Cameron-Liebler sets of k-spaces in PG(n, q)
A. Blokhuis
∗, M. De Boeck
†, J. D’haeseleer
†Abstract
Cameron-Liebler sets ofk-spaces were introduced recently in [13]. We list several equivalent definitions for these Cameron-Liebler sets, by making a generalization of known results about Cameron-Liebler line sets inPG(n, q)and Cameron-Liebler sets ofk-spaces inPG(2k+ 1, q).
We also present some classification results.
Keywords: Cameron-Liebler set, Grassmann graph.
MSC 2010 codes: 05B25, 51E20, 05E30, 51E14, 51E30.
1 Introduction
In [5] Cameron and Liebler introduced specific line classes inPG(3, q)when investigating the orbits of the projective groupsPGL(n+ 1, q). These line setsLhave the property that every line spread S in PG(3, q) has the same number of lines in common with L. A lot of equivalent definitions for these sets of lines are known. An overview of the equivalent definitions can be found in [10, Theorem3.2].
After a large number of results regarding Cameron-Liebler sets of lines in the projective space PG(3, q), Cameron-Liebler sets of k-spaces in PG(2k+ 1, q) [26], and Cameron-Liebler line sets in PG(n, q) [10] were defined. In addition, this research started the motivation for defining and investigating Cameron-Liebler sets of generators in polar spaces [8] and Cameron-Liebler classes in finite sets [9]. In fact Cameron-Liebler sets can be introduced for any distance-regular graph. This has been done in the past under various names: boolean degree1 functions, completely regular codes of strength0 and covering radius1, ... We refer to the introduction of [13] for an overview.
Note that the definitions do not always coincide, e.g. for polar spaces.
One of the main reasons for studying Cameron-Liebler sets is that there are several equivalent definitions for them, some algebraic, some geometrical (combinatorial) in nature. In this paper we investigate Cameron-Liebler sets ofk-spaces inPG(n, q). In Section 2 we give several equivalent definitions for these Cameron-Liebler sets ofk-spaces. Several properties of these Cameron-Liebler sets are given in the third section.
The main question, independent of the context where Cameron-Liebler sets are investigated, is always the same: for which values of the parameterxdo there exist Cameron-Liebler sets and which examples correspond to a given parameterx.
For the Cameron-Liebler line sets, classification results and non-trivial examples were discussed in [4,5,7,10,12,15,16,17,18,22,23,25]. The strongest classification results are given in [17,23], the latter of which proves that there exists a constantc >0so that there are no Cameron-Liebler line sets in PG(3, q) with parameter 2 < x < cq4/3. In [4, 6, 7, 10, 12, 16] the constructions of two non-trivial Cameron-Liebler line sets with parameter x= q22+1 and x= q22−1 were given.
Classification results for Cameron-Liebler sets of generators in polar spaces were given in [8] and for Cameron-Liebler classes of sets, a complete classification was given in [9]. Regarding the Cameron- Liebler sets ofk-spaces inPG(2k+ 1, q), the classification results are described in [21, 26].
If q ∈ {2,3,4,5} a complete classification is known for Cameron-Liebler sets of k-spaces in PG(n, q), see [13]. There the authors show that the only Cameron-Liebler sets in this context
∗Address: Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, the Netherlands
Email address: a.blokhuis@TUE.nl, Website: http://www.win.tue.nl/∼aartb/
†Address: Department of Mathematics, UGent, Krijgslaan 281, {S25,S8}, 9000 Gent, Flanders, Belgium Email address: {maarten.deboeck,jozefien.dhaeseleer}@ugent.be
are the trivial Cameron-Liebler sets. In the last section, we use the properties from Section3 to give the following classification result: there is no Cameron-Liebler set of k-spaces in PG(n, q), n >3k+ 1, with parameterxsuch that2≤x≤qn2−k
2
4−3k4−32(q−1)k
2
4−k4+12p
q2+q+ 1.
2 The characterization theorem
Note first that we will always work with projective dimensions and that vectors are regarded as column vectors. LetΠk be the collection of k-spaces inPG(n, q) for0≤k≤n and letA be the incidence matrix of the points and the k-spaces of PG(n, q): the rows of A are indexed by the points and the columns by thek-spaces.
We defineAi as the adjacency matrix of the relationRiwith
Ri={(π, π0)|π, π0∈Πk,dim(π∩π0) =k−i}, 0≤i≤k+ 1.
These relationsR0, R1, . . . , Rk+1form the Grassmann association schemeJq(n+ 1, k+ 1). Remark thatA0=I andPk+1
i=0 Ai =J whereI and J are the identity matrix and all-one matrix respec- tively. We denote the all-one vector byj. Note that the Grassmann graph fork-spaces inPG(n, q) has adjacency matrixA1.
It is known that there is an orthogonal decompositionV0⊥V1⊥ · · · ⊥Vk+1ofRΠk in maximal common eigenspaces ofA0, A1, . . . , Ak+1. In the following lemmas and theorems, we denote the disjointness matrixAk+1 also byK since the corresponding graph is a q-Kneser graph. For more information about the Grassmann schemes we refer to [2, Section9.3] and [19, Section9].
We will use theGaussian binomial coefficient a b
q fora, b∈Nand prime powerq≥2:
a b
q
= (qa−1)· · ·(qa−b+1−1) (qb−1)· · ·(q−1) . The Gaussian binomial coefficienta
b
q is equal to the number ofb-spaces of the vector spaceFaq, or in the projective context, the number of(b−1)-spaces in the projective space PG(a−1, q). If the field sizeqis clear from the context, we will writea
b
instead ofa b
q. The following counting result will be used several times in this article.
