Cameron-Liebler $k$-sets in $\text{AG}(n,q)$
Jozefien D'haeseleer Ferdinand Ihringer Jonathan Mannaert Leo Storme
math.CO


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theorem[St]Theorem
Le[St]Lemma
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Ex[St]Example
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Opm[St]Remark
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*ConsbConstruction









 * Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000 Gent, Flanders, Belgium (Email: jozefien.dhaeseleer@ugent.be, ferdinand.ihringer@ugent.be, leo.storme@ugent.be) (http://cage.ugent.be/∼ jmdhaese/, http://math.ihringer.org/, http://cage.ugent.be/∼ls).
    † Department of Mathematics and Data Science, University of Brussels (VUB),  Pleinlaan 2, Building G, 1050 Elsene, Flanders, Belgium  (Email: Jonathan.Mannaert@vub.be)(http://homepages.vub.ac.be/∼jonmanna/).




Practical Quantum Computing: solving the wave equation using a quantum approach
    J. D'haeseleer, F.Ihringer^*, J. Mannaert and L. Storme^*
    December 30, 2023
===============================================================================




        We study Cameron-Liebler k-sets in the affine geometry, so sets of k-spaces in (n, q). This generalizes research on Cameron-Liebler k-sets in the projective geometry (n, q).
        Note that in algebraic combinatorics, Cameron-Liebler k-sets of (n, q) correspond to certain equitable bipartitions of the association
        scheme of k-spaces in (n, q), while in the analysis of Boolean functions, they correspond to Boolean degree 1 functions of (n, q). 
        
        We define Cameron-Liebler k-sets in (n, q)  in a similar way as Cameron-Liebler k-sets in (n,q), such that its characteristic vector is a linear combination of point-pencils.
        In particular, we investigate the relationship between Cameron-Liebler k-sets in (n, q) and (n, q).
        As a by-product, we calculate the character table of the association scheme of affine lines.
        Furthermore, we characterize the smallest examples of Cameron-Liebler k-sets.
        
        This paper focuses on (n, q) for n > 3, while the case for Cameron-Liebler line classes in (3, q) was already treated separately.




§ INTRODUCTION


The investigation of Cameron-Liebler line classes in the projective geometry (n, q) 
goes back to Cameron and Liebler in 1982 <cit.>. 
Their motivation was the investigation of the subgroup structure of (n+1, q).
Particularly, a line orbit of a subgroup of (n+1, q) acting on (n, q) with the same number of point- and line-orbits is 
a Cameron-Liebler line class. 

The concept of Cameron-Liebler line classes was rediscovered several times, see the introduction of <cit.> for a short overview.
In particular, algebraic combinatorialists studied equitable bipartitions as a natural generalization of perfect codes 
under various names in several highly symmetric families of graphs such as hypercubes and Johnson graphs. Similarly,
they also correspond to Boolean degree 1 functions in the analysis of Boolean functions.

In the special case of (3, q), a Cameron-Liebler line class can be defined as 
a family of lines which intersects all spreads of (3, q) in exactly x lines for some constant x <cit.>.
We call x the parameter of the Cameron-Liebler line class.
In (3, q), there exists a list of examples which we refer to as trivial: (1) the empty set
with parameter x=0, (2) all lines through a fixed point with parameter x=1, 
(3) all lines in a fixed plane with parameter x=1, (4) the union of (2) and (3), when disjoint, with parameter x=2,
and (5)–(8) the complements of (1)–(4) with parameters x=q^2+1, q^2, q^2, q^2-1.
Cameron and Liebler conjectured that these are the only examples. This was disproven by Drudge who found an example with parameter x=5 in PG(3,3) <cit.>. Nowadays there are several infinite families of non-trivial examples known <cit.>.
In contrast to this, there are no non-trivial examples known for n > 3. Hence,
there is some difference in behaviour between n=3 and n > 3.
This carries over to (n, q), where this paper handles the case n>3, while we treat the case n=3 separately in <cit.>.

Cameron-Liebler line classes were generalized to k-spaces of (n, q) 
in <cit.>. These are families of k-spaces which lie in the span of the point-(k-space) incidence matrix.
We call such families Cameron-Liebler k-sets of (n, q).
Note that if  is a Cameron-Liebler k-set of (n, q), then its parameter x is defined by || = x nk_q where
nk_q denotes the  Gaussian binomial coefficient.
Analogously, we call a family ℒ of k-spaces which lies in the span of the (affine) point-(k-space) incidence matrix  a Cameron-Liebler k-set of (n, q). Moreover, if || = x nk_q, we say that x is the parameter of the Cameron-Liebler k-set .

After some preliminaries, we start our paper with some general properties of Cameron-Liebler k-sets in Section <ref>.
In particular, we show the following.


Let  be a Cameron-Liebler k-set with parameter x in PG(n,q) which does not contain k-spaces in some hyperplane H. 
Then  is a Cameron-Liebler k-set with parameter x of (n, q) ≅(n, q) ∖ H.
























We also obtained a result on the converse of Theorem <ref>. In particular, we show the following.


   
    If  is a Cameron-Liebler k-set of (n, q) with parameter x, then  is a Cameron-Liebler k-set of (n, q)
    with parameter x in the projective closure (n, q) of (n, q).



 We prove this theorem in general, but for the case k=1 we give an alternative proof using the character table of the association scheme of affine lines.
This association scheme has been investigated before as it is a well-known 3-class association scheme, see <cit.> for a more detailed study.
We could not find the character table of the affine lines scheme in the literature, so we provide the latter in Section <ref>. While for (n, q) the character tables of the association scheme 
of k-spaces is explicitly known due to Delsarte <cit.> and Eisfeld <cit.>, the determination of the character tables of the association scheme of k-spaces in (n, q)
is still open.

A 3-class association scheme has four common eigenspaces V_0, V_1, V_2, V_3, where V_0 is spanned
by the all-ones vector. In our ordering, we provide explicit bases for V_0 + V_1 and V_0 + V_3, and we give a spanning set for V_0 + V_2 + V_3.


An immediate consequence of Theorem <ref> is that the following results for Cameron-Liebler k-sets 
of (n, q) are also valid for Cameron-Liebler k-sets of (n, q).






<cit.>
	
There are no Cameron-Liebler k-sets in (n,q), with n≥ 3k+2 and q≥ 3, of parameter 
	
    2≤ x≤1/√(2)q^n/2-k^2/4-3k/4-3/2(q-1)^k^2/4-k/4+1/2√(q^2+q+1).


 The statement in <cit.> is slightly different and its proof currently contains an error. 
The authors of <cit.> have submitted an erratum. The version in this article is based on the recent corrected version on arXiv.[See arXiv:1805.09539v3 [math.CO]]
For n ≥5/2 k + 3/2, a similar bound is given in <cit.>.
For n = 2k+1, there is a better bound.

<cit.>
 For k≥ 3, 
 there are no non-trivial Cameron-Liebler k-sets with parameter x in (2k+1, q) for 2 ≤ x ≤ q/5 and q ≥ q_0
 for some universal constant q_0.


Note that <cit.> requires that k < q log q - q - 1. This condition can be removed, see <cit.>. Complementary to the two previous results, Theorem <ref> also implies that the situation is known for small q.

<cit.>
  Let n ≥ 2k+1 > 3. 
Then there are no non-trivial Cameron-Liebler k-sets in (n, q) for q ≤ 5.



We conclude with Section <ref>, where we obtain a classification of the smallest Cameron-Liebler k-sets of (n, q).


  All Cameron-Liebler k-sets of (n, q) with parameter x ≤ 2 are trivial.


Note that this cannot be deduced from the literature on (n, q). We are also able to classify all Cameron-Liebler sets of hyperplanes in AG(n,q), this will be done in Section <ref>. We conclude with suggestions for future work in Section <ref>.

	

§ PRELIMINARIES

	Consider a prime p and let q=p^h, with h≥ 1. Then consider (n,q), and (n,q), for n>2, as the n-dimensional projective, and affine, space over 𝔽_q respectively. 
 Suppose that we consider a hyperplane π_∞ in (n,q), which from now on will be called the hyperplane at infinity. Then we can consider all the points, lines, planes, and other spaces that do not lie inside π_∞. In this way we obtain the affine space (n,q). 
The following notation will be used throughout this article.

 For a,b ∈ℕ, we denote the Gaussian binomial coefficient by
  
    ab_q= (q^a-1)⋯(q^a-b+1-1)/(q^b-1)⋯ (q-1).

  The Gaussian binomial coefficient ab_q equals the number of (b-1)-spaces in  (a-1,q). Here we define that  ab_q=0 if b>a.



 In general we take k≥ 1 with n≥ k+1, unless otherwise stated. Note that if we also ask that (k+1)| (n+1), then it automatically follows that n≥ 2k+1. 


Consider (n,q), or (n,q) respectively.


  * A partial k-spread is a set of pairwise disjoint k-spaces.

  * A conjugated switching k-set is a pair of disjoint partial k-spreads that cover the same set of points.

  * A k-spread is a partial k-spread that partitions the point set of (n,q), or (n,q) respectively.




Since there is a lot of interaction between (n,q) and (n,q) we want to clarify some formulations. 


