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We analyze the asymptotic symmetry of higher spin gravity with $M \times M$matrix valued fields, which is given by rectangular W-algebras with su$(M)$symmetry. The matrix valued extension is expected to be useful for the relationbetween higher spin gravity and string theory. With the truncation of spin as$s=2,3,\ldots , n$, we evaluate the central charge $c$ of the algebra and thelevel $k$ of the affine currents with finite $c,k$. For the simplest case with$n=2$, we obtain the operator product expansions among generators by requiringtheir associativity. We conjecture that the symmetry is the same as that ofGrassmannian-like coset based on our proposal of higher spin holography.Comparing $c,k$ from the both theories, we obtain the map of parameters. Weexplicitly construct low spin generators from the coset theory, and, inparticular, we reproduce the operator product expansions of the rectangularW-algebra for $n=2$. We interpret the map of parameters by decomposing thealgebra in the coset description.

Rectangular W-algebras, extended higher spin gravity and dual coset CFTs