Quasisymmetric Schur functions and variants

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#Quasisymmetric Schur functions

The quasisymmetric Schur functions, $\{\qSchur_\alpha \},$ were introduced by J. Haglund, K. Luoto, S. Mason, and S. v. Willigenburg [HLMW11]. These refine the classical Schur functions in the sense that \[\schurS_\lambda(\xvec) = \sum_{\alpha \sim \lambda} \qSchur_\alpha(\xvec),\] where the sum is taken over all compositions $\alpha$ whose parts rearrange to the parts of $\lambda.$ The $\{\qSchur_\alpha \}$ constitute a basis for the space of quasisymmetric functions. S. Assaf and D. Searles use the name quasi-Schur functions for this quasisymmetric basis and introduce quasi-key polynomials as a polynomial lift [AS18]. The stable limits of the quasi-key polynomials are the quasi-Schur functions, and their Kohnert-tableau model lifts the expansion of Schur functions into quasi-Schur functions and then into fundamental quasisymmetric functions.

C. Bessenrodt, K. Luoto, and S. v. Willigenburg introduce skew quasisymmetric Schur functions and show that they expand positively into quasisymmetric Schur functions [BLW11].

The quasisymmetric Schur functions can be described as certain characters of the 0-Hecke algebra; see [TW15].

The original definition of the quasisymmetric Schur functions is in terms of Demazure atoms:

Definition

Let $\alpha$ be a composition. Then \[\qSchur_\alpha(\xvec) \coloneqq \sum_{\gamma : \mathrm{comp}(\gamma)=\alpha} \atom_{\gamma}(\xvec)\] where the sum ranges over all weak compositions $\gamma$ obtained by inserting $0$s between parts of $\alpha.$

We can compute the expansion of $\qSchur_\alpha$ in terms of fundamental quasisymmetric functions by using standard reverse composition tableaux. A semistandard reverse composition tableau (SSRCT) of shape $\alpha \vDash n$ is a filling of the diagram $\alpha$ (English notation) with positive integers satisfying:

rows are weakly decreasing;
the first column is strictly increasing with the row index;
for every triple $a,$ $b,$ $c$ arranged as \[\begin{array}{cc} b & c\\[-2pt] & \vdots\\[-2pt] & a \end{array}\]

the condition $a \leq b$ implies that the box $c$ is present and $a\lt c.$

A standard reverse composition tableau (SRCT) uses each integer $1,2,\dotsc,n$ exactly once in the filling. The descent set $\des(T)$ of a SRCT is defined as \[\des(T) \coloneqq \{ i : i+1 \text{ appears weakly to the right of } i \}.\]

Example (Example of a SRCT and its descent set).

(From [TW15]) The tableau below is an element in $SRCT(3,4,3,2)$ and has descent set $\{1,2,5,8,9,11\}.$

$5$ $4$ $2$ $ 8$ $7$ $6$ $3$ $ 11 $ $ 10 $ $ 1$ $ 12 $ $ 9$

Theorem

The fundamental quasisymmetric expansion of $\qSchur_\alpha$ is \[\qSchur_\alpha(\xvec) = \sum_{T \in \mathrm{SRCT}(\alpha)} \gessel_{\DES(T)}(\xvec).\]

Skew quasisymmetric Schur functions have a similar expansion.

Example (Fundamental expansion of $\qSchur_{(2,1,3)}$).

This example is from [Ex. 2.7, TW15]. The expansion \[\qSchur_{(2,1,3)} = \gessel_{(2,1,3)}+\gessel_{(2,2,2)}+\gessel_{(1,2,1,2)}\] corresponds to the three SRCTs

$\mathbf{2}$ $1$ $ \mathbf{3}$ $ 6$ $ 5 $ $ 4$ $\mathbf{2}$ $1$ $ \mathbf{4}$ $ 6$ $ 5 $ $ 3$ $\mathbf{3}$ $\mathbf{1}$ $ \mathbf{4}$ $ 6$ $ 5 $ $ 2$

where the descents are the bold entries.

