The structure of the endomorphism ring End_{𝔽_{q}[\overline{End(V)}]}𝔽_{q}[\overline{End (V) /GL (V)}] of a finite

vector space

Nondas E. Kechagias

Abstract. Let V be a finite dimensional vector space over a finite field 𝔽_{q}. The endomorphism ring

End_{𝔽_{q}[\overline{End(V)}]}𝔽_{q}[\overline{End (V) /GL (V)}] is studied. It is a local ring and their structure is described along with its invertible elements.

Definitions-Results

Let V be a vector space of dimension n over 𝔽_{q}. Here q is a power of a prime p. Let End (V) be its set of endomorphisms. We view End (V) as a monoid with respect to composition of linear maps. It admits a left and a right action by itself. Moreover, GL (V) acts on the left g · f or right f · g for g ∈ GL (V) and f ∈ End (V).

Definition 1. i) Let End (V) /GL (V) denote the set of orbits under the right action of GL (V) on End (V). We write \overline{f}. ii) Let 𝔽_{q}[End (V)] be the associated monoid algebra.

iii) Let 𝔽_{q}[\overline{End (V)}] = 𝔽_{q}[End (V)]/ ⟨0_{V} ⟩. Here 0_{V} stands for the zero map.

In iii), 𝔽_{q}[\overline{End (V)}] is generated by the set End (V) − \{0_{V} \} and its neutral element ∑_{f} 0f is identified with the zero map.

2000 Mathematics Subject Classification. Primary 15A04, 13H99; Secondary 13A50 .

Key words and phrases. Finite vector space, Endomorphism rings, Orbit spaces, Invariant orbits.

This paper is in final form and no version of it will be submitted for publication elsewhere.

Definition 2. i) Let 𝔽_{q}[End (V) /GL(V)] be the corresponding vector space.

ii) Let 𝔽_{q} [\overline{End (V) /GL (V)}] = 𝔽_{q}[End (V) /GL(V)]/ ⟨\overline{0_{V}} ⟩

Here again the zero orbit is identified with the neutral element

∑_{\overline{f}} 0\overline{f}.

The ring End (V) acts on the left of End (V) /GL (V) by

h · \overline{f} = \overline{hf}

Here f, h ∈ End (V). Because of the left action above 𝔽_{q}[End (V) /GL(V)] becomes an 𝔽_{q} [End (V)]-module. Moreover, 𝔽_{q} [\overline{End (V) /GL (V)}] is an 𝔽_{q} [End (V)]-module:

h (\overline{f} + ⟨\overline{0}_{V} ⟩) = \overline{hf} + ⟨\overline{0}_{V} ⟩

And it is also an 𝔽_{q} [\overline{End (V)}]-one:

(h + ⟨0_{V} ⟩) (\overline{f} + ⟨\overline{0}_{V} ⟩) = \overline{hf} + ⟨\overline{0}_{V} ⟩

Lemma 3. The 𝔽_{q} [End (V)]-module 𝔽_{q}[\overline{End (V) /GL(V)}] is isomorphic with the 𝔽_{q} [\overline{End (V)}]-one.

We are interested in studying the ring of 𝔽_{q}[End (V)]-endomorphisms of 𝔽_{q}[End (V) /GL(V)] and the corresponding one of 𝔽_{q} [\overline{End (V)}]-

endomorphisms of 𝔽_{q}[\overline{End (V) /GL(V)}].

In [2] we proved the next Theorem. Let the Dickson algebra, D (V), be the ring of GL (n, 𝔽_{p}) invariants of an exterior tensor a polynomial algebra:

D (V) := (E (x_{1}, ..., x_{n}) ⊗ P [y_{1}, ..., y_{n}])^{GL(n,𝔽_{p})}

For topological reasons, it is an algebra over the Steenrod algebra A, the algebra of Steenrod cohomology operations. \overline{D (V)} stands for its augmentation ideal. U stands for the category of unstable A-modules. End_{U} (\overline{D (V)}) stands for A-linear endomorphisms of degree 0.

Theorem 4. End_{U} (\overline{D (V)}) ≅ End_{𝔽_{p}[End(V)]} (𝔽_{p}[\overline{GL (V) \End (V)}])

In the last Theorem right orbits are considered.

In this work the following Theorem is proved.

