THE ELEMENTARY DIVISORS OF THE INCIDENCE

MATRICES OF POINTS AND LINEAR SUBSPACES IN ℙ^{n}(𝔽_{p}).

Peter Sin

University of Florida

Abstract. The elementary divisors of the incidence matrices between points and linear subspaces of fixed dimension in ℙ^{n}(𝔽_{p}) are computed.

Introduction

Let V be an (n + 1)-dimensional vector space over 𝔽_{p}. Let L_{r} denote the set of r-dimensional subspaces of V. Then L_{1} is the set of points of the projective space ℙ(V) and L_{n} the set of hyperplanes. The group G = GL(V) acts transitively on each of the sets L_{r}. Between any two of these sets we have an incidence relation given by inclusion of subspaces. This information can be encoded in an incidence matrix, a 0 − 1 matrix which can be read in any commutative ring. Thus, it is natural to ask for the elementary divisors of this matrix as an integer matrix.

The incidence relation can be interpreted as the map

ℤ^{L_{r}} ⟶ ℤ^{L_{1}}  (1)

between the associated ℤG-permutation modules which sends an r-subspace to the (formal) sum of the 1-subspaces it contains. This homomorphism has a finite cokernel and finding the elementary divisors of the incidence matrix is equivalent to finding a cyclic decomposition of the cokernel. The problem falls naturally into the two separate parts of describing the p-torsion and the p′-torsion.

The p′-torsion can be obtained as a corollary of James’ theory [6] of crosscharacteristic modular representations of GL(n, q), where q is a power of p. It is a cyclic group of order \frac{p^{r}−1}{p−1}, the number of 1-subspaces in an r-subspace. To see this consider the map ϵ : ℤ^{L_{1}} ⟶ ℤ sending each 1-subspace to 1. The image of the incidence map (1) is mapped by ϵ onto \frac{p^{r}−1}{p−1}ℤ. The result will therefore follow if we show that the intersection of the image of (1) with Ker ϵ has index a power of p in Ker ϵ. Since Ker ϵ is a pure subgroup of ℤ^{L_{1}}, this is equivalent to the

Typeset by AMS-TEX

Supported by NSF grant DMS9701065

statement that for each prime l ≠ p, the reduction mod l of Ker ϵ is in the image of the reduction mod l of the map (1). It is this last fact which has been proved in [6, Theorem 13.3 and 11.1, Submodule Theorem]. The same argument works with p replaced by q, showing that the cokernel of the map from r-spaces to 1-spaces of a finite vector space over 𝔽_{q} is the product of a cyclic group of order (q^{r} − 1)/(q − 1) with a p-group.

In this paper we concentrate on the p-torsion.

Let d_{i} be the coefficient of i(p − 1) in the expansion of (∑_{j=0}^{p−1} t^{j})^{n+1}. Explicitly,

d_{i} = ∑_{j=0}^{⌊ \frac{i(p−1)}{p} ⌋}(−1)^{j}\binom{n + 1}{j}\binom{n + i(p − 1) − jp}{n},  (2)

where the upper limit of the sum is the integer part of \frac{i(p−1)}{p}. We can now state our main result.

Theorem 1. The p-elementary divisors the incidence matrix between L_{1} and L_{r} are p^{r−i} with multiplicity d_{i} for 1 ≤ i ≤ r − 1.

The case r = n of point-hyperplane incidence was first computed by Black and List [1]. I thank R. Liebler for this reference and for useful discussions.

Our approach is to study ℤG-module structure of the cokernels of the incidence maps. In §1, we recall some well known facts about eigenvalues of incidence matrices. In §2, we localize at p and examine closely the permutation modules and incidence maps over ℤ_{p} and 𝔽_{p}. The principal objects of study are certain ℤ_{p}-forms M_{r} in ℚ_{p}G-modules isomorphic to the nontrivial component of ℚ_{p}^{L_{1}}. The submodule structure of the mod p reductions \overline{M}_{r} of these lattices is the essential ingredient in the proof Theorem 1, which is given in §3.

§1. Incidence maps

Here we collect together some standard facts about incidence matrices in a form convenient for our use later. Let η_{r,s} : ℤ^{L_{r}} ⟶ ℤ^{L_{s}} be the map sending an rsubspace to the (formal) sum of all s-subspaces incident with it. This is obviously a homomorphism of ℤG-modules. Then with respect to the bases L_{r} and L_{s} the matrices of η_{r,s} and η_{s,r} are transposes of each other.

