Quasisymmetric Schur functions
The authors define the quasisymmetric Schur functions as a basis for quasisymmetric functions. The functions arise from a specialization of nonsymmetric Macdonald polynomials developed by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [``Keys \& standard bases,'' Invariant theory and tableaux, Proc. Workshop, Minneapolis/MN (USA) 1988, IMA Vol.\ Math.\ Appl.\ 19, 125--144 (1990; Zbl 0815.20013)]; the authors call the functions Demazure atoms, because they decompose Demazure characters into their smallest parts. The construction of quasisymmetric Schur functions is fully combinatorial; semistandard (reverse) tableaux indexed by partition diagrams are generalized to tableaux indexed by composition diagrams, and the quasisymmetric Schur function is defined as a generating function indexed by an enumeration of these tableaux. These quasisymmetric Schur functions have many of the properties of the classical Schur functions. There are analogues of the Kostka numbers for expressing a quasisymmetric Schur function as a sum of monomial quasisymmetric functions and a similar rule for fundamental quasisymmetric functions, and of the Pieri rule for multiplying a quasisymmetric Schur function by a row or column quasisymmetic Schur function. The parameter \(t\) can be incorporated into the construction, providing a decomposition of the Hall-Littlewood polynomial into quasisymmetric functions; this is compared to a decomposition of the Macdonald polynomials into the fundamental quasisymmetric functions by \textit{J. Haglund} [The \(q,t\)-Catalan numbers and the space of diagonal harmonics. With an appendix on the combinatorics of Macdonald polynomials. University Lecture Series 41. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1142.05074)].
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