Theorem 18 End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{End (V) /GL (V)}]) is a local ring

with basis \{\overline{ψ}_{k}|k = 1, ..., n\} and \overline{ψ}_{k} is a non-invertible element for 1 ≤ k < n.

Proposition 5

End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{GL (V) ∖ End (V)}]) ≅ End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{End (V) /GL (V)}])

The structures of the rings above are described in proposition 15. Finally,

Corollary 23 Let n = n_{l}p^{l}+...+n_{0} > p^{l} be its p-adic decomposition, then the order of an automorphism of

End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{End (V) /GL (V)}])

is a divisor of (q − 1) p^{l+1}. Moreover, |Aut (𝔽_{q}[\overline{End (V) /GL (V)}])| = (q − 1) q^{n−1}.

Although, End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{End (V) /GL (V)}]) is a local ring, End_{𝔽_{q}[End(V)]} (𝔽_{q}[End (V) /GL (V)]) is not.

The bulk of the work is contained in the next section which is the main part. There is one more complementary section for the benefit of the reader: "Counting subspaces of V ". In that section we include all technical results needed in the main section.

1. End (V) /GL (V) and subspaces of V

We start with a basis of V. Given a subspace W ≤ V, a subspace W^{c} is defined uniquely in the next section such that W ⊕ W^{c} = V (proposition 33).

Let f : V → V and Im f = Z = ⟨z_{1}, ...z_{k}⟩. Let W = ⟨w_{1}, ..., w_{k}⟩ such that f (W) = Z.

Let Z^{c} = ⟨z_{1}^{c}, ...z_{n−k}^{c}⟩ and W^{c} = ⟨w_{1}^{c}, ..., w_{n−k}^{c}⟩ be their direct complements

f : W ⊕ W^{c} → Z

Let I_{(W,Z)} : V → V be the map given by

I_{(W,Z)} (∑ a_{i}w_{i} + ∑ b_{j}w_{j}^{c}) = ∑ a_{i}z_{i}

If f (w_{i}) = ∑ b_{ij}z_{j}, let g ∈ GL_{n} such that g (w_{i}) = ∑ b_{ij}w_{j}. Then

f = I_{(W,Z)}g ⇒ \overline{f} = Ī_{(W,Z)}

Let

I_{Z} : Z ⊕ Z^{c} → V

be the linear map given by I_{Z} (z + z^{c}) = z. For all g ∈ GL(V), Im (I_{Z}g) = Z and ker (gI_{Z}) = Z^{c}. Of course, I_{Z} depends also on Z^{c}, but the right orbit \overline{I_{Z}} does not. There exists a g′ ∈ GL (V) such that I_{Z} = I_{(W,Z)}g′.

Moreover,

\overline{f} = Ī_{Z} ⇔ Im f = Z

For left orbits of the form gf for g ∈ GL (V), the discussion above implies

l\overline{f} =^{l} Ī_{W} ⇔ ker f = W^{c}

Proposition 5. The transpose induces an 𝔽_{q}[\overline{End (V)}]-isomorphism

𝔽_{q}[\overline{End (V) /GL (V)}] ≅ 𝔽_{q}[\overline{GL (V) \End (V)}]

There is a relation between orbits and subspaces of V.

Definition 6. i) Let S (V) = {W |W ≤ V } and 𝔽_{q}[S (V)] the vector space over 𝔽_{q} on S (V).

ii) Let 𝔽_{q}[\overline{S (V)}] = 𝔽_{q}[S (V)]/ ⟨0⟩.

End (V) acts on the left of S (V) by f · W = f (W). Now 𝔽_{q}[S (V)] becomes an 𝔽_{q}[End (V)]-module and 𝔽_{q}[\overline{S (V)}] becomes an 𝔽_{q}[\overline{End (V)}]- one.

Lemma 7. 𝔽_{q}[\overline{S (V)}] is an 𝔽_{q}[\overline{End (V)}]-directly indecomposable module.

Proof. For every W and U there exists an endomorphism f such that f (W) = U or f (U) = W.

Corollary 8. End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{S (V)}]) is a directly indecomposable ring and 0 and 1 are its only idempotent elements.

Proof. This is an application of theorem 7.2.5 in ([1]).

Theorem 9. End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{S (V)}]) is a local ring and its noninvertible elements are exactly its nilpotent elements.