  * Suppose that we have a set of k-spaces ℒ in (n,q). Then the restriction of ℒ to (n,q) is the set of k-spaces of ℒ that are not contained in π_∞.

  * We say that two k-spaces  are projectively (or affinely) disjoint if they intersect in the empty set in (n,q) (or (n,q) respectively).



We will give some examples of k-spreads in (n,q), which we will denote by respectively type I, II and III for future purposes.

			Consider the affine space (n,q) and the corresponding projective space (n,q). Then the following k-sets 𝒮 are  k-spreads in (n,q).
			
			
				
  * (Type I) 
Every  k-spread in (n,q) restricted to the affine space.
				
  * (Type II) Consider a (k-1)-space K in π_∞ and define	the set 𝒮 as the set of all affine k-spaces through K.
				
  * (Type III) 	Consider an (n-2)-space π_n-2 in π_∞, then there are exactly q other hyperplanes through π_n-2 not equal to π_∞. Call these hyperplanes π_i, for i ∈{1,...,q}. If we select for every hyperplane π_i a (k-1)-space τ_i⊆π_n-2 (not all equal), then we can define the k-spread
				 
    𝒮:={K | K  a k-space in (n,q),    τ_i ⊆ K⊆π_i  for some  i ∈{1,...,q}}.
		
			 
	
 
 	
 		
  * Consider a projective k-spread 𝒮, then it is clear that every two k-spaces of 𝒮 are affinely disjoint.   Secondly, 𝒮 restricted to (n,q) partitions the point set of (n,q), since its extension reaches every  (affine) point in (n,q).
 		
  * Trivial.
 		
  * It is clear that all these elements are disjoint. Thus we only need to prove that for every affine point p there exists an element of 𝒮 that contains it. Consider for this point p the hyperplane ⟨ p, π_n-2⟩, then this is a hyperplane through π_n-2. Without loss of generality we may assume that it is π_i. Such that ⟨ p, τ_i ⟩ is a k-space in 𝒮 which contains p. This proves that 𝒮 is indeed a  k-spread.
 	




The size of a k-spread in (n,q) is equal to q^n/q^k=q^n-k, where q^k is the number of points in an affine k-space and q^n is the total number of points in (n,q). An analogous result can be obtained in (n,q), where the size of a k-spread is known to be q^n+1-1/q^k+1-1. Note that this number is only an integer if (k+1)| (n+1), so it follows that this is a necessary condition for the existence of k-spreads in (n,q). It is proven in <cit.> that this is also a sufficient condition. 

 


Let us denote the set of k-spaces in (n,q), and (n,q), by Π_k, and Φ_k, respectively. If we number the points and the k-spaces in these spaces, then we can define the point-(k-space) incidence matrix P_n and A_n respectively. These matrices are 0,1-valued matrices with a 1 on position (i,j) if and only if point i lies on k-space j.

We now give a special construction for the matrix P_n.
[Incidence matrix]
Consider now the point-(k-space) incidence matrix P_n of (n,q), where the rows correspond to the points and the columns correspond to the elements of Π_k. We order the rows and columns in such a way that the first rows and columns correspond to the affine points and  affine k-spaces respectively.  Then P_n is of the following form:

    P_n= [   A_n    0̅;   B_2 P_n-1 ],

where A_n is the incidence matrix of (n,q), where again the rows  correspond to the points and the columns correspond to the elements of Φ_k. The matrix 0̅ is the zero-matrix and the part that remains unnamed, we call B_2.


We will use the notation of Construction <ref> in the following results. These results give some information about the characteristic vector of Cameron-Liebler k-sets. This characteristic vector is a 0,1 valued vector, which contains a 1 on position i if and only if the ith k-space belongs to the Cameron-Liebler k-set.

<cit.>
	Consider a set ℒ of k-spaces in (n,q) such that χ_ℒ∈ ((A_n))^⊥=Im(A_n^T). 
Then it follows for every affine k-spread 𝒮 that
	
    |ℒ∩𝒮|=x,

	for a fixed integer x.



We should also note that in general it holds that Im(P_n^T)=((P_n))^⊥ and Im(A_n^T)=((A_n))^⊥.

<cit.>
	Consider the projective space (n,q) and consider a set of k-spaces ℒ. If its characteristic vector χ_ℒ∈ ((P_n))^⊥ and ℒ also contains no k-spaces at infinity, then  χ_ℒ restricted to the affine space belongs to ((A_n))^⊥. 







 §.§ Cameron-Liebler k-sets in (n,q)

Our goal here is to state some important results that are known for Cameron-Liebler k-sets in (n,q).
We start with the definition of Cameron-Liebler k-sets in (n,q). 


 A Cameron-Liebler k-set ℒ in (n,q) is a set of k-spaces such that for its characteristic vector χ_, it holds that χ_∈ Im(P_n^T). We say that ℒ has parameter x:=|ℒ|/nk.



The fact that χ_∈ Im(P_n^T) states that χ_ is a linear combination of the rows of P_n^T. In some literature, for example <cit.>,  the characteristic vector χ_ is called a Boolean degree 1 function in (n,q). Similarly we can consider χ_∈ Im(A_n^T) for the incidence matrix A_n in (n,q).


In this section we list some results on Cameron-Liebler k-spaces in PG(n,q). We refer to <cit.> for more information. 
 

<cit.>
	Let ℒ be a non-empty set of k-spaces in (n,q), n ≥ 2k+1, with characteristic vector χ, and x so that 
	|ℒ|= xnk_q. Then the following properties are equivalent.
	
		
  * The set ℒ is a Cameron-Liebler k-set. 
		
  * For every k-space K, the number of elements of ℒ disjoint from K is equal to (x-χ(K))n-k-1k_q q^k^2+k.
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		 
  * For every pair of conjugated switching sets ℛ and ℛ', |ℒ∩ℛ|=|ℒ∩ℛ'|.
		 
		
	
If (n,q) has a k-spread, then the following property is equivalent to the previous ones.


  4 For every k-spread 𝒮, |ℒ∩𝒮|=x.



<cit.>
The following k-sets are examples of Cameron-Liebler k-sets in (n,q).

 
  * The set of all the k-spaces through a fixed point is an example of a Cameron-Liebler k-set of parameter x=1.

 
  * If we consider the set of k-spaces inside a fixed hyperplane, then this is a Cameron-Liebler k-set of parameter x=q^(n-k)-1/q^(k+1)-1. Note that x is only an integer if and only if (k+1)| (n+1).


In order to give some context on the study of Cameron-Liebler k-sets in (n,q), we now give some classification results
<cit.>
Let q ∈{2,3,4,5}. Then all Cameron-Liebler k-sets in (n,q) are of the form of Example <ref>, if k,n-k ≥ 2 and either (a) n ≥ 5 or (b) n = 4 and q = 2.


<cit.> 
Let ℒ be a Cameron-Liebler k-set  with parameter x = 1 in (n,q), n ≥ 2k+1. Then ℒ consists out of all the k-spaces through a fixed point or n = 2k+1 and ℒ is the set of all the k-spaces in a hyperplane of PG(2k + 1,q).


<cit.>
 There are no Cameron–Liebler k-sets in PG(n,q) with parameter x ∈ ] 0,1[ and if n ≥ 3k + 2, then there are no Cameron–Liebler k-sets with parameter x∈ ]1,2[.








§ CAMERON-LIEBLER K-SETS IN (N,Q)


 
A Cameron-Liebler k-set ℒ in (n,q)  is a set of k-spaces such that for its characteristic vector χ_ℒ, it holds that  χ_ℒ∈Im(A_n^T). Here we say that ℒ has parameter x:=|ℒ|/nk.


Due to Lemma <ref>, we know that for every value of n, the affine space (n,q) contains k-spreads. 
By Lemma <ref>, we find that Cameron-Liebler k-sets have a constant intersection number with k-spreads. This number will be equal to the parameter x of the Cameron-Liebler k-set. This is proven in the next lemma. 



Suppose that ℒ is a Cameron-Liebler k-set in (n,q), then it holds for every k-spread 𝒮 that

    |ℒ∩𝒮|=x,

where x is the parameter of ℒ.







 Due to Lemma <ref>, it follows that for every k-spread 𝒮 it holds that |∩𝒮|=c, for a certain fixed integer c. To prove that c=x, we double count the pairs  (K,𝒮), where 𝒮 is a k-spread of type II and K ∈ℒ∩𝒮. So we obtain that

    xnk_q· 1=nk_q· c.

This proves the lemma.


Remark that this statement holds for every n and k, since in each case we have these k-spreads. We now give some basic properties.

	Consider ℒ and ℒ' to be Cameron-Liebler k-sets with parameter x and x' both in (n,q) or both in (n,q) respectively, then the following properties hold.
	
		
  * If  is a Cameron-Liebler k-set in (n,q) or (n,q), then we have that 0 ≤ x≤ q^n-k or 0 ≤ x≤q^n+1-1/q^k+1-1 respectively.
		
  * If ℒ∩ℒ'= ∅, then ℒ∪ℒ' is a Cameron-Liebler k-set of parameter x+x'.
		
  * If ℒ' ⊆ℒ, then ℒ∖ℒ' is a Cameron-Liebler k-set of parameter x-x'.
		
