Articulo 5.dvi CUBO A Mathematical Journal Vol.12, No¯ 02, (53–75). June 2010 Generalized quadrangles and subconstituent algebra 1 Fernando Levstein FaMAF-CIEM,UNC, Universidad Nacional de Córdoba Medina Allende y Haya de la Torre. CP 5000 - Córdoba, Argentina email: levstein@famaf.unc.edu.ar and Carolina Maldonado Departamento de Matemática, Centro de Ciências Exatas e da Natureza, Universidade Federal de Pernambuco Av. Prof. Luiz Freire, s/n Cidade Universitária - Recife, Brasil email: cmaldona@famaf.unc.edu.ar ABSTRACT The point graph of a generalized quadrangle GQ(s, t) is a strongly regular graph Γ = srg(ν, κ, λ, µ) whose parameters depend on s and t. By a detailed analysis of the adjacency matrix we compute the Terwilliger algebra of this kind of graphs (and denoted it by T ). We find that there are only two non-isomorphic Terwilliger algebras for all the generalized quadrangles. The two classes correspond to wether s2 = t or not. We decompose the algebra into direct sum of simple ideals. Considering the action T × CX −→ CX we find the decomposition into irreducible T -submodules of CX (where X is the set of vertices of the Γ). RESUMEN El grafo de puntos de un cuadrángulo generalizado GQ(s, t) es un grafo fuertemente regular Γ = srg(ν, κ, λ, µ) cuyos parámetros dependen de s y t. Mediante un análisis detallado de 1This work supported by FACEPE, CCEN UFPE and CIEM-FaMAF UNC, CONICET. 54 F. Levstein and C. Maldonado CUBO 12, 2 (2010) la matriz de adyacencia, calculamos el álgebra de Terwilliger (T -álgebra) de esta familia de grafos. Encontramos que para todos los cuadrángulos generalizados, existen solo dos tipos no isomorfos de T -álgebras asociadas. Dichas clases dependen de si s2 = t o no. Descomponemos el álgebra en suma directa de ideales simples. Considerando la acción T × CX −→ CX encontramos la descomposición de CX en T -submódulos irreducibles. (X es el conjunto de vértices de Γ) Key words and phrases: strongly regular graphs, generalized quadrangles, Terwilliger algebra. AMS (MOS) Subj. Class.: 05E30 1 Introduction The subconstituent algebra was first introduced by P. Terwilliger in his paper [13]. It was defined on a class of combinatorial objects known as association schemes (see also [2, 3]). It is a noncommutative, finite dimensional, semisimple C algebra. We will denote it by T . It has been studied for many examples such as P - and Q- polynomial association schemes [6], distance-regular graph that supports a spin model [7], group association schemes [4, 5], strongly regular graphs [17]. In [8] it was given an explicit description of the T -algebra of the hypercube and more generally in [10] of a Hamming scheme H(d; q). The case of the Johnson schemes it was analyzed in [9]. In this paper we focus on the T -algebra of a special family of strongly regular graphs, which are examples of association schemes: generalized quadrangles GQ(s, t) . They are indeed a subfamily of partial geometries defined in [1]. A strongly regular graph is associated to them, so we can study the T -algebra of such a family. We show that there are only two non-isomorphic T -algebras for all the generalized quadrangles. The two classes correspond to whether s2 = t or not. We obtain the dimension of T in both cases. This is in agreement with the result expected from [17] that gives dimensions of the T -algebra attached to a strongly regular graph. The particular class of GQ(s, s2) has a combinatorial characterization given by J.A. Thas in [16]. With a detailed analysis of the adjacency matrix, we obtain restriction on the parameters (s, t) (also given in 1.2.2 of [12]). The paper is organized as follows: in section 2 we give the basic definitions and comment on some known basic results of algebraic combinatorics. In section 3 we analyze the blocks of the matrices in T and we give a basis of T in Proposition 3.21. In section 4 we find the simple