Proof. Since 𝔽_{q}[\overline{S (V)}] is of finite length as an 𝔽_{q}[\overline{End (V)}]-module, the assertion is an application of Theorem 7.2.7 in ([1]).

For the rest of this work, we further investigate the relation between

End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{S (V)}]) and End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{End (V) /GL (V)}])

There is a bijection between S (V) and End (V) /GL (V) given by the image of a representative of the orbit \overline{I_{W}}. The next corollary follows.

Corollary 10.

End_{𝔽_{q}[End(V)]} (𝔽_{q}[End (V) /GL (V)]) ≅ End_{𝔽_{q}[End(V)]} (𝔽_{q}[S (V)])

End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{End (V) /GL (V)}]) ≅ End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{S (V)}])

Definition 11. i) Let ϕ_{i} : 𝔽_{q}[S (V)] → 𝔽_{q}[S (V)] be the 𝔽_{q}[End (V)]- map given by ϕ_{i} (V) = ∑_{dim U=i} U. Here i = 0, ..., n.

ii) Given ϕ ∈ 𝔽_{q}[S (V)], let \overline{ϕ} be its image in 𝔽_{q}[\overline{S (V)}].

Since g · ϕ_{i} = ϕ_{i} for all g ∈ GL (V), ϕ_{i} is a GL (V)-invariant map.

Proposition 12. i) End_{𝔽_{q}[End(V)]} (𝔽_{q}[S (V)]) ≅< ϕ_{i}|i = 0, ..., n > has dimension n+1 as a vector space and its neutral element is ∑_{0≤i≤n} ϕ_{i}.

ii) End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{S (V)}]) ≅< \overline{ϕ}_{i}|i = 1, ..., n > has dimension n

as a vector space and its neutral element is \overline{ϕ}_{0} = 0.

Proof. i) Let ϕ be an 𝔽_{q}[End (V)]-map: f · ϕ(W) = ϕ (f(W)). Suppose that ϕ has been defined on V, then ϕ(W) = ϕ(f(V)) = fϕ(V) for some f such that f (V) = W. Let ϕ be given by

ϕ (V) = ∑_{i} ∑_{j} λ_{ij}W_{ij}

Here dim (W_{ij}) = i for all j.

Since gϕ =ϕ for all g ∈ GL (V), ϕ is a GL (V)-invariant map. Thus λ_{i_{0}j_{0}} = λ_{i_{0}j} for all j. Now

ϕ (V) = ∑_{i} λ_{i} ∑_{j} W_{ij}

and ∑_{j} W_{ij} is a GL (V)-invariant element as well. Therefore

∑_{j} W_{ij} = ϕ_{i} (V) and ϕ (V) = ∑_{i} λ_{i}ϕ_{i} (V)

ii) As above.

Definition 13. Let ψ_{k} : 𝔽_{q}[End (V) /GL (V)] → 𝔽_{q}[End (V) /GL (V)] be the 𝔽_{q}[End (V)]-linear map given by

ψ_{k} (\overline{I_{V}} ) = ∑_{dim W =k} \overline{I_{W}} for k = 0, ..., n

and ψ_{k} (\overline{f}) = fψ_{k} (\overline{I_{V}} ) for all orbits.

Note that ψ_{k} is a GL (V)-invariant map.

Corollary 14. i) End_{𝔽_{q}[End(V)]} (𝔽_{q}[End (V) /GL (V)]) ≅< ψ_{k}|k = 0, ...n >.

ii) End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{End (V) /GL (V)}]) ≅< \overline{ψ}_{k}|k = 1, ...n >.

The structure of the rings above is described in the following proposition.

Proposition 15. i) ψ_{n} = \overline{I_{V}} = 1.

ii) Let n > l ≥ k > 0, then ψ_{k}ψ_{l} = ψ_{max(0,k+l−n)} = ψ_{l}ψ_{k}.

