Racah Problems for the Oscillator Algebra, the Lie Algebra $\mathfrak {sl}_n$, and Multivariate Krawtchouk Polynomials

Abstract

The oscillator Racah algebra $\mathcal {R}_n(\mathfrak {h})$ is realized by the intermediate Casimir operators arising in the multifold tensor product of the oscillator algebra $\mathfrak {h}$. An embedding of the Lie algebra $\mathfrak {sl}_{n-1}$ into $\mathcal {R}_n(\mathfrak {h})$ is presented. It relates the representation theory of the two algebras. We establish the connection between recoupling coefficients for $\mathfrak {h}$ and matrix elements of $\mathfrak {sl}_n$-representations which are both expressed in terms of multivariate Krawtchouk polynomials of Griffiths type.

Appendix A: Calculation of overlap coefficients

Let V be a finite-dimensional representation of $\mathfrak {sl}_2$ and $\tilde{.}$ an automorphism of $\mathfrak {sl}_2$. The element h is a Cartan generator of $\mathfrak {sl}_2$. Let $\{ \psi _k \}$ be an eigenbasis of h and $\{ \phi _s \}$ be an eigenbasis for $\tilde{h}$. The indices k and s run from 0 to N with $\dim (V)=N+1$. We are interested in the overlap coefficients $B_{ks}$ between these bases:

$$\begin{aligned} \phi _s= & {} \sum _{k=0}^N B_{sk}\psi _k.\nonumber \\ h\psi _k= & {} \mu _k\psi _k \qquad \tilde{h}\phi _s=\nu _s\phi _s. \end{aligned}$$

(38)

The algebra $\mathfrak {sl}_2$ has algebra relations $[h,e]=2e$ and $[h,f]=-2f$ with e the raising operator and f the lowering operator on $\{ \psi _k \}$:

$$\begin{aligned} e\psi _k=e_{kk+1}\psi _{k+1} \quad f \psi _k=f_{kk-1} \psi _{k-1} \end{aligned}$$

and $\mu _k=\mu _0+2k$. From the algebra relation $[e,f]=h$ it follows that

$$\begin{aligned} f_{kk-1}e_{k-1k}- e_{kk+1}f_{k+1k}=\mu _k. \end{aligned}$$

Let $A_k:=e_{kk-1}f_{k-1k}$. Then, we have

$$\begin{aligned} A_{k}-A_{k+1}=2k+\mu _0. \end{aligned}$$

From this we find

$$\begin{aligned} A_k=-k(k-1)-\mu _0 k-\Omega \end{aligned}$$

with $\Omega \in \mathbb {R}$. We express $\tilde{h}$ as a linear combination of h, e and f.

$$\begin{aligned} \tilde{h}=R_h h +R_e e+ R_f f \end{aligned}$$

with $R_eR_f+R_h^2=1$. We have set up everything we need to find the overlap coefficients. Let the operator $\tilde{h}$ act on both sides of equality (38).

$$\begin{aligned} \tilde{h}\psi _s=\sum _{k=0}^N B_{sk}(R_h h +R_e e+ R_f f) \psi _k. \end{aligned}$$

This gives

$$\begin{aligned} \nu _s\phi _s=\sum _{k=0}^N B_{sk}(R_h\mu _k\psi _k+R_e e_{kk+1}\psi _{k+1}+R_f f_{kk-1}\psi _{k-1}). \end{aligned}$$

We expand the left-hand side into the basis $\psi _k$ and we gather the terms on the right-hand side:

$$\begin{aligned} \sum _{k=0}^N \nu _s B_{sk} \psi _k=\sum _{k=0}^N (B_{sk}R_h\mu _k+B_{sk-1}R_e e_{k-1k}+B_{sk+1}R_f f_{k+1k})\psi _k. \end{aligned}$$

From this we find the recurrence relation

$$\begin{aligned} \nu _s B_{sk}=B_{sk+1}R_f f_{k+1k}+ B_{sk}R_h\mu _k+B_{sk-1}R_e e_{k-1k}. \end{aligned}$$

We want to recognize this recurrence relation as one of the family of orthogonal polynomials. Let

$$\begin{aligned} \tilde{B}_{ks}=\left( \prod _{t=2}^k f_{tt-1}R_f\right) B_{sk} \end{aligned}$$

to find

$$\begin{aligned} \nu _s \tilde{B}_{sk}=\tilde{B}_{sk+1}+R_h\mu _k\tilde{B}_{sk}+R_eR_f e_{k-1k}f_{k-1k}\tilde{B}_{sk-1}. \end{aligned}$$

We write the coefficients as polynomials in $x=\nu _s$:

$$\begin{aligned} x\tilde{B}_k(x)=\tilde{B}_{k+1}(x)+R_h\mu _k\tilde{B}_k(x)+R_eR_fA_k\tilde{B}_{k-1}(x). \end{aligned}$$

(39)

We want to compare this with the recurrence relation of the normalized Krawtchouk polynomials as defined in [38]:

$$\begin{aligned} xp_n(x)=p_{n+1}+(n(1-2r)+rN)p_n(x)+r(1-r)n(N+1-n)p_{n-1}(x) \end{aligned}$$

with $n=0,1, \dots , N$. Let $x=\alpha y+\beta$ and introduce $q_n(y)=p_n(\alpha y+\beta )/\alpha ^n$. The polynomial $q_n(x)$ satisfies the following recurrence relation:

$$\begin{aligned} yq_n(y)=q_{n+1}+\frac{n(1-2r)+rN-\beta }{\alpha }q_n(y)+\frac{r(1-r)n(N+1-n)}{\alpha ^2}q_{n-1}(y). \end{aligned}$$

We retrieve Eq. (39) if we set

$$\begin{aligned} \alpha =\frac{1}{2}, \quad r=\frac{1-R_h}{2}, \quad \beta =\frac{N}{2}, \quad k=n, \quad \Omega =0, \quad \mu _0=-N. \end{aligned}$$

We explicitly write down the polynomials $\tilde{B}_k(x)$.

$$\begin{aligned} \tilde{B}_k(x)&=2^kp_k\left( \frac{x+N}{2}\right) \\&=(-N)_k (1-R_h)^k K_k\left( \frac{x+N}{2}; \frac{1-R_h}{2},N\right) \\&=(-N)_k (1-R_h)^k {}_2F_1\left( \begin{array}{c} -k,-\frac{x+N}{2}\\ -N \end{array}\big | \frac{2}{1-R_h}\right) . \end{aligned}$$

The overlap coefficients are the Krawtchouk polynomials $K_k(x)$ (defined in the last line of the equation above) up to a normalization factor.

$$\begin{aligned} B_{sk}=\left( \prod _{t=2}^k f_{tt-1}R_f\right) (-N)_k (1-R_h)^k K_k\left( \frac{\nu _s+N}{2}; \frac{1-R_h}{2},N\right) . \end{aligned}$$