The degree four quasisymmetric Schur functions have the following fundamental quasisymmetric expansions. The subscript $112$ denotes the composition $(1,1,2).$ $ \alpha $ $ \qSchur_\alpha(\xvec) $ $ 1111 $ $ \gessel_{1111} $ $ 112 $ $ \gessel_{112} $ $ 121 $ $ \gessel_{121} $ $ 13 $ $ \gessel_{13}+\gessel_{22} $ $ 211 $ $ \gessel_{211} $ $ 22 $ $ \gessel_{121}+\gessel_{22} $ $ 31 $ $ \gessel_{31} $ $ 4 $ $ \gessel_{4} $

#Young quasisymmetric Schur functions

The Young quasisymmetric Schur functions , introduced in [LMW13], are obtained from the quasisymmetric Schur functions via the relation $\yqSchur_\alpha = \rho(\qSchur_{\rev(\alpha)})$ where $\rho$ is the involution on quasisymmetric functions given by $\rho\gessel_\beta=\gessel_{\beta^r}.$ They also refine Schur functions: \[\schurS_\lambda(\xvec) = \sum_{\alpha \sim \lambda} \yqSchur_\alpha(\xvec).\]

The degree four Young quasisymmetric Schur functions have the following fundamental quasisymmetric expansions. $ \alpha $ $ \yqSchur_\alpha(\xvec) $ $ 1111 $ $ \gessel_{1111} $ $ 112 $ $ \gessel_{112} $ $ 121 $ $ \gessel_{121} $ $ 13 $ $ \gessel_{13} $ $ 211 $ $ \gessel_{211} $ $ 22 $ $ \gessel_{121}+\gessel_{22} $ $ 31 $ $ \gessel_{22}+\gessel_{31} $ $ 4 $ $ \gessel_{4} $

S. Brauner, Z. Daugherty, S. Mason, and A. Schilling construct quasicrystal skeletons whose connected components have Young quasisymmetric Schur characters [BDMS26]. Contracting these skeletons recovers Bruhat-order structures and gives applications to Stanley symmetric functions.

#Quasisymmetric Schur $Q$-functions

The quasisymmetric Schur $Q$-functions were introduced by N. Jing and Y. Li [JL15] as lifts of Schur $Q$-functions to the peak algebra. They are indexed by peak compositions. E. K. O{\u{g}}uz [Og19] gave a tableau formula showing that these functions expand positively in the peak-function basis.

S.-I. Choi, S.-Y. Nam, and Y.-T. Oh [CNO25], and independently D. Searles and M. Slattery-Holmes [SS26], prove that quasisymmetric Schur $Q$-functions expand positively in the peak Young quasisymmetric Schur basis.

Example (A standard peak composition tableau).

Rows in a peak composition diagram are indexed from bottom to top. Thus the following displayed tableau has peak shape $(4,3,1):$

$8$ $ 5$ $6$ $7$ $ 1$ $2$ $3$ $4$

It is a standard peak composition tableau. Its upward descent set is $\{4,7\},$ so its peak composition is $(4,3,1).$ Hence this tableau contributes the summand $K_{431}$ to $\widetilde{Q}_{431}.$

Writing $K_\beta$ for the peak function indexed by the peak composition $\beta,$ the degree four quasisymmetric Schur $Q$-functions expand as follows. $ \alpha $ $ \widetilde{Q}_\alpha(\xvec) $ $ 4 $ $ K_{4} $ $ 22 $ $ K_{22} $ $ 31 $ $ K_{22}+K_{31} $

#Peak Young quasisymmetric Schur functions

The peak Young quasisymmetric Schur functions were introduced by D. Searles [Sea25] through diagram supermodules for $0$-Hecke–Clifford algebras. They form a basis of the peak algebra and are a peak-algebra analogue of the Young quasisymmetric Schur functions. Like the quasisymmetric Schur $Q$-functions, they are indexed by peak compositions. The construction also contains the Schur $Q$-functions as a distinguished subfamily.