Proof. ii) Let W = ⟨w_{1}, ...., w_{k}⟩ and Z = ⟨z_{1}, ..., z_{l}⟩ with l > k. Let Z^{c} = ⟨z_{1}^{c}, ..., z_{n−l}^{c}⟩ then

w_{i} = ∑_{1≤s≤l} a_{is}z_{s} + ∑_{1≤t≤n−l} b_{it}z_{t}^{c} for 1 ≤ i ≤ k

I_{Z} (w_{i}) = ∑_{1≤s≤l} a_{is}z_{s}

Let ψ_{k}ψ_{l} :

ψ_{k}ψ_{l} (\overline{I_{V}} ) = ψ_{k} ∑_{Z} \overline{I_{Z}} = ∑_{Z} ψ_{k}\overline{I_{Z}} = ∑_{Z} I_{Z} ∑_{W} \overline{I_{W}} = ∑_{Z} ∑_{W} \overline{I_{Z} (W)}

First case: k + l ≤ n. If I_{Z} (W) = \left⟨{ \atop ¯0}\right⟩, then a_{is} = 0 for all i and s. Thus W ≤ Z^{c} and there exist #_{n−l} (k) such subspaces for each Z. In total #_{n} (l) #_{n−l} (k). Second case: k +l > n. Then 0 < k +l −n ≤ dim (Z ∩ W). Let Y such that dim (Y) = m = k + l − n, then there exist Z and W such that I_{Z} (W) = Y ≤ Z. There exist #_{n−m} (l − m) such subspaces Z and for each such Z there exist a unique W such that W = Y ⊕ Z^{c}. In total #_{n−m} (l − m).

Given Y such that max (0, k + l − n) < dim (Y) = r < k, let I_{Z} (W) = Y ≤ Z. There exist #_{n−r} (l − r) such subspaces Z. We consider a basis for such a subspace W.

w_{i} = ∑_{1≤s≤l} a_{is}z_{s} + ∑_{1≤t≤n−l} b_{it}z_{t}^{c} for 1 ≤ i ≤ r

w_{i} = ∑_{1≤t≤n−l} b_{it}z_{t}^{c} for 1 + r ≤ i ≤ k

Since ⟨w_{r+1}, ...w_{k}⟩ ≤ Z^{c}, there exist #_{n−l} (k − r) such subspaces. On the other hand there exist q^{n−l−(k−r)} choices for {b_{i1}, ...b_{i,n−l}} for 1 ≤ i ≤ r. In total there exist #_{n−l} (k − r) q^{(n−l−(k−r))r} subspaces W.

Now let I_{Z} (W) = W′ ≤ Z, then W′ = ⟨w′_{1}, ..., w′_{k}⟩ such that

w_{i} = w′_{i} + a_{i1}z_{1}^{c} + ... + a_{in−l}z_{n−l}^{c}

It follows that there exist q^{(n−l)k} subspaces W′.

Finally,

ψ_{k}ψ_{l} = ψ_{max(0,k+l−n)}

Let k ≤ l. Let m = max (0, k + l − n). In a completely analogous way, there exist #_{n−m} (k − m) #_{n−k} (l − m) subspaces Y of dimension m.

Let I_{W} (Z) = Y ≤ W, dim (Y) = r ≤ k. There exist #_{n−r} (k − r) subspaces W. We consider a basis for such a subspace Z.

z_{i} = ∑_{1≤s≤k} a_{is}w_{s} + ∑_{1≤t≤n−k} b_{it}w_{t}^{c} for 1 ≤ i ≤ r

z_{i} = ∑_{1≤t≤n−l} b_{it}w_{t}^{c} for 1 + r ≤ i ≤ l

Since ⟨z_{r+1}, ...z_{l}⟩ ≤ W^{c}, there exist #_{n−k} (l − r) such subspaces. On the other hand there exist q^{n−k−(l−r)} choices for {b_{i1}, ...b_{i,n−k}} for 1 ≤ i ≤ r. In total there exist #_{n−k} (l − r) q^{(n−k−(l−r))r} subspaces Z.

