Dual immaculate quasisymmetric functions expand positively into Young quasisymmetric Schur functions

Abstract

We describe a combinatorial formula for the coefficients when the dual immaculate quasisymmetric functions are decomposed into Young quasisymmetric Schur functions. We prove this using an analogue of Schensted insertion. Using this result, we give necessary and sufficient conditions for a dual immaculate quasisymmetric function to be symmetric. Moreover, we show that the product of a Schur function and a dual immaculate quasisymmetric function expands positively in the Young quasisymmetric Schur basis. We also discuss the decomposition of the Young noncommutative Schur functions into the immaculate functions. Finally, we provide a Remmel-Whitney-style rule to generate the coefficients of the decomposition of the dual immaculates into the Young quasisymmetric Schurs algorithmically and an analogous rule for the decomposition of the dual bases.

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

“…The first condition and the triple condition guarantee that the entries in any given column of a PCT are all distinct. Figure 2 gives a PCT of shape (1,3,2,4) Let PCT σ (α) denote the set of all PCTs of shape α and type σ all of whose entries are weakly less than |α|. Additionally, let…”

“…Composition tableaux were introduced in [19] to define the basis of quasisymmetric Schur functions for the Hopf algebra of quasisymmetric functions. These functions are analogues of the ubiquitous Schur functions [41], have been studied in substantial detail recently [8,19,20,30,31,45], and have consequently been the genesis of an active new branch of algebraic combinatorics discovering Schur-like bases in quasisymmetric functions [1,5], type B quasisymmetric Schur functions [27,37], quasi-key polynomials [2,42] and quasisymmetric Grothendieck polynomials [35]. Just as Young tableaux play a crucial role in the combinatorics of Schur functions [40,43], composition tableaux are key to understanding the combinatorics of quasisymmetric Schur functions.…”

Recent Trends in Quasisymmetric Functions

“…For example, the coefficient of x 1 x 2 x 3 x 2 4 x 5 in I * 2,1,3 is 4 since there are four immaculate tableaux of shape (2,1,3) and weight x 1 x 2 x 3 x 2 4 x 5 . (These immaculate tableaux are given in Figure 4.1.…”

“…Grinberg recently proved Zabrocki's conjecture that the dual immaculate quasisymmetric functions can also be constructed using a variation on Bernstein's creation operators [47]. The dual immaculate quasisymmetric functions expand into positive sums of the monomial quasisymmetric functions, the fundamental quasisymmetric functions, and, recently shown in [3], the Young quasisymmetric Schur functions. The latter expansion is not at all obvious given the very different methods used to generate these two bases, and therefore provides further justification that both of these families of functions are interesting and natural objects of study.…”

“…The five standard Young tableau of shape(3,2).For example, if λ = (3, 2), then the standard Young tableaux of shape (3, 2) are shown inFigure 1.4, with respective descent sets {3}, {2, 4}, {2}, {1, 4}, and {1, 3}. Therefore…”