Let x ∈ L_{1}. Then in the equation

η_{r,1}η_{1,r}(x) = ∑_{x′} a_{xx′}x′  (3)

the coefficients are

a_{x,x′} = |\{h ∈ L_{r} | x, x′ ⊆ h\}| = \{ \begin{matrix}[ \begin{matrix}n−1 \\ r−2 \end{matrix} ] & p & if x ≠ x′ \\ [ \begin{matrix}n \\ r−1 \end{matrix} ] & p & if x = x′. \end{matrix}  (4)

In (4) we are using the p-binomial coefficients

[ \begin{matrix}m \\ s \end{matrix} ]_{p} = ∏_{i=1}^{s} \frac{p^{m−i+1} − 1}{p^{i} − 1}  (5)

This the number of s-dimensional subspaces in an m-dimensional vector space over 𝔽_{p}.

Thus, the matrix A = (a_{x,x′}) can be written as

A = p^{r−1}[ \begin{matrix}n−1 \\ r−1 \end{matrix} ]_{p}I + [ \begin{matrix}n−1 \\ r−2 \end{matrix} ]_{p}J,  (6)

where I is the identity and J is the matrix with all entries 1.

It follows that the eigenvalues of η_{r,1}η_{1,r} are p^{r−1}[ \begin{matrix}n−1 \\ r−1 \end{matrix} ]_{p}, with multiplicity |L_{1}|−

1 and [ \begin{matrix}n \\ r−1 \end{matrix} ]_{p}[ \begin{matrix}r \\ 1 \end{matrix} ]_{p}, with mutiplicity one, with 1_{1} as eigenvector.

Let r = n. If we fix bases of ℤ^{L_{1}} and ℤ^{L_{n}} then η_{n,1} is represented by a square matrix. The absolute value of the determinant of such a matrix does not depend on the bases chosen. It follows from (6) that

| det(η_{n,1})| = \frac{p^{n} − 1}{p − 1}.p^{\frac{(n−1)(|L_{1}|−1)}{2}} .  (7)

We consider the exponent of p in (7). Since d_{i} = d_{n−i+1} and ∑_{i=i}^{n} d_{i} = |L_{1}| − 1, we obtain

ν_{p}(det(η_{n,1})) = \frac{(n − 1)(|L_{1}| − 1)}{2} = ∑_{i=1}^{n}(n − i)d_{i}  (8)

The following simple observation will be used frequently. Let r ≥ s ≥ t. Then it is easy to check that we have a commutative diagram:

\xymatrix{ℤ^{L_{r}} \ar[rr]^{[ \begin{matrix}r−t \\ s−t \end{matrix} ]_{p}η_{r,t}} \ar[dr]_{η_{r,s}} &  & ℤ^{L_{t}} \\  & ℤ^{L_{s}} \ar[ur]_{η_{s,t}}}  (9)

§2. Related p-adic and p-modular representations

2.1. The modules Y_{r}.

Let ℤ_{p} denote the ring of p-adic integers and ℚ_{p} the field of p-adic numbers. For a ℤ_{p}-module A we shall use the notations ℚ_{p}A = ℚ_{p}⊗_{ℤ_{p}}A and \overline{A} = 𝔽_{p}⊗_{ℤ_{p}}A = A/pA.

Let ℤ_{p}^{L_{r}} denote the ℤ_{p}G-permutation module with basis L_{r} and let

Y_{r} = {y = ∑_{z∈L_{r}} b_{z}z ∈ ℤ_{p}^{L_{r}} | ∑_{z} b_{z} = 0}  (10)

Since p does not divide |L_{r}| , we have

ℤ_{p}^{L_{r}} = ℤ_{p}1_{r} ⊕ Y_{r},  and 𝔽_{p}^{L_{r}} = 𝔽_{p}1_{r} ⊕ \overline{Y}_{r}.  (11)

We shall continue to use η_{r,s} for the incidence map over ℤ_{p} and we will use \overline{η}_{r,s} for the maps over 𝔽_{p}. Since η_{r,s}(Y_{r}) ⊆ Y_{s} and, consequently, \overline{η}_{r,s}(\overline{Y}_{r}) ⊆ \overline{Y}_{s}, we will use the same notations for these restricted maps.

