Gelfand–Tsetlin-Type Basis in Lie Algebra Representations

• Gelfand–Tsetlin-type bases are explicitly constructed bases that use multiplicity-free branching rules and combinatorial patterns to index modules in representation theory.
• Their construction employs recursive algorithms, invariant theory, and Z-invariants to derive explicit formulas for matrix elements and special functions.
• These bases underpin applications in quantum information, numerical spectral analysis, and differential equation solutions by offering structured, orthogonal representations.

A Gelfand-Tsetlin–type basis is a highly structured, often explicit basis for modules over Lie algebras, group algebras, or solutions spaces of certain differential systems, characterized by their adaptation to multiplicity-free branching rules along chains of nested subalgebras, parabolic subgroups, or symmetry reductions. This basis provides a powerful computational and conceptual tool in the paper of representation theory, special function theory, and mathematical physics, as it encodes the hierarchy of the symmetry in a combinatorially rich, recursive fashion. The classical context is the construction by Gelfand and Tsetlin for the Lie algebra $\mathfrak{gl}_n$, with dramatic generalizations for other classical and exceptional Lie algebras, symmetric and alternating groups, Brauer and walled Brauer algebras, rational Galois orders, and spaces of differential equation solutions.

1. Foundational Principles and General Framework

A Gelfand-Tsetlin–type (GT-type) basis arises from the recursive application of multiplicity-free branching rules through a chain of subalgebras or subgroups. In the basic case of $\mathfrak{gl}_n$, the chain $$\mathfrak{gl}_n \supset \mathfrak{gl}_{n-1} \supset \cdots \supset \mathfrak{gl}_1$$ decomposes each irreducible module into a sequence of irreducible modules over smaller subalgebras, where each step in the chain introduces additional labeling data (“pattern rows”) encoding the branching (Delanghe et al., 2010, Artamonov et al., 2013).

Each basis vector is then indexed by a combinatorial object—a GT “pattern” or tableau—tracking the path along the branching. The essential requirement is that the restriction at each step is multiplicity-free, so the basis is well-defined and compatible with the subalgebra chain (no ambiguity in the labeling).

Extensions of the GT construction encompass:

• Orthogonal and symplectic Lie algebras, where the method of Z-invariants and highest vector analysis generalizes the tableau indexing and branching (Artamonov et al., 2013, Artamonov, 2016);
• Representation categories for diagram algebras (e.g., symmetric, alternating, and walled Brauer algebras), with GT-type bases corresponding to paths in a Bratteli diagram or branching graph (Geetha et al., 2016, Grinko et al., 2023);
• Weight bases for spaces of solutions to differential systems (Dirac, Hodge-de Rham, GMT systems), where the basis arises through symmetry-adapted methods such as Cauchy–Kovalevskaya (CK) extension or hypergeometric (A–GKZ) systems (Delanghe et al., 2010, Artamonov, 2020, Artamonov, 13 Oct 2025).

The GT-type basis is not unique—its precise form depends on the chosen chain and the branching conventions—but intrinsic structures such as the Kazhdan–Lusztig basis triangulate every such basis, demonstrating robustness across chains (Haidar et al., 7 Nov 2024).

2. Construction Techniques: Branching, Invariants, and Recursion

The principal methodologies for constructing Gelfand-Tsetlin-type bases exploit branching and symmetry reduction. The standard approach is as follows:

1. Branching via Subalgebra Chains:
   • Decompose the representation via a sequence of subalgebras, ensuring each restriction is multiplicity-free.
   • At each step, represent basis vectors for the previous subalgebra as linear combinations parametrized by combinatorial data (e.g., tableau rows, subset selections).
2. Invariant Theory and Z-Invariants:
   • For orthogonal and symplectic types, realize the representation in a function space (often on group cells) and extract subalgebra highest weight vectors by imposing invariance under subgroup actions (the “Z-invariants” of Zhelobenko) and solving indicator (PDE) systems (Artamonov et al., 2013, Artamonov, 2016).
3. Recursive and Algorithmic Lifting:
   • In Clifford analysis, recursively lift lower-dimensional bases to higher dimensions via the CK extension, with explicit expressions involving special functions (often Gegenbauer polynomials) (Delanghe et al., 2010).
   • For symmetric groups and analogs, a recursive path-lifting in the Bratteli graph produces the GT basis, with adjustments for self-conjugacy or additional symmetries (Geetha et al., 2016).
4. Functional, Determinantal, and Hypergeometric Realizations:
   • Some modern constructions realize representations in spaces of functions of matrix entries, using determinants or minors as variables; the basis is then built as explicit combinations or hypergeometric series over points in a pattern-indexed lattice (Artamonov, 2022, Artamonov, 2020, Artamonov, 13 Oct 2025).

These methodologies often yield bases that are orthogonal with respect to natural invariant inner products (e.g., Fischer, $L^2$), and respect further properties such as the Appell property (especially in Clifford and Clifford analysis contexts) (Bock et al., 2010, Delanghe et al., 2010).

3. Algebraic, Combinatorial, and Analytic Structure

Combinatorial Labeling and Patterns

The indexing set of a GT-type basis encodes the branching structure in the form of “patterns.” For $\mathfrak{gl}_n$ and its classical generalizations, these are arrays (GT tableaux) with interlacing or betweenness conditions dictated by the possible highest weights at each intermediate step.

For diagram algebras (e.g., symmetric, alternating, Brauer), paths in the branching graph (Young or Bratteli diagrams) index the basis, possibly with extra data such as signs or regularization in the presence of self-conjugate shapes (Geetha et al., 2016, Grinko et al., 2023).