  * If  is a Cameron-Liebler k-set in (n,q) or (n,q), then the complement of ℒ in (n,q) or (n,q) is a Cameron-Liebler k-set with parameter q^n-k-x or q^n+1-1/q^k+1-1-x respectively.
	


	This lemma follows due to Lemma <ref> and <cit.> for the projective case.


We now give some general results that will give connections between Cameron-Liebler k-sets in (n,q) and (n,q). These results will be similar to the results obtained in <cit.>. 




Here we prove Theorems <ref> and <ref> from the introduction.

Suppose that ℒ is a Cameron-Liebler k-set in (n,q) that misses the set of k-spaces in π_∞.  So the characteristic vector of  in (n,q) lies inside Im(P_n^T)= ((P_n))^⊥. Then, by Theorem <ref>, we obtain that for its characteristic vector χ_ℒ in (n,q) it holds that χ_ℒ∈ ((A_n))^⊥= Im(A_n^T). Here A_n is the point-(k-space) incidence matrix of the affine space from Construction <ref>.  Due to the size of , the parameter remains the same. This proves the assertion. 





Consider the characteristic vector χ_ℒ of the k-set  ℒ, then we know that χ_ℒ∈ Im(A_n^T). Here A_n is the point-(k-space) incidence matrix of (n,q). Due to Construction <ref>, we know that 

    [ χ_ℒ;  0̅ ]∈ Im(P_n^T),

with 0̅ the vector of the correct dimension that only contains zeroes. Note that χ_ℒ0̅ is in fact the characteristic vector of ℒ in (n,q). So ℒ is by definition a Cameron-Liebler k-set in (n,q). Due to the size of , the parameter remains the same.



	Suppose that ℒ is  a Cameron-Liebler k-set of parameter x in (n,q). Then   ℒ is a Cameron-Liebler k-set in (n,q) of the same parameter x if and only if ℒ is skew to the set of k-spaces in the hyperplane at infinity π_∞ of the affine space.


	Suppose that ℒ is a Cameron-Liebler k-set in (n,q) that misses the set of k-spaces in π_∞. Then, by Theorem <ref>, we obtain that ℒ is a Cameron-Liebler k-set in (n,q) that has the same parameter x.
	
	
Let ℒ be a Cameron-Liebler k-set in (n,q) whose restriction is a Cameron-Liebler k-set in (n,q) of the same parameter x. Then we can define the restriction of ℒ to (n,q) by ℒ'. Using Theorem <ref>, we know that ℒ' is a Cameron-Liebler k-set in (n,q). So, by Lemma <ref>, it follows that ℒ∖ℒ' is a Cameron-Liebler k-set of parameter x=0 in (n,q). This Cameron-Liebler k-set would only contain k-spaces in the hyperplane at infinity. So clearly ℒ∖ℒ'= ∅. Thus ℒ does not contain k-spaces at infinity.







	If there exists an affine Cameron-Liebler k-set with parameter x in (n,q), then there exists a Cameron-Liebler k-set of parameter x+q^n-k-1/q^k+1-1 in the projective closure (n,q).


Due to Theorem <ref>, we know that ℒ is a Cameron-Liebler k-set in (n,q).
Using Lemma <ref> and Example <ref>, we can extend every Cameron-Liebler k-set in (n,q) as follows: ℒ':= ℒ∪{K ∈Π_k   | K ⊆π_∞}. This set is a Cameron-Liebler k-set in (n,q) of parameter x+q^n-k-1/q^k+1-1.


 If we now combine Theorems  <ref> and <ref>, we find another interesting observation. Recall that every Cameron-Liebler k-set of parameter x in (n,q) can be extended by adding all k-spaces at infinity to a Cameron-Liebler k-set of parameter x+q^n-k-1/q^k+1-1 in (n,q). But the other way also holds. Suppose we have a Cameron-Liebler k-set of parameter x+q^n-k-1/q^k+1-1 in (n,q) that contains all the k-spaces at infinity. Then, by Lemma <ref>, we can remove all the k-spaces at infinity and obtain a Cameron-Liebler k-set of parameter x in (n,q).






 §.§ Equivalent definitions

Our goal will be to prove the following theorem, which presents equivalent definitions for Cameron-Liebler k-sets in (n,q).

	Consider the affine space (n,q), for n≥ 2k+1, and let ℒ be a set of k-spaces such that |ℒ|=x for a positive integer x. Then the following properties are equivalent.
	
		
  * ℒ is a Cameron-Liebler k-set in (n,q).
		
  * For every k-spread 𝒮, it holds that |ℒ∩𝒮|=x.
		
  * For every pair of conjugated switching k-sets ℛ and ℛ', |ℒ∩ℛ |= |ℒ∩ℛ'|.

If k=1 and we thus consider Cameron-Liebler line classes, then the following property is equivalent to the previous ones.

          
  4. For every line ℓ, the number of elements of ℒ affinely disjoint to ℓ is equal to
	
    ( q^2[ n-2;   1 ]_q +1 ) (x- χ_ℒ(ℓ))

	and through every point at infinity there are exactly x lines of ℒ.

 
	

To do this, we will need the following statement. Note that for every subspace τ of (n,q), we will denote [τ]_k:={K | K ∈Φ_k, K⊆τ}.


Let ℒ be a collection of k-spaces in (n,q) such that property (2) of Theorem <ref> holds. Suppose that τ_A is an arbitrary i-dimensional subspace in AG(n,q), with i≥max{k+1, 3}. Then property (2) also holds for ℒ∩ [τ_A]_k  in the (affine) subspace τ_A with respect to the k-spreads in τ_A.


Consider the projective closure PG(n,q) of AG(n,q) and let π_∞ be the hyperplane at infinity.  Let τ_A be an i-dimensional space in (n,q) and let τ be its projective closure, hence (τ∩π_∞)≥ k. 

 Our goal is to prove that ℒ restricted to τ_A satisfies property (2) with x. Pick a (k-1)-space I in τ∩π_∞ and denote E as the set of affine k-spaces through I and not in τ_A. Then it is clear that every k-space K∈ E is affinely disjoint towards every k-space in τ_A. It is also true that no two k-spaces in E share an affine point, and yet as a set they cover all affine points not in τ_A. So if we would choose a k-spread 𝒮 in τ_A, then we can always extend this k-spread to a k-spread in AG(n,q) in the following way

    𝒮':=𝒮∪ E.

Note that for every k-spread 𝒮 in τ_A, we can use the same E. So we have that 

    x=|ℒ∩𝒮'|=|ℒ∩𝒮|+|ℒ∩ E|.


Hence, we have for every k-spread 𝒮 in τ_A that |(ℒ∩[τ_A]_k)∩𝒮|=x-|ℒ∩ E|. But this last term is a constant, due to the fact that E was fixed for every k-spread 𝒮. This proves the theorem.



 We now can prove the main theorem.

We first prove the equivalence between the first 3 statements and then we prove the equivalence with statement 4.


  * From (1) to (3): Suppose that  is a Cameron-Liebler k-set  with characteristic vector χ_. Let ℛ and ℛ' be a pair of conjugated switching sets with characteristic vectors χ_ℛ and χ_ℛ' respectively. Then it holds, due to the definition of a pair of conjugated switching sets, that 
    χ_ℛ-χ_ℛ'∈(A_n),

where A_n is the point-line incidence matrix of (n,q). Since χ_∈Im(A_n^T)=((A_n))^⊥, we have that 
    χ_· (χ_ℛ-χ_ℛ')=0.

This concludes the statement.


  * From (3) to (2): Since for every pair of k-spreads 𝒮_1 and 𝒮_2, it holds that 𝒮_1∖𝒮_2 and 𝒮_2∖𝒮_1 are a pair of conjugated switching sets, we know that |ℒ∩𝒮_1|=c=|ℒ∩𝒮_2|. So we only need to show that c=x. To obtain this we double count the pairs (K, 𝒮), with K∈𝒮∩ and 𝒮 a k-spread of type II. Hence, due to the fact that ||=xnk and the number of k-space of type II through a fixed k-spread equals 1, we get that

    xnk· 1= nk· c.

Thus x=c, which completes the assertion.


  * From (2) to (1): If n=2k+1, then we know, due to Lemma <ref> (1) and Theorem <ref>, that  is a Cameron-Liebler k-set of parameter x in (n,q). Hence due to Theorem <ref>, the assertion follows.

Suppose now that n>2k+1. 
Here we will use similar techniques as in the proof of Theorem 2.9 in <cit.>, (more specifically in the step (7) to (1)). Let τ be an arbitrary (2k+1)-dimensional subspace, then we can consider ∩ [τ]_k in the affine space τ. Due to Theorem <ref>, we obtain that ∩ [τ]_k satisfies Property (2) in τ. Hence, using the previous observation, we obtain that ∩ [τ]_k is a Cameron-Liebler k-set of a certain parameter in τ. Note that this space τ was chosen arbitrarily. Thus it follows for every (2k+1)-dimensional subspace of (n,q) that, for the characteristic vector of ∩ [τ]_k, it holds that 

    χ_∩ [τ]_k∈Im(A_τ^T),

with A_τ the incidence matrix of τ. So we have that χ_∩ [τ]_k is a linear combination of the rows of A_τ. Note that due to the fact that A_τ has full row rank, it holds that this linear combination is unique. We only need to show that χ_ is uniquely defined by the vectors χ_∩ [τ]_k, with τ varying over all (2k+1)-spaces in (n,q). 