Lemma 2.1([27, Section 170]). The number ofj-spaces disjoint from a fixedm-space inPG(n, q) equalsq(m+1)(j+1)n−m
j+1
.
To end the introduction of this section, we give the definition of a k-spread and a partial k-spread ofPG(n, q).
Definition 2.2. A partial k-spread of PG(n, q) is a collection of k-spaces which are mutually disjoint. A k-spread in PG(n, q)is a partial k-spread in PG(n, q) that partitions the point set of PG(n, q).
Remark that ak-spread ofPG(n, q)exists if and only if k+ 1divides n+ 1, and necessarily contains qqn+1k+1−1−1 elements ([28]). A regular k-spread is a k-spread that can be constructed using field reduction.
Before we start with proving some equivalent definitions for a Cameron-Liebler set ofk-spaces, we give some lemmas and definitions that we will need in the characterization Theorem2.9.
Lemma 2.3 ([11]). Consider the Grassmann schemeJq(n+ 1, k+ 1). The eigenvaluePji of the distance-i relation forVj is given by:
Pji=
min (j,k+1−i)
X
s=max (0,j−i)
(−1)j+s j
s
n−k+s−j n−k−i
k+ 1−s i
qi(i+s−j)+(j−s2 ).
Lemma 2.4. IfP1i, i≥1, is the eigenvalue ofAi corresponding toVj, thenj= 1.
Proof. We need to prove thatP1i 6=Pji for qa prime power and j >1. We will first introduce φi(j) = max{a|qa|Pji}, which is the exponent ofq in the factorization ofPji. Remark thata
b
equals1 moduloq and note that it is sufficient to show that φi(j),j >1, is different fromφi(1) for alli. By Lemma2.3we see that
φi(j) = min
i(i+s−j) + j−s
2
|max{0, j−i} ≤s≤min{j, k+ 1−i}
unless there are two or more terms with a power of qwith minimal exponent as factor and that have zero as their sum. Ifsis the integer in{max{0, j−i}, . . . ,min{j, k+ 1−i}}closest toj−i−12, thenfij(s) =i(i+s−j) + j−s2
is minimal.
• Ifj ≤i, we see thatfij(s)is minimal fors= 0. Then we findφi(j) = 12j2−(i+12)j+i2. We see that for a fixedi,φi(k−1)> φi(k), k≤i. Note that the minimal value forfij(s)is reached for only ones.
• Ifj ≥i, we see thatfij(s)is minimal fors=j−i. Then we findφi(j) = 2i
. Again we note that the minimal value forfij(s)is reached for only one s.
We can conclude the following inequality for a giveni≥1:
φi(1)> φi(2)>· · ·> φi(i) =φi(i+ 1) =· · ·=φi(k+ 1). This implies the statement fori6= 1.
Fori= 1we haveP11=−k+1 1
+n−k 1
k 1
qandPj1=−j 1
k−j+2 1
+n−k 1
k+1−j 1
q, so we can see that they are different ifj6=n+ 1. This is always true sincej∈ {1, . . . , k+ 1} andk < n.
Note that for j ≥ 1 it was already known that |Pji| ≤ |P1i|. This result was shown in [3, Proposition5.4(ii)].
Lemma 2.5. Let π be a k-dimensional subspace in PG(n, q)with χπ the characteristic vector of the set {π}. IfZ is the set of all k-spaces in PG(n, q) disjoint from π with characteristic vector χZ, then
χZ−qk2+k
n−k−1 k
n k
−1
j−χπ
!
∈ker(A).
Proof. Letvπbe the incidence vector ofπwith its positions corresponding to the points ofPG(n, q).
Note thatAχπ=vπ. We have thatAχZ=qk2+kn−k−1 k
(j−vπ)sinceZis the set of allk-spaces disjoint from π and every point not in π is contained in qk2+kn−k−1
k
k-spaces skew to π (see Lemma2.1). The lemma now follows from
χZ−qk2+k
n−k−1 k
n k
−1
j−χπ
!
∈ker(A)
⇔ AχZ=qk2+k
n−k−1 k
n k
−1
Aj−Aχπ
! .
Definition 2.6. Aswitching setis a partialk-spreadRfor which there exists a partialk-spreadR0 such thatR ∩ R0=∅, and∪π∈Rπ=∪π∈R0π, in other words,RandR0 have no common members and cover the same set of points. We say thatRandR0 are a pair of conjugate switching sets.
The next lemma is a classical result in design theory.
Lemma 2.7. The incidence matrix of a2-design has full row rank.
The following lemma gives the relation between the common eigenspaces V0 and V1 of the matricesAi, i∈ {0, . . . , k+ 1}, and the row space of the matrixA. For the proof we refer to [19, Theorem 9.1.4].
Lemma 2.8. For the Grassmann scheme Jq(n+ 1, k+ 1)we have that Im(AT) =V0 ⊥V1 and V0=hji.
We want to make a combination of a generalization of Theorem3.2 in [10] and Theorem3.7in [26] to give several equivalent definitions for a Cameron-Liebler set ofk-spaces inPG(n, q).
Theorem 2.9. Let L be a non-empty set ofk-spaces inPG(n, q), n≥2k+ 1, with characteristic vectorχ, andxso that|L|=x
n k
. Then the following properties are equivalent.
1. χ∈ Im(AT).
2. χ∈ker(A)⊥.
