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The Eigenvalues of the Graphs D(4,q) G. Eric Moorhousea, , Shuying Sunb, Jason Williforda ∗ 7 aDepartment of Mathematics, University of Wyoming, Laramie WY 82071 USA 1 bDepartment of Mathematical Sciences, Universityof Delaware, Newark DE19716 USA 0 2 n a J Abstract 3 ThegraphsD(k,q)haveconnectedcomponentsCD(k,q)givingthebestknown 1 bounds on extremal problems with forbidden even cycles, and are denser than ] the well-knowngraphs of Lubotzky, Phillips, Sarnak [15] and Margulis [16, 17]. O Despitethis,littleaboutthespectrumandexpansionpropertiesofthesegraphs C is known. In this paper we find the spectrum for k =4, the smallest open case. . For each prime power q, the graph D(4,q) is q-regular graph on 2q4 vertices, h t all of whose eigenvalues other than q are bounded in absolute value by 2√q. a ± Accordingly,these graphsaregoodexpanders,infactverycloseto Ramanujan. m [ Keywords: expander graph, Cayley graph, graph spectrum 1 v 1. Introduction 5 8 LetΓbeagraphwithvertexsetV. (Allourgraphsareundirectedandhave 6 3 no loops or multiple edges. See e.g. [6, 3] for standard terminology and theory 0 ofgraphs.) GivenasetofverticesF V,wedefine∂F tobe thesetofvertices ⊂ . in V rF which are adjacent to some vertex of F. The isoperimetric constant 1 0 of Γ is defined to be 7 ∂F V 1 h(Γ)=min | | :F V and F 6 | | . : (cid:26) F ⊂ | | 2 (cid:27) v | | i An infinite family of d-regular graphs whose isoperimetric constants are uni- X formly bounded away from 0 is an expander family. The best known general r a bounds on h(Γ) are expressed in terms of the spectrum of Γ, i.e. the multi- set of eigenvalues of its adjacency matrix. In particular, if Γ is q-regular with second-largesteigenvalue λ (Γ)<λ =q, then 2 1 1 q λ (Γ) 6h(Γ)6 2q(q λ (Γ)); 2 − 2 − 2 (cid:0) (cid:1) p see e.g. [9, Prop.1.84]. (The second-largest eigenvalue is denoted differently in some sources, including [9].) Thus to certify an infinite family of q-regular ∗Correspondingauthor Email addresses: [email protected] (G.EricMoorhouse),[email protected] (Shuying Sun),[email protected] (JasonWilliford) Preprint submitted toElsevier January 16, 2017 graphsasanexpander family, we requirea uniformlowerbound onthe spectral gap q λ (Γ). Aq-regularconnectedgraphΓisRamanujan ifλ (Γ)62√q 1; 2 2 − − by the Alon-Boppana Theorem(see e.g. [9, Ch.3]) this bound is asymptotically best possible for any infinite family of q-regular graphs. In searching for good families of explicitly defined graphs with good expansion, a particularly promising infinite family of graphs is the sequence D(5,q) D(4,q) D(3,q) D(2,q) ···→ → → → defined by Lazebnik and Ustimenko [10] for each prime power q. Each graph D(k,q)inthissequenceisbipartiteq-regularon2qk verticeshavinggirth>k+4 (or k+5, when k is even); and each connecting map ‘ ’ is a graph-theoretic → cover (see [12, Sec.3B]). The graphs D(k,q) are connected for k 6 5 and q odd; see [11]. The covering property ensures that the girth of D(k,q) is weakly increasingask ,andthespectrumofD(k,q)embedsinthatofD(k+1,q); →∞ see [12, Sec.3C]. The graphs CD(k,q) are important in the study of Turán type problems on even cycles, giving better lower bounds on the maximum number of edges in graphs of girth g > 6 than the well-known Ramanujan graphs of Lubotzky, PhillipsandSarnak[15]. Similarly,thegraphsLD(q,r)ofAlonetal.