By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg
An advent to Quasisymmetric Schur Functions is aimed toward researchers and graduate scholars in algebraic combinatorics. The objective of this monograph is twofold. the 1st aim is to supply a reference textual content for the fundamental idea of Hopf algebras, specifically the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric features and connections among them. the second one objective is to provide a survey of effects with recognize to a thrilling new foundation of the Hopf algebra of quasisymmetric features, whose combinatorics is comparable to that of the popular Schur functions.
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Additional info for An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux
2. Let n be a nonnegative integer. Then the n-th elementary noncommutative symmetric function, denoted by en , is the indeterminate en where we set e0 = 1. The n-th complete homogeneous noncommutative symmetric function, denoted by hn , is defined by h0 = 1 ∑ hn = (−1)n−m ey1 ey2 · · · eym . ,ym ) n Let α = (α1 , . . , αk ) be a composition. The elementary noncommutative symmetric function eα is defined by eα = eα 1 · · · eα k and the complete homogeneous noncommutative symmetric function hα is defined by hα = hα1 · · · hαk = ∑ (−1)|α |− (β )eβ .
K ) be a partition. ,λk ) hn = ∑ (−1)n− (β )eβ . (β ) eβ . 7. Note that h1 = e1 = m(1) = x1 + x2 + x3 + · · · while h2 = m(2) + m(1,1) = x21 + x22 + x23 + · · · + x1 x2 + x1 x3 + x2 x3 + · · · and e2 = m(1,1) = x1 x2 + x1 x3 + x2 x3 + · · · . 2 The Hopf algebra of symmetric functions 27 e(2,1) = e2 e1 = (x1 x2 + x1x3 + x2x3 + · · ·)(x1 + x2 + x3 + · · ·). These bases are also of interest. For example, the fundamental theorem of symmetric functions states that Sym is a polynomial algebra in the elementary symmetric functions, that is Sym = Q[e1 , e2 , .
3. 4. 5. i ∈ Des(T ). i + 1 is weakly to the left of i in T . n − i is weakly to the left of n − i + 1 in Γˇ (T ). n − i + 1 is weakly to the right of n − i in Γˇ (T ). n − i ∈ Des(Γˇ (T )). This establishes the claim. 6 Schensted insertion Schensted insertion is an algorithm with many interesting combinatorial properties and applications to representation theory. For further details see [31,72,81]. We will also use this algorithm and the variation below in Chapter 5. In particular, Schensted insertion inserts a positive integer k1 into a semistandard or standard Young tableau T and is denoted by T ← k1 .