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A subalgebra $\mathcal{G} = \operatorname { Fun } _ { q } ( G ( k , n ) )$ in the algebra $\mathcal{A} = \operatorname { Fun } _ { q } ( \operatorname{SL} ( n , \mathbf{C} ) )$ of regular functions on the quantum group $\operatorname{SL} ( n , \mathbf{C} )$ (cf. [[Quantum groups|Quantum groups]]). $\mathcal{G}$ is generated by quantum minors $| T _ { i _ { 1 } , \ldots , i _ { k } } ^ { 1 , \ldots , k } | _ { q }$, with $1 \leq i _ { 1 } < \ldots < i _ { k } \leq n$ and with $T \in \operatorname { Mat } ( n ) \otimes \mathcal{A}$ the vector co-representation of $\mathcal{A}$ [[#References|[a1]]]. The $q$-minors satisfy quadratic relations, which turn into the Plücker relations (Young symmetries, cf. also [[Grassmann manifold|Grassmann manifold]]) when the deformation parameter $q$ is specialized to $1$. Since classically the Grassmannian, as a complex submanifold in the projective space $\mathbf{P} ( \wedge ^ { k } \mathbf{C} ^ { n } )$, is the common zero locus of the Plücker relations, one interprets $\mathcal{G}$ as a quantization of the complex Poisson manifold $G ( k , n )$ (cf. [[Symplectic structure|Symplectic structure]]). The co-multiplication $\Delta$ in $\mathcal{A}$ induces a right co-action $R = \Delta |_{\cal G} :\cal G \rightarrow G \otimes A$ and so $\mathcal{G}$ is a [[Quantum homogeneous space|quantum homogeneous space]].

A more general construction of (generalized) quantum flag manifolds exists for the group $\operatorname{SL} ( n , \mathbf{C} )$ [[#References|[a1]]], as well as for other simple complex Lie groups $G$ having quantum counterparts [[#References|[a2]]]. Another description was given in [[#References|[a3]]]. Both approaches [[#References|[a2]]], [[#References|[a3]]] also allow one to define quantum Schubert varieties.

Since $G ( k , n )$ is compact, the only holomorphic functions defined globally on it are the constants. But one can work instead with holomorphic coordinates $z _ { s t }$, $1 \leq s \leq k$, $1 \leq t \leq n - k$, on the big cell $C$, the unique Schubert cell of top dimension. The standard choice of coordinates is given via the Gauss decomposition of $T$. For the algebra $\mathcal{G}$ this means in fact a localization by allowing the $q$-minor $| T _ { 1, \dots, k } ^ { 1 , \ldots , k } | _ { q }$ to be invertible. The generators $z _ { s t }$ of the quantum big cell $\mathcal{C} = \operatorname { Fun } _ { q } ( C )$ satisfy the relations [[#References|[a4]]]

$$ z_{st} z_{uv} - z_{uv} z_{st} = (q^{\operatorname{sgn}(s-u)} - q^{\operatorname{sgn}(t-v)}) z_{ut} z_{sv}. $$

The [[Symplectic manifold|symplectic manifold]] $G ( k , n )$ can be realized as an orbit of the dressing transformation of $\operatorname{SU} ( n )$ acting on its dual [[Poisson Lie group|Poisson Lie group]]. The transformation can be also viewed as the right $\operatorname{SU} ( n )$-action on the manifold $M$ of $( n \times n )$ unimodular positive matrices: $( m , u ) \mapsto u ^ { * } m u$. The orbits are determined by sets of eigenvalues and $G ( k , n )$ corresponds to a two-point set $\{ \lambda _ { 1 } , \lambda _ { 2 } \}$ with multiplicities $k$ and $n - k$, respectively. There exists a quantum analogue as a right co-action

\begin{equation*} \operatorname { Fun } _ { q } ( M ) \rightarrow \operatorname { Fun } _ { q } ( M ) \otimes \operatorname { Fun } _ { q } ( \operatorname {SU} ( n ) ). \end{equation*}

$\operatorname { Fun } _ { q } ( M )$ is endowed with a $*$-involution and, correspondingly, one can turn $\mathcal{C}$ into a $*$-algebra by determining the commutation relations between $z _ { s t }$ and $z ^ { *} _{ u v}$ in dependence on the parameters $\lambda _ { 1 }$ and $\lambda _ { 2 }$ [[#References|[a4]]].

Similarly as for quantum spheres (cf. [[Quantum sphere|Quantum sphere]]), other types of quantum Grassmannians have been defined, distinguished by possessing classical points, ''i.e.'' one-dimensional representations $\operatorname {Fun}_q ( G ( k , n ) ) \rightarrow \mathbf C $ [[#References|[a5]]].

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  E. Taft,   J. Towber,   "Quantum deformations of flag schemes and Grassmann schemes I. A $q$-deformation for the shape algebra $\operatorname{GL} ( n )$"  ''J. Algebra'' , '''142'''  (1991)  pp. 1–36</td></tr>
<tr><td valign="top">[a2]</td> <td valign="top">  Ya.S. Soibelman,   "On the quantum flag manifold"  ''Funct. Anal. Appl.'' , '''26'''  (1992)  pp. 225–227</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  V. Lakshmibai,   N. Reshetikhin,   "Quantum deformations of flag and Schubert schemes"  ''C.R. Acad. Sci. Paris'' , '''313'''  (1991)  pp. 121–126</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  P. Šťovíček,   "Quantum Grassmann manifolds"  ''Comm. Math. Phys.'' , '''158'''  (1993)  pp. 135–153</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  M. Nuomi,   M.S. Dijkhuizen,   T. Sugitani,   "Multivariable Askey–Wilson polynomials and quantum complex Grassmannians"  M.E.H. Insmail (ed.)  et al. (ed.) , ''Special Functions, $q$-Series and Related Topics'' , ''Fields Inst. Commun.'' , '''14''' , Amer. Math. Soc.  (1997)  pp. 167–177</td></tr>
</table>