[1],another expander family with fixed degree, have girth 3 (after removing loops). By comparison,therefore,onemightexpectthegraphsCD(k,q)to haveverygood expansion properties. However, little is known about the eigenvalues of these graphs. In fact, to date only the spectrum of D(2,q) and D(3,q) are known, their characteristic polynomials being (x2 q2)(x2 q)q 1x2q(q 1) − − − − and (x2 q2)(x2 2q)q(q−1)2/2(x2 q)2q(q−1)xq3−2q2+3q−2 − − − respectively; see [13, Sec.5]. In particular, these graphs are Ramanujan. How- ever,Reichard[19]andThomason[20]independently showedbycomputer that the graphs D(4,q) are not Ramanujan for certain q, refuting the claim of [21]; see also the final note in this paper where we investigate this question more closely. The same statements apply to D(k,q) for all k >4, since the spectrum of D(4,q) is embedded in that of D(k,q) for k >4. It was later claimed in [22] that the eigenvalues of D(k,q) other than q ± are bounded by 2√q. However,a flaw was later found in the argument,leaving the problem open; see the Math Review MR2048644for [22]. To date, we have not found any counterexample to this statement, so we list it as a conjecture. Following [10], we denote by CD(k,q) a connected component of D(k,q); and we note that CD(4,q)=D(4,q) whenever q / 2,4 . ∈{ } Conjecture 1.1 (Ustimenko). Forall(k,q),CD(k,q)hassecondlargesteigen- value less than or equal to 2√q. In this paper we verify Conjecture 1.1 for k =4: 2 Theorem 1.2. The second largest eigenvalue of CD(4,q) is less than or equal to 2√q. This implies that these graphsare veryclose to Ramanujan. Our proofis given inSection5forevenq,andinSection6foroddq. Amoreexplicitdetermination of the spectrum is given in Section 7 for prime values q =p. Our approach is similar to [4], in that we first realize the halved (point) graphof D(k,q) as a Cayley graphof a certainp-groupG. Unlike the situation for the Wenger graphs in [4], or the graphs D(2,q) and D(3,q), our group G is nonabelianwheneverqisodd,thusrequiringmoreextensiveuseoftherepresen- tationtheoryofG. Finally,ourboundsoneigenvaluesareobtainedusingWeil’s bound for exponential sums over F , or over Galois rings of characteristic 3 in q the case q =3e. 2. The Graphs D(4,q) and their Point Collinearity Graphs Γ(4,q) Throughout, we take F = F where q is a prime power. The graph D(4,q) q is bipartite and q-regular with 2q4 vertices. These include q4 vertices P = P(p ,p ,p ,p ) called points, and q4 vertices L = L(ℓ ,ℓ ,ℓ ,ℓ ) called lines, 1 2 3 4 1 2 3 4 whereallcoordinatesareinF; andthe pointP andline L (with coordinatesas above) are incident iff p +ℓ =p ℓ , p +ℓ =p ℓ and p +ℓ =p ℓ . 2 2 1 1 3 3 1 2 4 4 2 1 AlsodenotebyΓ=Γ(4,q)thepointcollinearitygraphofD(4,q),i.e.thegraph whose vertices are the points of D(4,q), two points being adjacent in Γ iff they are distinct but collinear in D(4,q); see e.g. [3, Sec.14.2.2]. One checks that two verticesP(p ,p ,p ,p ),P(p ,p ,p ,p ) are adjacentin Γ (i.e. distinct and 1 2 3 4 ′1 ′2 ′3 ′4 collinear in D(4,q)) iff p1 6=p′1, (p1−p′1)(p4−p′4)=(p2−p′2)2 and p3−p′3 =p2p′1−p1p′2. TheadjacencymatrixofΓ hasthe formA=B BT qI whereB isaq4 q4 1 1 − q4 1 × matrix for which 0 B B := 1 (cid:20) BT 0 (cid:21) 1 is the adjacency matrix of D(4,q) (with the first q4 rows and columns indexed by points, and the last q4 rows and columns indexed by lines). Note that Γ is a q(q 