Skew Schur Functions Overview

• Skew Schur functions are symmetric functions defined by pairs of partitions and semistandard Young tableaux, generalizing classical Schur functions.
• They can be expressed through Jacobi–Trudi and dual Jacobi–Trudi determinants, linking combinatorial constructions with algebraic identities.
• Applications span representation theory, algebraic geometry, integrable systems, and noncommutative settings, underpinning diverse combinatorial structures.

A skew Schur function is a symmetric function associated to a pair of partitions $\lambda$, $\mu$ with $\mu \subseteq \lambda$, defined combinatorially as the generating function over semistandard Young tableaux of skew shape $\lambda/\mu$ and equivalently as a Jacobi–Trudi determinant involving complete homogeneous symmetric functions. Skew Schur functions generalize the ordinary Schur functions and retain a central role across representation theory, algebraic geometry, and symmetric function theory, with deep combinatorics (such as the Littlewood–Richardson rule), extensions to noncommuting and quasisymmetric settings, and connections to Plücker relations, integrable systems, and algebraic K-theory.

1. Classical Definition and Fundamental Properties

Let $\lambda=(\lambda_1,\dots,\lambda_\ell)$ and $\mu=(\mu_1,\dots,\mu_\ell)$ be integer partitions with $\mu\subseteq\lambda$. The skew diagram $\lambda/\mu$ is the difference of the Ferrers diagrams, and a semistandard Young tableau (SSYT) of shape $\lambda/\mu$ is a filling by positive integers, weakly increasing along rows and strictly increasing down columns. The tableau definition is

$$s_{\lambda/\mu}(x) = \sum_{T \in \mathrm{SSYT}(\lambda/\mu)} x^T,$$

where $\mu$0 (Aokage et al., 11 Apr 2025, Foley et al., 2020).

The Jacobi–Trudi formula provides a determinantal expression: $\mu$1 with $\mu$2 the complete homogeneous symmetric function ( $\mu$3 for $\mu$4 ) (Aokage et al., 11 Apr 2025, Okada, 2017).

2. Expansion, Littlewood–Richardson Rule, and Interval Support

Skew Schur functions expand linearly in the Schur basis: $\mu$5 where $\mu$6 counts Littlewood–Richardson tableaux of shape $\mu$7 and content $\mu$8 (Azenhas et al., 2010, Aokage et al., 11 Apr 2025).

The support of $\mu$9 is always contained in the dominance order interval $\mu \subseteq \lambda$0 determined by the minimal and maximal LR-values (the most column- and most row-packed tableaux). For certain “basic” shapes, the expansion is multiplicity-free and fills this interval, corresponding to a classification of seven explicit families of skew shapes (Azenhas et al., 2010).

3. Determinantal and Pfaffian Identities

Every skew Schur function admits several determinantal representations:

• Classical Jacobi–Trudi: as above.
• Dual Jacobi–Trudi: in terms of elementary symmetric functions.
• Giambelli-type formula: For partitions in Frobenius notation ($\mu \subseteq \lambda$1, $\mu \subseteq \lambda$2), one has

$\mu \subseteq \lambda$3

with blocks of hook-shaped skew Schurs and $\mu \subseteq \lambda$4 functions (Okada, 2017, Foley et al., 2020).

Pfaffian analogues yield formulas for skew Schur Q-functions, generalizing Schur's Pfaffian identity (Okada, 2017, Foley et al., 2020).

4. Advanced Combinatorial and Algebraic Structures

Plücker Relations and Integrable Hierarchies

Skew Schur functions satisfy generalized Plücker-type (Hirota bilinear) relations mirroring those of Schur functions and relating to the geometry of flag varieties and KP/mKP hierarchies. The skew Plücker relations involve quadratic relations among skew Schur functions indexed by sets of partitions and hook shapes, extending classical relations and encoding the projective geometry of appropriate embeddings (Aokage et al., 11 Apr 2025).

Extensions: Grothendieck, Stable, and Quasisymmetric

Grothendieck polynomials $\mu \subseteq \lambda$5 for skew shapes $\mu \subseteq \lambda$6 expand as alternating sums over skew Schur functions, with combinatorial coefficients given by set-valued tableaux counts with specified row bounds. Conversely, skew Schur functions expand as signed sums of Grothendieck polynomials, with dual tableau-counting rules (Chan et al., 2019).

