A Schubert class in the Grassmannian is rigid if the only proper subvarieties representing that class are Schubert varieties. The hyperplane class σ 1 is not rigid because a codimension one Schubert cycle can be deformed to a smooth hyperplane section. In this paper, we show that this phenomenon accounts for the failure of rigidity in Schubert classes. More precisely, we prove that a Schubert class in G(k, n) is not rigid if and only if the partition λ = (λ 1 , . . . , λ k ) defining the class

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... as a part λ i such that n − k ≥ λ i−1 > λ i and λ i = λ i+1 + 1. Under these assumptions on λ, the parts λ i and λ i+1 determine a partial flag isomorphic to one defining a hyperplane class in another Grassmannian G(k , n ). Using a deformation of the hyperplane in G(k , n ), we can deform the Schubert cycle Σ λ . Otherwise, the Schubert class σ λ is rigid. We also show that if the partition λ contains the partition defining a rigid and singular Schubert cycle in some Grassmannian as a sub-partition, then σ λ cannot be represented by a smooth subvariety of G(k, n). More precisely, if λ does not have the form λ 1 = · · · = λ j 1 = n − k, λ i = λ i+1 + 1 for j 1 < i < j 2 and λ i = λ j 2 for i ≥ j 2 , then the Schubert class σ λ cannot be represented by a smooth subvariety of G(k, n).