Generalised Schur Partition Functions

• Generalised Schur partition functions are families of symmetric generating functions that extend classical Schur polynomials through deformations and combinatorial data.
• They unify multiple series, determinant formulas, and partition identities, serving as powerful tools in algebraic combinatorics and the analysis of integrable models.
• These functions play a critical role in quantum field theory by providing RG-invariant indices and modular invariants that capture deep symmetry and scaling properties.

Generalised Schur Partition Functions encompass a broad class of symmetric function partition structures and series expansions, with deep connections to algebraic combinatorics, representation theory, integrable lattice models, and quantum field theory. They interpolate key families such as Schur polynomials, symplectic and orthogonal characters, and serve as invariants for supersymmetric gauge theories under RG flows and deformations. The generalisations span analytic, combinatorial, plethystic, and algebraic aspects, and manifest as explicit determinant formulas, multiple-series generating functions, modular objects, and special characters of quantum groups.

1. Definitions and Principal Frameworks

Generalised Schur partition functions refer to families of generating functions or polynomials indexed by combinatorial data (partitions, plane partitions, strict partitions, multiset partitions) and equipped with deformation parameters, symmetries, or auxiliary variables beyond the standard Schur framework.

Key Types:

• Generalised Determinants: Hamel–King’s two-parameter symmetric functions $H^{(z)}(x;q)$ generalise Schur’s Jacobi–Trudi determinants to $z$-asymmetric shapes and parametric deformations, using

$$H^{(z)}(x;q) = \det_{1\leq i,j\leq k}\left(h_{\lambda_i - i + j}(x) + [j > -z] \, q \, h_{\lambda_i - i - j - 1}(x)\right)$$

with summations over $z$-asymmetric subshapes (Albion, 30 Jan 2025).

• Multiple Series for Partition Statistics: Positive multiple series (triple, quadruple, quintuple) generating functions enumerate classes of partitions with difference and congruence conditions, as in Schur’s partition theorem, and refinements by parity and residue (Kurşungöz, 2018, Andrews et al., 2014).
• Ice Model Partition Functions: Wavefunctions and boundary partition functions in type-$C$ (reflecting) and type-$B$ ice models can be exactly expressed as generalized symplectic Schur functions and Whittaker functions, with explicit Korepin-type determinant/factorisations (Motegi et al., 2018).
• RG-Invariant Gauge Theory Integrals: In conformal field theory, “generalized Schur partition functions” $\hat{\mathcal{Z}}(q;\alpha)$ encapsulate double-scaled limits of indices, exhibiting invariance under RG flows and matching between non-trivially related $\mathcal{N}=2$ SCFTs (Deb et al., 16 Jun 2025).
• Partition Function Families: Six-parameter families of double series unify classical partition identities of Schur, Göllnitz–Gordon, and related theorems, as explicit hypergeometric $q$-series (Andrews et al., 2014).

2. Algebraic Combinatorics and Symmetric Function Theory

Generalised Schur partition functions are rooted in advanced algebraic combinatorics:

• Cores and Quotients: The decomposition of partitions into $t$-cores and $t$-quotients controls factorisations under Verschiebung (plethysm-adjoint) operators. For Schur functions, plethysm by $p_t$ yields the SXP-rule, factoring as a product over $t$-quotients when the $t$-core vanishes (Albion, 30 Jan 2025).
• $z$-Asymmetric Classes: Extending to $z$-asymmetric shapes, generalized determinants and associated factorisations link to symplectic and orthogonal universal characters and rational deformations thereof (Albion, 30 Jan 2025).
• Positive Multiple Series: Bijective and “move”-based combinatorics (Kurşungöz) classify partitions by statistic vectors (pairs, singletons) and construct triple, quadruple, and quintuple series reflecting the finer congruence and parity conditions (Kurşungöz, 2018).
• Cellular Algebras and Plethystic Generating Functions: Multiset partition algebras $\mathcal{MP}_\lambda(\xi)$ act as centralizers for the $S_n$ action on tensor powers, yielding generating functions for irreducible multiplicities via plethysms of Schur functions:

$$\sum_{\lambda_1,\ldots,\lambda_s \geq 0} m_\mu(\lambda,n) x_1^{\lambda_1} \cdots x_s^{\lambda_s} = s_\mu[1 + h_1(x) + h_2(x) + \ldots]$$

(Narayanan et al., 2019).