Writing $K_\beta$ for the peak function indexed by the peak composition $\beta,$ the degree four peak Young quasisymmetric Schur functions expand as follows. $ \alpha $ $ \widetilde{S}_\alpha(\xvec) $ $ 4 $ $ K_{4} $ $ 22 $ $ K_{22} $ $ 31 $ $ K_{22}+K_{31} $

The filling in the preceding example is also a standard peak Young composition tableau of shape $(4,3,1).$ Its left descent set is again $\{4,7\},$ so it contributes $K_{431}$ to $\widetilde{S}_{431}.$

#Row-strict quasisymmetric Schur functions

The row-strict quasisymmetric Schur functions were introduced by S. Mason and J. Remmel [MR13]. These quasisymmetric functions are indexed by compositions, and refine the Schur functions as \[\schurS_\lambda(\xvec) = \sum_{\alpha \sim \lambda'} \rsqSchur_\alpha(\xvec).\]

They are related to the quasisymmetric Schur functions by \[\qSchur_\alpha = \omega(\rsqSchur_\alpha),\] where $\omega$ is the involution on quasisymmetric functions which extends the usual involution on $\spaceSym.$ Equivalently, $\rsqSchur_\alpha=\omega(\qSchur_\alpha).$

The degree four row-strict quasisymmetric Schur functions have the following fundamental quasisymmetric expansions. $ \alpha $ $ \rsqSchur_\alpha(\xvec) $ $ 1111 $ $ \gessel_{4} $ $ 112 $ $ \gessel_{13} $ $ 121 $ $ \gessel_{22} $ $ 13 $ $ \gessel_{112}+\gessel_{121} $ $ 211 $ $ \gessel_{31} $ $ 22 $ $ \gessel_{121}+\gessel_{22} $ $ 31 $ $ \gessel_{211} $ $ 4 $ $ \gessel_{1111} $

A product rule for multiplying a row-strict quasisymmetric Schur function with an ordinary Schur function, and expanding into row-strict quasisymmetric Schur functions, is given in [Thm. 13, Fer11]. See also [Thms. 10 and 11, MN15].

#Row-strict Young quasisymmetric Schur functions

The row-strict Young quasisymmetric Schur functions and their skew versions are introduced in [MN15]. They can be defined via the relation with Young quasisymmetric Schur functions: $\rsyqSchur_\alpha = \omega(\yqSchur_\alpha),$ see [Thm. 12, MN15]. They refine Schur functions by summing over the compositions whose parts rearrange to the conjugate partition: \[\schurS_\lambda(\xvec) = \sum_{\alpha \sim \lambda'} \rsyqSchur_\alpha(\xvec).\]

The degree four row-strict Young quasisymmetric Schur functions have the following fundamental quasisymmetric expansions. $ \alpha $ $ \rsyqSchur_\alpha(\xvec) $ $ 1111 $ $ \gessel_{4} $ $ 112 $ $ \gessel_{13} $ $ 121 $ $ \gessel_{22} $ $ 13 $ $ \gessel_{112} $ $ 211 $ $ \gessel_{31} $ $ 22 $ $ \gessel_{121}+\gessel_{22} $ $ 31 $ $ \gessel_{121}+\gessel_{211} $ $ 4 $ $ \gessel_{1111} $

J. Bardwell and D. Searles construct 0-Hecke algebra modules whose quasisymmetric characteristics are the Young row-strict quasisymmetric Schur functions [BS22].

The four families \[\qSchur_\alpha,\qquad \yqSchur_\alpha,\qquad \rsqSchur_\alpha,\qquad \rsyqSchur_\alpha\] are related by the involutions $\omega$ and $\rho.$ If $\alpha^r$ denotes the reverse composition, then \[\begin{array}{rcl} \yqSchur_\alpha &=& \rho(\qSchur_{\alpha^r}),\\ \rsqSchur_\alpha &=& \omega(\qSchur_\alpha),\\ \rsyqSchur_\alpha &=& \omega(\yqSchur_\alpha),\\ \rsyqSchur_\alpha &=& \rho(\rsqSchur_{\alpha^r}). \end{array}\] The last identity follows from $\psi=\omega\circ\rho=\rho\circ\omega$ on quasisymmetric functions.