ideals of T (Propositions 4.3, 4.4) and in Theorem 4.5 we decompose T into direct sum of simple ideals. Finally in section 5 we give the irreducible T -submodules of the action T × CX −→ CX (where X is the set of vertices of the Γ). CUBO 12, 2 (2010) Generalized quadrangles and subconstituent algebra 55 2 Definitions 2.1 Strongly regular graphs Definition 2.1. (see [11]) A strongly regular graph Γ = srg(ν, κ, λ, µ) is a graph with ν vertices that is regular of degree κ and that has the following properties: • for any two adjacent vertices x, y there are exactly λ vertices adjacent to x and to y • for any two nonadjacent vertices x, y there are exactly µ vertices adjacent to x and to y 2.2 Generalized Quadrangles Definition 2.2. (see [1] , [12]) A generalized quadrangle GQ(s, t) is a system of points and lines with an incidence relation satisfying the axioms (1) − (4) below. We will use standard geometric language. A point incident with a line is said to lie on the line and the line is said to pass through the point. If two lines are incident with the same point, we say that they intersect. Axioms 1. for any two distinct points there is at most one line passing through them; 2. there are exactly r = t + 1 lines passing for each point; 3. there are exactly k = s + 1 points lying on each line; 4. if a point p does not lie on the line l, then there is exactly one line passing through p and intersecting l If two points lie on a common line, we say that they are collinear and we write x ∼ y. The point graph of a generalized quadrangle is the graph with the points of the quadrangle as vertices, and edges {x, y} such that x ∼ y. It is well known by [1, 12] that the point graph of a GQ(k − 1, r − 1) is a (possibly trivial) Γ = srg(ν, κ, λ, µ) with: ν = k (1 + (k − 1)(r − 1)) , κ = r(k − 1), λ = k − 2, µ = r (1) 2.3 Bose-Mesner algebra Let Γ = srg(ν, κ, λ, µ) be a strongly regular graph, X be the set of vertices and ∂ : X × X → {0, 1, 2} be the path-length distance for Γ. Let M atX (C) denote the C-algebra of matrices with complex entries, where the rows and columns are indexed by X. 56 F. Levstein and C. Maldonado CUBO 12, 2 (2010) Definition 2.3. The adjacency matrix of Γ of is the following (0, 1)-matrix in M atX (C): (A)xy = { 1 if ∂(x, y) = 1 0 otherwise Proposition 2.4. (see [11]) Let Γ = srg(ν, κ, λ, µ) be a strongly regular graph, A the adjacency matrix of Γ and I, J ∈ M atX (C) the identity and the full ones matrix respectively. Then AJ = κJ (2) A2 + (µ − λ)A + (µ − κ)I = µJ (3) Proof. By definitions 2.1 and 2.3; A is a symmetric matrix with κ 1’s on each row and column. This proves equation (2). To prove (3) we observe that defining (A2)xy = { 1 if ∂(x, y) = 2 0 otherwise , axioms of definition 2.1 imply that I + A + A2 = J (A2 6= J − I) otherwise Γ would be a complete graph). Computing: (A2)xy = Σz∈X AxzAzy = |{z : ∂(x, z) = 1 and ∂(z, y) = 1}| =       κ if x = y λ if ∂(x, y) = 1 µ if ∂(x, y) = 2 Therefore A2 = κI + λA + µA2 = κI + λA + µ(J − I − A) which implies the (3). Definition 2.5. (see [2], [3] ) The Bose-Mesner algebra of a strongly regular graph Γ is the 3-dimensional algebra of matrices in M atX (C) which are linear combinations of I, J and A. We denoted it by A. That this is indeed an algebra is a consequence of equations (2) and (3) in Proposition 2.4. The following facts are well known in algebraic combinatorics (see [2, 3]). CUBO 12, 2 (2010) Generalized quadrangles and subconstituent algebra 57 The algebra A consists of symmetric commuting matrices and identifying C X = {f : X → C} we can consider for all M ∈ A the action: M × CX → CX . Since {I, J, A} consists of symmetric commuting matrices , they are diagonalyzed simultaneously by a unitary matrix. That is, we have a decomposition of CX into common eingenspaces of I, J, A. The number of eigenspaces is 2 + 1 since any strongly regular graph has diameter= 2 (diameter:= the greatest distance in the graph). Therefore, let Γ be a strongly regular graph, C X = V0 ⊕ V1 ⊕ V2 be such a decomposition and let Ei, i = 0, 1, 2 be the orthogonal projections Ei : C X → Vi expressed in matrix form with respect to the canonical basis {ei} i = 1...