1)-regular graph on q4 vertices. The spectra of A and B are in − direct relationship. Indeed, elementary methods yield the following, which is also implicit in [4, 13]: Lemma 2.1. Denote the characteristic polynomial of A, the adjacency matrix of Γ(4,q), by φ(x) = det(xI A). Then the characteristic polynomial of B, q4 − the adjacency matrix of D(4,q), is det(xI B)=φ(x2 q). (cid:3) 2q4 − − 3 Equivalently,everyeigenvalueλofA,withmultiplicitym,correspondstoapair of eigenvalues √q+λ of B, eachwith multiplicity m (or a single eigenvalue 0 ± of multiplicity 2m in case λ = q). The remainder of this paper is devoted to − proving Theorem 2.2. The graph Γ = Γ(4,q) is connected except for q 2,4 , when ∈ { } the graph has 4 connected components. When q is odd, the adjacency matrix A of Γ has characteristic polynomial φ(x)=det(xI A) of the form q − φ(x)=(x q(q 1))(x+q)(q 1)(q2 q+1)x3q(q 1)(x q)q(q 1)2φ(x) − − − − − − − e where all roots of φ(x) Z[x] have the form λ= q+ε2 where ε 62√q. Each ∈ − | | such value ε lies the ring Z 2cos2π , or Z 2cos2π if p=3. e p 9 (cid:2) (cid:3) (cid:2) (cid:3) A complete determination of φ(x) is given in Theorem 5.1 when q is even, and in Theorem 7.6 when q =p is prime. Now using Lemma 2.1 we obtain Theorem 2.3. The graph D(4,q) has eigenvalues q, each of multiplicity 1 ± (unless q 2,4 when each of the eigenvalues q has multiplicity 4). All ∈ { } ± remaining eigenvalues have the form ε where ε 62√q. ± | | Once again,the eigenvalues ε of Theorem2.3 are cyclotomic integers satisfying the conclusion of Theorem 2.2. In Theorem 2.2 the multiplicity of the eigen- value0mayactuallyexceed3q(q 1);inparticularthishappenswheneverq 2 − ≡ mod 3. We find explicit formulas for the actual eigenvalues, by expressing the ‘error’terms ε as exponential sums defined over finite fields (or over the Galois ring GR(9,e) of order 9e = q2 and characteristic 9, in the case q = 3e). This leads to our bound ε 6 2√q, using the Hasse-Davenport-Weil bound when | | q =pe, p>5; or the analogous bound of Kumar, Helleseth and Calderbank [8] in the case p=3. Ourstrategyforprovingthis result(see [2]fordetails)istofirstrealizeΓ as aCayleygraphCay(G,S)foranonabeliangroupGoforderq4, andconnection set S G. (Thus Γ has vertices labeled by elements of G; and two vertices ⊂ g,g G are adjacent in Γ iff g g 1 S). Since our graph Γ is undirected and ′ ′ − ∈ ∈ connected with no loops or multiple edges, we will have S = G, 1 / S, and h i ∈ g S iff g 1 S. We then determine the number k of conjugacy classes of G, − ∈ ∈ and a complete set (up to equivalence) of irreducible ordinary representations π : G GL (C) for i = 1,2,...,k. For each i, we compute the complex i → ni n n matrix π (S):= π (g). i× i i g S i ∈ P Theorem 2.4 ([2, 5]; see also [9]). The characteristic polynomial of A, the adjacency matrix of Γ, is given by k φ(x)=det(xI A)= det[xI π (S)]ni. |G|− iY=1 ni − i 4 Note that this gives k n2 = G eigenvalues (counting according to their i=1 i | | respective multiplicities)Pas required. In those cases where G is abelian, the eigenvalues are simply the character values χ (S). A similar simplification is i possible whenS is a unionof conjugacyclassesofG, but this does not apply in our case. When G is nonabelian and the full matrices of the representations π i arenotexplicitlyknown,determiningtheeigenvaluesofπ (S)fromthecharacter i values alone