#Dual immaculate Schur functions

The dual immaculate Schur functions (and their skew version), $\{\diSchur_\alpha\},$ were introduced in [BBSS+14]. They constitute a basis for the space of quasisymmetric functions. The original definition is that they are dual to immaculate Schur functions in noncommutative symmetric functions.

They are defined by \[\diSchur_\alpha = \sum_{T \in IT(\alpha)} \xvec^T\] where we sum over all immaculate tableaux of shape $\alpha.$ These are fillings of the diagram $\alpha$ whose rows are weakly increasing and whose first column is strictly increasing with respect to the row indexing. The weight $\xvec^T$ is computed in the same manner as for semistandard Young tableaux.

By [Prop. 3.37, BBSS+14], \[\diSchur_\alpha = \sum_{\beta \leq_\ell \alpha} L_{\alpha,\beta} \gessel_\beta\] where $L_{\alpha,\beta}$ are certain nonnegative coefficients, and $\leq_\ell$ denotes lexicographic ordering. E. Allen, J. Hallam, and S. Mason prove that the dual immaculate functions expand positively in the Young quasisymmetric Schur basis [AHM18]. S.-Y. Lee and Y.-T. Oh later give a $0$-Hecke-module interpretation of this positivity [LO25].

Example (A small dual immaculate expansion).

For $\alpha=(2,1),$ the standard immaculate tableaux are the following two tableaux.

$1$ $2$ $3$ $1$ $3$ $2$

Their descent sets are $\{2\}$ and $\{1\},$ respectively, so their descent compositions are $(2,1)$ and $(1,2).$ Hence \[\diSchur_{(2,1)}(\xvec) = \gessel_{(1,2)}(\xvec)+\gessel_{(2,1)}(\xvec).\]

The degree four dual immaculate functions expand in the monomial quasisymmetric basis as follows. As above, the subscript $112$ denotes the composition $(1,1,2).$ $ \alpha $ $ \diSchur_\alpha(\xvec) $ $ 1111 $ $ \qmonom_{1111} $ $ 112 $ $ \qmonom_{1111}+\qmonom_{112} $ $ 121 $ $ 2\qmonom_{1111}+\qmonom_{112}+\qmonom_{121} $ $ 13 $ $ \qmonom_{1111}+\qmonom_{112}+\qmonom_{121}+\qmonom_{13} $ $ 211 $ $ 3\qmonom_{1111}+\qmonom_{112}+\qmonom_{121}+\qmonom_{211} $ $ 22 $ $ 3\qmonom_{1111}+2\qmonom_{112}+2\qmonom_{121} +\qmonom_{13}+\qmonom_{211}+\qmonom_{22} $ $ 31 $ $ 3\qmonom_{1111}+2\qmonom_{112}+2\qmonom_{121} +\qmonom_{13}+2\qmonom_{211}+\qmonom_{22}+\qmonom_{31} $ $ 4 $ $ \qmonom_{1111}+\qmonom_{112}+\qmonom_{121}+\qmonom_{13} +\qmonom_{211}+\qmonom_{22}+\qmonom_{31}+\qmonom_{4} $

Note: The Schur functions are not positive in the dual immaculate basis.

S. Mason and D. Searles construct polynomial lifts of the dual immaculate functions [MS21]. Their lifts are designed so that the stable limit recovers the dual immaculate basis, in the same spirit that slide and related polynomial bases lift quasisymmetric families. S. Mason and D. Searles also study the broader Young and reverse dichotomy of polynomial bases [MS22], organizing several quasisymmetric and polynomial families into parallel Young and reverse versions.