|X|. Then, E0 = 1 |X|J (J the matrix of all 1,s) E0 + E1 + E2 = I EiEj = δij Ei The Ei are called the primitive idempotents of Γ. 2.4 Dual Bose-Mesner algebra Definition 2.6. (see [13]) The ith dual idempotent with respect to the vertex x denoted by E∗i := E ∗ i (x) is the diagonal matrix in M atX (C) defined by (E∗i )yy = { 1 if ∂(x, y) = i 0 if ∂(x, y) 6= i Lemma 2.7. The matrices {E∗i }2i=0 satisfy the following equations: E∗0 + E ∗ 1 + E ∗ 2 = I (4) E∗i t = E∗i (5) E∗i E ∗ j = δij E ∗ i (6) Proof. Its follows straightforward from definition above. Definition 2.8. Let Γ be a strongly regular graph. For x ∈ X, the Dual Bose-Mesner algebra of Γ with respect to x, is the 3-dimensional algebra of matrices in M atX (C) which are linear combinations of {E∗i }2i=0. We denoted it by A∗ := A∗(x). 58 F. Levstein and C. Maldonado CUBO 12, 2 (2010) That this is indeed an algebra is a consequence of equations (4),(5) and (6) in the previous Lemma. 2.5 Terwilliger algebra Definition 2.9. (see [13]) Let Γ be a strongly regular graph and X be its set of vertices. The sub- constituent or Terwilliger algebra of Γ with respect to the vertex x ∈ X is the algebra generated by the Bose-Mesner algebra A := A(x) and the dual Bose-Mesner algebra A∗ := A∗(x). We denote this algebra by T := T (x). Remark 2.10. T is closed under the conjugate-transpose map, so it is semi-simple. 3 T -algebra of GQ(k − 1, r − 1). In this section we consider a connected strongly regular graph Γ = srg(ν, κ, λ, µ) coming from a generalized quadrangle GQ(k − 1, r − 1). We fix x0 ∈ X and we analyze the associated T (x0)-algebra . In the following we analyze the structure of the matrices belonging to T in a more detailed way . Lemma 3.1. For all T ∈ T , T is generated by A, E∗0 , E∗1 , E∗2 Proof. By definition T is generated by the algebras A = 〈{I, J, A}〉 and A∗ = 〈{E∗0 , E∗1 , E∗2 }〉. That is T consist on sum and products of matrices in {I, J, A, E∗0 , E∗1 , E∗2 }. Equation (3) shows that J can be obtained as a linear combination of A2, A, I and equation (4) shows that the identity is the sum of {E∗i }2i=0. Remark 3.2. It is well known that for the point graph of a generalized quadrangle the isomorphism class of T (x) is independent on the vertex x, since the group of automorphism of the graph Γ acts transitively on X preserving the distance. Then any automorphism g : X → X x → y, induces an isomorphism Tg : T (x) → T (y). M x → M y where M yuv := M x g−1ug−1v , for M x ∈ T (x), M y ∈ T (y); u, v ∈ X and then T (x) ≃ T (y) In view of Lemma 3.1 we consider the products E∗i AE ∗ j i, j = 0, 1, 2 where A is the adjacency matrix and E∗i the dual idempotents of definitions 2.3 and 2.6 respectively. CUBO 12, 2 (2010) Generalized quadrangles and subconstituent algebra 59 3.1 Block analysis We will use an order of the set of vertices X that allows us to analyze the matrices in T (x0) in a convenient way. Let x0 be a fixed vertex of X. Take Ω0 = {x0}, Ωi = {y ∈ X | ∂(x0, y) = i} We consider the matrices in M atX (C) indexed by the blocks Ωi × Ωj . Just to give examples, we have: E∗0 =       x0 Ω1 Ω2 x0 1 0 0 Ω1 0 0 0 Ω2 0 0 0       E∗1 =       x0 Ω1 Ω2 x0 0 0 0 Ω1 0 I 0 Ω2 0 0 0       E∗1 AE ∗ 2 =       x0 Ω1 Ω2 x0 0 0 0 Ω1 0 0 A|Ω1×Ω2 Ω2 0 0 0       We will denote P := A|Ω1×Ω1 Q := A|Ω1×Ω2 S := A|Ω2×Ω2 and Iik := I|Ωi×Ωk , Jik := J|Ωi×Ωk , that is the submatrix of I or J of size Ωi × Ωk. Then A|x0×Ω1 = J01 = (1, ..., 1) and since A is symmetric we have A|Ω2×Ω1 = Q t , A|Ω1×x0 = J t 01 = (1, ..., 1) t . Then A looks like: 60 F. Levstein and C. Maldonado CUBO 12, 2 (2010) A =                x0 Ω1 Ω2 x0 0 1...1 0 1 Ω1 ... P Q 1 Ω2 0 Q t