The number of invariant subspaces under a linear operator on finite vector spaces

Let V be an n-dimensional vector space over the finite field Fq and T a linear operator on V . For each k ∈ {1, . . . , n} we determine the number of k-dimensional T -invariant subspaces of V . Finally, this method is applied for the enumeration of all monomially nonisometric linear (n, k)-codes over Fq . 0. Introduction Let q be a power of a prime p, Fq the finite field with q elements and n a positive integer. Consider V an n-dimensional vector space over Fq, without loss of generality V = Fq , and a linear operator T on V . A subspace U of V is called T -invariant if TU is contained in U . It is well known that the T -invariant subspaces of V form a lattice, the lattice L(T ) of T -invariant subspaces. We show how to determine the polynomial σ(T ) = ∑n k=0 σk(T )x k ∈ Q[x], where σk(T ) is the number of k-dimensional, T -invariant subspaces of V . According to [2] the lattice L(T ) is self-dual, which means that the coefficients of σ(T ) satisfy σk(T ) = σn−k(T ) for 0 ≤ k ≤ n. In the sequel we use basic facts about the decomposition of a vector space into primary components or the decomposition of a primary vector space as a direct sum of cyclic subspaces. The corresponding theory can be found in textbooks on algebra, e.g. in [8, mainly chapter III]. In the final section we apply our method to the enumeration of monomially nonisometric linear codes. 1. V as an Fq[x]-module The Fq-vector space V is a left Fq[x]-module when we define the product fv of f = ∑r i=0 aix i ∈ Fq[x] and v ∈ V by fv := ∑r i=0 aiT v. The polynomial f annihilates v if fv = 0. The monic polynomial of least degree which annihilates v is called the minimal polynomial of v. There exists a monic polynomial g ∈ Fq[x] of least degree which annihilates all vectors in V . It is called the minimal polynomial of T . 2000 Mathematics Subject Classification: Primary: 05E18; Secondary: 47A46.