Finally,

ψ_{k}ψ_{l} = ψ_{l}ψ_{k} = ψ_{max(0,k+l−n)}

Proposition 16. i) The endomorphism ψ is invertible iff it is of

the form a_{n} (ψ_{n} + ∑_{t=0}^{n−1} a_{t}ψ_{t}) for a_{n} ≠ 0 mod p and 1+ ∑_{0}^{n−1} a_{i} ≠ 0 mod p.

ii) The endomorphism \overline{ψ} is invertible iff it is of the form a_{n} (\overline{ψ}_{n} + ∑_{t=1}^{n−1} a_{t}\overline{ψ}_{t})

for a_{n} ≠ 0 mod p. If a_{n}^{−1} (\overline{ψ}_{n} + ∑_{t=1}^{n−1} b_{t}\overline{ψ}_{t}) is its inverse, then their co-

efficients satisfy the following relation: (1)

∑_{2i≥j≥i} a_{n−j+i}b_{n−i} + ∑_{j≥t≥1} a_{n−j}b_{n−j+t} + ∑_{2i−1>j−1≥i} a_{n−i}b_{n−j+i} = 0 mod p

Proof. i) This is an application of last proposition. The matrix of our linear system is of the form

( \begin{matrix}1 + ∑_{0}^{n−1} a_{i} & ∑_{0}^{n−1} a_{i} & ∑_{0}^{n−2} a_{i} & ⋯ & a_{1} + a_{0} & a_{0} \\ 0 & 1 & a_{n−1} & ⋯ & a_{2} & a_{1} \\ & & 1 & ⋯ & a_{3} & a_{2} \\ ⋮ & & ... & & & ⋮ \\ & & & 1 & a_{n−1} & a_{n−2} \\ & & & & 1 & a_{n−1} \\ 0 & & ⋯ & & 0 & 1 \end{matrix} )

ii) In this case the matrix decomes

( \begin{matrix}1 & a_{n−1} & ⋯ & a_{2} & a_{1} \\ 0 & 1 & ⋯ & a_{3} & a_{2} \\ ⋮ & ... & & & ⋮ \\ & & 1 & a_{n−1} & a_{n−2} \\ & & & 1 & a_{n−1} \\ 0 & ⋯ & & 0 & 1 \end{matrix} )

Example 17. Let p = 3 and n = 3. i) ψ_{3} + ψ_{2} + ψ_{1} and ψ_{2} + ψ_{1} are non-invertible elements. ii) (\overline{ψ}_{2} + \overline{ψ}_{1}) is non-invertible and (\overline{ψ}_{2} + \overline{ψ}_{1})^{3} = \overline{ψ}_{0} = 0.

Theorem 18. End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{End (V) /GL (V)}]) is a local ring

with basis \{\overline{ψ}_{k}|k = 1, ..., n\} and \overline{ψ}_{k} is a non-invertible element for 1 ≤

k < n. Each ∑_{t=1}^{n−1} a_{t}\overline{ψ}_{t} is a nilpotent element.

Corollary 19. End_{𝔽_{q}[End(V)]} (𝔽_{q}[End (V) \GL (V)]) is not a local ring.

For the rest of this section, we study the corresponding group of automorphisms.

Definition 20. Let i = 1, ..., n. For k_{i}, a_{i} ∈ ℕ, let [k_{n}, ..., k_{1}] denote

\frac{(∑ k_{i})!}{∏ k_{i}!} ∏ a_{i}^{k_{i}}

Proposition 21. Let

A = ( \begin{matrix}a_{n} & a_{n−1} & ⋯ & & a_{1} \\ 0 & a_{n} & ⋯ & & a_{2} \\ 0 & 0 & a_{n} \\ ⋮ & & & ... & ⋮ \\ 0 & & ⋯ & 0 & a_{n} \end{matrix} ) and A^{m} ( 0 ⋮ 0 1 ) = (c_{s}^{(m)})_{1≤s≤n}

Then c_{s}^{(m)} is the sum of all [k_{n}, ..., k_{1}] such that ∑ k_{i} (n − i) = n − s and ∑ k_{i} = m. Here m > 1. Moreover,

(1, ..., 1) ( \begin{matrix}a_{n} & a_{n−1} & ⋯ & & a_{1} \\ a_{1} & a_{n} & ⋯ & & a_{2} \\ a_{2} & a_{1} & a_{n} \\ ⋮ & & & ... & ⋮ \\ a_{1} & & ⋯ & a_{n−1} & a_{n} \end{matrix} )^{m} ( 0 ⋮ 0 1 ) = (a_{n} + ... + a_{1})^{m}

Proof. We proceed by induction on m > 1. Let [k′_{n}, ..., k′_{1}] ∈ c_{s}^{(m+1)} such that ∑ k′_{i} (n − i) = n − s and ∑ k′_{i} = m + 1. For each i such that k′_{t} > 0, let