S                The following lemma gives some descriptions of blocks of A. Lemma 3.3. Let Γ = srg(ν, κ, λ, µ) be a srg associated to a generalized quadrangle GQ(k − 1, r − 1) (that is the parameters (ν, κ, λ, µ) satisfy equations in (1)). Let Jkl, P, Q, S be defined as above. Then 1. A|x0×Ω1 = J01 2. J10 = J t 01 3. |Ω1| = r(k − 1); |Ω2| = (r − 1)(k − 1)2 4. P is a block of size |Ω1| × |Ω1| with (k − 2) 1′s on each row and column, 5. Q is a block of size |Ω1| × |Ω2| with (r − 1)(k − 1) 1′s on each row and r 1′s on each column and 6. S has size |Ω2| × |Ω2| with r(k − 2) 1′s on each row and column. Proof. • (1) holds since by definition of A, the block indexed by x0 × Ω1 is the set of neighbors of x0. • (2) holds since A is symmetric. • (3) holds since |Ω1| = κ (the degree of Γ) and 1 + |Ω1| + |Ω2| = ν (the number of vertices of Γ). Parameters κ, ν are given in Equations (1). • Assertion (4) holds since for a fixed x ∈ Ω1 there are λ = k − 2 neighbors of x in Ω1. • On the same way for a fixed x ∈ Ω2 there are µ = r neighbors of x in Ω1 which implies that Q has r 1’s on each column. The number of 1’s on each row of Q is |Ω1| − (k − 2) − 1. • The number of 1’s on each row and column of S is |Ω1| − r. CUBO 12, 2 (2010) Generalized quadrangles and subconstituent algebra 61 Remark 3.4. We have already discussed that in order to describe T we should analyze the prod- ucts among the matrices in {E∗i AE∗j }i,j=0,1,2. That is essentially the products among the blocks J01, J10, P, Q, Q t and S. In the following subsections we analyze the structure of each block Ωi × Ωj and finally we give a basis for each one. 3.2 Ω1 × Ω1-block We start giving expressions for some products belonging to the Ω1 × Ω1-block: {P n, QQt, P J11, J11P, J10J01}. We describe the powers of P . Lemma 3.5. P satisfies P 2 = (k − 3)P + (k − 2)I11 Proof. The Ω1 × Ω1-block has size r(k − 1) × r(k − 1) and P has (k − 2) 1’s on each row and column. It is indexed by the vertices in Ω1. It has a one in the (xi, xj ) entry if and only if the common neighbors xi, xj of x0 form an edge of the graph Γ. As the equation for P does not depend on the order of the vertices of Ω1 we will consider a special ordering in which P has a simple form. We label the vertices in the following way: Ω0 = {x0} and l1, l2...lr the r lines passing through the point x0. We call x1,1, x1,2, ...x1,k−1 the (k − 1) points lying on the l1 \ {x0}; x2,1, x2,2, ...x2,k−1 the points lying on the l2 \ {x0} and so on. All the points lying on the same line are collinear points. Then any two of them form an edge on the point graph of the generalized quadrangle. If we order the vertices of the Ω1 × Ω1-block with the order of the lines, that is l1 l2 . . . lr ︷ ︸︸ ︷ x1,1, x1,2, ...x1,k−1; ︷ ︸︸ ︷ x2,1, x2,2, ...x2,k−1; . . . ; ︷ ︸︸ ︷ xr,1, xr,2, ...xr,k−1 P has the form: P =                              J-I 0 ... ... ... 0 0 J-I ... ... ... 0 ... ... ... ... ... ... ... ... ... ... ... ... 0 ... ... ... J-I 0 0 ... ... ... 0 J-I                              and is not difficult to see that P 2 = (k − 3)P + (k − 2)I11, which implies the lemma. 62 F. Levstein and C. Maldonado CUBO 12, 2 (2010) Corollary 3.6. The matrices P, I11 and J11 are independent and P 2 depends on P and I11. Proof. P, I11 and J11 are independent, otherwise the relation among them should be P = J11−I11. But this would imply that the graph is not connected. Since we omit these cases we have the conclusion. Lemma 3.7. Using the same ordering as above for Ω1 and any order for Ω2 we have QQt = (r − 1)(k − 2)I11 − (r − 1)P + (r − 1)J11 J10J01 = r(k − 1)J11 P J11 = (k − 2)J11 Proof. Equating the Ω1 × Ω1-block of (3) we have J10J01 + P 2 + QQt + (µ − λ)P + (µ − κ)I11 = µJ11. Replacing the parameters λ, µ, κ by Equation (1) and P 2 as in the previous lemma, we get J11 + (k − 3)P + (k − 2)I11 + QQt + (r − k + 2)P − r(k − 2)I11 = rJ11. which implies the expression for QQt. The other equations are easy to check. Proposition 3.8. The products P n, QQt, J10J01 y P J11 can be expressed as a linear combinations of P, I11, J11 and they are linearly independent. Proof. It follows directly from lemmas 3.5 and 3.7. 