  We study the 3-parametric family of vertex operator algebras based on the
unitary Grassmannian coset CFT $\mathfrak{u}(M+N)_k/(\mathfrak{u}(M)_k \times
\mathfrak{u}(N)_k)$. This VOA serves as a basic building block for a large
class of cosets and generalizes the $\mathcal{W}_\infty$ algebra. We analyze
representations and their characters in detail and find surprisingly simple
character formulas for the representations in the generic parameter regime that
admit an elegant combinatorial formulation. We also discuss truncations of the
algebra and give a conjectural formula for the complete set of truncation
curves. We develop a theory of gluing for these algebras in order to build more
complicated coset and non-coset algebras. We demonstrate the power of this
technology with some examples and show in particular that the $\mathcal{N}=2$
supersymmetric Grassmannian can be obtained by gluing three bosonic
Grassmannian algebras in a loop. We finally speculate about the tantalizing
possibility that this algebra is a specialization of an even larger
4-parametric family of algebras exhibiting pentality symmetry. Specialization
of this conjectural family should include both the unitary Grassmannian family
as well as the Lagrangian Grassmannian family of VOAs which interpolates
between the unitary and the orthosymplectic cosets.
