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The Number of Invariant Subspaces Under a Linear Operator on Finite Vector Spaces

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The Number of Invariant Subspaces Under a Linear Operator on Finite Vector Spaces Advances in Mathematics of Communications Web site: http://www.aimSciences.org Volume 5, No. 2, 2011, 407–416 THE NUMBER OF INVARIANT SUBSPACES UNDER A LINEAR OPERATOR ON FINITE VECTOR SPACES This work is dedicated to Professor Adalbert Kerber in deep gratitude and adoration. Harald Fripertinger Institut f¨urMathematik Karl-Franzens Universit¨atGraz Heinrichstr. 36/4, A–8010 Graz, Austria (Communicated by Marcus Greferath) Abstract. 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 n 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. Pn k We show how to determine the polynomial σ(T ) = k=0 σk(T )x ∈ 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 Pr i Pr i of f = i=0 aix ∈ Fq[x] and v ∈ V by fv := 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. Key words and phrases: Invariant subspaces, enumeration, finite field, finite vector space, cyclic vector space, monomially nonisometric linear codes. 407 c 2011 AIMS-SDU 408 Harald Fripertinger Qs ci Let g = i=1 fi be the factorization of g into its irreducible divisors fi with ci ci ∈ N≥1, 1 ≤ i ≤ s. Furthermore, let Vi := {v ∈ V | fi v = 0}, then Vi is a T -invariant subspace of V and s M V = Vi. i=1 This is the primary decomposition of V into its primary components Vi, 1 ≤ i ≤ s. We call a vector space primary if its minimal polynomial is the power of an irreducible polynomial. Denote by Ti the restriction of T to Vi, then Ti is a linear operator on Vi. According to [2] the lattice L(T ) is the direct product of the lattices L(Ti), i.e. s Y L(T ) = L(Ti). i=1 This means that for each U ∈ L(T ) there exists exactly one sequence (U1,...,Us) ∈ Qs i=1 L(Ti), so that U = U1 ⊕· · ·⊕Us. Therefore, knowing the polynomials σ(Ti) = Pdim Vi k Qs k=0 σk(Ti)x we can compute σ(T ) as the product i=1 σ(Ti). Consequently, it is enough to study the lattices L(Ti) of the primary components Vi, 1 ≤ i ≤ s. 2. Cyclic vector spaces For v ∈ V let [v] := Fq[x]v = {fv | f ∈ Fq[x]}. Then [v] is a T -invariant subspace. It is called the cyclic subspace generated by v. If d denotes the degree of the minimal polynomial of v, then [v] = hv, T v, . , T d−1vi and its dimension is equal to d. In general, a vector space U is called cyclic if there exists some v ∈ U, so that U = [v]. If U is a T -invariant vector space and if v belongs to U, then [v] is a subspace of U, thus, [v] is the smallest T -invariant subspace of V containing v. 3. Decomposition of a primary space into cyclic subspaces Let V be primary, i.e. V is an n-dimensional vector space with minimal polyno- c mial f where f is irreducible over Fq and c ∈ N≥1. According to [2] the lattice L(T ) is either simple (which means that the only lattice homomorphisms of L(T ) are isomorphisms or constant mappings) or it is a chain. Moreover, L(T ) is a chain if and only if V is cyclic. Let I(h) denote the ideal generated by h in Fq[x]. In the present situation V can c be seen as an Fq[x]/I(f )-module. It is well known that there exists a decomposition of V as a direct sum of cyclic subspaces, i.e. r M V = Ui, i=1 ti where Ui = [vi] ' Fq[x]/I(f ) and c = t1 ≥ ... ≥ tr ≥ 1. For v ∈ V define the height of both v and [v] by dim[v] h([v]) := h(v) := . deg f Then f h(v) is the minimal polynomial of v. Each v ∈ V has a unique representation v = u1 + ··· + ur with ui ∈ [vi], 1 ≤ i ≤ r, and h(v) = max {h(ui) | 1 ≤ i ≤ r}. We collect properties of a cyclic vector space in t Lemma 1. Let T be a linear operator on V and let U = [v] ' Fq[x]/I(f ), where f is irreducible, t ∈ N≥1, be a cyclic subspace of V . Advances in Mathematics of Communications Volume 5, No. 2 (2011), 407–416 The number of invariant subspaces 409 1. L(T |U ) is the chain U = [v] ⊃ [fv] ⊃ ... ⊃ [f t−1v] ⊃ {0}. 2. Consider some τ ∈ {0, . , t}, then f t−τ is the minimal polynomial of f τ v and of T restricted to [f τ v]. 3. dim U = t deg f. 4. Consider some τ ∈ {1, . , t}. The elements of height τ in U are the elements which belong to [f t−τ v] \ [f t−τ+1v]. If Q := qdeg f , then there are Qτ − Qτ−1 vectors of height τ in U. Each vector of height τ in U generates the cyclic vector space [f t−τ v]. t s 5. Consider a polynomial g ∈ Fq[x] with gcd(g, f ) = f , and an integer τ ∈ N with s + τ ≤ t. Then {gv | v ∈ [f τ v]} = [f s+τ v]. The proof is left to the reader. Lr Consider the decomposition V = i=1[vi] from above. Then (cf. [10]) the species of this decomposition is the vector λ = (λ1, . , λc) of nonnegative integers, where λj is the number of summands [vi] of height j, i.e. λj = |{i ∈ {1, . , r} | h(vi) = j}| . Pc Pc Consequently, λc ≥ 1, j=1 λj = r, and j=1 jλj deg f = dim V = n. In general the decomposition of a primary vector space V as a direct sum of cyclic subspaces is not unique. However two different decompositions of V have the same species. Therefore, λ = (λ1, . , λc) is the species of V and each decomposition of V as a direct sum of cyclic subspaces has the species λ. Lemma 2. Let V be a primary vector space of species λ = (λ1, . , λc), λc 6= 0, and assume that there exists some t ∈ {1, . , c} so that λj = 0 for all j < t. Then c t−s for each u ∈ V with h(u) = s ≤ t and for each g ∈ Fq[x] with gcd(g, f ) = f there exists some v ∈ V with h(v) = t and gv = u. Proof. Each u ∈ V has the unique representation u = u1 + ··· + ur with ui ∈ [vi]. The height of u is equal to s if and only if h(ui) ≤ s for all i ∈ {1, . , r} and there exists some i0 ∈ {1, . , r} so that h(ui0 ) = s. Hence, there exist polynomials c ai gi ∈ Fq[x] so that ui = givi, gcd(gi, f ) = f where ai ≥ ti − s ≥ t − s and t−s bi ai0 = ti0 − s. Consequently, gi =g ˜if f where bi = ai − (t − s) ≥ 0 and t−s bi t−s bi ui =g ˜if (f vi) ∈ g˜if [f vi]. Consider a polynomial g of the formgf ˜ t−s whereg ˜ and f are relatively prime. bi Then by Lemma 1.5 there existsu ˜i ∈ [f vi] so that ui = gu˜i for 1 ≤ i ≤ r. Pr Pr Thus u = i=1 ui = g i=1 u˜i. Moreover h(˜ui) = ti − bi = ti − ai + t − s ≤ Pr ti − (ti − s) + t − s = t and h(˜ui0 ) = t. If we set v = i=1 u˜i, then h(v) = t and gv = u. The next example shows how to generalize the formula for the enumeration of all k-dimensional subspaces to the enumeration of the k-dimensional T -invariant subspaces of V . n Example 1. Let V = Fq , T = idV , then the minimal polynomial of T is f = x − 1, thus c = 1. Each 1-dimensional subspace is a cyclic one, hence hvi = [v] for all (i) n v ∈ V .
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