3.3 Ω1 × Ω2- block Now we give expressions for the products P Q, QS, J11Q, QJ22, J12S Lemma 3.9. Using the same ordering for Ω1 as in the Lemma 3.5 the following equation holds: P Q = J12 − Q Proof. The Ω1 × Ω2 -block has size r(k − 1) × (r − 1)(k − 1)2. From Lemma 3.3, we now that Q has (r − 1)(k − 1) 1’s on each row and r 1’s on each column. By hypothesis, the rows of Q are indexed by the vertices of the lines l1, l2, ...lr. The columns are indexed by the set Ω2 (the vertices which are not neighbors of x0). Let (xij , y) be an entry of the product P Q where y ∈ Ω2 and xij is the ith vertex of the line lj. Then (P Q)(xij ,y) = r∑ m=1 k−1∑ n=1 P(xij ,xmn)Q(xmn,y). Since P vanishes on the vertices lying on different lines (P(xij ,xkl) = 0 for i 6= k), (P Q)(xij ,y) = k−1 ∑ n=1 P(xij ,xin)Q(xin,y). CUBO 12, 2 (2010) Generalized quadrangles and subconstituent algebra 63 Each vertex of Ω2 has exactly one neighbor on the line li (fourth axiom of definition 2.2). Therefore for y ∈ Ω2 there exist a unique xiny ∈ li such that Q(xij , y) = { 1 if j = ny 0 if j 6= ny Then (P Q)(xij ,y) = ∑k−1 n=1 P(xij ,xin)Q(xin,y) = P(xij ,xiny ) = (J − I)(xij ,xiny ) = { 0 if j = ny 1 if j 6= ny = (J − Q)(xij ,xiny ), which proves the lemma. Lemma 3.10. Q and S satisfy: QS = (r − 1)J12 + (k − 1 − r)Q, J11Q = rJ12, QJ22 = (r − 1)(k − 1)J12, J12S = r(k − 2)J12 Proof. The Ω1 × Ω2-block of identity (3) for A gives P Q + QS + (r − k + 2)Q = rJ12. Replacing P Q by the result of the lemma 3.9 we have the first equation. For the other equations, we use that Q has (r − 1)(k − 1) 1′s on each row and r 1′s on each column, and S has r(k − 2) 1′s on each row and column. Proposition 3.11. The products P nQ, SnQ, J11Q, QJ22, J12S can be expressed as linear combi- nations of Q and J12. Proof. Using lemmas 3.5 and 3.9 we can prove inductively that P nQ is a linear combination of Q and J12. On the same way Lemma 3.10 proves inductively the assertion for S nQ . The other equations were also proved in Lemma 3.10. 3.4 Ω2 × Ω2-block In the following, we give an expression for Sn, QtQ and J22S. Lemma 3.12. QtQ = −S2 + r(k − 2)I22 + (k − 2 − r)S + rJ22, SJ22 = r(k − 2)J22 (7) Proof. The Ω2 × Ω2-block of identity (3) for A gives the first equation. The matrix S has r(k − 2) 1′s on each row and column thus we get the second equation. 64 F. Levstein and C. Maldonado CUBO 12, 2 (2010) Proposition 3.13. S3 = ((k − 1 − r) + (k − 2 − r)) S2 + (r(k − 2) − (k − 1 − r)(k − 2 − r)) S − ((k − 1 − r) + r(k − 2)) I22 + (r(r − 1)(k − 2)) J22. Equivalently if we denote λ1 = k − r − 1, λ2 = k − 2, λ3 = −r, then S satisfies the equation (S − λ1I22) (S − λ2I22) (S − λ3I22) = r(r − 1)J22 (8) Proof. Postmultiplying QtQ given in (3.12) by S we have Q t QS = −S3 + r(k − 2)S + (k − 2 − r)S2 + r2(k − 2)J22 Replacing QS by the expression given in the lemma 3.10 Q t ((k − 1 − r)Q + (r − 1)J22) = −S3 + r(k − 2)S + (k − 2 − r)S2 + r2(k − 2)J22, S 3 = −(k − 1 − r)QtQ − r(r − 1)J22 + r(k − 2)S + (k − 2 − r)S2 + r2(k − 2)J22. Replacing QtQ by 3.12 we have the first equation, that is equivalent to S3 − ((k − 1 − r) + (k − 2 − r)) S2 − (r(k − 2) − (k − 1 − r)(k − 2 − r)) S + ((k − 1 − r) + r(k − 2)) I22 = (r(r − 1)(k − 2)) J22 At this moment we can not tell whether S2, S, I22 and J22 are independent or not. In what follows we are going to show that S2 depends on S I22 and J22 if and only if the parameters of the generalized quadrangle satisfy (k − 1)2 = r − 1. Corollary 3.14. Denoting λ0 = r(k − 2), λ1 = k − r − 1, λ2 = k − 2, λ3 = −r S satisfies the equation (S − λ0I22) (S − λ1I22) (S − λ2I22) (S − λ3I22) = 0 Proof. By Lemma The Ω2 ×Ω2-block has size (r−1)(k−1)2 ×(r−1)(k−1)2. S