We first want to show that for every two (2k+1)-spaces τ and τ' the coefficients of the row corresponding to a point in τ∩τ' in the linear combination of χ_∩ [τ]_k and χ_∩ [τ']_k are equal.

Consider the subspace τ∩τ', and consider the corresponding columns of A_n. Then using the fact that A_τ∩τ' also has full row rank, we conclude that the linear combination of the rows that give χ_∩ [τ∩τ']_k is unique. Note that this unique linear combination has the same coefficients for the rows corresponding
with points in τ∩τ' as χ_∩ [τ]_k and χ_∩ [τ']_k has respectively. Here we also used the fact that an entry of A_n corresponding with a point of τ∖τ' or τ'∖τ and a k-space in τ∩τ' is zero. Thus we may conclude that the common rows in χ_∩ [τ]_k and χ_∩ [τ']_k have the same coefficient.

Using all of these (2k+1)-spaces, we have that χ_ is uniquely defined and χ_∈Im(A_n^T). This proves the assertion.



  * Equivalence between (4) and the rest, when k=1:
First if ℒ is an affine Cameron-Liebler line class with parameter x, then, by Theorem <ref>, we get that ℒ is a Cameron-Liebler line class in (n,q). Here we know that for every (affine) line ℓ, there are exactly
	
    q^2 [ n-2;   1 ]_q (x-χ_ℒ(ℓ))

	lines of ℒ projectively disjoint to ℓ. So we only still need to consider the lines of ℒ through the point ℓ∩π_∞. But since this is a point at infinity, which gives a line spread of type II, we have a total of x lines of ℒ through this point. Thus if we add those x-χ_ℒ(ℓ) lines of ℒ not equal to ℓ, we get a total of
	
    ( q^2 [ n-2;   1 ]_q +1 ) (x-χ_ℒ(ℓ))

	lines of ℒ disjoint to ℓ in AG(n,q).
	
	 Conversely, suppose that Property (2) holds, then we look at the corresponding projective space (n,q).  We can see that of the 
	
    ( q^2 [ n-2;   1 ]_q +1 ) (x-χ_ℒ(ℓ))

	lines of ℒ that are disjoint in (n,q) to an affine line ℓ, there are
	
    q^2 [ n-2;   1 ]_q (x-χ_ℒ(ℓ))

	elements of ℒ projectively disjoint to ℓ. 

If we now pick a line ℓ in π_∞, then there are
	 
    q^n-1/q-1-(q+1)= q^2 [ n-2;   1 ]_q

	 points in π_∞ not in ℓ. Through every such point, there are exactly x lines of ℒ that are disjoint to ℓ. If we combine these results we obtain that, by Theorem <ref>, it follows that ℒ is a  Cameron-Liebler line class in (n,q) with parameter x. Using  Theorem <ref>, we see that ℒ is also a Cameron-Liebler line class in (n,q) with the same parameter x.	







		



There is also another way to prove the equivalence of (1) and (2) for  k=1. For this, we will use association schemes. This will be done in Section 5.



	
Suppose that ℛ and ℛ' are a pair of conjugated switching k-sets in (n,q). If we define ℛ_A (and ℛ'_A) as the set of affine k-spaces of ℛ (and ℛ' respectively), then ℛ_A and ℛ_A' are conjugated switching k-sets in (n,q).


	 Since ℛ∩ℛ'=∅, it is clear that
	
    ℛ_A∩ℛ_A' = ∅.

	Since ℛ_A  and ℛ_A' arose from ℛ and ℛ', we know that no two k-spaces in the same set intersect. Thus both are still partial k-spreads. So we only need to show that they still cover the same set of points. If an affine point p is covered by ℛ_A, then this point (which is also a projective point) is also covered by ℛ and, hence, by ℛ'. Since this point was affine, the corresponding k-space of ℛ' is contained in ℛ'_A and, hence, the point is covered by ℛ'_A.


This lemma also shows that there exist conjugated switching sets in (n,q), since they exist in (n,q). This fact implies that Theorem <ref> does not have a trivial assumption.

































§ THE ASSOCIATION SCHEME OF AFFINE LINES

Our goal in this section is that we want to investigate the association scheme of lines in (n,q). We start with repeating some definitions of association schemes. If the reader is not familiar with association schemes, we refer to <cit.>.
 <cit.>
	Let X be a finite set. A d-class   association scheme is a pair (X, ℛ), where ℛ={ℛ_0,ℛ_1,..., ℛ_d} is a set of binary symmetrical relations with the following properties:
	
		
  * {ℛ_0,ℛ_1,..., ℛ_d} is a partition of X× X.
		
  * ℛ_0 is the identity relation.
		
  * There exist constants p_ij^l such that for x,y∈ X, with (x,y)∈ℛ_l, there are exactly p_ij^l elements z with (x,z)∈ℛ_i and (z,y)∈ℛ_j. These constants are called the intersection numbers of the association scheme.
		
		
	
	
	

In such a d-class association scheme we can define adjacency matrices as follows.
 
Consider a d-class association scheme (X,ℛ), where ℛ={ℛ_0,ℛ_1,..., ℛ_d} and X={x_1,...,x_n}. Then we can define d+1 matrices B_0,...,B_d of dimension n × n, such that

    (B_k)_ij= { 1,     if  (x_i,x_j)∈ℛ_k 
     0,     if  (x_i,x_j)∉ℛ_k. .

These matrices are called the adjacency matrices of the association scheme.

An important property of these adjacency matrices is that they can be diagonalized simultaneously, so we obtain maximal common (right) eigenspaces V_0,...,V_d.
 It is also known that these adjacency matrices span a (d+1)-dimensional commutative ℂ-algebra 𝒜. This algebra is called the Bose-Mesner algebra, which has a basis of idempotents {E_i | 0 ≤ i ≤ d}. One can prove that every matrix E_i is the orthogonal projection to the eigenspace V_i. If we would consider the common eigenspaces, we can denote all the eigenvalues in a matrix. This matrix is called the eigenvalue matrix.


Consider a d-class association scheme  (X,ℛ), where ℛ={ℛ_0,ℛ_1,..., ℛ_d} and X={x_1,...,x_n}. Let B_0,...,B_d be the adjacency matrices and {E_i | 0≤ i≤ d} be the idempotent basis of the Bose-Mesner algebra. Then the eigenvalue matrix P=[P_ij] and the dual eigenvalue matrix Q=[Q_ij] are the matrices for which it holds that

    B_j= ∑_i=0^d P_ijE_i    and   E_j= 1/n∑_i=0^d Q_ijB_i.

Here 0 ≤ i,j ≤ d.

Since every E_i in the idempotent basis gives an orthogonal projection onto V_i, it is indeed true that the values P_ij are the eigenvalues. Another important fact is that PQ= n I_d+1=QP. 

We now give a well-known example of such an association scheme.

<cit.>
Consider the set of lines in (n,q), with n≥ 3. Then this is a finite set, which we will call Π_1. Consider now the following set of relations ℛ'={ℛ_0', ℛ_1',ℛ_2'}. Then for ℓ and ℓ' in Π_1, we have that


  * (ℓ,ℓ')∈ℛ_0' if ℓ=ℓ'.
	
  * (ℓ,ℓ')∈ℛ_1' if they meet in a point.
	
  * (ℓ,ℓ')∈ℛ_2' when they do not meet at all.

It is well-known that Δ'=(Π_1, ℛ') gives an association scheme. This concept can be generalized to k-spaces in (n,q).


We try to define a similar association scheme for lines in (n,q). Note that due to the fact that there exists a concept of infinity in (n,q), this will lead to an increase of relations. Here we see that relation ℛ_1' will split into two separate relations.

Consider the set Φ_1 of lines of (n,q), with n≥ 3. Then we can define a 3-class association scheme 
Δ=(Φ_1, ℛ), where we denote the following relations ℛ={ℛ_0,ℛ_1,ℛ_2,ℛ_3} as follows. Pick ℓ,ℓ' ∈Φ_1, then

	
  * (ℓ,ℓ')∈ℛ_0 if ℓ=ℓ'.
	
  * (ℓ,ℓ')∈ℛ_1 if they meet in an affine point.
	
  * (ℓ,ℓ')∈ℛ_2 if they meet in a point at infinity.
	
  * (ℓ,ℓ')∈ℛ_3 when they do not meet in the corresponding projective space.

In order to prove that this is an association scheme, we can refer to <cit.>, where the intersection numbers were explicitly calculated. Another way to view this, is as a semilattice and conclude, due to <cit.>, that Δ is indeed an association scheme.