3. Integrable Models and Quantum Invariant Structures

Generalised Schur partition functions provide the algebraic backbone for exactly solvable lattice models and quantum correlators:

• Ice Models: The free-fermionic six-vertex type-$C$ model with reflecting boundaries defines partition functions as factorizable objects, and $N$-particle wavefunctions expressible as generalized symplectic Schur determinants with inhomogeneity and factorial parameters (Motegi et al., 2018).
• Izergin–Korepin Techniques: The Korepin-style recursion properties, determinant characterisations, and dual Cauchy identities underpin the polynomial uniqueness and factorization structure for such wavefunctions and model partition functions (Motegi et al., 2018).
• Tetrahedron Equation and Schur Presentation: 3D integrable vertex models yield explicit algebraic formulas for partition functions in terms of Schur polynomials via the Zamolodchikov–Faddeev relations and commutation algebra, with loop-elementary symmetric function generalizations in inhomogeneous settings (Iwao et al., 16 May 2024).

4. Applications in Quantum Field Theory and RG Flows

Generalised Schur partition functions have transformative roles in quantum field theory, connecting four-dimensional superconformal theories with modular and representation-theoretic invariants:

• RG-Invariant Families: The double-scaled limit of the superconformal index defines $\hat{\mathcal{Z}}(q;\alpha)$, interpolating between Higgs and Coulomb branch indices. This function is invariant under mass and vev deformations that preserve certain gauge-theoretic symmetries (Deb et al., 16 Jun 2025).
• Spectral Parameter Matching: For distinct theories on different Coulomb branch loci, partition functions match exactly upon non-trivial redefinition of the spectral parameter $\alpha$, determined from central charge and scaling dimension relations. This mechanism generates the indices for all theories in the Deligne–Cvitanović series by special values of $\alpha$ (Deb et al., 16 Jun 2025).
• Modular Differential Equations: The partition functions $\mathcal{Z}_G(q;\alpha)$ for $USp(2N)$ theories provide contour-integral realisations of vector-valued modular forms solving monic MLDEs of fixed order, with vanishing Wronskian index (Chandra et al., 1 Dec 2025). For rational $\alpha$, one recovers RCFT characters and interprets MLDE solutions in terms of physical observables (e.g., BPS monodromy traces at $\alpha=-k$).

5. Higher Dimensional Generalisations and Symmetric Function Deformations

Beyond classical partitions, various higher-dimensional and deformed Schur-type partition functions have been constructed:

• Plane Partition ‘3-Schur’ Functions: Recursive definitions and orthogonality in the space of time variables indexed by plane partitions fundamentally generalise Schur functions to higher dimensions, with non-abelian cut-and-join algebras and connections to Macdonald polynomials via duality and projection (Morozov, 2018).
• Crystal Structures and Modular Representation Theory: Generalized Schur partitions for odd $p$ form $A^{(2)}_{p-1}$ crystal structures, which encode modular branching rules for spin representations of Schur covers, with closed form generating functions paralleling Rogers–Ramanujan and Göllnitz–Gordon identities (Tsuchioka et al., 2016).
• Overlap and Littlewood–Schur Functions: Overlap identities for Littlewood–Schur functions (indexed by partition pairs with staircase interleaving) expand the class of generalised Schur partition functions and yield determinantal expansions, bijective characterisations, and dual Cauchy identities (Riedtmann, 2018).

6. Analytic Series and Product Identities

Classical and generalised Schur partition functions admit hypergeometric and $q$-series representations, often as double or multiple summations with combinatorial interpretations:

• Double-Series Expansions: Families $R(s,t,\ell,u,v,w)$ encompass Schur, Göllnitz–Gordon, and other partition identities; analytic proofs employ $q$-difference recurrences, bijective mappings, and transformation formulas (Andrews et al., 2014).
• Multiple-Series Positive Expansions: Explicit multiple-series formulae for enumerative coefficients connect to refinement by parity/congruence, with combinatorial algorithms for base-partition selection and step-wise moves reconstructing the full series (Kurşungöz, 2018).
• Product-Side Results: Specialized cases recapture famous product identities, e.g., Schur’s theorem:

$f_S(1;q) = (-q;q^3)_\infty(-q^2;q^3)_\infty$

and new infinite families arise in the generalized context (Andrews et al., 2014, Tsuchioka et al., 2016).

7. Outlook and Open Questions

Active areas of research include: closed-form characterizations and expansion theorems for all generalised Schur bases; extension of integrable structures and modular invariants to broader classes of symmetric and non-symmetric functions; deepening links between RG-invariants and MLDEs in quantum field theory; and additional analytic proofs for multi-parameter series/generalised partition identities in the spirit of Rogers–Ramanujan–Gordon and Andrews–Gordon style constructions (Kurşungöz, 2018, Deb et al., 16 Jun 2025, Chandra et al., 1 Dec 2025).