A note on Erd\"os-Ko-Rado sets of generators in Hermitian polar spaces

The size of the largest Erd\H os-Ko-Rado set of generators in the finite classical polar space is known for all polar spaces except for $H(2d-1,q^2)$ when $d\ge 5$ is odd. We improve the known upper bound in this remaining case by using a variant of the famous Hoffman's bound.


Introduction
An Erdős-Ko-Rado set of generators in a finite classical polar space is a set of generators of the polar space that have mutually non-trivial intersection. The largest size of an Erdős-Ko-Rado set of generators in a finite classical polar space was determined in [7] for all finite classical polar spaces except for the hermitian polar space H(2d − 1, q 2 ) of odd rank d ≥ 5. Here rank means vector space rank and not projective dimension. The best known upper bound for this remaining case was proved in [4]. The idea of the proof was to formulate a linear optimization problem whose solution gives an upper bound. This idea goes back to Delsarte and uses the primitive idempotents of the associations scheme related to set of generators of a polar space, see [1]. In [4] we were however not able to determine the optimal solution of the optimization problem. Using a slightly different approach, the previous bound can be improved as follows.
Remarks 1. The set consisting of all generators of H(2d − 1, q 2 ), d odd, on a point has size roughly q d 2 whereas the bound given in the theorem has size roughly q d 2 +1 .
2. It can be shown that equality can not occur in the theorem. At the end of Section 2 we sketch a proof of this fact.
3. For small d it can be checked by computer that the given bound is also the solution of the optimization problem mentioned above. I guess this is true for all d, but I did not try to show this.

Proof of the theorem
Consider the graph whose vertices are the generators of the hermitian polar space H(2d − 1, q 2 ) of odd rank d ≥ 3. Let N be the number of generators and number them as G 1 , . . . , G N . For 0 ≤ i ≤ d, let A i be the real symmetric (N × N )-matrix whose (r, s)-entry is 1, if G r ∩ G s has rank d − i, and 0 otherwise. These real matrices are symmetric and commute pairwise, so they can simultaneously be diagonalized. It is known that there are exactly d + 1 common eigenspaces V 0 , . . . , V d of these matrices. Also one of the eigenspaces is j where j is the all one vector of length N . We choose notation so that V 0 = j . If P i,j denotes the eigenvalue of A j on V i , then with a suitable ordering of the eigenspaces we have, see [8], Theorem 4.3.6 We want to apply Hoffman's bound (see blow) to the generalized adjacency where we use for f the value for which the smallest eigenvalue of A is as large as possible. The eigenvalues for A are of course which results in the following definition.
Proof. The row sum of A is the eigenvalue of A on the eigenspace j . Using f < q 2 − 1, it follows that K > 0.
Proof. It follows from the list of eigenvalues that λ is the eigenvalue of Also, the way we determined f shows that the eigenvalue of A on V 1 is also λ. A straightforward calculation shows that and, using this expression, it is easy to see that λ < −q d 2 −2d+2 .
The eigenvalue of A on V 0 is K and the previous lemma shows that K > 0. For and we show in the remaining part of the proof that this eigenvalue is larger than λ.
First consider the case when i is odd. Then in the above formula for P i,d−2 as a sum over u ∈ {0, 1, 2}, only the term corresponding to u = 1 is positive. Hence Using f < q 2 − 1, we obtain the following bound for the eigenvalue of A on V i .
Here we have used that 3 ≤ i ≤ d − 2 (since i is odd).
If i is even, then P i,d > 0 and it is not difficult to see that P i,d−2 < 0. In this case the eigenvalue P i − f P i,d−2 of A on V i is positive.

Proof.
We denote by f 1 and f 2 the nominator and denominator in the definition of f . We have .
An easy calculation gives .
, the assertion follows.
Let G be a simple and non-empty graph with N vertices v 1 , . . . , v N . A real symmetric N × N matrix A with diagonal entries zero is called an extended weight matrix of G if A rs ≤ 0 whenever r = s and {v r , v s } is not an edge of the graph G, and if A rs = 0 for at least one edge {v r , v s } of the graph. It is called a K-regular extended weight matrix, if it has in addition constant row sum K. The following result appeared in various forms in the literature, we present it in the form of Corollary 3.3 in [2] but it already appeared in Lemma 6.1 of [3] when applied to the matrix A − λI, and it was also mentioned in [5].
For later application, we scetch the easy proof. Here I stand for the N × N identity matrix and J for the all-one matrix of the same size.
Result 2.4. Let Γ be a finite simple and non-empty graph with N vertices and suppose that A is a K-regular generalized weight matrix of G with least eigenvalue λ. Then every independent set S of G satisfies |S| · (K + |λ|) ≤ |λ| · N.
The vectors v 1 and v d are also eigenvectors of A 1 , . . . , A d and the eigenvalues are known. Since S is an Erdős-Ko-Rado set, the entries of A d χ corresponding to elements of S are zero. This gives a linear equation for the entries a 1 and a d of v 1 and v d corresponding to some element of S. A second linear equation comes from χ = |S| N j + v 1 + v d , namely 1 = |S| N + a 1 + a d . The two equations are linearly independent, so a 1 and a d can be calculated and are of course independent of the element of S. With this information the entries of A i χ corresponding to elements of S can be calculated for all i, which gives the number of elements of S that meet a given element of S in dimension d − i. It turns out that not all these numbers are integers, which is the desired contradiction. The same argument was used in the last section of [6].