Theorem

Let $\alpha=(\alpha_1,\dotsc,\alpha_\ell)$ be a composition of $n.$ For a cell $c=(i,j)$ in the diagram of $\alpha,$ define its immaculate hook by \[H_{i,j} \coloneqq \begin{cases} \{(i',j') : i \leq i' \leq \ell,\ 1 \leq j' \leq \alpha_{i'}\}, & \text{if } j=1,\\ \{(i,j') : j \leq j' \leq \alpha_i\}, & \text{if } j\gt{}1. \end{cases}\] Let $h_{i,j}\coloneqq |H_{i,j}|.$ The number $f^\alpha$ of standard immaculate tableaux of shape $\alpha$ is \[f^\alpha = \frac{n!}{\prod_{(i,j)\in\alpha} h_{i,j}}.\] This hook-length formula was proved in [Prop. 3.13, BBSS+14]; see also the bijective proof of E. L. L. Gao and A. L. B. Yang [Thm. 1.1, GY16]. Equivalently, \[\frac{(n-1)!}{ \prod_{k=1}^{\ell-1}(n-\alpha_1-\alpha_2-\dotsb-\alpha_k) \prod_{k=1}^{\ell}(\alpha_k-1)! }.\]

Example (Immaculate hook lengths for $(2,1,2)$).

For $\alpha=(2,1,2),$ the immaculate hook lengths are \[\begin{array}{cc} 5 & 1\\ 3 & \\ 2 & 1 \end{array}\] and therefore \[f^{(2,1,2)} = \frac{5!}{5\cdot 1\cdot 3\cdot 2\cdot 1}=4.\] Indeed, the four standard immaculate tableaux of this shape are

$1$ $2$ $3$ $4$ $5$ $1$ $3$ $2$ $4$ $5$ $1$ $4$ $2$ $3$ $5$ $1$ $5$ $2$ $3$ $4$

S. Mason and T. Xie classify when skew immaculate functions are nonzero [MX26]. Their results compare the matrix definition of skew immaculate functions with the Hopf-algebraic definition, and give necessary and sufficient conditions for nonzero terms to survive cancellation.

S. Daugherty generalizes the immaculate and dual immaculate bases to colored algebras in partially commutative variables [Dau24]. The colored dual immaculate functions are defined by tableaux, while the colored immaculate functions are defined by creation operators. The paper also studies skew functions, Hopf-algebra structure, Pieri rules, and colored row-strict analogues. J. M. Campbell and S. Daugherty introduce lexical tableaux, a cyclic-word generalization of immaculate tableaux [CD25]. Standard lexical tableaux are counted by unsigned Stirling numbers of the first kind, and the resulting tableau models define dual bases of $\spaceQSym$ and $\mathrm{NSym}$ with expansions in the monomial/fundamental and ribbon/complete bases. S.-Y. Lee and Y.-T. Oh give a $0$-Hecke-module interpretation of the positive expansion of dual immaculate quasisymmetric functions into Young quasisymmetric Schur functions [LO25]. They prove that the dual-immaculate module has a distinguished filtration and construct an indecomposable $0$-Hecke module for the Young quasisymmetric Schur function itself.

N. Bergeron, J. S{\'a}nchez-Ortega, and M. Zabrocki prove a Pieri rule for the dual immaculate basis [BSZ16]. For every positive integer $s,$ the product $\gessel_{(s)}\diSchur_\beta$ has a signed multiplicity-free expansion in the dual immaculate basis. By duality, this is equivalent to their explicit adjoint formula for $F_s^\perp$ on the noncommutative immaculate basis; see [Lem. 3.1 and Thm. 4.4, BSZ16].

