Burge Correspondence: Orthosymplectic Type

Updated 29 October 2025

The Burge correspondence of orthosymplectic type is a combinatorial bijection that unifies representation theory and geometry by matching partitions and weights across orthogonal, symplectic, and superalgebra frameworks.
Its algorithmic procedures, including modified insertion and border strip removal, enable explicit construction of crystal bases and character ring isomorphisms in Lie (super)algebras.
The correspondence underpins geometric Satake equivalences, quantum module constructions, and quiver stratifications, linking algebraic, combinatorial, and categorical invariants.

The Burge correspondence of orthosymplectic type is a pivotal combinatorial and representation-theoretic structure that unifies the interplay between orthogonal and symplectic Lie algebras (type B, C, D) and their superalgebra generalizations, supergroups, and quantum versions. This correspondence emerges in the explicit construction and matching of representations, moduli spaces, combinatorial data, and crystal bases across diverse algebraic and geometric frameworks, including classical, super, quantum, and geometric representation theory.

1. Definition and Fundamental Properties

The Burge correspondence of orthosymplectic type generalizes the classical Burge correspondence (a variant of the Robinson–Schensted–Knuth algorithm for type D) to settings involving both Lie superalgebras and their quantum analogues, as well as categories, sheaves, and crystals indexed by orthosymplectic data.

Its defining feature is a combinatorial bijection or character-theoretic correspondence relating:

Certain partitions (with prescribed arm-leg statistics) and weights for orthogonal (type B, D) and symplectic (type C) Lie algebras.
Simple objects (representations, modules, sheaves, crystal elements) in paired categories: for example, representations of $O(2m+1)$ (type B) and $\mathrm{SpO}(2m|1)$ (type BC), or weights/classes for pure free resolutions and Schur modules over quadric rings (Sam et al., 24 Mar 2025).
Explicit formulas for characters, intertwining the Weyl group and spectral flow data, and governed by alternations and border strip modifications.

The correspondence is realized in multiple settings:

Algebraic: Character rings, Grothendieck groups, and highest weight modules.
Combinatorial: Modification rules for partitions and border strips, matching partitions to matrix or tableau data via insertion algorithms of Burge or RSK type.
Geometric: Parametrization of orbits and closures in affine Grassmannians or quiver moduli spaces, explicitly reflected in stalk computations of perverse sheaves and IC complexes via orthosymplectic Kostka polynomials (Braverman et al., 2022, Braverman et al., 2019).
Crystals: Isomorphisms between crystal bases for quantum orthosymplectic superalgebras, explicit embeddings, and combinatorial algorithms (Jang et al., 28 Oct 2025).

2. Character Ring Isomorphisms and Representation Theory

A fundamental instance is the established isomorphism between the character rings of $O(2m+1)$ and $\mathrm{SpO}(2m|1)$ (Sam et al., 24 Mar 2025): $K(O(2m+1)) \cong K(\mathrm{SpO}(2m|1))$ This bijection preserves highest weight indexing (subject to admissibility conditions) and matches Schur functor classes: $[S^d(U)] \mapsto [S^d(\mathbb{C}^{2m+1})] - [S^{d-2}(\mathbb{C}^{2m+1})]$ Combinatorial rules, such as border strip removal and modification, play a critical role in specifying which partitions (representations) survive under specialization, directly analogized to Burge's scheme for relating type B and type C representations.

Moreover, the functorial construction of Schur modules for quadric hypersurface rings leverages this character coincidence:

Explicit objects $Z_{V, \mathbb{C}^n}$ over $A = S^\bullet(V)/(\text{quadric }q)$ have characteristics matching the orthosymplectic and orthogonal sides, under the isomorphism.

3. Crystal Theoretic Realization

The Burge correspondence attains a precise combinatorial model in the context of crystal bases for quantum orthosymplectic superalgebras (Jang et al., 28 Oct 2025):

The crystal base $\mathcal{B}((g)^-)$ for the negative half of a quantum orthosymplectic superalgebra $g = \mathfrak{d}_{m|n}$ is built as a tensor of crystal bases for the nilpotent and Levi parts.
The embedding of oscillator module crystals $\mathcal{B}(V(\lambda,\ell))$ into $\mathcal{B}((g)^-)$ factors through tableau separation (body/tail) and, crucially, the Burge correspondence algorithm, mapping matrix data to semistandard tableaux in type D.

The central theorem (Thm 5.5 and Thm 6.13 (Jang et al., 28 Oct 2025)) asserts the crystal isomorphism: $\kappa^{\mathfrak{d}_{m|n}}: \mathbf{M}^{\mathfrak{d}_{m|n}} \longrightarrow \bigsqcup_{\delta} SST_{m|n}(\delta^\pi)$ where the correspondence commutes with all crystal operators, giving a canonical embedding of combinatorial data under crystal morphisms.

