By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg
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Additional info for An Introduction to Quasisymmetric Schur Functions (September 26, 2012)
5 we see that m(2,1) = M(2,1) + M(1,2) . 18) β where the sum is over all compositions β |λ /µ| and d(λ /µ)β = the number of SYTs T of shape λ /µ such that des(T ) = set(β ). 19) β where the sum is over all compositions β |λ /µ| and d(λ /µ)β = the number of SRTs Tˇ of shape λ /µ such that des(Tˇ ) = set(β ). 11. We have s(3,2) = F(3,2) + F(2,2,1) + F(2,3) + F(1,3,1) + F(1,2,2) from the following SYTs. 2 P-partitions We now use the theory of P-partitions to describe the product of quasisymmetric functions.
Let n be a nonnegative integer, then recall that a permutation of [n] is a bijection σ : [n] → [n], which we may write as an n-tuple σ (1) · · · σ (n). The set of all permutations of [n] is denoted by Sn , and the union n 0 Sn by S∞ . We identify a permutation σ ∈ Sn with a bijection of the positive integers by defining σ (i) = i if i > n. Then S0 ⊂ S1 ⊂ · · · and S∞ becomes a group, known as a symmetric group, with the operation of map composition. The identity element is the unique permutation of 0.
28) χ(rα ) = rα . 5 Relationship between Sym, QSym, and NSym We summarize the relationship between Sym, QSym, and NSym with the following diagram. 6 Automorphisms The notions of complement, reversal, and transpose of compositions correspond to well-known involutive automorphisms of QSym, which can be defined in terms of the fundamental basis as follows. 32) Note that these automorphisms commute, and that ω = ρ ◦ ψ = ψ ◦ ρ. Moreover, considering QSym as a Hopf algebra, the antipode S is also given by S(Fα ) = (−1)|α| ω(Fα ).
An Introduction to Quasisymmetric Schur Functions (September 26, 2012) by Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg