(2) +s(2,1)/(1,1) ⊗s(1,1) +1⊗s(2,1).

The two upper maps in diagram ii. are given by (1⊗)(c) = 1 ⊗ c and (⊗1)(c) = c ⊗ 1 for c ∈ C. We may omit the indexing required to express a coproduct Δ (c) as an element of C ⊗ C and use Sweedler notation to write Δ (c) = ∑ c1 ⊗ c2 . Thus, in Sweedler notation, diagrams i. and ii. 1) ∑ ε (c1 )c2 = c = ∑ ε (c2 )c1 . 2) We say the coproduct Δ is cocommutative if c ⊗ c is a term of Δ (c) whenever c ⊗ c is. A submodule I ⊆ C is a coideal if Δ (I ) ⊆ I ⊗ C + C ⊗ I and ε (I ) = {0}. 22 3 Hopf algebras A map f : C → C , where (C , Δ , ε ) is another coalgebra over R, is a coalgebra morphism if Δ ◦ f = ( f ⊗ f ) ◦ Δ and ε = ε ◦ f .

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