has r(k−2) 1’s on each row and on each column. So we have SJ22 = r(k − 2)J22. Thus, if we multiply (8) by S − r(k − 2)I22 we have the corollary. CUBO 12, 2 (2010) Generalized quadrangles and subconstituent algebra 65 This corollary implies that S has at most four different eigenvalues. We know that r(k − 2) is an eigenvalue associated to the one dimensional eigenspace generated by (1, 1, ..., 1). then by Perron- Frobenious Theorem it has multiplicity one. Let di = dim Vλi , where Vλi is the eigenspace corresponding to λi. We have the following linear system of equations on d0 and the unknowns: {di}3i=1 trI = ∑3 i=0 di = (r − 1)(k − 1)2, trS = ∑3 i=0 λidi = 0 and trS2 = ∑3 i=0 λi 2 di = r(k − 2)(r − 1)(k − 1)2, then trI = ∑3 i=1 di = (r − 1)(k − 1)2 − 1, trS = ∑3 i=1 λidi = −r(k − 2) and trS2 = ∑3 i=1 λi 2 di = r(k − 2)(r − 1)(k − 1)2 − (r(k − 2))2 , with set of solutions d1 = r(k − 2) d2 = r(k−1)2(r−2) (k+r−2) and d3 = (k−2)(r−1)((k−1)2−(r−1)) (k+r−2) . As the dimensions are non negative integers we have (k−1)2 ≥ (r−1), which is known as the inequality of D.G. Higman.(page 3 of [12]) In general k + r − 2 must divide both (k − 2)(r − 1)((k − 1)2 − (r − 1)) and r(k − 1)2(r − 2) if the parameters correspond to a generalized quadrangle. Dimensions {di}3i=1 are always positive integers unless (k − 1)2 = r − 1, in which case d3 = 0 and λ3 is not an eigenvalue. Thus we have the following: Proposition 3.15. S has λ3 = −r as eigenvalue if and only if the parameters r and k satisfy (k − 1)2 > r − 1. Proof. It follows by the comments above. Corollary 3.16. The matrices S, I22, J22 are linearly independent. S 2 depends on such matrices if and only if (k − 1)2 = r − 1 Proof. We have seen in Proposition 3.13 that the vector space generated by {Sn}n≥0 has dimension 3 or 4. This depends on the minimal polynomial of S and we have shown it has 3 different eigenvalues if and only if (k − 1)2 = r − 1. Proposition 3.17. The products {QtQ, J22S, {Sn}n≥0} can be expressed as a linear combinations of S, I22 and J22 , if and only if the parameters r, k of the generalized quadrangle satisfy (k − 1)2 = r − 1. Otherwise S2, S, I22 and J22 span these products. 66 F. Levstein and C. Maldonado CUBO 12, 2 (2010) Proof. Follows directly from Lemma 3.10 and Corollary 3.16 . Theorem 3.18. The following spanning set are basis for the corresponding blocks. {x0} × Ωi = 〈J0i〉 i = 0, 1, 2 Ω1 × Ω1 = 〈{I11, J11, P }〉 Ω1 × Ω2 = 〈{J12, Q}〉 Ω2 × Ω2 = 〈{I22, J22, S}〉 ⇔ (k − 1)2 = r − 1 = 〈{ I22, J22, S, S 2 }〉 ⇔ (k − 1)2 6= r − 1 Proof. It follows straightforward from Propositions 3.8, 3.11 and 3.17. 3.5 Basis for T as a vector space The previous block-analysis allows to give a basis (as a vector space) of the T -algebra attached to a GQ(k − 1, r − 1). Actually we have analyzed the blocks of arbitrary matrices in T . To be rigorous we should embed each block in M atX (C) . To do this we propose the following Definition 3.19. Let B an arbitrary block indexed by the vertices in {Ωi × Ωj} i, j = 0, ...2. We identify the block B with a matrix ι(B) in M atX (C) in the following way: ι(B)xy = { Bxy if (x, y) Ωi × Ωj 0 otherwise Example 3.20. Let B be a block-matrix indexed by Ω2 × Ω1. Then ι(B) =       x0 Ω1 Ω2 x0 0 0 0 Ω1 0 0 0 Ω2 0 B 0       Proposition 3.21. If the parameters of GQ(k − 1, r − 1) satisfy (k − 1)2 6= r − 1 then T = 〈{ {ι(Jij )}2i,j=0, {ι(Ijj )}2j=1, ι(P ), ι(Q), ι(Qt), ι(S), ι(S2) }〉 otherwise T = 〈{ {ι(Jij )}2i,j=0, {ι(Ijj )}2j=1, ι(P ), ι(Q), ι(Qt), ι(S) }〉 . Therefore dim(T ) = 16 or dim(T ) = 15 respectively. Proof. By Theorem 3.18 ,the matrices {ι(Jmn)}2m,n=0, {ι(Imm)}2m=1, ι(P ), ι(Q), ι(Qt), ι(S) CUBO 12, 2 (2010) Generalized quadrangles and subconstituent algebra 67 and eventually ι(S2) (when (k − 1)2 6= r − 1) give a basis (as a vector space) of a subalgebra of T . This subalgebra contains the adjacency matrix A and the dual idempotents {E∗i } since A = ι(J00) + ι(J10) + ι(J10) t + ι(P ) + ι(Q) + ι(Qt) + ι(S) E∗m = ι(Imm). Therefore it coincides with T . 