Let us consider Δ. If we number the lines of (n,q) in a fixed order 
    {ℓ_i | i ∈{ 0,..., q^n-1(q^n-1)/(q-1)-1}},

then we can define the adjacency matrices as B_0,B_1,B_2 and B_3. We know that these are q^n-1(q^n-1)/(q-1)×q^n-1(q^n-1)/(q-1) matrices  over ℂ that have common (right) eigenspaces.
If we define these common (right) eigenspaces  by  V_0,V_1,V_2 and V_3, then we know that ℂ^Φ_1= V_0 ⊥ V_1 ⊥ V_2 ⊥ V_3. Consider now the Bose-Mesner algebra 𝒜 of the association scheme Δ, which will be a 4-dimensional ℂ-algebra. Then we know that 𝒜 has  a basis of idempotents {E_i | 0≤ i≤ 3}, such that every E_i is the orthogonal projection onto V_i.




 §.§ Calculating the eigenvalue matrix and dual eigenvalue matrix of Δ

In order to find the eigenvalue matrix P and the dual eigenvalue matrix Q, we need to define some other matrices known as the intersection matrices.

	Consider a d-class association scheme with intersection numbers p_ij^k. Then we can define the following (d+1)× (d+1) matrices for i ∈{0,...,d}
	
    𝒫_i=[p_ij^k]_k,j,

	hence the (k,j)-entry is 𝒫_i(k,j)=p_ij^k. These matrices are known as intersection matrices. 

These intersection matrices for the association scheme of Construction <ref> can be calculated:

    𝒫_0 = [ 1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1 ],


    𝒫_1= [                       0         q( q^n-1/q-1-1)                       0                       0;                       1 (q-1)^2+( q^n-1/q-1-2 )                     q-1  (q-1)( q^n-1/q-1-1-q );                       0                     q^2                       0      q( q^n-1/q-1-1-q );                       0                     q^2                       q  q( q^n-1/q-1-1-(q+1) );                         ],


    𝒫_2= [         0         0   q^n-1-1         0;         0       q-1         0   q^n-1-q;         1         0   q^n-1-2         0;         0         q         0 q^n-1-1-q;           ]

and

    𝒫_3=([                                                                     0                                                                     0                                                                     0                                  q^2 - (q + 1) q^n +
    		q^2  n-1/q - 1;                                                                     0                                                            -q^2 + q^n                                                            -q + q^n-1         q^3
    		+ q^2 - (2   q^2 + q - 1) q^n-1 - q + q^2 
    			n-1/q - 1;                                                                     0                                                  -q^3 - q^n + 1/q - 1                                                                     0                         q^3 +
    		q^2 - (2   q + 1) q^n + q^2 n-1/q - 1;                                                                     1                                        -q^3 + q^2 - q - q^n + 1/q - 1                                                      -q
    		- 1 + q^n-1 q^3 + 3   q^2 - (2   q^2 + 2
    			  q - 1) q^n-1 - 2   q+ q^2 n-1/q - 1 ]).

For these calculations we refer to <cit.>.
<cit.>
	Consider a d-class association scheme together with the eigenvalue matrix P and the intersection matrices 𝒫_i, for i ∈{0,...,d}. Then
	
    P ·𝒫_i · P^-1= [ P_0i    0    0  ...    0;    0 P_1i    0  ...    0;    ⋮    ⋮    ⋮    ⋱    ⋮;    0    0    0  ... P_di ].

	Consequently, 𝒫_i and the adjacency matrix B_i have the same eigenvalues.
	

This lemma implies that the intersection matrices can be diagonalized simultaneously.
In order to find P, we use the following theorem.

	
	
	 

	
	
	
	
	
	
	

<cit.>
Consider a d-class association scheme and let u_i, for i ∈{0,...,d}, be the set of common left normalized (column) eigenvectors of the intersection matrices. Here we mean with normalized, that (u_i)_0=1 for every i. Then  the rows of the eigenvalue matrix P are the elements (u_i)^T.

This lemma together with the following left (normalized) eigenvectors of the intersection matrices above, will give us the eigenvalue matrix.


    u_0= ( 1 , -q + q^n + 1 - 1/q - 1 - 1 , q^n - 1 - 1 , q
    	+ q^n + q^2 n - 1 - 1/q - 1 - 2  (q^n + 1
    			- 1)/q - 1 + 1)^T,


    u_1=( 1 ,-q + q^n - 1/q - 1 - 1 , -1 , q - q^n
    		- 1/q - 1 + 1)^T,


    u_2=( 1,-q,-1,q)^T ,


    u_3=(  1 , -q , q^n - 1 - 1 , q - q^n - 1)^T.


These left eigenvectors were calculated by using Sage <cit.>. From this together with the lemma above, we can obtain the eigenvalue matrix P of the association scheme Δ, see Construction <ref>. So we conclude that

    P=   
    ([                                  1               -q^2 - q^n + 1/q - 1                          q^n-1 - 1 q^2 - (q + 1) q^n + q^2  n-1/q - 1;                                  1                   -q^2 - q^n/q - 1                                 -1                   q^2 -
    	q^n/q - 1;                                  1                                 -q                                 -1                                  q;                                  1                                 -q                          q^n-1 - 1                          q - q^n-1;                                    ])

and due to PQ=q^n-1( q^n-1/q-1) I_4= |Φ_1| I_4=QP, we obtain that

    Q=([                                                     1                                               q^n - 1              -(q^2 + 1) q^n - q^2 - q^2  
    		n/q^2 - q                                         q^n - q/q - 1;                                                     1          (q^2 + 1)
    	q^n - q^2 - q^2   n/q^2 - q^n + 1        -(q^2 + 1) q^n - q^2 - q^2  
    		n/q^2 - q^n + 1                                                    -1;                                                     1                                   q - q^n + 1/q^n - q (q^2 + 1) q^n - q^2 - q^2  
    		n/(q - 1) q^n - q^2 + q                                        q^n - q/q -
    	1;                                                     1                                 q - q^n +
    		1/q^n - q                                   q^n + 1 - q/q^n - q                                                    -1 ]).





§ CAMERON-LIEBLER LINE CLASSES IN (N,Q)



In this section we will give an alternative proof for the following statement, which is a special case of Theorem <ref>.


Suppose that ℒ is a set of lines in (n,q), n≥ 3, such that for every line spread 𝒮 it holds that 

    |ℒ∩𝒮|=x.

Then ℒ is a Cameron-Liebler line class of parameter x.

To prove this theorem, we make use of the association scheme of Section 4. Let us recall Δ from Construction <ref>, then we first start with the concept of inner distributions.




 §.§ Inner distribution

(<cit.> and <cit.>)
	Consider a d-class association scheme (X, ℛ) and let  be a subset of X, then we can consider its characteristic vector χ_. For this vector we can define its inner distribution as the row vector u=( u_0,   u_1,   u_2,   ...,   u_d) with elements in ℝ, for which it holds that
	
    u_i= 1/|||ℛ_i ∩ (×)|.
 
	
	


	Note that for the inner distribution u=( u_0,   u_1,   u_2,   ...,   u_d) of a certain characteristic vector χ_, it holds that
	
    u_i= 1/||χ_^T · B_i ·χ_,

	for 0 ≤ i≤ d.

The following theorem will give us a way to observe in which eigenspaces of Δ a characteristic vector lies in.
(<cit.>)
	Consider a d-class association scheme Γ=(X, ℛ) and let 𝒜 be its Bose-Mesner algebra. Denote the idempotent basis of 𝒜 by {E_i | 0≤ i≤ d}, with common eigenspaces V_0,...,V_d. Then it follows for every subset  of X, that its characteristic vector χ_∈ℝ^d can be written as follows
		
    χ_= a_0 v_0+a_1v_1+⋯+a_dv_d,

	with v_i ∈ V_i and a_i ∈ℝ for each 0≤ i ≤ d. If u is the inner distribution of χ_, then the following properties are equivalent for fixed 0≤ i ≤ d


  * (u · Q)_i=0, with Q the dual eigenvalue matrix of Γ.

  * E_i·χ_=0.

This last property implies that the projection of χ_ onto the eigenspace V_i is zero, thus a_i=0.

Now we mention the next very useful theorem stated in <cit.>. Our formulation is based on unpublished notes by Klaus Metsch.

<cit.>
	Let Γ=(X, ℛ) be a d-class association scheme, with {E_i | 0≤ i≤ d} the idempotent basis of the Bose-Mesner algebra. Suppose G is a subgroup of Aut(Γ) that acts transitively on X and whose orbits on X× X are the relations ℛ_0,..., ℛ_d.  Let χ and ψ be vectors of ℝ^|X|. Then the following two statements are equivalent.
	
		
  * For all k≥ 1, we have E_k·χ=0 or E_k·ψ=0.
		
  * χ·ψ^g is constant for all g ∈ G.
	
	


	We know that property (1) is equivalent with the fact that both vectors lie in opposite (common) eigenspaces besides V_0. 	


	A second observation is that in the 3-class association scheme Δ, the group (n,q) acts indeed transitively on pairs of lines of the same type in (n,q). It is also clear that elements of (n,q) send line spreads of type I and type II to line spreads of type I and type II respectively.

The same happens for line spreads of type III, we explicitly proved this fact. But first we give a definition.

Let 𝒮 be a line spread of type III, with the property that all the chosen points p_i in π_n-2 are chosen differently. Then we call 𝒮 a line spread of type III^+.