3. For everyk-spaceπ, the number of elements ofLdisjoint from πis(x−χ(π))n−k−1 k
qk2+k. 4. The vector χ−xqqn+1k+1−1−1j is a vector inV1.
5. χ∈V0⊥V1.
6. For a giveni∈ {1, . . . , k+ 1}and anyk-spaceπ, the number of elements ofL, meetingπin a(k−i)-space is given by:
(x−1)qqk−i+1k+1−1−1+qi qn−kqi−1−1
qi(i−1)
"
n−k−1 i−1
# "
k i
#
ifπ∈ L x
"
n−k−1 i−1
# "
k+ 1 i
#
qi(i−1) ifπ /∈ L
.
7. for every pair of conjugate switching sets RandR0, we have that|L ∩ R|=|L ∩ R0|.
IfPG(n, q)admits ak-spread, then the following properties are equivalent to the previous ones.
8. |L ∩ S|=xfor everyk-spreadS inPG(n, q).
9. |L ∩ S|=xfor every regulark-spread S in PG(n, q).
Proof. We first prove that properties1,2,3,4,5,6are equivalent by proving the following implica- tions:
• 1⇔2: This follows sinceIm(BT) = ker(B)⊥ for every matrixB.
• 2⇒3: We assume thatχ∈ker(A)⊥. Letπ∈Πk andZ the set ofk-spaces disjoint fromπ.
By Lemma2.5, we know that
χZ−qk2+k
n−k−1 k
n k
−1 j−χπ
!
∈ker(A).
Sinceχ∈ker(A)⊥, this implies
χZ·χ−qk2+k
n−k−1 k
n k
−1
j·χ−χπ·χ
!
= 0
⇔ |Z ∩ L| −qk2+k
n−k−1 k
n k
−1
|L| −χ(π)
!
= 0
⇔ |Z ∩ L|= (x−χ(π))qk2+k
n−k−1 k
.
The last equality shows that the number of elements ofLdisjoint fromπis(x−χ(π))qk2+kn−k−1 k
.
• 3⇒4: By expressing property3in vector notation, we find thatKχ= (xj−χ)n−k−1 k
qk2+k and since by Lemma 2.1 we have Kj =q(k+1)2n−k
k+1
, we see that v =χ−xqqk+1n+1−1−1j is an
eigenvector ofK:
Kv=K
χ−xqk+1−1 qn+1−1j
= (xj−χ)
n−k−1 k
qk2+k−xqk+1−1 qn+1−1q(k+1)2
n−k k+ 1
j
=
n−k−1 k
qk2+k
xj−χ−xqn+1−qk+1 qn+1−1 j
= −
n−k−1 k
qk2+k
χ−xqk+1−1 qn+1−1j
=P1,k+1v .
By Lemma2.4fori=k+ 1, we know thatv∈V1.
• 4⇒5: This follows sinceV0=hji(see Lemma2.8).
• 5⇒1: This follows from Lemma2.8.
• 4 ⇒ 6: Denote χ−xqqk+1n+1−1−1j by v. The matrix Ai corresponds to the relation Ri. This implies that(Aiχ)π gives the number ofk-spaces inL that intersectπin a(k−i)-space.
Aiχ=Aiv+xqk+1−1
qn+1−1Aij=P1iv+xqk+1−1 qn+1−1P0ij
=
−
n−k−1 i−1
k+ 1 i
qi(i−1)+ n−k
i k
i
qi2 χ−xqk+1−1 qn+1−1j
+xqk+1−1 qn+1−1
n−k i
k+ 1 i
qi2j
=
n−k i
k i
qi2− k+ 1
i
n−k−1 i−1
qi(i−1)
χ
+xqk+1−1 qn+1−1qi(i−1)
n−k−1 i−1
k+ 1 i
− n−k
i k
i
qi+ n−k
i
k+ 1 i
qi
j
=
n−k i
k i
qi2− k+ 1
i
n−k−1 i−1
qi(i−1)
χ
+xqk+1−1 qn+1−1qi(i−1)
n−k−1 i−1
k i
qk+1−1
qk−i+1−1−qn−k−1 qi−1 qi
1− qk+1−1 qk−i+1−1
j
=
n−k i
k i
qi2− k+ 1
i
n−k−1 i−1
qi(i−1)
χ+x
n−k−1 i−1
k+ 1 i
qi(i−1)j
Remark that this proves the implication for everyi∈ {1, . . . , k+ 1}.
• 6⇒4: We follow the approach of Lemma 3.5 in [26] where we look for an eigenvalue ofAi
and we defineβi=xk+1 i
n−k−1
i−1
qi(i−1). From property6 we know that
Aiχ=x k+ 1
i
n−k−1 i−1
qi(i−1)(j−χ) +
(x−1) qk+1−1
qk−i+1−1+qiqn−k−1 qi−1
qi(i−1)
n−k−1 i−1
k i
χ
=
n−k i
k i
qi2− k+ 1
i
n−k−1 i−1
qi(i−1)
χ+x
n−k−1 i−1
k+ 1 i
qi(i−1)j
=P1iχ+βij.
Then we can see thatvi=χ+P βi
1i−P0ij is an eigenvector for Ai with eigenvalueP1i: Ai
χ+ βi P1i−P0i
j
=P1iχ+βij+ βi P1i−P0i
P0ij
=P1i
χ+ βi
P1i−P0ij
.
By Lemma2.4we know thatχ+P βi
1i−P0ij=χ−xqqn+1k+1−1−1j ∈V1.
