Tridiagonal PairEdit
Tridiagonal pairs occupy a central niche in linear algebra and algebraic combinatorics, describing a special relationship between two linear operators on a finite-dimensional vector space. The core idea is that each operator is diagonalizable, and with respect to an eigenbasis of one, the other acts in a tridiagonal fashion. This dual tridiagonal action, and the irreducibility condition that rules out nontrivial common invariant subspaces, give tridiagonal pairs their characteristic structure. The concept generalizes Leonard pairs and connects to a web of ideas including distance-regular graphs, orthogonal polynomials, and certain quantum-algebraic relations.
Definition
Let V be a finite-dimensional vector space over a field F, and let A and A* be linear operators on V. The pair (A, A*) is called a tridiagonal pair if the following conditions hold:
A and A* are diagonalizable on V.
There exists a basis of V in which A is diagonal and A* is irreducible tridiagonal (that is, A* has nonzero entries on the sub- and super-diagonal positions and no row is entirely zero off the main diagonal).
There exists a basis of V in which A* is diagonal and A is irreducible tridiagonal.
The pair is irreducible in the sense that V has no nonzero proper subspace W that is invariant under both A and A*.
Equivalently, in an eigenbasis of A, A* acts in a tridiagonal fashion, and symmetrically in an eigenbasis of A*, A acts tridiagonally. The eigenrank data (the dimensions of the eigenspaces for A and for A*) and the way A and A* couple those eigenspaces encode the essential structure of the pair. In many treatments, A and A* are written with standard notations for their eigenvalue sequences and the corresponding eigenprojectors, and the interplay between these sequences highlights the tridiagonal nature of the action.
For readers familiar with broader operator-algebra language, a tridiagonal pair is often studied as an irreducible module for the algebra they generate, subject to the tridiagonal interaction rules described above. See Leonard pair for a related, more restrictive notion, and Terwilliger algebra for the algebraic framework in which many tridiagonal pairs arise.
Structure and representations
Eigenspace structure: A and A* each have a full set of eigenvectors, and one can index the eigenpaces as V_i for A and V^_i for A. The action of A* on the A-eigenspace decomposition satisfies A* V_i ⊆ V_{i-1} + V_i + V_{i+1}, with a similar relation obtained by exchanging the roles of A and A*.
Irreducible module and split decomposition: If (A, A*) is irreducible, V carries a decomposition that reflects the tridiagonal coupling in a controlled way (often called a split decomposition). In particular, there is a filtration or direct-sum decomposition that is stabilized by both A and A*, and the action of each operator moves you at most one step up or down in the index, reflecting the tridiagonal constraint.
Askey–Wilson relations: The pair (A, A*) often satisfies a pair of polynomial relations of degree three known as the Askey–Wilson relations. These relations tie the pair to a family of orthogonal polynomials and to the representation theory of certain quantum-algebraic structures. See Askey-Wilson polynomials for the analytic side of the connection and Terwilliger algebra for the operator-algebra viewpoint.
Connections to orthogonal polynomials: When the eigenvalue sequences and the tridiagonal couplings align in particular ways, the eigenvalue data of A and A* encode recurrence relations that mirror families of orthogonal polynomials. This is a central theme linking tridiagonal pairs to classical and q-orthogonal polynomials.
Relationship to Leonard pairs: A Leonard pair is a more restrictive instance in which both A and A* are diagonalizable with each acting tridiagonally in a basis that diagonalizes the other, and where the eigenvalue multiplicities are all one. Every Leonard pair is a tridiagonal pair, but not every tridiagonal pair is a Leonard pair. See Leonard pair for the specialized case and its rich polynomial-orthogonality interpretation.
Examples and realizations
General two-operator framework: On a vector space V of dimension d+1, choose A to be diagonalizable with distinct eigenvalues θ_0, θ_1, …, θ_d, and pick A* so that, in the basis that diagonalizes A, A* is irreducible tridiagonal. If, conversely, A* is diagonalizable with distinct eigenvalues and A is irreducible tridiagonal in the A*-basis, you have a tridiagonal pair. This dual presentation is the defining feature and is easy to test in concrete matrix realizations.
Leonard-pair specialization: If A and A* form a Leonard pair, then they automatically form a tridiagonal pair. Leonard pairs arise from certain distance-regular graphs with P- and Q-polynomial structure and are tightly connected to families of orthogonal polynomials. See Leonard pair and Distance-regular graph.
Quantum-algebra realizations: In finite-dimensional representations of certain quantum groups or related algebras (for example, realizations tied to q-Serre relations and the Askey–Wilson framework), natural pairs of generators can act as a tridiagonal pair on a module. This connects the abstract definition to explicit matrix models and to q-analogs of classical polynomials. See Askey-Wilson polynomials and Terwilliger algebra for the surrounding theory.
Association schemes and Terwilliger theory: In the setting of an association scheme, the adjacency algebra and a dual adjacency operator often form a tridiagonal pair on a standard module. This is a common way tridiagonal pairs appear in algebraic combinatorics and is a bridge to the study of distance-regular graphs. See Distance-regular graph and Terwilliger algebra for context.