  	The affine collineation group AGL(n,q) sends spreads of type III to spreads of type III. In particular, it sends spreads of type III^+ to spreads of type III^+.
 

	Consider 𝒮 to be a line spread of type III, defined by an (n-2)-space π_n-2⊆π_∞, the set of hyperplanes H= {π_i | i ∈{1,...,q }} and the q points p_i∈π_n-2. If we now consider θ∈ AGL(n,q), then π_n-2^θ⊆π_∞ and all the hyperplanes of H= {π_i | i ∈{1,...,q}} are sent to different hyperplanes through π_n-2^θ. Also all the points p_i are sent to points p_i^θ∈π_n-2^θ, which if they all are different points they shall remain so. We conclude that 
	
    𝒮^θ= {K^θ∈Φ_1 | p_i∈ K ⊆π_i  for some  i }={K' ∈Φ_1 | p_i^θ∈ K' ⊆π_i^θ for some  i },

	which is of the required form.





 §.§ About the common eigenspaces


In this section we give a basis for V_0⊥ V_1 and V_0⊥ V_3, and give a spanning set for V_0⊥ V_2 ⊥ V_3 in the association scheme Δ from Construction  <ref>.

A point-pencil in (n,q) or (n,q) is the set of lines through a fixed point in (n,q) or (n,q) respectively.

(<cit.>)
	The point-line incidence matrix of (n,q)  and (n,q) has full rank, which equals the number of points in (n,q) and (n,q) respectively. Hence the rows of these incidence matrices, which correspond to points and give point-pencils are linearly independent.

<cit.>
		Consider the dual eigenvalue matrix Q in an association scheme, then Q_0i= (V_i). 

We now prove the following theorem that characterizes the space V_0⊥ V_1.

 	Consider the affine space AG(n,q) and the 3-class association scheme Δ (see Construction <ref>). Then the point-pencils form a basis of the space V_0⊥ V_1.
 

	Let us first find the inner distribution of a point-pencil. It can be seen that this is equal to
 	
    u=(1, q^n - q/q - 1, 0, 0).

Thus we obtain that 
    u· Q=(q^n - 1/q - 1,  -(q + 1)
    	q^n-1 - 1 - q^2  n-1/q - 1,   0, 0).
 Hence these first two entries will never be zero for n>1 and q a prime power. So Theorem <ref> shows that all the point-pencils lie inside V_0⊥ V_1. 
 


 From Lemma <ref> and the description of Q in Equation (<ref>), we obtain that (V_0⊥ V_1)=1+ ( q^n-1)=q^n. This number is equal to the number of point-pencils in (n,q). Together with Lemma <ref>, we have that the point-pencils form a basis for the space V_0⊥ V_1.


We now give a second result on these eigenspaces.

In the affine space AG(n,q) with association scheme Δ (see Construction <ref>), we have the following:


  * The line spreads of type II form a basis for the space V_0⊥ V_3.


  * The space V_0 ⊥ V_2 ⊥ V_3 is spanned by line spreads of type III^+  and for the characteristic vector χ_𝒮 of such a line spread 𝒮, it holds that E_2 ·χ_𝒮≠ 0 ≠ E_3 ·χ_𝒮.





  * This is done in a similar way as the previous lemma. The inner distribution of a line spread 𝒮_1 of type II is equal to
 
    s_1=(1, 0, q^n - 1 - 1, 0).

From this we obtain that 
    s_1· Q=(q^n - 1, 0, 0, q^2 n-1 - q^n/q -
     	1).

The first and last entry will never be zero for n>1 and q a prime power.
 So from Theorem <ref>, we obtain that
 χ_𝒮_1∈ V_0⊥ V_3. Note that these line spreads are in fact subsets of point-pencils in the hyperplane at infinity in (n,q). But due to the fact that no two subsets contain the same line, we know that these line spreads are also linearly independent. From Lemma <ref> and the description of Q in Equation (<ref>), we obtain that
	
    (V_0⊥ V_3)= 1+q^n-q/q-1= q^n-1/q-1.

	This dimension is equal to the number of spreads of type II, which proves that these line spreads form a basis.


  * Analogously for a line spread 𝒮_2 of type III^+. The inner distribution is equal to

    s_2=(1, 0, q^n - 2 - 1, q^n - 1 - q^n - 2),

such that 
    s_2 · Q =(q^n - 1, 0, q^2n-2 - q^n-2, -(q^2
    		- q + 1) q^n-2 - q^2  n-2/q - 1).
 
The first and third entry will never be zero for n>1 and q a prime power. The last entry needs some arguments. If (q^2-q+1)q^n-2-q^2n-2=0, then q=0 or q(q-1)=q^n-1 and thus q=0 or q^n-1+...+q^2+1=0. This statement is never true if n>1 and q a prime power. Hence, using Theorem <ref>, we obtain that χ_𝒮_2∈ V_0⊥ V_2⊥ V_3 and especially we have that E_2 ·χ_𝒮_2≠ 0 ≠ E_3 ·χ_𝒮_2. 

To show that V_0⊥ V_2 ⊥ V_3 is spanned by line spreads of type III^+, we use Theorem <ref>. Suppose that these line spreads would span V_0⊥ W_1, with V_2 ⊥ V_3= W_1 ⊥ U_1, then we want to show that W_1=V_2 ⊥ V_3. If there exists a ψ∈ U_1∖{0}, then we know that E_2 ·ψ≠ 0 or E_3 ·ψ≠ 0. Let us now consider a line spread 𝒮 of type III^+, then we know that its characteristic vector χ_𝒮∈ V_0⊥ W_1 ⊆ V_0⊥ V_2 ⊥ V_3. Hence χ_𝒮 lies in the complementary space V_0⊥ W_1 of U_1, thus χ_𝒮·ψ=0. Due to Lemma <ref>, we have that for every θ∈(n,q) it holds that χ_𝒮^θ·ψ=0. So from Theorem <ref>, we obtain that  E_2 ·χ_𝒮  = 0 or E_3 ·χ_𝒮= 0. This is a contradiction with the end of the preceding paragraph.








 §.§ The proof of Theorem <ref>


Consider the association scheme Δ from Construction <ref> and let  ℒ be a line set in (n,q) such that for every line spread 𝒮 it holds that |ℒ∩𝒮|=x. 
Then our goal is to prove that χ_ℒ∈ V_0⊥ V_1, since, from Theorem <ref>, it then follows that χ_ℒ∈ Im(A_n^T)  and hence ℒ is a Cameron-Liebler line class of parameter x. 

Consider 𝒮 to be a line spread of type III^+. Such a line spread exists if we can choose q different points in π_n-2. This is clearly the case if n≥ 3. If we denote the characteristic vector of 𝒮 by χ_𝒮, we know by the definition of ℒ that

    χ_ℒ·χ_𝒮=x.

In combination with Lemma <ref>, we know that

    χ_ℒ·χ_𝒮^θ=x,

for all θ∈(n,q). Hence, from Theorem <ref> and Lemma <ref> (Property (2)) which states that E_2·χ_𝒮≠0 and that E_3·χ_𝒮≠0, we may conclude that E_2·χ_ℒ=0=E_3·χ_ℒ. Thus using Theorem <ref>, we obtain that

    χ_ℒ∈ V_0⊥ V_1.

This proves the theorem.

















































































§ CLASSIFICATION RESULTS

In this section we will focus on some classification results of Cameron-Liebler k-sets in (n,q) with certain parameters. From now on we will use all the equivalent definitions of Theorem <ref> to describe Cameron-Liebler k-sets. In order to obtain a classification result, we will need the following result.
 <cit.>
	
	Let 0 ≤ t ≤ k be positive integers. Let 𝒮 be a set of k-spaces in (n,q), pairwise intersecting in at least a t-space. If n≥ 2k+1, then
	
    |𝒮|≤[ n-t; k-t ]_q.

	Equality holds if and only if 𝒮 is the set of all k-spaces through a fixed t-space, or n=2k+1 and 𝒮 is the set of all k-spaces inside a fixed (2k-t)-space. 
	
	
	
	
	


Before we give the classification results, the reader should keep Example <ref> in mind, where we gave some examples of Cameron-Liebler k-sets in (n,q). Note that by restriction to (n,q) we actually obtain fewer examples or stronger conditions on Cameron-Liebler k-sets. We first give the following lemma.

A non-empty set of k-spaces contained in a hyperplane of (n,q), is not a Cameron-Liebler k-set in AG(n,q).


Let ℒ be a Cameron-Liebler k-set in (n,q), that consists out of a set of k-spaces inside a hyperplane π. Pick a k-space K∈ℒ, which we can consider in the projective closure PG(n,q). Then we can define a type II k-spread 𝒮_1 as the set of affine k-spaces through  K∩π_∞. Analogously we can define  𝒮_2 as the set of affine k-spaces through another (k-1)-space at infinity that does not lie in π. It is clear that

    |ℒ∩𝒮_1|≠ |ℒ∩𝒮_2|=0.

This is a contradiction with Lemma <ref>.

This lemma gives the following classification result.
<cit.>
Let q ∈{2,3,4,5}. Then all Cameron-Liebler k-sets in (n,q) consist out of all the k-spaces through a fixed point, if k+1,n-k ≥ 2 and either (a) n ≥ 5 or (b) n = 4 and q = 2.