We show that properties8 and9 are equivalent with the previous properties ifPG(n, q)admits a k-spread.
• 2⇒8: LetS be ak-spread inPG(n, q)andχS its characteristic vector. Then we know that χS−n
k
−1
j∈ker(A). Sinceχ∈ker(A)⊥ we have that
0 =χ· χS− n
k −1
j
!
=|L ∩ S| − |L|
n k
−1
,
so|L ∩ S|=|L|n k
−1
=x.
• 8⇒9: Trivial.
• 9 ⇒ 3: Suppose that PG(n, q) containsk-spreads, hence also regular k-spreads. We know that the group PGL(n+ 1, q)acts transitively on the pairs of pairwise disjointk-spaces. Let ni, for i = 1,2, be the number of regular k-spreads that contain i fixed pairwise disjoint k-spaces. This number only depends on i, and not on the chosenk-spaces.
Letπbe a fixedk-space. The number of pairs(π0,S), withSa regulark-spread that contains πandπ0 is equal toq(k+1)2n−k
k+1
·n2=n1·qn+1−1 qk+1−1 −1
, son1/n2=qk(k+1)n−k−1 k
. By counting the number of pairs(π0,S), withπ0 ∈ LandSa regulark-spread that contains πandπ0, we find that the number ofk-spaces inL, disjoint from a fixedk-spaceπ, is given by(x−χ(π))n1/n2= (x−χ(π))qk(k+1)n−k−1
k
.
To end this proof, we show that property7is equivalent with the other properties.
• 2⇒7: LetχRandχR0be the characteristic vectors of the pair of conjugate switching setsR andR0 respectively. AsRandR0 cover the same set of points, we find: χR−χR0 ∈ker(A).
This implies0 =χ·(χR−χR0) =χ·χR−χ·χR0, so thatχ·χR=|L ∩ R|=|L ∩ R0|=χ·χR0.
• 7 ⇒1: We first show that property 7 implies the other properties if n= 2k+ 1. For any twok-spreadsS1,S2, the setsS1\ S2andS2\ S1form a pair of conjugate switching sets. So
|L ∩(S1\ S2)|=|L ∩(S2\ S1)|, which implies that|L ∩ S1|=|L ∩ S2|=c.
Now we prove that this constantcequalsx=|L|2k+1 k
−1
. Letni, fori= 0,1, be the number of k-spreads containing i fixed pairwise disjoint k-spaces. This number only depends on i, and not on the chosenk-spaces. The number of pairs(π,S), withS ak-spread that contains π, is equal to2k+2
k+1
·n1=n0· qq2k+2k+1−1−1, which implies thatn0/n1=2k+1 k
.
By counting the number of pairs(π,S), withSak-spread that containsπ, and whereπ∈ L, we find, that the number of k-spaces in L ∩ S equals |L|n1/n0 = |L|2k+1
k
−1
= x. This implies property8, and hence, property1.
Now we prove that implication 7 ⇒ 1 also holds if n > 2k+ 1. Given a subspace τ in PG(n, q), we will use the notationA|τ for the submatrix ofA, where we only have the rows, corresponding with the points ofτ, and the columns corresponding with thek-spaces inτ.
We know that the matrixA|τ has full rank by Lemma2.7.
Let Π be a (2k+ 1)-dimensional subspace in PG(n, q). By property 7, we know that for every two k-spreads R,R0 in Π, we have |L ∩ R| = |L ∩ R0| since R \ R0 and R0\ R are conjugate switching sets. This implies thatχL|Π∈Im
AT|Π
by the arguments above applied for the(2k+ 1)-spaceΠ. So, there is a linear combination of the rows ofA|Π equal toχL|Π. This linear combination is unique since A|Π has full row rank. Now we will show that the linear combination of χL is uniquely defined by the vectors χL|Π, with Π varying over all (2k+ 1)-spaces inPG(n, q).
We show, for every two(2k+ 1)-spacesΠ,Π0, that the coefficients of the row corresponding to a point inΠ∩Π0 in the linear combination ofχL|Πand in the linear combination ofχL|Π0
are equal.
SupposeχL|Π=a1r1+a2r2+· · ·+alrl+al+1rl+1+· · ·+amrmandχL|Π0 =bl+1rl+1+· · ·+ bmrm+bm+1rm+1+· · ·+bsrs, wherer1, . . . , rl, . . . , rmandrl+1, . . . , rm, . . . , rs are the rows
corresponding with the points ofΠ and Π0, respectively. Remark that we only look at the columns corresponding with thek-spaces inΠ andΠ0, respectively.
We now look at the spaceΠ∩Π0, and at the corresponding columns inA. Recall thatA|Π∩Π0 also has full row rank, so the linear combination that givesχL|(Π∩Π0) is unique, and equal to the ones corresponding withΠ andΠ0, restricted to Π∩Π0. This proves thatai =bi for l+ 1≤i≤m. Here we also used the fact that the entry inAcorresponding with a point of Π\Π0 orΠ0\Π and ak-space in Π∩Π0 is zero.
By using all(2k+ 1)-spaces, we see thatχLis uniquely defined, and by construction we have χL ∈ Im(AT). Note that we only used that property 7 holds for conjugate switching sets inside a(2k+ 1)-dimensional subspace.
Definition 2.10. A setL of k-spaces inPG(n, q) that fulfills one of the statements in Theorem 2.9 (and consequently all of them) is called a Cameron-Liebler set of k-spaces in PG(n, q) with parameterx=|L|n
k
−1
.