may require substantial additional work (see [2]); but for us, the group G is sufficiently nice that explicit descriptions of the full matrices of the representations π are easily available, making our job much easier. i 3. Background on Finite Fields Generalresults on finite fields canbe found in [14]. Let F =F be a field of q order q =pe where e>1 and p is prime. The absolute trace map is tr :F F , tr(a)=a+ap+ap2 + +ape−1. p → ··· We also fix a primitive p-th root of unity ζ =ζ C; here it suffices to assume p ∈ that ζ = e2πi/p. We define the exponential sum of an arbitrary function f : F F as the cyclotomic integer → ε = ζtr[f(a)] Z[ζ]. f ∈ aXF ∈ Lemma 3.1. For every polynomial of the form f(t)=bt+c F[t] we have ∈ 0, if b=0; εf =(cid:26) qζtr(c), othe6rwise. Proof. See [14, Ch.5]. (cid:3) Lemma 3.2. Let k be a non-negative integer. Then 1, if k =(q 1)k for some integer k >1; ak = − − 1 1 (cid:26) 0, otherwise. aXF ∈ Proof. See [14, p.271]. (cid:3) Lemma 3.3. (i) Let n be the number of nonzero polynomials a t2+a t+ k 2 1 a F[t] having exactly k distinct roots in F. Then 0 ∈ n = 1(q 1)(q2 q+2); n =2q(q 1); n = 1q(q 1)2 0 2 − − 1 − 2 2 − andn =0otherwise. Heren +n +n =q3 1;andfor k =0weinclude k 0 1 2 − 1q(q 1)2 irreducible quadratics and q 1 nonzero constant polynomials. 2 − − (ii) For q even, let n be the number of nonzero polynomials a t3+a t+a k 3 1 0 ∈ F[t] having exactly k distinct nonzero roots in F. Then n = 1(q 1)(q2+8); n = 1(q 1)2(q+4); n = 1(q 1)2(q 2) 0 3 − 1 2 − 3 6 − − and n =0 otherwise. Here n +n +n =q3 1. k 0 1 3 − 5 Proof. Every nonzero polynomial of degree 6 2 with a single root has the form a (t t ) or a (t t )2, giving n =2q(q 1). Every nonzero polynomial 1 1 2 1 1 − − − ofdegree 2 having two distinct roots has the forma (t t )(t t ) with a =0 1 1 2 2 − − 6 and t = t ; and there are n = 1q(q 1)2 such polynomials. This leaves 1 6 2 2 2 − n = q3 1 n n = 1(q 1)(q2 q+2), and the remaining assertions of 0 − − 1− 2 2 − − (i) follow. Now suppose q is even, and consider a nonzero polynomial f(t) = a t3 + 3 a t + a F[t]. If f(t) = a (t + t )(t + t )(t + t ) then t + t + t = 0; 1 0 3 1 2 3 1 2 3 ∈ so in characteristic 2, the number of distinct nonzero roots must be 0, 1 or 3. There are n = 1(q 1)2(q 2) nonzero polynomials of the form f(t) = 3 6 − − a (t+t )(t+t )(t+t +t ) where t ,t are nonzero and distinct. There are 3 1 2 1 2 1 2 (q 1)2 cubics of the form a t(t +t )2 where t = 0; and by (i), there are 3 1 1 − 6 1q(q 1)2 cubicsoftheform(t+t ) a t2+a t t+a0 forwhicht =0andthe 2 − 1 3 3 1 t1 1 6 quadratic factor is irreducible. The(cid:0)se, together with(cid:1)the (q 1)2 polynomials − a t+a having a ,a =0, give 1 0 1 0 6 n =(q 1)2+ 1q(q 1)2+(q 1)2 = 1(q 1)(q2+4). 1 − 2 − − 2 − This leaves n =q3 1 n n = 1(q 1)(q2+8). 0 − − 1− 3 2 − One checksthat this includes 1(q 1)(q2+2)irreducible cubics ofthe required 3 − form, together with q 1 polynomials of the form a t with a = 0, and q 1 1 1 − 6 − nonzero constant polynomials. (cid:3) 4. A Regular Group of Automorphisms of Γ For all t,u,v,w F we define the matrix ∈ 1 t u v+tu w  1 0 u 0  − g =g(t,u,v,w)= 1 t 0 .    1 0     1    These q4 matrices form a subgroup G<GL (F) acting regularly on points via 5 (1,p ,p ,p ,p ) (1,p ,p ,p ,p )g(t,u,v,w). 1 2 3 4 1 2 3 4 7→ which can be written simply as P Pg 7→ after a slight abuse of notation by which we identify P =P(p ,p ,p ,p )=(1,p ,p ,p ,p ). 