By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg
An creation to Quasisymmetric Schur Functions is geared toward researchers and graduate scholars in algebraic combinatorics. The target of this monograph is twofold. the 1st aim is to supply a reference textual content for the fundamental conception of Hopf algebras, specifically the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric features and connections among them. the second one aim is to provide a survey of effects with recognize to an exhilarating new foundation of the Hopf algebra of quasisymmetric capabilities, whose combinatorics is comparable to that of the popular Schur functions.
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Extra resources for An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux
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8. Let H = n 0 H n be a connected, graded Hopf algebra over R, such that each homogeneous component H n is finite-dimensional. Define the module H ∗ by H∗= (H n )∗ , n 0 n )∗ where (H denotes the set of all linear maps f : H n → R. ∗ Then H is a Hopf algebra with 1. product m : H ∗ ⊗ H ∗ → H ∗ induced by the convolution product f ∗ g = mR ◦ ( f ⊗ g) ◦ ΔH , where mR is the product of R, and ΔH is the coproduct of H (in Sweedler notation, the convolution product is given by ( f ∗ g)(h) = ∑ f (h1 )g(h2 ) for all h ∈ H ), 2.
Then D(w, γ ) = set(α ), since 1. i, i + 1 ∈ set(α ) implies γ (wi ) > γ (wi+1 ), 2. j, j + 1 ∈ [n] − set(α ) implies γ (w j ) < γ (w j+1 ), 3. i ∈ set(α ) and j ∈ [n] − set(α ) implies γ (wi ) > γ (w j ). 21. If α is the composition (3, 2, 4) 9, then set(α ) = {3, 5} and [n] − set(α ) = {1, 2, 4, 6, 7, 8, 9}. Let w be the chain with order w1 < · · · < w9 and labelling γ that respectively maps w3 , w5 → 9, 8 and w1 , w2 , w4 , w6 , w7 , w8 , w9 → 1, 2, 3, 4, 5, 6, 7. 40 3 Hopf algebras Then γ respectively maps w1 , .
Ik ) of indices i1 < · · · < ik . We define M0/ = 1. 5. We have M(2,1) = x21 x12 + x21 x13 + x21 x14 + x22 x13 + · · · while M(1,2) = x11 x22 + x11 x23 + x11 x24 + x12 x23 + · · · . Since the Mα are independent we have QSymn = span{Mα | α n}. A closely related basis is the basis of fundamental quasisymmetric functions. 6. Let α be a composition. Then the fundamental quasisymmetric function Fα is defined by Fα = ∑ Mβ . β α If α n, then in terms of the variables x1 , x2 , . . we have Fα = ∑ xi1 · · · xin , where the sum is over all n-tuples (i1 , .