∑_{i≠t} k′_{i} (n − i) + (k′_{t} − 1) (n − t) = n − s − (n − t) and ∑ k′_{i} = m

Then

∑_{t=n,k′_{t}>0}^{n−s} a_{t} [k′_{n}, ..., k′_{t+1}, k′_{t} − 1, k′_{t−1}, ..., k_{1}] =

∑_{t=n,k′_{t}>0}^{n−s} \frac{m!k′_{t}}{∏ k′_{i}!} ∏ a_{i}^{k′_{i}} = [k′_{n}, ..., k′_{1}]

Each [k′_{n}, ..., k′_{t+1}, k′_{t} − 1, k′_{t−1}, ..., k_{1}] ∈ c_{n−(n−s+t)}^{(m)}. Thus [k′_{n}, ..., k′_{1}] is a summand in

(a_{n}, ..., a_{s}) ( c_{s}^{(m)} ; ⋮ ; c_{n}^{(m)} )

The second assertion is obvious.

Corollary 22. i) Let m > 1 and n > m. Then

A^{m} ( 0 ⋮ 0 1 ) ≠ ( 0 ; ⋮ ; 0 ; a_{n}^{m} ) mod p

ii) Let m = m_{l}p^{l} + ... + m_{0} ≥ n > p^{l}, then

\binom{m}{p^{l}}a_{n}^{m−p^{l}}a_{n−1}^{p^{l}} = m_{l}a_{n}^{m−p^{l}}a_{n−1}^{p^{l}} mod p

is a summand in c_{n−p^{l}}^{(m)}.

Proof. i) Since n > m, a_{n−1}^{m} is a summand in c_{n−m}^{(m)}.

ii) This is a consequence of the formula

\binom{m_{l}p^{l} + ... + m_{0}}{p^{l}} = \binom{m_{l}}{1} mod p

and the fact that n > p^{l}.

Corollary 23. Let n = n_{l}p^{l} + ... + n_{0} be its p-adic decomposition, then the order of an invertible element of

End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{End (V) /GL (V)}])

is a divisor of (q − 1) p^{l+1}.

Proof. If a_{n} ≠ 1, then a_{n}ψ_{n} + ∑_{t=1}^{n−1} a_{t}ψ_{t} has order a multiple of

q − 1. Let us consider ψ_{n} + ∑_{t=1}^{n−1} a_{t}ψ_{t}. The coefficients of

(ψ_{n} + ∑_{t=1}^{n−1} a_{t}ψ_{t})^{m}

are given by the matrix

( \begin{matrix}1 & a_{n−1} & ⋯ & & a_{1} \\ 0 & 1 & ⋯ & & a_{2} \\ 0 & 0 & 1 \\ ⋮ & & & ... & ⋮ \\ 0 & & ⋯ & 0 & 1 \end{matrix} )^{m} ( 0 ⋮ 0 1 )

The assertion follows from corollary 22.

Corollary 24. The order of the group of automorphisms of

End_{𝔽_{q}[\overline{End(V)}]} (𝔽_{q}[\overline{End (V) /GL (V)}])

is (q − 1) q^{n−1}.

2. Counting subspaces of V

We include this section for completeness and we do not claim originality. The book of Roman is an excellent companion [3].

Definition 25. Let #_{n} (k) denote the number of k-dimensional subspaces in V of an n-dimensional vector space over 𝔽_{q}.

Lemma 26. #_{n} (k) = ( ∏_{0}^{k−1} (q^{n−i} − 1)) / ( ∏_{0}^{k−1} (q^{k−i} − 1)) ≡ 1 mod p.

Proof. Let W be a subspace of dimension k. W contains q^{k} − 1 non-zero vectors. Given one such a vector u_{1} there are q^{k} − q linearly independent vectors with u_{1} in W. Then given two linearly independent vectors u_{1} and u_{2} there are q^{k} − q^{2} linearly independent vectors in W.

