Overview
This work establishes a unified algebraic framework for two distinct six-parameter families of bivariate $q$-Racah type functions. This framework utilizes the representation theory of the quantum loop algebra $\mathcal{L} U_q sl_2$ of $sl_2$. The bivariate $q$-Racah type functions are interpreted as overlap coefficients. These coefficients relate six 'distinguished' eigenbases that are parameterized by $a,b \in {\mathbb C}^*$ and derived from the tensor product (evaluation) representations of $\mathcal{L} U_q sl_2$, which are themselves labeled by evaluation parameters $u_1,u_2$. The analysis identifies conditions on $a,b,u_1,u_2$ under which a subset of pairs of elements behave as tridiagonal pairs of type I, also known as $q$-Racah type. Separately, for the condition $u_1/u_2=1$, another subset of element pairs is conjectured to act as factorized Leonard pairs.
Research Context
The research is grounded in the representation theory of the quantum loop algebra $\mathcal{L} U_q sl_2$. This algebraic structure provides the foundation for unifying the description of specific bivariate functions. The study initiates its analysis from two left and right coideal subalgebras embedded within $\mathcal{L} U_q sl_2$. Furthermore, six commutative subalgebras are constructed from eight specific elements within $\mathcal{L} U_q sl_2 \otimes \mathcal{L} U_q sl_2$. These eight elements depend on two scalar parameters, $a,b \in {\mathbb C}^*$. The elements are diagonalized on a finite-dimensional vector space, forming a critical component of the algebraic setup.
Approach
The approach involves leveraging the representation theory of $\mathcal{L} U_q sl_2$ to algebraically characterize the bivariate $q$-Racah type functions. The identified families of functions are then viewed as overlap coefficients. These coefficients describe the relationship between different eigenbases. Specifically, the process involves:
- Identifying two left and right coideal subalgebras of $\mathcal{L} U_q sl_2$.
- Constructing six commutative subalgebras from eight elements in $\mathcal{L} U_q sl_2 \otimes \mathcal{L} U_q sl_2$.
- Defining these eight elements in terms of two scalars $a,b \in {\mathbb C}^*$.
- Diagonalizing these elements on a finite-dimensional vector space.
- Interpreting the bivariate $q$-Racah type functions as overlap coefficients connecting six 'distinguished' eigenbases. These eigenbases are parameterized by $a,b$ and originate from the tensor product (evaluation) representations of $\mathcal{L} U_q sl_2$, which are themselves labeled by the evaluation parameters $u_1,u_2$.
Findings
- A unified algebraic framework was established for two six-parameter families of bivariate $q$-Racah type functions using the representation theory of the quantum loop algebra $\mathcal{L} U_q sl_2$.
- The starting point of the analysis included two left and right coideal subalgebras of $\mathcal{L} U_q sl_2$ and six commutative subalgebras, which were built from eight elements in $\mathcal{L} U_q sl_2 \otimes \mathcal{L} U_q sl_2$.
- These eight elements, dependent on two scalars $a,b \in {\mathbb C}^*$, were diagonalized on a finite-dimensional vector space.
- The bivariate $q$-Racah type functions are interpreted as the overlap coefficients that relate six 'distinguished' eigenbases. These eigenbases are parameterized by $a,b$ and are associated with the tensor product (evaluation) representations of $\mathcal{L} U_q sl_2$, which are labeled by evaluation parameters $u_1,u_2$.
- Under specific conditions on $a,b,u_1,u_2$, a subset of pairs of these eight elements were shown to act as tridiagonal pairs of type I (also termed $q$-Racah type).
- For the condition $u_1/u_2=1$, another subset of pairs of elements are conjectured to act as factorized Leonard pairs.
- In both cases (tridiagonal pairs and conjectured factorized Leonard pairs), corresponding overlap coefficients relating the various eigenbases associated with different pairs were obtained.
- Some properties of these overlap coefficients were discussed.
- A relation between these overlap coefficients and known bivariate polynomials of Tratnik type and the rank 2 Askey--Wilson algebra was also discussed.