#Row-strict dual immaculate Schur functions

The row-strict dual immaculate Schur functions $\{\rsdiSchur_\alpha\},$ were introduced in [NSWV+23] and are defined similarly to the dual immaculate Schur functions. They constitute a basis for the space of quasisymmetric functions. For an introduction, see S. Sundaram’s AlCoVE 2022 talk [Sun22]. The paper of S. Niese, S. Sundaram, S. v. Willigenburg, J. Vega, and B. Wang relates this basis to the $\psi$ involution on the dual immaculate functions, gives Bernstein-like operators, and introduces skew and hook variants. More precisely, \[\psi(\diSchur_\alpha)=\rsdiSchur_\alpha.\]

They are defined by \[\rsdiSchur_\alpha = \sum_{T \in \mathrm{RSIT}(\alpha)} \xvec^T\] where we sum over all row-strict immaculate tableaux of shape $\alpha.$ These are fillings of the diagram $\alpha$ whose rows are strictly increasing and whose first column is weakly increasing with respect to the row indexing.

The functions $\rsdiSchur_\alpha$ are positive in the fundamental quasisymmetric basis.

Example (Row-strict dual immaculate example).

For $\alpha=(1,2),$ the two packed row-strict immaculate tableaux, using entries $\{1,\dotsc,k\}$ for some $k,$ are the following.

$1$ $1$ $2$ $1$ $2$ $3$

They have contents $(2,1)$ and $(1,1,1),$ respectively, and hence \[\rsdiSchur_{(1,2)}(\xvec) = \qmonom_{(2,1)}(\xvec)+\qmonom_{(1,1,1)}(\xvec) = \gessel_{(2,1)}(\xvec).\] This is the small example in [Ex. 3.10, NSWV+23] and is consistent with $\rsdiSchur_\alpha=\psi(\diSchur_\alpha).$

The degree four functions have the following expansions in the monomial quasisymmetric basis. In this table, for instance, the subscript $112$ denotes the composition $(1,1,2).$ $ \alpha $ $ \rsdiSchur_\alpha(\xvec) $ $ 1111 $ $ \qmonom_{1111}+\qmonom_{112}+\qmonom_{121}+\qmonom_{13} +\qmonom_{211}+\qmonom_{22}+\qmonom_{31}+\qmonom_{4} $ $ 112 $ $ \qmonom_{1111}+\qmonom_{121}+\qmonom_{211}+\qmonom_{31} $ $ 121 $ $ 2\qmonom_{1111}+\qmonom_{112}+\qmonom_{121} +2\qmonom_{211}+\qmonom_{22}+\qmonom_{31} $ $ 13 $ $ \qmonom_{1111}+\qmonom_{211} $ $ 211 $ $ 3\qmonom_{1111}+2\qmonom_{112}+2\qmonom_{121} +\qmonom_{13}+2\qmonom_{211}+\qmonom_{22}+\qmonom_{31} $ $ 22 $ $ 3\qmonom_{1111}+\qmonom_{112}+\qmonom_{121} +2\qmonom_{211}+\qmonom_{22} $ $ 31 $ $ 3\qmonom_{1111}+\qmonom_{112}+\qmonom_{121}+\qmonom_{211} $ $ 4 $ $ \qmonom_{1111} $

#Extended Schur functions

The extended Schur functions were introduced by S. Assaf and D. Searles in [AS22]. They are obtained by taking the stable limit of lock polynomials. Alternatively, the extended Schur functions are dual to the shin polynomials (see [CFLS+14]) which constitute a basis for the ring of noncommutative symmetric functions. S. Daugherty studies extended Schur functions and related quasisymmetric bases connected by involutions [Dau24]. In particular, the extended Schur functions sit in a four-family system closed under the $\psi,$ $\rho,$ and $\omega$ involutions. The other three bases in this system are the row-strict extended Schur functions, the flipped extended Schur functions, and the backward extended Schur functions. The row-strict family is obtained from the extended Schur family by $\psi,$ while Daugherty’s flipped and backward families are obtained from it by $\rho$ and $\omega,$ respectively, with the corresponding reversal of the indexing composition.