Thus W contains ( ∏_{0}^{k−1} (q^{k} − q^{i})) sets of size k of linearly independent

vectors. This is actually the size of GL(W). The number #_{n} (k) equals the number of k-tuple linearly independent vectors in V divided by the

number ( ∏_{0}^{k−1} (q^{k} − q^{i})). Thus

#_{n} (k) = ( ∏_{0}^{k−1} (q^{n} − q^{i})) / ( ∏_{0}^{k−1} (q^{k} − q^{i}))

Definition 27. Let S_{k} be the following set

\{ \begin{matrix}\{(0, ..., 0, 1, a_{l_{1},l_{1}+1}, ..., a_{l_{1},n}) , ..., (0, ..., 0, 1, a_{l_{k},l_{k}+1}, ..., a_{l_{k},n})\} | \\ 1 ≤ l_{1} < ... < l_{k} ≤ n, a_{l_{i},l_{j}} = 0, a_{l_{i},l_{i}+t} ∈ 𝔽_{q} for 1 ≤ i < j ≤ k, 1 ≤ t ≤ n − l_{i} \end{matrix} \}

and 𝔽_{q}^{n} denote the standard n-dimensional vector space over 𝔽_{q}.

There exists a bijection between S_{k} and the set of matrices of rankk in reduced echelon form.

Proposition 28. There exists a bijection between S (𝔽_{q}^{n}) and ⋃_{0}^{n} S_{k}.

Proof. Each set {(0, ..., 0, 1, a_{l_{1},l_{1}+1}, ..., a_{l_{1},n}) , ..., (0, ..., 0, 1, a_{l_{k},l_{k}+1}, ..., a_{l_{k},n})} is linearly independent. Two different such sets provide different subspaces. Now the claim follows.

Lemma 29. Let W be a subspace of dimension k and Y ≤ W of dimension r. There are

q^{(l−r)(k−r)} ∏_{0}^{l−r−1} (\frac{q^{k} − q^{i}}{q^{l−r} − q^{i}} ) = q^{(l−r)(k−r)}#_{n−k} (l − r)

different subspaces W′ of dimension l ≤ k of V such that W ∩W′ = Y.

Proof. We start with a basis of Y and extend it into a basis of W. Let A (W, Y) denote the number of such subspaces W′ of V such that W ∩ W′ = Y. Given a basis of Y, we count the number A (W, Y).

There are ( ∏_{0}^{l−r−1} (q^{l−r} − q^{i})) bases for each such subspace. The total

number of subspaces of dimension k − r is #_{n} (k − r) and each one

contains ( ∏_{0}^{k−r−1} (q^{k−r} − q^{i})) bases. On the other hand V contains

( ∏_{0}^{k+l−r−1} (q^{n} − q^{i})) linearly independent subsets of size l+k−r. Given

a basis of W/Y and a basis of W′, such a subset of V is obtained. Thus

#_{n} (k − r) ∏_{0}^{k−r−1} (q^{k−r} − q^{i}) A (W, Y) ∏_{0}^{l−r−1} (q^{l−r} − q^{i}) = ∏_{0}^{k+l−r−1} (q^{n} − q^{i})

Corollary 30. Let W be a subspace of dimension k ≥ n − k. There are

q^{k(n−k)}

different subspaces W′ of V such that W ⊕ W′ = V.

Corollary 31.

( ∏_{0}^{k−1} (q^{n} − q^{i})) / ( ∏_{0}^{k−1} (q^{k} − q^{i})) = p^{kn− \frac{k(k−1)}{2}} ∑_{1≤l_{1}<...<l_{k}≤n} p^{− ∑_{1}^{k} l_{i}}

Proof. The right hand side of the equation above is the cardinality of S_{k}.

Lemma 32. i) #_{n} (k) = #_{n} (n − k). ii) #_{n} (k) = #_{n−1} (k − 1) + p^{k}#_{n−1} (k).

The next proposition is an application of last lemma and proposition 28.

Proposition 33. There is a bijection

Φ : U_{k} → U_{n−k}

such that W ⊕ Φ (W) = V.

References

[1] F. Kasch, Modules and Rings, ed. London Mathematical Society Monograph No.17 (Academic Press, 1982).

[2] Nondas Kechagias, A Steenrod-Milnor action ordering on Dickson invariants, preprint 2008, 16 pages.

[3] S. Roman, Advanced Linear Algebra, G.T.M. No.135 (Springer Verlag, 1992).

Department of Mathematics, University of Ioannina, Ioannina, 45110, Greece

E-mail address: nkechag@uoi.gr
URL: http://www.math.uoi.gr/nondas_k