Remark 2.11. Cameron-Liebler sets ofk-spaces inPG(n, q)were introduced before in [13] as we mentioned in the introduction. Remark that the definition we present here is consistent with the definition in [13] since the definition given in that article is property 5. from the previous theorem.
Note that the parameter of a Cameron-Liebler set of k-spaces in PG(n, q) is not necessarily an integer, while the parameter of Cameron-Liebler line sets in PG(3, q) and the parameter of Cameron-Liebler sets of generators in polar spaces are integers (see [8, Theorem 4.8]).
We end this section with showing an extra property of Cameron-Liebler sets of k-spaces in PG(n, q).
Lemma 2.12. LetLbe a Cameron-Liebler set ofk-spaces inPG(n, q), then we find the following equality for everyj-dimensional subspace αand every i-dimensional subspace τ, with α⊂τ and j < k < i:
|[k]α∩ L|+ n−j−1
k−j
(qk−j−1) i
k
(qi−k−1) |[k]τ∩ L|= n−j−1
k−j
i−j−1
k−j
|[k]τα∩ L|+ n−j−1
k−j−1
n
k
|L|.
Here [k]α,[k]τ and [k]τα denote the set of all k-spaces through α, the set of all k-spaces in τ and the set of allk-spaces in τ through α, respectively.
Proof. Letχ[α],χ[τ]andχ[α,τ]be the characteristic vectors of[k]α,[k]τ and[k]τα, respectively, and define
v=χ[α]+ n−j−1
k−j
(qk−j−1) i
k
(qi−k−1) χ[τ]− n−j−1
k−j
i−j−1
k−j
χ[α,τ]− n−j−1
k−j−1
n
k
j.
By calculating(Av)P0 for every pointP0, we see thatAv = 0. This implies thatv∈ker(A). Let χbe the characteristic vector ofL. By Definition 2 in Theorem2.9we know thatχ∈ker(A)⊥, so by calculatingχ·v the lemma follows.
Fork= 1, Drudge showed in [10] that this property is an equivalent definition for a Cameron- Liebler line set inPG(n, q). Fork >1 we pose it as an open problem to show that this property is also an equivalent definition.
3 Properties of Cameron-Liebler sets of k-spaces in PG(n, q )
We start with some properties of Cameron-Liebler sets ofk-spaces in PG(n, q)that can easily be proved.
Lemma 3.1. Let L andL0 be two Cameron-Liebler sets ofk-spaces inPG(n, q) with parameters xandx0 respectively, then the following statements are valid.
1. 0≤x≤ qqn+1k+1−1−1.
2. The set of all k-spaces in PG(n, q) not in L is a Cameron-Liebler set of k-spaces with pa- rameter qqn+1k+1−1−1 −x.
3. If L ∩ L0=∅, then L ∪ L0 is a Cameron-Liebler set of k-spaces with parameterx+x0. 4. If L0⊆ L, thenL \ L0 is a Cameron-Liebler set ofk-spaces with parameterx−x0.
We present some examples of Cameron-Lieblerk-sets inPG(n, q).
Example 3.2. The set of allk-spaces through a pointP is a Cameron-Liebler set ofk-spaces with parameter1 since the characteristic vector of this set is the row ofAcorresponding to the pointP. We will call this set ofk-spaces the point-pencil throughP.
Example 3.3. By property 3 in Theorem 2.9, we can see that the set of all k-spaces in a fixed hyperplane is a Cameron-Liebler set ofk-spaces in PG(n, q)with parameter qqn−kk+1−1−1. Remark that this parameter is not an integer ifk+ 1- n+ 1, or equivalently, if PG(n, q) does not contain a k-spread.
In [21] several properties of Cameron-Liebler sets ofk-spaces inPG(2k+ 1, q)were given. We will first generalize some of these results to use them in Section4.
Lemma 3.4. Let π andπ0 be two disjointk-spaces inPG(n, q)with Σ = hπ, π0i, and let P be a point inΣ\(π∪π0)and letP0 be a point not inΣ. Then the number ofk-spaces disjoint fromπand π0 equalsW(q, n, k), the number ofk-spaces disjoint from π and π0 through P equals WΣ(q, n, k) and the number ofk-spaces disjoint from πandπ0 through P0 equalsWΣ¯(q, n, k).
Here,W(q, n, k), WΣ(q, n, k), WΣ¯(q, n, k)are given by:
W(q, n, k) =
k
X
i=−1
Wi(q, n, k)
WΣ(q, n, k) = 1 (qk+1−1)2
k
X
i=0
Wi(q, n, k)(qi+1−1)
WΣ¯(q, n, k) = 1 qn+1−q2k+2
k−1
X
i=−1
Wi(q, n, k)(qk+1−qi+1)
Wi(q, n, k) = (
q2k2+k+3i22−2i−3ikn−2k−1 k−i
k+1 i+1
Qi
j=0(qk−j+1−1) if i≥0 q2(k+1)2n−2k−1
k+1
if i=−1 .
Proof. To count the number of k-spaces π00, that are disjoint from π and π0, we first count the number of possible intersectionsπ00∩Σ.
We count the number ofi-spaces inΣ, disjoint fromπandπ0, by counting((P0, P1, . . . , Pi), σi) in two ways. Hereσiis ani-space inΣ, disjoint fromπandπ0, and the pointsP0, P1, . . . , Pi form a basis ofσi. For the ordered basis (P0, P1, . . . , Pi)we have Qi
j=0
q2j(qk−j+1−1)2
q−1 possibilities since there are2k+2
1
−2k+j+1 1
+2j 1
=q2j(qk−j+1q−1 −1)2 possibilities forPj ifP0, P1, . . . , Pj−1 are given.