Skew quasisymmetric Schur functions generalize the theory to the quasisymmetric Hopf algebra (QSym), defined via semistandard composition tableaux and satisfying dual Littlewood–Richardson rules and representation-theoretic correspondences with noncommutative symmetric functions (Bessenrodt et al., 2010).

Noncommuting Variables and NCSym

In the Hopf algebra of symmetric functions in noncommuting variables (NCSym), skew Schur functions are defined via a noncommutative Jacobi–Trudi determinant, with additional indexing by permutations in $\mu \subseteq \lambda$7. The product and coproduct structures mimic the commutative case but retain more refined information, splitting the classical product into a sum over concatenation types and lifting the classical Littlewood–Richardson rule to a noncommutative context (Aliniaeifard et al., 2021). A recent classification for equality of skew Schur functions in NCSym reveals that two such functions are equal only under antipodal ribbon conjugation and a strict block-preserving property of the underlying permutations (Jin et al., 2024).

5. Variational Theory: Factorial, Ninth, and Supersymmetric Skew Schur Functions

The “ninth-variation” skew Schur function replaces variables by an array indexed by diagonal content and admits a combinatorial sum over “supertableaux” with row/column content constraints. This variation supports a rich family of determinantal identities attached to outside decompositions of the skew shape, including row, column, Giambelli, outer rim, and inner rim variants—all generalizing classical form (Foley et al., 2020, Foley et al., 2020).

Factoring-in “factorial” parameters (Macdonald’s sixth variation) recovers supersymmetry—restoring (separate) symmetry in the x- and y-blocks and vanishing under x=y substitution. This demonstrates that certain determinantal identities and combinatorial rules are intrinsic to the skew shape and independent of the ordering of primed and unprimed variables (Foley et al., 2020).

6. Positivity, Specializations, and Additional Structures

Skew Schur functions corresponding to staircase shapes exhibit Schur P-positivity: they expand nonnegatively in the Schur P-function basis, with combinatorial interpretations in terms of compatible shifted tableaux. In special cases, these coefficients relate to enumerative invariants such as Eulerian numbers (Ardila et al., 2011).

Further, the skew quantum Murnaghan–Nakayama rule governs the expansion of products of skew Schur functions with quantum power sums in terms of other skew Schur functions and broken ribbons, interpolating the classical Pieri and Murnaghan–Nakayama rules and connecting via combinatorial insertion processes and involutions (Konvalinka, 2011).

7. Structural Identities and Equalities

The Hopf algebraic structure and cocommutativity of the coproduct in the symmetric functions force various identities among skew Schur functions. For example, under stipulated conditions involving ribbon-compositions and shape compositions, two skew Schur functions indexed by shapes related by 180° rotation are equal (Yeats, 2015). This reflects deep symmetries and extends to compound determinant and Pfaffian identities for various generalizations.

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References

• (Aokage et al., 11 Apr 2025) Skew Plücker relations
• (Chan et al., 2019) Combinatorial relations on skew Schur and skew stable Grothendieck polynomials
• (Aliniaeifard et al., 2021) Schur functions in noncommuting variables
• (Jin et al., 2024) Equality of skew Schur functions in noncommuting variables
• (Azenhas et al., 2010) Multiplicity-free Skew Schur functions with full interval support
• (Okada, 2017) Generalized Sylvester Formulas and skew Giambelli Identities
• (Foley et al., 2020) Determinantal and Pfaffian identities for ninth variation skew Schur functions and Q-functions
• (Foley et al., 2020) Factorial supersymmetric skew Schur functions and ninth variation determinantal identities
• (Bessenrodt et al., 2010) Skew quasisymmetric Schur functions and noncommutative Schur functions
• (Ardila et al., 2011) Staircase skew Schur functions are Schur P-positive
• (Konvalinka, 2011) Skew quantum Murnaghan-Nakayama rule
• (Yeats, 2015) A Hopf algebraic approach to Schur function identities