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Extra info for An Introduction to Quasisymmetric Schur Functions (September 26, 2012)
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 , . ]. However, the most important basis of Sym is considered to be the basis of Schur functions, due to its connections to other areas of mathematics such as representation theory and algebraic geometry. 8. Let λ be a partition. Then the Schur function sλ is defined to be s0/ = 1 and sλ = ∑ x T T where the sum is over all SSYTs (or equivalently SSRTs) T of shape λ .
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 ). The automorphism ρ restricts to the identity on Sym, and hence both ω and ψ restrict to the well-known “conjugating” automorphism of Sym, which is usually denoted by ω and satisfies ω(hr ) = er .
In particular, (i, j) < (i, j + 1) and (i, j) < (i + 1, j), so elements of Pλ increase in order as we move from left to right in rows and bottom to top in columns. Let γλ be the labelling that assigns the numbers 1, 2, . . to the elements of Pλ in the following order: first the coordinates of the cells in the first column, from top to bottom; then the coordinates of the cells in the second column, from top to bottom; and so on. 22. Let λ = (5, 4, 2, 2). Below is the Young diagram of λ , with the label γλ (i, j) written in the cell with coordinate pair (i, j).