By a similar argument, we find that the number of ordered bases(P0, P1, . . . , Pi)for a givenσi isQi
j=0
qj(qi−j+1−1)
q−1 .
In this way we find that the number ofi-spaces inΣ, disjoint fromπandπ0, is given by:
Qi j=0
q2j(qk−j+1−1)2 q−1
Qi j=0
qj(qi−j+1−1) q−1
=
i
Y
j=0
qj(qk−j+1−1)2
qi−j+1−1 =q(i+12 )k+ 1 i+ 1
i Y
j=0
(qk−j+1−1).
Now we count, for a given i-space σi in Σ, the number of k-spaces π00 through σi such that π00∩Σ = σi. This equals the number of (k−i−1)-spaces in PG(n−i−1, q), disjoint from a (2k−i)-space. This number isq(k−i)(2k−i+1)n−2k−1
k−i
by Lemma 2.1. By this lemma we also see that the number of k-spaces disjoint from Σis given by q(k+1)(2k+2)n−2k−1
k+1
. This implies that Wi(q, n, k),−1≤i≤k, is the number ofk-spaces disjoint fromπandπ0, and intersectingΣin an i-space.
Now we have enough information to count the number ofk-spaces disjoint fromπandπ0: W(q, n, k) =
k
X
i=−1
Wi(q, n, k).
We use the same arguments to calculateWΣ(q, n, k)andWΣ¯(q, n, k). By double counting(P, π00), withπ00a k-space throughP ∈Σdisjoint fromπandπ0, and double counting(P0, π00), withπ00 a k-space throughP0∈/Σdisjoint fromπandπ0, we find:
2k+ 2 1
−2 k+ 1
1
·WΣ(q, n, k) =
k
X
i=0
Wi(q, n, k)· i+ 1
1
and n+ 1
1
−
2k+ 2 1
·WΣ¯(q, n, k) =
k−1
X
i=−1
Wi(q, n, k)·
k+ 1 1
− i+ 1
1
.
This implies:
WΣ(q, n, k) = 1 (qk+1−1)2
k
X
i=0
Wi(q, n, k)(qi+1−1)
WΣ¯(q, n, k) = 1 qn+1−q2k+2
k−1
X
i=−1
Wi(q, n, k)(qk+1−qi+1).
From now on we denote Wi(q, n, k), WΣ(q, n, k) and WΣ¯(q, n, k) by Wi, WΣ and WΣ¯ if the dimensionsn,kand the field size qare clear from the context.
Lemma 3.5. LetL be a Cameron-Liebler set of k-spaces in PG(n, q)with parameter x.
1. For everyπ∈ L, there ares1 elements ofLmeeting π.
2. For skew π, π0 ∈ L and a k-spread S0 in Σ =hπ, π0i, there exist exactly d2 subspaces in L that are skew to both πandπ0 and there exists2 subspaces inL that meet both πandπ0. Here,d2,s1 ands2 are given by:
d2(q, n, k, x,S0) = (WΣ−WΣ¯)|S0∩ L| −2WΣ+xWΣ¯
s1(q, n, k, x) =x n
k
−(x−1)
n−k−1 k
qk2+k s2(q, n, k, x,S0) =x
n k
−2(x−1)
n−k−1 k
qk2+k+d2(q, n, k, x,S0),
whereWΣ andWΣ¯ are given by Lemma3.4.
3. Defined02(q, n, k, x) = (x−2)WΣands02(q, n, k, x) =xn k
−2(x−1)n−k−1
k
qk2+k+d02(q, n, k, x).
If n > 3k+ 1, then |S0 ∩ L| ≤ x for every k-spread S0 in Σ. Moreover we have that d2(q, n, k, x,S0)≤d02(q, n, k, x) ands2(q, n, k, x,S0)≤s02(q, n, k, x).
Proof.
1. This follows directly from Theorem2.9(3)and|L|=xn k
.
2. Letχπ andχπ0 be the characteristic vectors of{π}and{π0}, respectively, and letZ be the set of allk-spaces inPG(n, q)disjoint fromπandπ0, and letχZ be its characteristic vector.
Furthermore, let vπ and vπ0 be the incidence vectors of π and π0, respectively, with their positions corresponding to the points ofPG(n, q). Note that Aχπ =vπ andAχπ0 =vπ0. By Lemma 3.4we know the numbers WΣand WΣ¯ ofk-spaces disjoint from πand π0, through a point P, ifP ∈Σ and P /∈Σ respectively. LetS0 be ak-spread in Σand let vΣ be the incidence vector ofΣ(as a point set). We find:
AχZ=WΣ(vΣ−vπ−vπ0) +WΣ¯(j−vΣ)
=WΣ(AχS0−Aχπ−Aχπ0) +WΣ¯
n k
−1
Aj−AχS0
!
⇔ χZ−WΣ(χS0−χπ−χπ0)−WΣ¯
n k
−1
j−χS0
!
∈ker(A).
We know that the characteristic vectorχofLis included inker(A)⊥. This implies:
χZ·χ=WΣ(χS0·χ−χ(π)−χ(π0)) +WΣ¯(x−χS0·χ)
⇔ |Z ∩ L|=WΣ(|S0∩ L| −2) +WΣ¯(x− |S0∩ L|)
⇔ |Z ∩ L|= (WΣ−WΣ¯)|S0∩ L| −2WΣ+xWΣ¯ ,
which gives the formula for d2(q, n, k, x). The formula for s2(q, n, k, x) follows from the inclusion-exclusion principle.