The following result, which is central to our method, is a reformulation of the well known structure of generalized projective Reed-Muller codes.

Let S_{i} be the degree i(p − 1) component of the graded G-algebra S^{∗}(V)/(V^{p}), the quotient of the symmetric algebra on V by the ideal generated by p-th powers.

Theorem 2 (cf. [2, Theorem A ]). The 𝔽_{p}G-module \overline{Y}_{1} is uniserial with a composition series

0 = W_{n+1} ⊂ W_{n} ⊂ ⋯ ⊂ W_{1} = \overline{Y}_{1}  (12)

such that for each i = 1, …, n the simple factor W_{i}/W_{i+1} is isomorphic to S_{i}.

We observe that the dimension of S_{i} is equal to the number d_{i} in Theorem 1. Theorem 2 also gives the structure of \overline{Y}_{n} by geometric duality; G has an outer

automorphism (inverse-transpose) of order two which interchanges the stablizers of points with the stabilizers of hyperplanes, and interchanges S_{i} with S_{n−i+1}. Therefore, it follows from Theorem 2 that the module \overline{Y}_{n} is uniserial, with the same composition factors a \overline{Y}_{1}, but in the opposite order. Thus we have

0 = U_{0} ⊂ U_{1} ⊂ ⋯ ⊂ U_{n} = \overline{Y}_{n},  with U_{i}/U_{i−1} ≅ S_{i} ≅ W_{i}/W_{i+1}.  (13)

We need in addition the following fact.

Lemma 1. For each r, \overline{Y}_{r} has a unique maximal submodule, with simple quotient isomorphic to S_{r}.

Proof. Let L be a simple module. By Frobenius reciprocity,

Hom_{𝔽_{p}G}(𝔽_{p}^{L_{r}}, L) ≅ Hom_{𝔽_{p}G_{r}}(𝔽_{p}, L)  (14)

where G_{r} is the stabilizer of an r-space. Thus if L is a simple quotient of 𝔽_{p}^{L_{r}}, it contains a nonzero vector fixed by G_{r}. The general theory of modular representations of finite groups of Lie type [3, 7, 10] then says that such a vector exists only in the trivial module and in the simple module of highest weight (p − 1)ω_{r}, where ω_{r} is the r-th fundamental weight (the highest weight of ∧^{r}(V)). Moreover, the fixed vector is the highest weight vector so is unique up to scalars. A routine computation of weights in S_{r} shows that its highest weight is indeed (p − 1)ω_{r}, so L ≅ S_{r}, proving the lemma.

Lemma 2.

(a) Hom_{ℤ_{p}G}(Y_{1}, Y_{r}) ≅ ℤ_{p}, for all r.

(b) ℚ_{p}Y_{1} is simple and each ℚ_{p}Y_{r} has a unique simple summand isomorphic to ℚ_{p}Y_{1}.

(c) ℚ_{p}Y_{n} ≅ ℚ_{p}Y_{1} and Ker η_{r,1} = Ker η_{r,n}. The latter module is a pure submodule of Y_{r}.

Proof. Since the stabilizer of a 1-dimensional subspace of V has two orbits on L_{r}, we have

rank_{ℤ_{p}} Hom_{ℤ_{p}G}(ℤ_{p}^{L_{1}}, ℤ_{p}^{L_{r}}) = 2,  (15)

and (a) now follows from (11). Part (b) follows immediately from (a), as does the first part of (c), since dim_{ℚ_{p}} ℚ_{p}Y_{n} = dim_{ℚ_{p}} ℚ_{p}Y_{1}. Kernels of homomorphisms between ℤ_{p}-lattices are always pure. Therefore, to show Ker η_{r,1} = Ker η_{r,1} it is enough to show the equality of ℚ_{p} Ker η_{r,1} and ℚ_{p} Ker η_{r,1}. But the latter are the kernels of the maps over ℚ_{p} , and their equality follows from (a).

