  We consider the decomposition into irreducible components of the exterior
algebra $\bigwedge\left(\mathbb{C}^{n}\otimes
\left(\mathbb{C}^{k}\right)^{*}\right)$ regarded as a $GL_{n}\times GL_{k}$
module. Irreducible $GL_{n}\times GL_{k}$ representations are parameterized by
pairs of Young diagrams $(\lambda,\bar{\lambda}')$, where $\bar{\lambda}'$ is
the complement conjugate diagram to $\lambda$ inside the $n\times k$ rectangle.
We set the probability of a diagram as a normalized specialization of the
character for the corresponding irreducible component. For the principal
specialization we get the probability that is equal to the ratio of the
$q$-dimension for the irreducible component over the $q$-dimension of the
exterior algebra. We demonstrate that this probability distribution can be
described by the q-Krawtchouk polynomial ensemble. We derive the limit shape
and prove the central limit theorem for the fluctuations in the limit when
$n,k$ tend to infinity and $q$ tends to one at comparable rates.