3. Suppose Σ is a (2k+ 1)-space in PG(n, q), and suppose S0 is a k-spread in Σ such that
|S0∩ L| > x. By property 1 in Theorem2.9 we know that the characteristic vector χ ofL can be written asP
P∈PG(n,q)xPrTP for somexP ∈RwhererP is the row ofAcorresponding to the point P. Letχπ be the characteristic vector of the set {π} with π a k-space, then χπ·χ =P
P∈πxP equals 1 ifπ ∈ L and 0 if π /∈ L. As χ·j =|L| = xn k
we find that P
P∈PG(n,q)xP =x.
If |S0∩ L| > x, then χ·χS0 = P
P∈ΣxP > x. From these observations, it follows that P
P∈PG(n,q)\ΣxP =P
P∈PG(n,q)xP−P
P∈ΣxP is negative. As n >3k+ 1, there exists a k-space τ in PG(n, q), disjoint from Σ, with χτ·χ = P
P∈τxP negative, which gives the contradiction.
It follows that|S0∩ L| ≤x. Since this is true for every k-spread S0 in every(2k+ 1)-space inPG(n, q), the statement holds.
Remark that we will use the upper boundd02(q, n, k, x)ands02(q, n, k, x)instead ofd2(q, n, k, x,S0) ands2(q, n, k, x,S0)respectively, since they are independent of the chosenk-spread S0.
The following lemma is a generalization of Lemma2.4in [21].
Lemma 3.6. Letc, n, k be nonnegative integers withn >3k+ 1 and (c+ 1)s1−
c+ 1 2
s02> x
n k
,
then no Cameron-Liebler set ofk-spaces inPG(n, q)with parameterxcontainsc+ 1mutually skew k-spaces.
Proof. Assume that PG(n, q) has a Cameron-Liebler set L of k-spaces with parameter x that containsc+ 1mutually disjoint k-spacesπ0, π1, . . . , πc. Lemma 3.5shows that πi meets at least s1(q, n, k, x)−is2(q, n, k, x) elements of L that are skew to π0, π1, . . . , πi−1. This implies that xn
k
=|L| ≥(c+ 1)s1−Pc
i=0is2≥(c+ 1)s1−Pc
i=0is02which contradicts the assumption.
4 Classification results
In this section, we will list some classification results for Cameron-Liebler sets of k-spaces in PG(n, q). First note that a Cameron-Liebler set ofk-spaces with parameter0is the empty set.
In the following lemma we start with the classification for the parametersx∈]0,1[∪]1,2[.
Lemma 4.1. There are no Cameron-Liebler sets ofk-spaces inPG(n, q)with parameterx∈]0,1[
and ifn≥3k+ 2, then there are no Cameron-Liebler sets of k-spaces with parameterx∈]1,2[.
Proof. Suppose there is a Cameron-Liebler setLofk-spaces with parameter x∈]0,1[. ThenLis not the empty set so supposeπ ∈ L. By property 3 in Theorem 2.9 we find that the number of k-spaces inLdisjoint fromπis negative, which gives the contradiction.
Suppose there is a Cameron-Liebler set L of k-spaces with parameter x∈ ]1,2[in PG(n, q), n≥3k+ 2. By property 3in Theorem2.9, we know that there are at least two disjointk-spaces π, π0 ∈ L. By Lemma3.5(2,3) we know that there ared2≤d02 elements of Ldisjoint fromπand π0. Sinced02 is negative forx∈]1,2[, we find a contradiction.
We continue with a classification result for Cameron-Lieblerk-sets with parameterx= 1, where we will use the following result, the so-called Erdős-Ko-Rado theorem for projective spaces.
Theorem 4.2 ([20, 24] ). If L is a set of pairwise non-trivially intersectingk-spaces inPG(n, q) withn≥2k+ 1, then|L| ≤n
k
, and equality holds if and only if L either consists of all k-spaces through a fixed point, orn= 2k+ 1 andL consists of allk-spaces in a fixed hyperplane.
Theorem 4.3. Let L be a Cameron-Liebler set of k-spaces with parameter x = 1 in PG(n, q), n≥2k+ 1. ThenLis a point-pencil or n= 2k+ 1andLis the set of allk-spaces in a hyperplane ofPG(2k+ 1, q).
Proof. The theorem follows immediately from Lemma4.2since, by Theorem2.9(3), we know that Lis a family of pairwise intersectingk-spaces of sizen
k
.
We continue this section by showing that there are no Cameron-Liebler sets of k-spaces in PG(n, q), n≥3k+ 2, with parameter2≤x≤qn2−k
2
4−3k4−32(q−1)k
2
4−k4+12p
q2+q+ 1. For this classification result, we will use the following theorem, the so called Hilton-Milner theorem for projective spaces.
Theorem 4.4 ([1, Theorem 1.4] ). Let k≥1 be an integer. If q≥3 andn≥2k+ 2, or ifq= 2 andn≥2k+ 3, then any familyF of pairwise non-trivially intersectingk-spaces ofPG(n, q), with
∩F∈FF =∅ has size at mostn k
−qk2+kn−k−1 k
+qk+1.
To simplify the notations, we denoteqn2−k42−3k4−32(q−1)k42−k4+12p
q2+q+ 1 byf(q, n, k).