2.2. The modules M_{r}.

Let M_{r} = Y_{r}/ Ker η_{r,1}, (1 ≤ r ≤ n). We have M_{1} = Y_{1} and M_{n} = Y_{n}. Since Ker η_{r,1} is pure, M_{r} is a ℤ_{p}-form in the ℚ_{p}G-module ℚ_{p}Y_{r}/ℚ_{p} Ker η_{r,1} ≅ ℚ_{p}Y_{1} so, by general theory [4], all the mod p reductions \overline{M}_{r} have the same composition factors S_{i}, 1 ≤ i ≤ n as \overline{Y}_{1} and \overline{Y}_{n}.

The commutativity of the diagram (9) implies that for r ≥ s, η_{r,s}(Ker η_{r,1}) ⊆ Ker η_{s,1}. Thus we have induced ℤ_{p}G-module homomorphisms

μ_{r,s} : M_{r} ⟶ M_{s},  and  \overline{μ}_{r,s} : \overline{M}_{r} ⟶ \overline{M}_{s}.  (16)

The next result gives the submodule structure of the \overline{M}_{r}.

Lemma 3. Assume 2 ≤ r ≤ n

(a) \overline{M}_{r} has a unique maximal submodule with simple quotient S_{r}.

(b) The maximal submodule of \overline{M}_{r} is the direct sum of two uniserial modules J_{r}^{+} = Ker \overline{μ}_{r,n}, which maps isomorphically under \overline{μ}_{r,1} to W_{r−1}, and J_{r}^{−} = Ker \overline{μ}_{r,1}, which maps isomorphically under \overline{μ}_{r,n} to U_{r+1}.

Proof. The property of in (a) is inherited from \overline{Y}_{r}. To prove (b), we observe that by (a), the image of the nonzero homomorphism \overline{μ}_{r,1} in \overline{Y}_{1} must be the module W_{r}. It follows that the kernel J_{r}^{−} must have composition factors S_{r−1}, …, S_{1}. Likewise, \overline{μ}_{r,n} has image U_{r} and kernel J_{r}^{+}, with composition factors S_{r+1}, …, S_{n}. Since the kernels have no composition factors in common, it follows that each map is injective when restricted to the kernel of the other. Thus, J_{r}^{−} is isomorphic to a submodule of U_{r}, which from inspection of composition factors must be U_{r−1}. Similiarly J_{r}^{+} ≅ W_{r+1} and the lemma is proved.

It may help to visualize these modules as shown in the picture (for n = 5) below.

\overline{M}_{1}: \begin{matrix}S_{1} \\ S_{2} \\ S_{3} \\ S_{4} \\ S_{5} \end{matrix}  \overline{M}_{2}: \begin{aligned}& S_{2} \\ S_{1} & & S_{3} \\ & & S_{4} \\ & & S_{5} \end{aligned}  \overline{M}_{3}: \begin{aligned}& S_{3} \\ S_{2} & & S_{4} \\ S_{1} & & S_{5} \end{aligned}  \overline{M}_{4}: \begin{aligned}& S_{4} \\ S_{3} & & S_{5} \\ S_{2} \\ S_{1} \end{aligned}  \overline{M}_{5}: \begin{matrix}S_{5} \\ S_{4} \\ S_{3} \\ S_{2} \\ S_{1} \end{matrix}

For notational convenience, we make the conventions that J_{n}^{−} = U_{n−1}, J_{n}^{+} = 0, J_{1}^{−} = 0 and J_{1}^{+} = W_{2}.

Lemma 4. For 1 ≤ s < r ≤ n, we have Ker \overline{μ}_{r,s} = J_{r}^{−}.

Proof. It suffices to show J_{r}^{−} ⊆ Ker \overline{μ}_{r,r−1}. The image of \overline{M}_{r} in \overline{M}_{r−1} has a unique maximal submodule with top composition factor S_{r}, so by Lemma 3 must be J_{r−1}^{+}. Since the latter has no composition factors in common with J_{r}^{−}, by Lemma 3, the result follows.

§3 Proof of Theorem 1

The proof of Theorem 1 will proceed as follows. We shall construct a basis of M_{r} containing a subset {x_{ij}} (1 ≤ i ≤ r − 1, 1 ≤ j ≤ d_{i}) of ∑_{i=1}^{r−1} d_{i} elements such that for each i, the d_{i} elements x_{ij} are mapped under μ_{r,1} into p^{r−i}Y_{1}. This will establish that μ_{r,1} can be represented by a matrix which is the product AD of two square matrices, the matrix D being diagonal with elementary divisors as stated in Theorem 1.