Here we use the combination of Theorem <ref>, Theorem <ref> and Lemma <ref>.




 §.§ Cameron-Liebler k-sets with parameter x=1 in (n,q)


Consider ℒ  as the set of k-spaces through a fixed affine point in (n,q). Then ℒ is a Cameron-Liebler k-set in (n,q) of parameter x=1. This can be seen from  Corollary <ref> together with the fact that ℒ also is a Cameron-Liebler k-set in (n,q) of parameter x=1.
	

Using Theorem <ref>, we know that for n> 2k+1 this example is the only example of a Cameron-Liebler k-set of parameter x=1  in PG(n,q). If n=2k+1, the set of all k-spaces in a hyperplane also gives an example of a Cameron-Liebler k-set with parameter x=1. This fact gives the following theorem. 
















































The following theorem also proves the first part of Theorem <ref> from the introduction.

	Consider the affine space (n,q) and let ℒ be a Cameron-Liebler k-set with parameter x=1 in this affine space. If also n ≥ 2k+1, then ℒ consists of all the k-spaces through an affine point.



Using Theorem <ref>, we obtain that every Cameron-Liebler k-set in (n,q) is a Cameron-Liebler k-set in (n,q). The latter will have the same parameter x=1. From Theorem <ref> and Theorem <ref>, the assertion follows.








We also will be able to improve this result for n≥ k+2, see Corollary <ref>.



 §.§ Cameron-Liebler line classes of parameter x=2 in (n,q)






In this section, our goal will be to exclude the parameter x=2 for Cameron-Liebler line classes in (n,q), with n≥ 3. To do this, we will need the following lemma.

	Consider an affine Cameron-Liebler line class ℒ with parameter x=2 in (n,q), with n≥ 4. Then for every two points p_1 and p_2 in π_∞, there are two lines of ℒ through each of them. These 4 lines generate at most a 3-space.


	Denote the lines of ℒ through p_1 by ℓ_1 and ℓ_2, and denote the lines of ℒ through p_2 by r_1 and r_2.
	
	We start by considering ⟨ℓ_1,ℓ_2,r_1 ⟩, then we know that (⟨ℓ_1,ℓ_2,r_1 ⟩) ∈{2,3,4}. Suppose now that (⟨ℓ_1,ℓ_2,r_1 ⟩)=4. If we now call the plane ⟨ℓ_1,ℓ_2⟩=τ, then we know that 
	(τ∩π_∞)=1.
	This intersection line we call t_1, see Figure <ref>. Now we can find a (n-2)-space π⊆π_∞ such that 
	
    ⟨ t_1, p_2⟩⊆π

	and 
    ⟨ℓ_1,ℓ_2,r_1 ⟩∩π_∞⊈π.

	This is possible, since first ⟨ℓ_1,ℓ_2,r_1 ⟩∩π_∞ is a 3-dimensional space that contains ⟨ t_1, p_2⟩ as a 2-dimensional subspace. And, secondly, since n≥ 4, we know that n-2 ≥ 2.

 Thus with the identity of Grassmann, we obtain that
	
    (⟨π, τ⟩)=n-2+ 2-( π∩τ )=n-1,


    (⟨π,r_1⟩)=n-2+1-(π∩ r_1)=n-1

	and
	
    (⟨ℓ_1, ℓ_2, r_1, π⟩)    = (⟨ℓ_1,ℓ_2, r_1 ⟩)+ (π)- (⟨ℓ_1,ℓ_2, r_1 ⟩∩π)
       = (⟨ℓ_1,ℓ_2, r_1 ⟩)+ (π)- ((⟨ℓ_1,ℓ_2, r_1 ⟩∩π_∞) ∩π)
       =4+(n-2)-2=n.

    Thus we can conclude that ⟨π, ℓ_1,ℓ_2 ⟩≠⟨π, r_1 ⟩. So we can define a line spread 𝒮 of AG(n,q) (of type[This exists due to n≥ 4.] III) that contains ℓ_1,ℓ_2 and r_1, such that |ℒ∩𝒮|≥ 3, which is a contradiction. Thus from this we can conclude that 
	(⟨ℓ_1,ℓ_2,r_1⟩)≤ 3. Analogously, we can obtain that ⟨ℓ_1,ℓ_2,r_2⟩ and in general every space generated by three of these four lines is at most a 3-dimensional space. To show that these four lines span at most a 3-space, we need to consider some cases.
	
		
  * First if p_2 ∉t_1, then ⟨ℓ_1,ℓ_2,r_2⟩ intersects ⟨ℓ_1,ℓ_2,r_1⟩ in at least the point p_2 and the plane τ. So, since p_2∉τ, both 3-spaces are the same. Hence, from now on, we assume that p_2 ∈ t_1.
		
  * If r_1 and/or r_2 are contained in τ, we are done, since these four lines span a plane or a 3-space.
		
  * If r_1 and r_2 are not contained in τ and ⟨ r_1, r_2 ⟩∩τ= t_1, then again we can conclude that all four lines lie in a 3-space.
		
  * If r_1 and r_2 are not contained in τ and ⟨ r_1, r_2 ⟩∩τ≠ t_1. Then ⟨ r_1,r_2 ⟩∩π_∞∩ t_1=p_2 and we analogously obtain from previous cases that  ⟨ r_1,r_2,ℓ_2⟩ and ⟨ r_1,r_2,ℓ_1⟩ are two 3-spaces that now contain p_1 ∉⟨ r_1,r_2 ⟩ and ⟨ r_1,r_2 ⟩.
	
	 This proves the lemma.


Let us now state the following known theorem.
[Folklore]
	Consider a set of k-spaces ℰ in (n,q), 1≤ k≤ n-1, such that every two k-spaces intersect in a (k-1)-space. Then ℰ consists out of a subset of all the k-spaces through a fixed (k-1)-space or all the k-spaces inside a (k+1)-space.

We are ready to state the main theorem. This theorem proves a second part of Theorem <ref> in the introduction.

	There does not exist a Cameron-Liebler line class ℒ of parameter x=2 in (n,q), n ≥ 3.


The case for n=3 is proven in <cit.>, so we may suppose that n≥ 4.

Suppose there exists a Cameron-Liebler line class ℒ of parameter x=2. Then we can define ℰ as the set of planes, such that each plane is defined by a point at infinity and the two corresponding lines of ℒ through this point. Due to Lemma <ref> we know that these planes pairwise intersect in a line or coincide.
Using Theorem <ref>, we can conclude that ℰ consists out of a subset of all the planes through a fixed line or all the planes in a 3-space σ. If ℰ would consist out of all the planes in a 3-space σ, then ℒ is a set of lines inside σ and thus inside a certain hyperplane. This is a contradiction with Lemma <ref>. So we conclude that ℰ consists out of planes through a fixed line ℓ.


 If ℓ would be a line at infinity then |ℒ|=2(q+1), since every point at infinity belongs to two lines of ℒ. Note that for n≥ 3 this number is strictly smaller than the size of a Cameron-Liebler line class in AG(n,q) of parameter x=2, which has size 2q^n-1/q-1. So the line ℓ should be affine. See Figure <ref>.
















Note that |ℰ|= q^n-1-1/q-1, since every point at infinity will define exactly one plane in ℰ by definition. Hence this is also the number of all planes through a line, such that we know that ℰ consists out of all the planes through ℓ. Let us denote s= ℓ∩π_∞.
Now we can use Theorem <ref>, where we have shown that being an affine Cameron-Liebler line class in (n,q) is equivalent with the following statement.
	For every affine line ℓ_1, the number of affine lines in ℒ  disjoint to ℓ_1 in (n,q) is equal to 
    (q^2q^n-2-1/q-1+1)(x-χ_ℒ(ℓ_1)
    	).
 Consider now an affine line ℓ' through s that is contained in ℒ and not equal to the intersection line ℓ. Note that this is always possible, since s belongs to exactly two lines of ℒ. Then all lines of ℒ except those in the plane ⟨ℓ,ℓ'⟩, are disjoint to ℓ'. Since every other plane through ℓ has 2q lines of ℒ skew to ℓ' and we also need to count the other line of ℒ through s,
this number is equal to 
	
    (|ℰ|-1)· 2q+1.

With some calculations, we find that this equals
	
    2q^2( q^n-2-1/q-1) +1.

	This number should be equal to Equation (<ref>). In this equation we fill in χ_(ℓ')=1, and we obtain that there should be  (q^2 q^n-2-1/q-1+1) (2-1
	) lines disjoint to ℓ'∈ℒ. These two numbers are not equal. So there does not exist Cameron-Liebler line classes with parameter x=2 in (n,q), n≥ 4 either.



















In contrast with the case for x=1, we have proven the preceding theorem without using Theorem <ref>. The reason for this choice was that we are not able to deduce the preceding theorem from results in (n,q). Reducing to the projective case and using Theorem <ref>, we observe that there do not exist Cameron-Liebler k-sets of parameter x=2 in (n,q) if q∈{2,3,4,5} together with n≥ 5 or n=4, but q=2. 