Recall that the set of all k-spaces in a hyperplane in PG(n, q) is a Cameron-Liebler set of k- spaces with parameterx= qqn−kk+1−1−1 (see Example3.3) and note thatf(q, n, k)∈ O(p
qn−2k)while
qn−k−1
qk+1−1 ∈ O(qn−2k−1).
We start with some lemmas.
Lemma 4.5. Forn≥2k+ 2, we have:
n k
>
n−k−1 k
qk2+k > WΣ. If also k≥2, then
n−k−1 k
qk2+k> qnk−k2+qnk−k2−1+qnk−k2−2. Proof. The first inequality follows since n
k
is the number of k-spaces through a fixed point in PG(n, q), n−k−1
k
qk2+k is the number of k-spaces through a fixed point disjoint from a given k- space not through that point (see Lemma2.1), andWΣis the number ofk-spaces through a fixed point and disjoint from two givenk-spaces not through that point.
The second inequality, fork >1, follows from n−k−1
k
qk2+k=
k−3
Y
i=0
qn−k−1−i−1 qk−i−1
!
qn−2k+1−1 q−1
qn−2k−1 q2−1
qk2+k
> q(n−2k−1)(k−2)(qn−2k+qn−2k−1+qn−2k−2)qn−2k−2qk2+k
=qnk−k2+qnk−k2−1+qnk−k2−2.
Lemma 4.6. LetL be a Cameron-Liebler set ofk-spaces inPG(n, q),n≥3k+ 2, with parameter 2≤x≤f(q, n, k), thenLcannot contain bxc+ 1 mutually disjoint k-spaces.
Proof. This follows from Lemma3.6, withc=bxc ≥2:
(bxc+ 1)s1−
bxc+ 1 2
s02> x
n k
⇔ (bxc+ 1)x n
k
−(bxc+ 1)(x−1)
n−k−1 k
qk2+k−(bxc+ 1)xbxc 2
n k
+ (bxc+ 1)(x−1)bxc
n−k−1 k
qk2+k−(bxc+ 1)(x−2)bxc 2 d02> x
n k
⇔ (1− bxc)xbxc 2
n k
+ (x−1)(bxc2−1)
n−k−1 k
qk2+k> (x−2)(bxc+ 1)bxc
2 WΣ
Asn k
≥n−k−1 k
qk2+k by the first inequality in Lemma4.5, the following inequality is sufficient.
bxc2x+bxcx
2 − bxc2−x+ 1 n−k−1 k
qk2+k >(x−2)(bxc+ 1)bxc
2 WΣ.
By the first inequality in Lemma4.5and since bxc2x+bxcx2 − bxc2−x+ 1> (x−2)(bxc+1)bxc
2 , we find that the above inequality is always valid.
Lemma 4.7. Ifx≤f(q, n, k)andn≥3k+ 2, then n−k−1
k
qk2+k−(x−2)s02>max
x n
k
−x
n−k−1 k
qk2+k,
n k
−
n−k−1 k
qk2+k+qk+1
.
Proof. Fork >1, we will prove the following inequalities:
n−k−1 k
qk2+k−(x−2)s02> x n
k
−x
n−k−1 k
qk2+k >
n k
−
n−k−1 k
qk2+k+qk+1. To prove the first inequality, we show thatx≤f(q, n, k)implies it. The first inequality is equivalent with
(2x2−5x+ 5)
n−k−1 k
qk2+k−(x2−x) n
k
−(x−2)2WΣ>0.
SinceWΣ≤n−k−1 k
qk2+k, the following inequality is sufficient:
(x2−x+ 1)
n−k−1 k
qk2+k−(x2−x) n
k
>0
⇔
n−k−1 k
qk2+k >(x2−x) n
k
−
n−k−1 k
qk2+k
.
Given ak-spaceπinPG(n−1, q)the number of(k−1)-spaces meetingπequalsn k
−n−k−1
k
qk2+k by Lemma2.1. We know that this number is smaller than the product of the number of points Q∈πand the number of(k−1)-spaces through Q. This implies that
n k
−
n−k−1 k
qk2+k ≤ k+ 1
1
n−1 k−1
≤ k+ 1
1
(qn−1−1)· · ·(qn−k+1−1) (qk−1−1)· · ·(q−1)
≤ qnk−k22−n+3k2+1 (q−1)k22−k2+1
.
By the second inequality in Lemma4.5we know thatn−k−1
k
qk2+k≥qnk−k2+qnk−k2−1+qnk−k2−2, which gives that the following inequality is sufficient:
(x2−x)< qn−k
2
2−3k2−3(q−1)k
2
2−k2+1(q2+q+ 1). The inequality(x2−x)≤ x−122
implies that x−1
2 < qn2−k
2
4−3k4−32(q−1)k
2
4−k4+12p
q2+q+ 1 =f(q, n, k)
is sufficient, which is a direct consequence of x≤f(q, n, k). We prove the second inequality in a similar way. We have
x n
k
−x
n−k−1 k
qk2+k>
n k
−
n−k−1 k
qk2+k+qk+1
⇔ (x−1) n
k
−
n−k−1 k
qk2+k
> qk+1.
The number of (k−1)-spaces in a hyperplane α of PG(n, q) meeting a k-space π in α equals n
k
−n−k−1 k
qk2+k by Lemma2.1. This number is larger than the number of(k−1)-spaces in α meeting thisk-spaceπexactly in one point, which equalsk+1
1
n−k−1 k−1
qk2−k, also by Lemma2.1.
We find that
(x−1) k+ 1
1
n−k−1 k−1
qk2−k> qk+1