4 Simple ideals of T In this section we decompose T as a direct sum of orthogonal simple ideals. We will guide us by the expression given by Proposition 3.21. There is one ideal present in every T -algebra: the ideal M linearly generated by {ι(Jmn)}2m,n=0. Definition 4.1. For m, n = 0, 1, 2 let Mmn ∈ M atX (C) be: Mmn = 1√ |Ωm||Ωn| ι(Jmn) Proposition 4.2. The vector subspace M = 〈 {Mmn}2m,n=0 〉 is a simple ideal of T and M ≃ End(C3). Proof. It not difficult to prove that MmnMpq = δnpMmq m, n, p, q = 0, 1, 2 which implies the proposition. Using standard techniques we compute the following basis for the second ideal. Let us denote N11 = 1 k−1 ι ((k − 2)I11 − P ) , N12 = 1 (k−1) √ (k−1)(r−1) ι ((k − 1)Q − J12) , N21 = N t 12, N22 = 1 (k−1)2(r−1) ι ((k − 1)QtQ − rJ22) We have the following Proposition 4.3. The vector subspace N = 〈 {Nmn}2m,n=1 〉 is a simple ideal of T orthogonal to the ideal M and N ≃ End(C2). Proof. It not difficult to prove that NmnNpq = δnpNmq m, n, p, q = 1, 2 M N = 0 ∀ M ∈ M, N ∈ N , which implies the proposition. 68 F. Levstein and C. Maldonado CUBO 12, 2 (2010) Now we give the expressions for the remaining one-dimensional ideals of T . One can easily prove the following: Proposition 4.4. The matrices P11 = 1 k−1 ι ( P + I11 − 1r J11 ) R22 = 1 (r−1)(k−2+r) ι ( S2 − (k − 1 − 2r)S − r(k − 1 − r)I22 − rJ22 ) S22 = 1 (k−1)(k−2+r) ι ( S 2 − (2k − r − 3)S + (k − 1 − r)(k − 2)I22 − (k−2)(r−1)(k−1) J22 ) are idempotents and orthogonal to the ideals M and N . Moreover, if (k − 1)2 = (r − 1) R22 = 1 r−1 ι ( S − (k − 1 − r)I22 − 1k−1 J22 ) , S22 = 0 If not, R22 y S22 are linearly independent and orthogonal. Then P = 〈P11〉, R = 〈R22〉, S = 〈S22〉 are ideals of T , orthogonal among them and orthogonal to M and to N . We get directly the following: Theorem 4.5. Let M, N , P, R, S ⊆ T be the simple ideals described above. Then, the T -algebra of a GQ(k−1, r−1) has the following decomposition as a direct sum of orthogonal simple ideals: T = M ⊕ N ⊕ P ⊕ R ⊕ S ≃ End(C3) ⊕ End(C2) ⊕ End(C1) ⊕ End(C1) ⊕ End(C1) ⇐⇒ (k − 1)2 6= r − 1 T = M ⊕ N ⊕ P ⊕ R ≃ End(C3) ⊕ End(C2) ⊕ End(C1) ⊕ End(C1) ⇐⇒ (k − 1)2 = r − 1 Proof. It follows straightforward from Propositions 4.2, 4.3 and 4.4. 5 Decomposition of CX into irreducible T -submodules In this section we consider the action of the T -algebra T × CX −→ CX (X is the set of vertices of the generalized quadrangle). We have that T CX ⊆ CX and since I ∈ T it holds T CX = CX . In the following we give a decomposition of CX into irreducible left T -submodules. CUBO 12, 2 (2010) Generalized quadrangles and subconstituent algebra 69 5.1 Isotypic left T -submodules Let T = M ⊕ N ⊕ P ⊕ R ⊕ S be the decomposition of Theorem 4.5. We can associate to each simple ideal a left T −submodule in the following way: {simple ideals of T } : → { left T -submodules of CX } Z → ZCX They are indeed left T -submodules since by the orthogonality of the simple ideals we have T ZCX ⊆ ZCX for any simple ideal Z ∈ {M, N , P, R, S} . We call them isotypic T -submodules. Then the decomposition of CX is : C X = MCX ⊕ N CX ⊕ PCX ⊕ RCX ⊕ SCX (9) SCX = 0 ⇐⇒ (k − 1)2 = r − 1 (10) 5.2 Irreducible left T -submodules In this section we decompose each of the left isotypic T -submodules into irreducible left T - submodules. To give the needed definitions we use as a guide the simple ideal N = {N11, N12, N21, N22} associated to the left isotypic T -submodule N CX . The matrices of the basis satisfy Nij Nkl = δjk Nil i, j, k, l = 1, 2 (11) In particular, {Nii}i=1,2 are idempotents and they have a (not unique) decomposition as a sum of rk(Nii) projectors of rank one.