Finally, we shall show det A is a unit, so Theorem 1 will be proved.

Let 2 ≤ r ≤ n. In the isomorphism J_{r}^{−} ≅ U_{r−1}, let K_{r,i} be the submodule of J_{r}^{−} mapping to U_{i}, (1 ≤ i ≤ r − 1). We have K_{r,r−1} = J_{r}^{−} and K_{r,i}/K_{r,i−1} ≅ S_{i}.

Lemma 5. There exists submodules F_{r,i} (1 ≤ i ≤ r − 1) in M_{r} with the properties (1) F_{r,i} maps onto K_{r,i} under the reduction map M_{r} ⟶ \overline{M}_{r};

(2) F_{r,i} has a unique maximal submodule with quotient S_{i}

(3) F_{r,i} ⊆ F_{r,j} whenever i < j.

Proof. We start by making the conventions that K_{r,r} = \overline{M}_{r}, and F_{r,r} = M_{r}. Then (1) and (3) hold trivially for i = r. We will construct modules F_{r,i} for 1 ≤ i ≤ r − 1 which satisfy (1) and (3) by iteration. Suppose F_{r,i+1} has already been constructed satisfying (1) and (3). Let P_{i} be the projective cover of S_{i} in the category of ℤ_{p}G-modules. Consider the following diagram.

\xymatrix{ & P_{i} \ar[ddl] \ar[d] \ar[dr] \\  & K_{r,i} \ar[d]^{∩} \ar[r] & S_{i} \ar[r] & 0 \\ F_{r,i+1} \ar[r] & K_{r,i+1} \ar[r] & 0}  (17)

6

By the projectivity of P_{i} the homomorphism P_{i} ⟶ S_{i} may be lifted to a homomorphism P_{i} ⟶ K_{r,i}, which must be surjective, by the uniseriality of K_{r,i}. Then the composite P_{i} ⟶ K_{r,i} ⊂ K_{r,i+1} can be lifted to a homomorphism P_{i} ⟶ F_{r,i+1}, as shown. Let F_{r,i} be the image of P_{i} in F_{r,i+1}. Then (1) and (3) hold by construction. Finally, we note that for 1 ≤ i ≤ r − 1, F_{r,i} is a quotient of P_{i} and hence (2) holds since P_{i} has this property [4]. The lemma is proved.

We can certainly find d_{i} elements \overline{x}_{ij} of K_{r,i} with the property that their images in S_{i} form a basis. By Lemma 5, we pick preimages x_{ij} ∈ F_{r,i} of these elements. We do this for all i with 1 ≤ i ≤ r − 1, obtaining a set 𝔛_{r} of ∑_{i=1}^{r−1} d_{i} elements whose images \overline{x}_{ij} form a basis of J_{r}^{−}. Therefore, by Nakayama’s Lemma, the set 𝔛_{r} can be extended to a basis of M_{r}. Since x_{ij} ∈ F_{r,i}, the next result shows that this basis will have the property mentioned above.

Lemma 6. μ_{r,1}(F_{r,i}) ⊂ p^{r−i}Y_{1} for 1 ≤ i ≤ r − 1.

Proof. We shall descend inductively starting at i = r − 1. We have a commutative diagram

\xymatrix{F_{r,r−1} \ar[d] \ar[r]^{⊂} & M_{r} \ar[d] \ar[r]^{μ_{r,1}} & Y_{1} \ar[d] \\ K_{r,r−1} \ar[r]^{⊂} & _{\overline{M}_{r}} \ar[r]^{_{\overline{μ}_{r,1}}} & _{\overline{Y}_{1}}}  (18)

Since K_{r,r−1} = J_{r}^{−} = ker \overline{μ}_{r,1}, the commutativity of the diagram shows that μ_{r,1}(F_{r,r−1}) ⊆ pY_{1}, So the result holds for i = r − 1. Assume the lemma holds for F_{r,i}. Then we have the following maps.