4. Geometric and Category-Theoretic Incarnations

The categorification of the Burge correspondence arises in geometric Satake equivalences for supergroups (Braverman et al., 2019, Braverman et al., 2022):

The equivalence between representations of degenerate orthosymplectic supergroups and ( $SO(N-1, \mathbb{C}[[t]])$ or $Sp(2n, \mathbb{C}[[t]])$ )-equivariant sheaves/modules on affine Grassmannians parametrizes irreducibles by pairs of partitions $(\lambda, \mu)$ .
Orbit closure properties, stalk computations, and fusion coefficients are governed by orthosymplectic Kostka polynomials, themselves a precise combinatorial realization of the Burge correspondence.

In these geometric frameworks, the labeling and ordering of orbits, closure relations, and the decomposition of convolution products match the partition combinatorics specified by Burge’s scheme, generalized to the super and quantum settings.

5. Quantum Affine and Quiver Theoretic Aspects

Quantum analogues of the Burge correspondence relate $q$ -oscillator representations and finite-dimensional modules of quantum affine (super)algebras via exact monoidal truncation functors (Kwon et al., 2023): $\mathcal{O}_{\mathrm{osc},X} \xleftarrow{\mathrm{tr}_1} \mathcal{O}_{\mathrm{osc},\epsilon} \xrightarrow{\mathrm{tr}_2} \mathcal{O}_{\mathrm{fd},Y}$ These functors categorically realize the Burge correspondence as an interpolation between different types (C/D, D/C), reflecting underlying Howe duality.

In orthosymplectic quotient quiver subtraction (Bennett et al., 25 Mar 2025), the palindromic structure and type-switching rules for framed quotient quivers encode the Burge correspondence at the level of magnetic quivers, mirroring the stratification of moduli spaces into unions of cones (nilpotent orbit closures or slices). The matching of Hilbert series (moduli, closure relation of orbits) and type-changing phenomena further reflect the combinatorial data of the Burge correspondence.

6. Algorithmic and Combinatorial Procedures

For type D (orthosymplectic scenario), the Burge insertion algorithm:

Translates biword matrix data into tableaux via a “column insertion” process with altered bumping rules compared to RSK, ensuring preservation of appropriate symmetries and isotropy conditions.
Specifies the precise correspondence between PBW/Lusztig data in the negative half algebra and the tableaux parametrizing crystal elements, representations, or orbit closures.

A summary diagram for the crystal-embedding process is: $\mathcal{B}(V(\lambda,\ell)) \longrightarrow (T^{body}, T^{tail}) \longrightarrow (\text{Matrix Lusztig data}, T^{tail}) \in \mathbf{M}^g_{\lambda}$ with the Burge algorithm giving $\mathbf{m} \leadsto$ semistandard tableau $T^{body}$ .

7. Broader Implications, Generalizations, and Significance

The Burge correspondence of orthosymplectic type binds together:

Transfer of character-theoretic, combinatorial, and functorial invariants in type B/BC/D/C representations, supergroups, and their quantum or K-theoretic variants.
Construction of analogues of Schur modules, resolutions (Littlewood complexes), and crystal bases with full combinatorial transparency.
Stratification and fusion rules in geometric representation theory (Satake equivalence, perverse sheaves, D-modules, IC complexes) via explicit combinatorial objects.
Spectral and categorical dualities between oscillator and finite-dimensional quantum modules (Howe duality) as arising from precise functorial correspondences matching Burge combinatorics.

The correspondence has implications for equivariant positivity, total positivity in quadric rings, stratification of moduli spaces, and the construction of pure free resolutions, with ongoing extensions to exceptional supergroups and quantum setups (Braverman et al., 2022).

Table: Appearance of Burge Correspondence of Orthosymplectic Type

Context	Mathematical Object	Role of Burge Correspondence
Character rings	$K(O(2m+1)), K(SpO(2m\|1))$	Bijective mapping preserves highest weight/partition structure
Crystals	PBW/Lusztig/Poincaré bases	Embedding, isomorphism with tableau models, crystal operators
Geometric Satake	Perverse sheaves, D-modules	Orbit parametrization, closure relations, Kostka polynomials
Quantum superalgebras	$q$ -oscillator, finite-dim modules	Functorial interpolation, weight matching, spectral decomposition
Quiver moduli spaces	Framed quotient quivers	Stratification by cones, type changing, palindromic structure

The Burge correspondence of orthosymplectic type is thus an organizing principle, operational algorithm, and combinatorial bridge across advanced representation theory, algebraic geometry, quantum algebra, and categorification programs relevant to type B, C, D Lie theory and supergroups.