Extended Schur Functions in QSym Combinatorics

• Extended Schur functions are quasisymmetric functions defined via standard extended tableaux that extend the classical Schur basis to arbitrary compositions.
• They serve as a QSym basis with positive fundamental and monomial expansions, with a dual “shin” basis in NSym that bridges symmetric and quasisymmetric theory.
• Representation-theoretically, they arise as the quasisymmetric characteristics of indecomposable 0-Hecke modules, with symmetry occurring only when the outer shape is a partition.

Extended Schur functions are quasisymmetric functions indexed by compositions that extend the classical Schur basis from the symmetric-function algebra $Sym$ to the full Hopf algebra $QSym$. In their straight-shape form they are defined from standard extended tableaux and, in skew form, from semistandard fillings of composition diagrams with partition inner shape. They coincide with ordinary Schur or skew Schur functions on partition shapes, form a basis of $QSym$, admit a dual “shin” basis in $NSym$, and arise as quasisymmetric characteristics of indecomposable $0$-Hecke modules. Recent work has also classified precisely when skew, row-strict, and advanced extended Schur-type functions are symmetric: in the extended Schur case, symmetry recovers only classical skew Schur theory (Searles, 2019, Esipova et al., 10 Jul 2025).

1. Definitions and tableau models

For a composition $\alpha=(\alpha_1,\dots,\alpha_k)\vDash n$, the composition diagram $D(\alpha)$ is the left-justified array with $\alpha_i$ boxes in row $i$, with row $1$ at the bottom. A standard extended tableau of shape $QSym$0 is a bijective filling of $QSym$1 by $QSym$2 such that rows strictly increase left to right and columns strictly increase bottom to top. If $QSym$3 is a partition, this is exactly a standard Young tableau of shape $QSym$4 (Searles, 2019).

The straight extended Schur function attached to $QSym$5 is defined by the fundamental-basis expansion

$QSym$6

where $QSym$7 is the descent composition determined by the descent set

$QSym$8

Thus extended Schur functions are quasisymmetric functions built from standard tableaux on composition shapes rather than partition shapes (Searles, 2019).

A skew, semistandard version uses a composition $QSym$9, a partition $QSym$0, and the skew composition diagram $QSym$1. The set

$QSym$2

consists of fillings by positive integers whose columns strictly increase bottom to top and whose rows weakly increase left to right. The skew extended Schur function is then

$QSym$3

When both $QSym$4 and $QSym$5 are partitions, one has

$QSym$6

so the skew construction is a genuine extension of skew Schur functions from partition shapes to composition shapes (Esipova et al., 10 Jul 2025).

2. Basis of $QSym$7, duality, and relation to Schur functions

The family $QSym$8 is a $QSym$9-basis of $NSym$0. In this sense extended Schur functions complete the Schur basis of $NSym$1 to a basis indexed by all compositions. A distinctive feature emphasized in the literature is that, unlike other Schur-like quasisymmetric bases, the extended Schur basis contains the ordinary Schur functions themselves as a subset: if $NSym$2 is a partition, then

$NSym$3

(Searles, 2019).

The straight functions admit positive expansions in both the fundamental and monomial quasisymmetric bases. Writing

$NSym$4

shows that the transition to the fundamental basis is governed by descent statistics on standard extended tableaux. The monomial positivity is also recorded through the dual theory in $NSym$5 (Searles, 2019).

Under the standard pairing between $NSym$6 and $NSym$7, the dual basis to $NSym$8 is the shin basis $NSym$9. Ribbon Schur functions expand positively in this basis: $0$0 This places extended Schur functions within the canonical Schur-like landscape alongside quasisymmetric Schur functions and dual immaculate functions, while preserving a direct link to the classical Schur basis (Searles, 2019).

3. $0$1-Hecke modules and indecomposability

Extended Schur functions admit a representation-theoretic realization via the $0$2-Hecke algebra $0$3. For a composition $0$4, one first considers the vector space $0$5 spanned by standard row-increasing tableaux of shape $0$6, with operators $0$7 defined by comparing the relative positions of $0$8 and $0$9. The subspace generated by tableaux violating column increase is a $\alpha=(\alpha_1,\dots,\alpha_k)\vDash n$0-submodule $\alpha=(\alpha_1,\dots,\alpha_k)\vDash n$1, and the quotient

$\alpha=(\alpha_1,\dots,\alpha_k)\vDash n$2

has basis naturally identified with $\alpha=(\alpha_1,\dots,\alpha_k)\vDash n$3 (Searles, 2019).

On this quotient, the $\alpha=(\alpha_1,\dots,\alpha_k)\vDash n$4-Hecke action becomes explicit: $\alpha=(\alpha_1,\dots,\alpha_k)\vDash n$5 A filtration of $\alpha=(\alpha_1,\dots,\alpha_k)\vDash n$6 by tableaux ordered under the $\alpha=(\alpha_1,\dots,\alpha_k)\vDash n$7-action has one-dimensional successive quotients isomorphic to the simple $\alpha=(\alpha_1,\dots,\alpha_k)\vDash n$8-modules indexed by the descent compositions of the tableaux. Consequently,

$\alpha=(\alpha_1,\dots,\alpha_k)\vDash n$9

so extended Schur functions are exactly the quasisymmetric characteristics of these modules (Searles, 2019).