\xymatrix{F_{r,i−1} ⊂ F_{r,i} \ar[r]^{μ_{r,1}} & p^{r−i}Y_{1} \ar[r] & p^{r−i}Y_{1}/p^{r−i+1}Y_{1} ≅ \overline{Y}_{1}}  (19)

By property (2) of Lemma 5, the image of F_{r,i} in \overline{Y}_{1} has a unique simple quotient, isomorphic to S_{i}. The image must therefore be zero or W_{i}. Likewise, the image of F_{r,i−1} must be zero or W_{i−1}, but the latter is impossible since W_{i−1} ⊈ W_{i}. Thus, μ_{r,1}(F_{r,i−1}) ⊆ p^{r−i+1}Y_{1}, completing the induction.

Lemma 6 shows that μ_{r,1} is represented using the above basis of M_{r} by a product AD of square matrices, where D is a diagonal matrix with the elementary divisors specified in Theorem 1. Thus ν_{p}(det(D)) = ∑_{i=1}^{r−1}(r − i)d_{i}. Therefore the following computation will conclude the proof of Theorem 1.

Lemma 7. ν_{p}(det(μ_{r,1})) = ∑_{i=1}^{r−1}(r − i)d_{i}.

Proof. By Lemma 4, for all s = 2, …,n we have J_{s}^{−} = Ker \overline{μ}_{s,s−1} so, since the image of 𝔛_{s} in \overline{M}_{s} lies in J_{s}^{−}, we have μ_{s,s−1}(𝔛_{s}) ⊆ pM_{s−1}. Since 𝔛_{s} forms part of a basis of M_{s} it follows that ν_{p}(det(μ_{s,s−1})) ≥ |𝔛_{s}| = ∑_{i=1}^{s−1} d_{i}.

Next, it will be helpful to consider the following commutative diagram of homomorphisms.

[figure: Y_{n}; Y_{n−1}; ⋯; Y_{s}; η_{s,s−1}; Y_{s−1}; ⋯; Y_{1}; Y_{n}; M_{n−1}; ⋯; M_{s}; μ_{s,s−1}; M_{s−1}; ⋯; Y_{1}; _{\overline{Y}_{n}}; _{\overline{M}_{n−1}}; ⋯; _{\overline{M}_{s}}; _{\overline{μ}_{s,s−1}}; _{\overline{M}_{s−1}}; ⋯; _{\overline{Y}_{1}}] (20)

By (9), the composite of the maps in the top row of (20) differs from η_{n,1} by a unit in ℤ_{p}.

Therefore, using (8),

∑_{i=2}^{n}(n−i)d_{i} = ν_{p}(det(η_{n,1})) = ∑_{s=2}^{n} ν_{p}(det(μ_{s,s−1})) ≥ ∑_{s=2}^{n} ∑_{i=1}^{s−1} d_{i} = ∑_{i=2}^{n}(n−i)d_{i}, (21)

so we must have equality throughout. Hence, ν_{p}(det(μ_{s,s−1})) = ∑_{i=1}^{s−1} d_{i} and, using (9) again, we have

ν_{p}(det(μ_{r,1})) = ∑_{s=2}^{r} ν_{p}(det(μ_{s,s−1})) = ∑_{s=2}^{r} ∑_{i=1}^{s−1} d_{i} = ∑_{i=1}^{r−1}(r − i)d_{i}.  (22)

The proof is finished.

Concluding Remarks.

1. If we replace the 𝔽_{p}-vector space V by one over a finite extension 𝔽_{q}, then, as we have already pointed out in the introduction, the problem of determining the elementary divisors of the incidence matrices for r-spaces versus 1-spaces is reduced to finding those which are powers of p. It is known [9, §1] that every conceivable power of p occurs as an elementary divisor. Also, Theorem 2, which is crucial in our proof of Theorem 1 has been generalized to all q in [2].

2. It is natural to consider the same problems for incidences of r-subspaces with ssubspaces. In this direction, little is known about the p-torsion, but when l ≠ p, the ranks of all such incidence maps over a field of characteristic l have been computed in [5], using [6].

3. Although Theorem 1 has been stated as a numerical result, a closer look at the proof reveals very detailed information about the ℤ_{p}G-module structure of the cokernel of η_{r,1}. Indeed, the method used here was partly inspired by the technique of Weyl module filtrations [7, II.4.19] in the representation theory of reductive groups, to which it bears some resemblance. In the context of codes of designs, a precursor of our method can be found in [8, p.73], where the example of points and hyperplanes is worked out for p = 2.

References

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Department of Mathematics, University of Florida, Gainesville, FL 32611, USA e-mail: sin@math.ufl.edu