The degree four extended Schur functions have the following expansions in the monomial quasisymmetric basis. As above, the subscript $112$ denotes the composition $(1,1,2).$ $ \alpha $ $ \extSchur_\alpha(\xvec) $ $ 1111 $ $ \qmonom_{1111} $ $ 112 $ $ \qmonom_{1111}+\qmonom_{112} $ $ 121 $ $ 2\qmonom_{1111}+\qmonom_{112}+\qmonom_{121} $ $ 13 $ $ \qmonom_{1111}+\qmonom_{112}+\qmonom_{121}+\qmonom_{13} $ $ 211 $ $ 3\qmonom_{1111}+\qmonom_{112}+\qmonom_{121}+\qmonom_{211} $ $ 22 $ $ 2\qmonom_{1111}+\qmonom_{112}+\qmonom_{121} +\qmonom_{211}+\qmonom_{22} $ $ 31 $ $ 3\qmonom_{1111}+2\qmonom_{112}+2\qmonom_{121} +\qmonom_{13}+2\qmonom_{211}+\qmonom_{22}+\qmonom_{31} $ $ 4 $ $ \qmonom_{1111}+\qmonom_{112}+\qmonom_{121}+\qmonom_{13} +\qmonom_{211}+\qmonom_{22}+\qmonom_{31}+\qmonom_{4} $ For instance, the same Rust check gives \[\extSchur_{211}(\xvec) = \gessel_{112}+\gessel_{121}+\gessel_{211}\] in the fundamental quasisymmetric basis.

For a connection with the 0-Hecke algebra, see [Sea20], where extended Schur functions are shown to be characters of certain 0-Hecke modules. S.-I. Choi, Y.-H. Kim, S.-Y. Nam, and Y.-T. Oh study projective covers of tableau-cyclic indecomposable $H_n(0)$-modules [CKNO20]. This gives another $0$-Hecke-algebra construction connected to extended Schur and related quasisymmetric functions.

Y.-H. Kim and S. Yoo construct weak Bruhat interval $0$-Hecke modules for genomic Schur functions [KY24]. The quasisymmetric characteristic of their module gives a homogeneous component of the genomic Schur function $U_\lambda,$ using an action on increasing gapless tableaux. Y.-H. Kim proves that, when $\lambda$ has two rows, the Schur-positive expansion of $U_\lambda$ is the characteristic of a filtration of these $0$-Hecke modules [Kim26]. This gives the conjectured representation-theoretic interpretation of the two-row coefficients.

M. Esipova, J. Liang, and S. v. Willigenburg classify when several quasisymmetric Schur-like functions are actually symmetric [ELW25]. Their results cover dual immaculate, row-strict dual immaculate, extended Schur, row-strict extended Schur, and skew and advanced variants; in every case the symmetric examples recover classical skew Schur functions.

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[BDMS26]Sarah Brauner, Zajj Daugherty, Sarah Mason and Anne Schilling. Contractions and applications of crystal skeletons: Young quasisymmetric and Stanley symmetric functions. arXiv:2607.12232v1, 2026.

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[CD25]John M. Campbell and Spencer Daugherty. Lexical tableaux and quasisymmetric functions. arXiv:2511.00713, 2025.

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[CFLS+14]John Maxwell Campbell, Karen Feldman, Jennifer Light, Pavel Shuldiner and Yan Xu. A Schur-like basis of NSym defined by a Pieri rule. The Electronic Journal of Combinatorics, 21(3), September 2014.

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[CKNO20]Seung-Il Choi, Young-Hun Kim, Sun-Young Nam and Young-Tak Oh. The projective cover of tableau-cyclic indecomposable $H_n(0)$-modules. arXiv:2008.06830, 2020.

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[CNO25]Seung-Il Choi, Sun-Young Nam and Young-Tak Oh. Quasisymmetric Schur Q-functions and peak Young quasisymmetric Schur functions. European Journal of Combinatorics, 130:104213, 2025.

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[Dau24]Spencer Daugherty. A generalization of the dual immaculate quasisymmetric functions in partially commutative variables. Electronic Journal of Combinatorics, 31(2):P2.7, 2024.

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[Dau24]Spencer Daugherty. Extended Schur functions and bases related by involutions. Advances in Applied Mathematics, 161:102770, 2024.