1 2 3 4 1 2 3 4 6 Onechecksthatthisactionpreservescollinearityofpoints,andsogivesagroup of automorphisms of Γ which is regular on the vertices. Thus Γ is a Cayley graph Cay(G,S) for the set of q(q 1) elements − S = g G : P(0,0,0,0)g is (distinct from and) collinear with P(0,0,0,0) { ∈ } = g(t,rt, rt2,r2t) : r,t F, t=0 . { − ∈ 6 } The commutator of two typical elements of G is [g(t,u,v,w),g(t,u,v ,w )]=g(0,0,2tu 2tu,0). ′ ′ ′ ′ ′ ′ − Atthis pointwemustconsiderseparatelythe casesq evenandq odd, forwhich G is abelian or nonabelian, respectively. 5. The case q even In this section we suppose q is even, so that g(t,u,v,w)g(t′,u′,v′,w′)=g(t+t′,u+u′,v+v′,w+w′). In this case G is elementary abelian, with q4 irreducible linear characters χ g(t,u,v,w) =( 1)tr(αt+βu+γv+ηw), α,β,γ,η F. α,β,γ,η − ∈ (cid:0) (cid:1) Theorem 5.1. Suppose q is even. Then the characteristic polynomial of A, the incidence matrix of Γ, is φ(x)=det(xI A) q4 − = x q(q 1) (x 3q)q(q−1)2(q−2)/24(x q)q(q−1)2(q+4)/4 − − − − (cid:0) x(q−1)(q3+8q(cid:1)+3)/3(x+q)3q(q−1)2(q+2)/8. × The graph Γ is connected for q > 8; while for q 2,4 , Γ has 4 connected ∈ { } components. Proof. By Theorem 2.4 and Lemma 3.1, we have φ(x)= x ( 1)tr(αt+βrt+γrt2+ηr2t) α,β,Yγ,η F(cid:16) − rX,t∈F − (cid:17) ∈ t6=0 =(x q(q 1))(x+q)q 1 x ( 1)tr(αt+βrt+γrt2+ηr2t) . − − − α,βY,γ,η∈F(cid:16) − rX,t∈F − (cid:17) (β,γ,η)6=(0,0,0) t6=0 Now using the fact that the map F F, r r2 is an automorphism (in → 7→ particular bijective and trace-preserving), ( 1)tr(αt+βrt+γrt2+ηr2t) = ( 1)tr(αt) ( 1)tr[(β2t+γ2t3+η)r2t] − − − rX,t∈F 0=Xt F rXF t6=0 6 ∈ ∈ =q ( 1)tr(αt). − 0X6=t∈F β2t+γ2t3=η 7 After re-indexing via (β,γ,η) (β1/2,γ1/2,η), 7→ φ(x)=(x q(q 1))(x+q)q 1 x q ( 1)tr(αt) . − − − (β,γ,ηY)=(0,0,0)Yα (cid:16) − 0X6=t∈F − (cid:17) 6 βt+γt3=η If the polynomial f(t)=γt3+βt+η F[t] has a unique nonzero root t F, 1 then the map F F , α tr(αt ) ta∈kes each of the values in 0,1 exact∈ly q → 2 7→ 1 { } 2 times, in which case x q ( 1)tr(αt) =(x2 q2)q/2. Yα (cid:16) − 0X6=t∈F − (cid:17) − βt+γt3=η Similarly, if f(t) (as above) has three distinct nonzero roots t ,t ,t F, then 1 2 3 ∈ t +t +t = 0 and the map F F3, α (tr(αt ),tr(αt ),tr(αt )) attains e1ach o2f the3 triples (0,0,0),(1,1,0→),(12,0,1)7→,(0,1,1)1exactly2q times3, in which 4 case x q ( 1)tr(αt) =(x 3q)q/4(x+q)3q/4. Yα (cid:16) − 0X6=t∈F − (cid:17) − βt+γt3=η Thus φ(x)=(x q(q 1))(x+q)q−1xn0 (x2 q2)q/2 n1 (x 3q)q/4(x+q)3q/4 n3 − − − − (cid:2) (cid:3) (cid:2) (cid:3) where n is given by Lemma 3.3(ii). Simplification yields the formula claimed k for φ(x). Now we simply read off the multiplicity of the largest eigenvalue to obtain the number of connected components of Γ (see e.g. [3, Prop.1.3.8]). (cid:3) 6. The case q is odd Here and for the remainder of this paper, we take q to be odd. From the generalformula for commutators in G given at the end of Section 4, we deduce the commutator subgroup and centre G = g(0,0,u,0):u F , Z =Z(G)= P(0,0,v,w):v,w F ; ′ { ∈ } { ∈ } alsothecentralizerofanoncentralelement(i.e.with(t,u)=(0,0))isasubgroup 6 C g(t,u,v,w) = g(ct,cu,v ,w ):c,v ,w F G ′ ′ ′ ′ ∈ (cid:0) (cid:1) (cid:8) (cid:9) of order q3. So G has q3+q2 q conjugacy classes (q2 of size 1, and q3 q of − − size q). There are G/G =q3 linear characters of G, given by ′ | | χ g(t,u,v,w) =ζtr(αt+βu+γw) α,β,γ (cid:0) (cid:1) where α,β,γ F. As