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

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 elemental conception of Hopf algebras, specifically the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric features and connections among them. the second one objective is to offer a survey of effects with appreciate to an exhilarating new foundation of the Hopf algebra of quasisymmetric capabilities, whose combinatorics is similar to that of the popular Schur functions.

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Extra info for An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux

Example text

For example, F(2,1) = M(2,1) + M(1,1,1) while F(1,2) = M(1,2) + M(1,1,1). 1 Products and coproducts As with symmetric functions, the product of two quasisymmetric functions when expressed in either of the bases introduced has a combinatorial description. Given compositions α = (α1 , . . , αk ) and β = (β1 , . . , β ), consider all paths P in the (x, y) plane from (0, 0) to (k, ) with steps (1, 0), (0, 1) and (1, 1). Let Pi be the pointwise sum of the first i steps of P where P0 = (0, 0). Then we define the composition corresponding to a path P with m steps, denoted by γP , to be γP = (γ1 , .

In this way, we establish a one-to-one correspondence φ between (P, γ )-partitions that map αi elements to ki and those that map αi elements to li . It follows that the coefficients of the monomials xαk11 · · · xαkmm and xαl11 · · · xαlmm are equal. Of particular interest is the weight enumerator of a labelled chain. 20. 3 The Hopf algebra of quasisymmetric functions 39 F(w, γ ) = Fα , the fundamental quasisymmetric function indexed by the composition α set(α ) = D(w, γ ). |w| with Proof. 6 we have Fα = ∑ xi1 · · · xin , where the sum is over all sequences (i1 , .

We follow the Cartesian or French convention, which means that we number the rows from bottom to top, and the columns from left to right. The cell in the i-th row and j-th column is denoted by the pair (i, j). K. 2. α = (2, 1, 4, 3, 1) We now define a poset to enable us to define skew versions of Young composition diagrams. 3. The Young composition poset Lcˆ is the poset consisting of all compositions in which α = (α1 , . . , α ) is covered by 1. (α1 , . . , α , 1), that is, the composition obtained by suffixing a part of size 1 to α .

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