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[ELW25]Maria Esipova, Jinting Liang and Stephanie Willigenburg. Symmetric quasisymmetric Schur-like functions. arXiv:2507.08083, 2025.

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[Fer11]Jeffrey Ferreira. A Littlewood–Richardson type rule for row-strict quasisymmetric Schur functions. 23th International conference on formal power series and algebraic combinatorics:329–338, 2011.

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[GY16]Emma L. L. Gao and Arthur L. B. Yang. A bijective proof of the hook-length formula for standard immaculate tableaux. Proceedings of the American Mathematical Society, 144(3):989–998, 2016.

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[HLMW11]James Haglund, Kurt W. Luoto, Sarah K. Mason and Stephanie Willigenburg. Quasisymmetric Schur functions. Journal of Combinatorial Theory, Series A, 118(2):463–490, 2011.

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[JL15]Naihuan Jing and Yunnan Li. A lift of Schur’s Q-functions to the peak algebra. Journal of Combinatorial Theory, Series A, 135:268–290, 2015.

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[Kim26]Young-Hun Kim. A representation-theoretic interpretation of the Schur expansion of two-row genomic Schur functions. arXiv:2604.24454v1, 2026.

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[KY24]Young-Hun Kim and Semin Yoo. Weak Bruhat Interval Modules for Genomic Schur Functions. The Electronic Journal of Combinatorics, 31(4), 2024.

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[LO25]So-Yeon Lee and Young-Tak Oh. Distinguished filtrations of the $0$-Hecke modules for dual immaculate quasisymmetric functions. arXiv:2501.11304, 2025.

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[LMW13]Kurt Luoto, Stefan Mykytiuk and Stephanie Willigenburg. An introduction to quasisymmetric Schur functions. Springer New York, 2013.

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[MN15]Sarah K. Mason and Elizabeth Niese. Skew row-strict quasisymmetric Schur functions. Journal of Algebraic Combinatorics, 42(3):763–791, May 2015.

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[MR13]Sarah Mason and Jeffrey Remmel. Row-strict quasisymmetric Schur functions. Annals of Combinatorics, 18(1):127–148, November 2013.

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[MS21]Sarah Mason and Dominic Searles. Lifting the dual immaculate functions. Journal of Combinatorial Theory, Series A, 184:105511, 2021.

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[MS22]Sarah Mason and Dominic Searles. The Young and Reverse dichotomy of polynomials. The Electronic Journal of Combinatorics, 29(3), 2022.

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[MX26]Sarah Mason and Tianhao Xie. A classification of nonzero skew immaculate functions. Involve, a Journal of Mathematics, 19(1):87–105, 2026.

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[NSWV+23]Elizabeth Niese, Sheila Sundaram, Stephanie Willigenburg, Julianne Vega and Shiyun Wang. Row-strict dual immaculate functions. Advances in Applied Mathematics, 149:102540, 2023.

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[Og19]Ezgi Kantarcı Oğuz. A note on Jing and Li’s type B quasisymmetric Schur functions. Annals of Combinatorics, 23(1):159–170, 2019.

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[Sea20]Dominic Searles. Indecomposable 0-Hecke modules for extended Schur functions. Proceedings of the American Mathematical Society, 148(5):1933–1943, February 2020.

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[Sea25]Dominic Searles. Diagram supermodules for 0-Hecke–Clifford algebras. Mathematische Zeitschrift, 310(3), 2025.

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[SS26]Dominic Searles and Matthew Slattery-Holmes. Expanding quasisymmetric Schur Q-functions into peak Young quasisymmetric Schur functions. Journal of Combinatorics, 17(1):121–143, 2026.

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[Sun22]Sheila Sundaram. The immaculate hecke poset. AlCoVE 2022 talk, 2022.

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[TW15]Vasu V. Tewari and Stephanie Willigenburg. Modules of the 0-Hecke algebra and quasisymmetric Schur functions. Advances in Mathematics, 285:1025–1065, November 2015.

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The symmetric functions catalog