# An Introduction to Quasisymmetric Schur Functions (September by Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

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Extra resources for An Introduction to Quasisymmetric Schur Functions (September 26, 2012)

Example text

12 Let α Then n, β ✁ 21 = {1243, 1423, 1432, 4123, 4132, 4312}. m and σ ∈ Sn , τ ∈ Sm , such that d(σ ) = set(α) and d(τ) = set(β ). Fα Fβ = ∑ Fcomp(d(π)) . 9. F(1,2) F(1) = F(1,3) + F(1,2,1) + F(2,2) + F(1,1,2) since 213 ∈ S3 with d(213) = {1} = set((1, 2)) 1 ∈ S1 with d(1) = 0/ = set((1)) and 213 ✁ 1 = {2134, 2143, 2413, 4213}. The coproduct on each of these bases is even more straightforward to describe ∆ (Mα ) = ∑ α=β ·γ Mβ ⊗ Mγ ∆ (Fα ) = ∑ Fβ ⊗ Fγ . 10. ∆ (M(2,1,3) ) = 1 ⊗ M(2,1,3) + M(2) ⊗ M(1,3) + M(2,1) ⊗ M(3) + M(2,1,3) ⊗ 1.

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. Then γ respectively maps w1 , . . , 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.

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 , .