in Section 3, ζ = ζ is a complex p-th root of unity p ∈ and tr : F F is the trace map. The remaining irreducible characters of p → G may be found by inducing linear characters of a subgroup of order q3 (thus 8 yieldingmonomialrepresentationsofdegreeq);butguidedbyalittlehindsight, we will instead directly exhibit the missing representations and show that they are irreducible and distinct. For each pair α,β F with α = 0, we define ∈ 6 M :G GL (C) by α,β q → M (g(t,u,v,w))= ζtr[α(v 2iu)+βw]δ α,β − i+t,j i,j F (cid:2) (cid:3) ∈ using the Kronecker delta notation δ = 0 or 1 according as i,j F either i,j ∈ differ or coincide. It is routine to check that M (g)M (g ) = M (gg ) α,β α,β ′ α,β ′ for all g,g G, and M (g(0,0,0,0)) = I ; so M is a representation of ′ α,β q α,β ∈ degree q. The associated character is found to be ζtr(αv+βw)q, if t=u=0; ψ g(t,u,v,w) =trM g(t,u,v,w) = α,β α,β (cid:26) 0, otherwise (cid:0) (cid:1) (cid:0) (cid:1) using the fact that α = 0. These q2 q characters of G are irreducible and 6 − inequivalent since 1 [ψ ,ψ ] = ψ g(t,u,v,w ψ g(t,u,v,w) α,β α′,β′ G q4 α,β α′,β′ t,u,Xv,w F (cid:0) (cid:1) (cid:0) (cid:1) ∈ 1 = ζtr[(α−α′)v+(β−β′)w]q2 q4 v,Xw F ∈ 1, if (α,β)=(α,β ); = ′ ′ (cid:26) 0, otherwise. These are also distinct from the characters χ and so we have the complete α,β,γ list of q3+q2 q irreducible characters of G. − Now by Theorem 2.4, the adjacency matrix A of Γ has characteristic polynomial φ(x)=det(xI A)= (x χ (S)) det[xI M (S)]q. q4 α,β,γ q α,β − − − α,βY,γ F αY,β∈F ∈ α6=0 Those eigenvalues of A obtained from the linear characters of G are χ (S)= χ (g)= ζtr[(α+βr+γr2)t] =(m 1)q α,β,γ α,β,γ α,β,γ − gXS rX,t∈F ∈ t6=0 where m is the number of values r F such that α+βr+γr2 = 0. By α,β,γ ∈ Lemma 3.3(ii), the first q3 factors of φ(x) are (x q(q 1)) x (m 1)q α,β,γ − − Y (cid:16) − − (cid:17) (α,β,γ)=(0,0,0) 6 =(x q(q 1))(x q)n2xn1(x+q)n0 − − − =(x q(q 1))(x q)q(q 1)2/2x2q(q 1(x+q)(q 1)(q2 q+2)/2; − − − − − − − 9 thus the characteristic polynomial φ(x)=det(xI A) has the form q4 − φ(x)=x2q(q 1)(x q(q 1))(x q)q(q 1)2/2(x+q)(q 1)(q2 q+2)/2 − − − − − − − q det(xI M (S)) . q α,β × − αY,β∈F(cid:2) (cid:3) α6=0 Now for α=0, 6 M (S)= M (g)= ζtr[βr2t α(t+2i)rt]δ α,β α,β − i+t,j i,j F gXS rX,t∈F(cid:2) (cid:3) ∈ ∈ t6=0 = ζtr[βr2(j i) αr(j2 i2)] qI − − − q hrP∈F ii,j∈F − =Uα,βUα∗,β−qIq where‘ ’denotesconjugate-transpose,andwehaveintroducedtheq qcomplex ∗ × matrices U = ζtr(αi2j βij2) . α,β − i,j F (cid:2) (cid:3) ∈ We first treat the cases β =0=α for which we obtain 6 M (S)= ζtr[αr(i2 j2)] qI . α,0 − q hrP∈F ii,j∈F − Denoting by e the standard basis of CF = Cq, we find a new basis r r F { } ∈ consisting of eigenvectors of M (S) as follows: α,0 1(q 1) eigenvectors of the form e +e where 0 = r F, each with • 2 − r −r 6 ∈ eigenvalue q; 1(q 1) eigenvectors of the form e e as r ranges over a set of rep- • 2 − r − −r resentatives of the distinct nonzero pairs r, r in F. Each such vector { − } has eigenvalue q; − M e =0. α,0 0 • After including the factors det(xI M (S)) q =xq(q 1)(x2 q2)q(q 1)2/2, q α,0 − − − − 0=Yα F(cid:2) (cid:3) 6 ∈ we update our formula for the characteristic polynomial of A as φ(x)=x3q(q−1)(x q(q 1))(x q)q(q−1)2(x+q)(q−1)(2q2−2q+1) − − − q det(xI M (S)) . q α,β × − αY,β∈F(cid:2) (cid:3) αβ6=0 Finally we describe the remaining q2(q 1)2 eigenvalues of A arising from − M (S) for αβ =0. α,β 6 10