Number of orbits for finite group action on finite vector space equals number of orbits on dual vector space
Suppose is a finite group, is a finite field, and is a finite-dimensional vector space over . Suppose we have a linear representation of on , i.e., a homomorphism . The contragredient representation is a linear representation of on , i.e., a homomorphism .
The claim is that the number of orbits of under the original -representation equals the number of orbits of under the contragredient representation.