While using Theorem <ref>, we obtain a non-existence result for n≥ 5 and q≥ 3. Both statements combined are weaker than the preceding theorem. This seems a small difference, but for general k the results are even stronger. This will be proven in Corollary <ref>.







 §.§ Characterisation of the parameter x of Cameron-Liebler k-sets in AG(n,q)


Our goal here is to prove that there do not exist Cameron-Liebler k-sets in (n,q) of parameter x=2, with n≥ k+2. As we already briefly discussed, we can not deduce this result so far by using Theorem <ref> and thus reducing to the projective case. If we would reduce to (n,q) and use Theorem <ref> and Theorem <ref>, we would obtain a non-existence result for (1) n≥ 3k+2 and q≥ 3, (2)  n≥ 5 with q∈{2,3,4,5} and (3) n=4 with q=2.  Note that this statement is significantly weaker than the previous claim. In achieving this goal we will also obtain a minor non-existence condition on the parameters of certain Cameron-Liebler k-sets in AG(n,q). We will do so by connecting every Cameron-Liebler k-set to a Cameron-Liebler line class of the same parameter. For this we will need the following observation.

	Let ℒ be a Cameron-Liebler k-set with parameter x in (n,q). Then the number of elements of ℒ through a fixed i-space at infinity, for -1 ≤ i ≤ k-2,  is equal to 
	
    [ n-i-1; k-i-1 ]_q x.



Consider an i-space I at infinity. Then we can count all the elements of ℒ through I, by counting the number of (k-1)-spaces through I inside π_∞ and multiplying this by the number of elements of ℒ through each (k-1)-space.  Both numbers are known, since through every (k-1)-space at infinity there are, by Lemma <ref>, in total x elements of ℒ.
The assertion follows

A remarkable observation is that n-i-1k-i-1_q x equals the size of a Cameron-Liebler (k-i-1)-set in (n-i-1,q). This observation will lead to the following construction.





Consider ℒ to be a Cameron-Liebler k-set in (n,q) of parameter x, and pick an i-space I at infinity, for 0≤ i≤ k-2 and n≥ k+2. Then, by Lemma <ref>, there are 
    [ n-i-1; k-i-1 ]_q x

k-spaces of ℒ that contain I. Pick now an (n-i-1)-space π in (n,q) skew to I. Then every k-space of ℒ through I will intersect π in a (k-i-1)-space in (n,q), see Figure 5. 

We shall denote this set of (k-i-1)-spaces by 𝒥. Remark that π is in fact a projective space of dimension n-i-1, which is an affine space π_A (≃(n-i-1,q)) of the same dimension with hyperplane at infinity equal to π∩π_∞.



We first give a result on k-spreads.

Consider Construction <ref>. Then every (k-i-1)-spread in π_A  can be extended to an affine k-spread in (n,q), such that all the k-spaces in this k-spread contain I.


Let 𝒮' be a (k-i-1)-spread in π_A, then  

    𝒮:={⟨ I, N ⟩| N ∈𝒮'}

is a k-spread in AG(n,q).



Suppose that n≥ 2k-i. Consider Construction <ref>. Then the set 𝒥 is a Cameron-Liebler (k-i-1)-spaces in π_A (≃(n-i-1,q)), which has the same parameter x.


Consider ℒ and 𝒥 as in Construction <ref>, then we need to prove that 𝒥 is a Cameron-Liebler (k-i-1)-set with the same parameter x in π_A. We know that, due to Lemma <ref>, every (k-i-1)-spread 𝒮' in π_A can be extended to a k-spread 𝒮 in (n,q) such that every element of 𝒮 contains I. Since ℒ is a Cameron-Liebler k-set in (n,q), it holds that |ℒ∩𝒮|=x. But since every k-space of ℒ∩𝒮 contains I, it projects to an element of 𝒥∩𝒮' and vice versa. So we have that |𝒥∩𝒮'|=x. Using Theorem <ref>, which gives the condition on n, we have proven the theorem.



Note that this construction cannot be done in a similar way for (n,q).

We now have found a way to reduce Cameron-Liebler k-sets to Cameron-Liebler line classes of the same parameter x. Hence, this will lead to a transfer of non-existence results.

Let ℒ be a Cameron-Liebler k-set in AG(n,q), with n≥ k+2. Suppose now that ℒ has parameter x, then x satisfies every condition which holds for Cameron-Liebler line classes in AG(n-k+1,q).


We can use Theorem <ref> for i=k-2, thus n≥ k+2, and obtain that there exists a Cameron-Liebler line class in (n-(k-2)-1,q) with the same parameter x. This proves the theorem.

This theorem has the following consequences.

	Suppose that ℒ is a Cameron-Liebler (n-2)-set with parameter x of (n,q), then it holds that
	
    x(x-1)/2≡ 0  (q+1).

 

Use Theorem <ref> for n-2=k and the modular equality from  <cit.> or <cit.>.

The following corollary completes the proof of x=2 in Theorem <ref> in the introduction.

There do not exist Cameron-Liebler k-sets in (n,q) with parameter x=2, with n≥ k+2.


If there would exist a Cameron-Liebler k-set ℒ of parameter x=2 in (n,q), we can use Theorem <ref> and obtain that there exists a Cameron-Liebler line class in (n-k+1,q) with parameter x=2. Since n-k+1≥ 3, we may use Theorem <ref> to obtain a contradiction.


We also have the following improvement of Theorem <ref> To conclude Theorem <ref>.

The only Cameron-Liebler k-set  of parameter x=1 in (n,q), with n≥ k+2, consists of the set of k-spaces through a fixed point.


Again we use Theorem <ref> to obtain a Cameron-Liebler line class ' of parameter x=1 in (n-k+1,q). But combined this with Theorem <ref> for k=1 in (n-k+1,q), we obtain that ' consists out of all lines through a fixed point. Using Construction <ref>, it is easy to see that  is the set of all k-spaces through a point.







§ CAMERON-LIEBLER SETS OF HYPERPLANES IN (N,Q)

In this section we study Cameron-Liebler sets of hyperplanes in (n,q). We will be able to give a complete classification. This will be done by giving a classification of affine (n-1)-spreads.

The only (n-1)-spreads in (n,q) are spreads of type II.


Let 𝒮 be an (n-1)-spread, then we need to prove that ℒ is of type II. Consider now the projective closure (n,q), then we know that every two hyperplanes of 𝒮 will intersect in an (n-2)-space. Due to the fact that 𝒮 is an affine (n-1)-spread, these intersections must lie at infinity. But since every affine hyperplane only has an (n-2)-dimensional intersection with infinity, all these (n-2)-spaces need to be the same.
Hence, we have that 𝒮 is of type II.

The fact that we are able to classify all (n-1)-spreads in (n,q), gives us information how we can construct these Cameron-Liebler sets of hyperplanes in (n,q).

Let ℒ be a  set of affine hyperplanes  in (n,q) and consider the projective closure (n,q). Then ℒ is a Cameron-Liebler set of hyperplanes of parameter x if and only if ℒ is a set of hyperplanes such that through every (n-2)-space at infinity we have chosen x arbitrary hyperplanes.


The proof is similar as we have done for lines in Section 5. Hence, we find a 2-class association scheme Δ=(Φ_n-1, ℛ), with ℛ:={ℛ_0, ℛ_1, ℛ_2}. Here ℛ_0 is the identical relation, while ℛ_1 is the relation that denotes that the two hyperplanes are disjoint and thus intersect in an (n-2)-space at infinity and ℛ_2 is the relation that denotes intersection.  Using similar techniques as in Section 5, we obtain that 

    P=[           1         q-1 q^n+1-1/q-1;           1         q-1          -1;           1          -1           0;  ] and  Q=[          1  q^n-q/q-1      q^n-1;          1  q^n-q/q-1 -q^n-1/q-1;          1         -1          0 ].

Due to Lemma <ref>, we have that the inner distribution of an (n-2)-spread is equal to the vector v=(1,q-1,0). Also point-pencils have inner distribution w=(1, 0, q^n-1/q-1-1). Using Theorem <ref>, we know that the (n-2)-spreads lie inside V_0⊥ V_1 and the point-pencils lie inside V_0⊥ V_2, in fact both sets will span these spaces. Combining this with Theorem <ref>, we find that a Cameron-Liebler (n-2)-set can be characterized by the constant intersection of (n-2)-spreads. Hence, due to  Lemma <ref>, the assertion follows.


















§ FUTURE RESEARCH
 
We want to end this paper with some suggestions for further research. 
 One could also attempt to classify more parameters of a Cameron-Liebler k-set in (n,q), since intuitively this will be less difficult than (n,q). We remind the reader of Theorem <ref>, which states that Cameron-Liebler k-sets in (n,q) are special cases of Cameron-Liebler k-sets in (n,q). Another interesting problem  is to look for examples of Cameron-Liebler k-sets in (n,q). Note that it would be enough to find a Cameron-Liebler k-set in (n,q) that does not contain any k-spaces inside a hyperplane, see Theorem <ref>.


  
Acknowledgement
The research of Jozefien D'haeseleer and Ferdinand Ihringer is supported by the FWO (Research Foundation Flanders).

The authors thank the referees for their suggestions to improve this article.
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