The same paper proves that each $D(\alpha)$0 is indecomposable. The crucial combinatorial object is the super-standard extended tableau $D(\alpha)$1, obtained by filling rows from bottom to top with consecutive integers. The module $D(\alpha)$2 is cyclically generated by $D(\alpha)$3, and an endomorphism analysis shows that the only idempotent endomorphisms are $D(\alpha)$4 and $D(\alpha)$5. This makes extended Schur functions the characteristic functions of indecomposable $D(\alpha)$6-Hecke modules, paralleling the role played by dual immaculate functions in earlier work (Searles, 2019).

4. Skew, row-strict, and involution-related variants

Several closely related families accompany the basic extended Schur construction. The row-strict skew extended Schur function

$D(\alpha)$7

is defined by weakly increasing columns and strictly increasing rows. Ehrenborg’s involution $D(\alpha)$8 on $D(\alpha)$9 satisfies

$\alpha_i$0

just as $\alpha_i$1 sends a skew Schur function to the Schur function of the transposed skew shape (Esipova et al., 10 Jul 2025).

Later work developed further bases related by involutions on $\alpha_i$2 and $\alpha_i$3. In particular, the paper “Extended Schur functions and bases related by involutions” introduces two new bases of $\alpha_i$4, the flipped extended Schur functions and the backward extended Schur functions, together with dual bases in $\alpha_i$5, the flipped shin functions and backward shin functions. These are obtained from the extended Schur and shin bases by the involutions $\alpha_i$6 and $\alpha_i$7, which generalize the classical involution $\alpha_i$8 on symmetric functions (Daugherty, 2024).

That same work defines skew extended Schur functions from the left action of $\alpha_i$9 on $i$0 and skew-II extended Schur functions from the right action. The left-action skew theory has a positive tableau model using skew shin-tableaux, while the skew-II theory need not have positive monomial expansion; negative coefficients occur, so there is no straightforward positive tableau model in general. The paper also proves Jacobi–Trudi-type formulas for certain shin functions via creation operators and transports these formulas to the involution-related bases (Daugherty, 2024).

A convenient summary of the main skew families is:

Family	Tableau conditions	Symmetric outcome
$i$1	rows weak, columns strict	iff $i$2 is a partition; then $i$3
$i$4	rows strict, columns weak	iff $i$5 is a partition; then $i$6
$i$7	rows strict, columns strict	only hook shapes $i$8; then $i$9
$1$0	rows weak, columns weak	only hook shapes $1$1; then $1$2

5. Symmetry classification

A central recent result is the complete classification of when skew extended Schur functions are symmetric. If $1$3 is a composition and $1$4 a partition with $1$5, then

$1$6

Moreover, in the symmetric case,

$1$7

Thus symmetry forces the outer composition shape to be an ordinary partition, and no genuinely new symmetric functions arise from nonpartition outer shapes (Esipova et al., 10 Jul 2025).

The proof uses coefficient comparison in the monomial quasisymmetric basis. For a symmetric quasisymmetric function, coefficients of $1$8 must agree whenever $1$9 and $QSym$00 have the same underlying partition. For nonpartition $QSym$01, the authors construct two contents $QSym$02 with the same sorted partition but different tableau counts, via an explicit injection between tableau sets that is not surjective. This shows that the monomial coefficients fail the symmetry criterion (Esipova et al., 10 Jul 2025).

By the involution relation $QSym$03, the same classification holds for row-strict extended Schur functions: $QSym$04 and then

$QSym$05

The transpose appears because $QSym$06 sends skew Schur functions to transposed skew Schur functions (Esipova et al., 10 Jul 2025).

The advanced variants are even more rigid. For the strictly advanced family $QSym$07 and weakly advanced family $QSym$08, symmetry occurs exactly when the skew shape is a hook

$QSym$09

for some $QSym$10. In those cases the functions collapse to a single Schur function: $QSym$11 A common misconception is that extended Schur-type constructions might yield many new symmetric functions on composition shapes; the classification shows the opposite. For extended Schur and row-strict extended Schur functions, the symmetric regime is exactly classical skew Schur theory, and for the advanced variants it is even more degenerate (Esipova et al., 10 Jul 2025).

6. Broader significance

Extended Schur functions occupy a distinctive position among Schur-like bases of $QSym$12. They refine Schur theory to composition shapes while retaining the Schur basis literally on partition indices, they possess a dual shin basis in $QSym$13, and they admit a concrete $QSym$14-Hecke realization by indecomposable modules (Searles, 2019). This combination of tableau-theoretic, Hopf-algebraic, and representation-theoretic features makes them a particularly rigid extension of Schur functions.

At the same time, the symmetry classification indicates the limits of that extension. Dual immaculate functions can be symmetric for many nonpartition shapes because their column conditions are imposed only on the leftmost column, but extended Schur functions impose column conditions on every column, and that rigidity forces symmetry back to the partition-shape setting. A plausible implication is that extended Schur functions are best regarded not as a source of new symmetric functions, but as intrinsically quasisymmetric objects whose symmetric part is already exhausted by classical skew Schur theory (Esipova et al., 10 Jul 2025).

The later involution-based theory reinforces this picture. Flipped and backward extended Schur functions, skew and skew-II variants, and the associated shin bases show that extended Schur theory is closed under a rich system of involutions and dualities, yet the classical Schur sector remains the only symmetric core. In that sense, extended Schur functions provide a precise boundary between classical Schur theory and genuinely quasisymmetric combinatorics (Daugherty, 2024).