For spaces tied to differential equations, basis elements are labeled by pattern-like data (multi-indices, lattice points), and the analytic properties (e.g., orthogonality, recursion) are encoded via recursive functional construction (e.g., CK extension, hypergeometric series) (Delanghe et al., 2010, Artamonov, 2020, Artamonov, 13 Oct 2025).

Explicitness and Algorithmic Realizability

A hallmark of GT-type bases is their explicitness:

• Closed, recursive formulas for vector construction, often reducing to explicit polynomials, rational functions, or sums over patterns;
• Explicit descriptions of the action of the algebra’s generators on the basis, frequently leading to factorized matrix element formulas (e.g., in terms of Wigner coefficients, binomial coefficients, or special functions) (Artamonov et al., 2013, Delanghe et al., 2010);
• Orthogonality and Appell properties, essential for numerical and symbolic computation in applications (e.g., in mathematical physics).

Triangularity and Canonical Relations

A recurring phenomenon in broad GT-type basis constructions is the existence of a transition (change-of-basis) matrix from the GT-type basis to canonical or geometric bases (Kazhdan–Lusztig, Chari–Loktev, canonical bases). These matrices are upper (or lower) triangular with respect to combinatorially-defined partial orders (e.g., row-wise dominance), and in many cases, the diagonal entries are given by explicit product formulas (Raghavan et al., 2019, Haidar et al., 7 Nov 2024). This reflects deep structural compatibilities between combinatorial and canonical bases.

4. GT-Type Bases in Representation Theory and Special Functions

The GT-type basis serves as a universal tool in representation theory:

• For Lie algebras, it encodes branching for chains of subalgebras (both for type A and all classical series, and, via functional realization, for exceptional types such as $\mathfrak{g}_2$) (Artamonov et al., 2013, Artamonov, 13 Oct 2025).
• In diagram algebras, it provides a combinatorial parameterization of irreducible modules, crucial for computational problems and for realizing connections between quantum information protocols and combinatorial representation theory (Grinko et al., 2023).
• In Clifford analysis and harmonic analysis, the GT basis yields explicit bases of polynomial solution spaces adapted to symmetry reductions, fundamental for analytic and computational work on Dirac, Hodge–de Rham, and GMT systems (Delanghe et al., 2010).
• In the theory of special functions, GT-type bases relate to families of orthogonal polynomials (Gegenbauer, hypergeometric, Appell sequences), and their construction is often intertwined with properties of hypergeometric systems (classical GKZ and their deformations) (Delanghe et al., 2010, Artamonov, 2022, Artamonov, 2020, Artamonov, 13 Oct 2025).

The construction techniques and explicit formulas (including CK extension, Wigner coefficients, Horn- and $A$-hypergeometric series) directly facilitate the computation of matrix elements, Clebsch–Gordan coefficients, and spectral decompositions.

5. Extensions: Singularities, Categorification, and New Algebraic Structures

Singular and Generalized Modules

The GT-type paradigm extends beyond the finite-dimensional, generic setting. For example:

• The introduction of derivative tableaux enables the description of singular GT modules for $\mathfrak{gl}_n$ and the classification of modules with singular characters and higher weight multiplicities (Futorny et al., 2014).
• GT modules associated with rational Galois orders are constructed using BGG differential operators and combinatorics of the Bruhat order (via Postnikov–Stanley polynomials), yielding universal structures for GT-type modules across broad algebraic contexts (Futorny et al., 2018).

Categorical and Diagrammatic Approaches

Methods from higher representation theory offer categorified realizations of the GT basis:

• The branching and recursion structure of the GT basis for type A Lie algebras is categorified via Khovanov–Lauda–Rouquier (KLR) algebras, with projection and restriction functors implementing the diagrammatic analog of classical branching rules (Vaz, 2013).
• The treatment of Specht and Verma modules through KLRW diagrammatic algebras connects the canonical and GT-type bases through categorical and algorithmic structure (Silverthorne et al., 2020).

Connections to Modern Symmetry and Duality

Recent work defines actions of large symmetry groups (such as the cactus group) on GT-type bases, especially for orthogonal Lie algebras, via crystal commutors and Howe duality (Svyatnyy, 19 Apr 2025). These constructions reinforce the centrality of GT-type bases as bridges between combinatorics, algebraic symmetry, and categorical actions.

6. Applications and Impact

The GT-type basis methodology is central to several domains:

• Symbolic and numerical computation: Algorithmic realizations of the bases, explicit matrix elements, and orthogonality make GT bases essential for computational approaches in representation theory, mathematical physics, and spectral analysis (Delanghe et al., 2010, Grinko et al., 2023).
• Quantum information theory: Recent applications encompass quantum circuits for the optimal port-based teleportation protocol and symmetry reduction in semidefinite programming, exploiting efficiency from GT bases and their associated transforms (Grinko et al., 2023).
• Differential equations and geometry: Orthogonal GT-type bases facilitate explicit description and numerical approximation of solution spaces to elliptic and overdetermined system in Clifford analysis, Dirac-type PDEs, and systems with group symmetry (Delanghe et al., 2010).
• Theoretical unification: The interplay between GT-type bases and canonical bases (Kazhdan–Lusztig, Chari–Loktev, Littelmann), as demonstrated through triangularity and positivity results, unifies approaches from geometry, combinatorics, and categorification (Molev et al., 2018, Haidar et al., 7 Nov 2024).

The rigorous structural understanding and explicitness of GT-type bases make them indispensable in both abstract and applied settings. The deep relations to special functions, categorical representation theory, and recent developments in computational mathematics indicate their continued importance and scope for further generalization.