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.

振荡器Racah代数\（\ mathcal {R} _n（\ mathfrak {h}）\）由中间Casimir算子实现，该中间Casimir算子出现在振荡器代数\（\ mathfrak {h} \）的张量积中。提出了李代数\（\ mathfrak {sl} _ {n-1} \）到\（\ mathcal {R} _n（\ mathfrak {h}）\）的嵌入。它涉及两个代数的表示理论。我们建立\（\ mathfrak {h} \）的重新耦合系数与\（\ mathfrak {sl} _n \） -表示的矩阵元素之间的联系，它们均用格里菲思类型的多元Krawtchouk多项式表示。