(Here rk(A) denote rank of A.) That is, there exist {N (j)11 } rk(N11) j=1 , {N (l) 22 } rk(N22) l=1 one-rank projectors such that (12) N11 = rk(N11)∑ j=1 N (j) 11 , N22 = rk(N22)∑ l=1 N (l) 22 which satisfy (13) N (j) ii N (k) ii = δjk N (j) ii for i = 1, 2 (14) 70 F. Levstein and C. Maldonado CUBO 12, 2 (2010) Remark 5.1. By equation (11) we have for example, N21 = N21N11 then N21 = rk(N11)∑ j=1 N21 N (j) 11 The remark carries out to define the following subspaces of N CX Definition 5.2. For i = 1, . . . , rk(N11) W N (i) 11 := { N (i) 11 v + (N21N (i) 11 )w v, w ∈ CX ; } Then we have: Proposition 5.3. For i = 1, . . . , rg(N11); WN (i)11 is an irreducible left T -submodule of dimension 2 and W N (i) 11 ≃ W N (j) 11 . Proof. Equation (9) and the fact that mutually different ideals are orthogonal implies that W N (i) 11 ⊆ N CX and that W N (i) 11 is a left T -submodule. W N (i) 11 is two dimensional since N (i) 11 is a one-rank projector ∀ i = 1, . . . , rk(N11). Therefore given {ej}|X|j=1 the canonical basis of C X , the subspace 〈 { N (i) 11 ej }|X| j=1 〉 has dimension one as well has 〈 { N21N (i) 11 ej }|X| j=1 〉 , which implies that W N (i) 11 has dimension two. It is irreducible since if we consider a one dimensional subspace, it should be of the form {( αN (i) 11 + βN21N (i) 11 ) v; v ∈ CX } but the following actions of T would imply N11 (αN (i) 11 + βN21N (i) 11 )C X ⊆ αN (i)11 CX ⇒ β = 0 N22 (αN (i) 11 + βN21N (i) 11 ) C X ⊆ βN21N (i)11 ⇒ α = 0 (which is a contradiction since it was a one dimensional subspace.) It is easy to check that W N (i) 11 ≃ W N (j) 11 considering the isomorphism: σN : ( N (i) 11 + N21N (i) 11 ) C X −→ ( N (j) 11 + N21N (j) 11 ) C X N (i) 11 v + N21N (i) 11 w −→ N (j) 11 v + N21N (j) 11 w which preserve the action of T . Proposition 5.4. N CX = rk(N11)⊕ j=1 W N (j) 11 CUBO 12, 2 (2010) Generalized quadrangles and subconstituent algebra 71 Proof. We have that ∑rk(N11) j=1 WN (j) 11 ⊆ N CX . Conversely, N11 C X ⊆ ∑rk(N11) j=1 WN (j)11 since by equation (13) N11 C X = ( ∑rk(N11) j=1 N (j) 11 ) C X ⊆ ∑rk(N11) j=1 WN (j)11 . Also N21C X ⊆ ∑rk(N11) j=1 WN (j)11 , since N21 C X = N21N11 C X = N21 ( ∑rk(N11) j=1 N (i) 11 ) C X = ( ∑rk(N11) j=1 N21N (i) 11 ) C X = ∑rk(N11) j=1 ( N21N (i) 11 ) C X But we also have N12C X ⊆ ∑rk(N11) j=1 WN (j)11 , since by equation (9) N12C X = N12N CX by equation (11) = N11N CX = N11C X Analogously N22C X = N22N CX = N21N CX = N21C X . which implies ∑ j W N (j) 11 ⊇ N CX and therefore the equality holds. We will prove that it is a direct sum by comparing dim   rk(N11)∑ j=1 W N (j) 11   with dimN CX . We have rk(N11) 2-dimensional subspaces. By equation (14) and by definition of WN (j)11 given in 5.2; it follows that rk(N11)∑ j=1 dimW N (j) 11 = 2 rk(N11) = 2 tr(N11) = 2 r(k − 2). 72 F. Levstein and C. Maldonado CUBO 12, 2 (2010) On the other hand, we obtain the dimension of N CX , computing the rank of the projection N : CX → N CX N = N11 + N22 which has the form =     0 0 0 0 ((k−2)I11−P ) k−1 0 0 0 ((k−1)Qt Q−rJ22) (k−1)2(r−1)     . It is easy to check that rk(N ) = tr(N ) = tr ( ((k−2)I11−P ) k−1 ) + tr ( ((k−1)QtQ−rJ22) (k−1)2(r−1) ) by Lemmas 3.3 and 3.5 = k−2 k−1 |Ω1| + (k−1)r−r(k−1)2(r−1) |Ω2| = k−2 k−1 r(k − 1) + (k − 2)r = 2r(k − 2) Analogously we can decompose the other isotypic left T -submodules. Considering the matrices Mij , P11, R22, S22 we define (the same way as for WN (i)11 ), Definition 5.5. WM00 := { M00u + M10M00v + M20M00w u, v, w ∈ CX } W P (i) 11 := { P (i) 11 u u ∈ CX , i = 1, . . . , rk(P11) } W R (i) 22 := { R (i) 22 u u ∈ CX , i = 1, . . . , rk(R22) } W S (i) 22 := { S (i) 22 u u ∈ CX , i = 1, . . . , rk(S22) } Then we have the following Theorem 5.6. C X = WM00 ⊕ r(k−2) j=1 WN (j)11 ⊕r−1 j=1 WP (j)11 ⊕dR j=1 WR(j)22 ⊕dS j=1 WS(j)22 and MCX = WM00 where WM00 is an irreducible left T -module of dimension 3 PCX = ⊕r−1j=1 WP (j)11 where WP j11 are irreducible left T -modules of dimension 1 RCX = ⊕dRj=1 WR(j)22 where WR(j)22 are irreducible left T -modules of dimension 1 SCX = ⊕dSj=1 WS(j)22 where WS(j)22 are irreducible left T -modules of dimension 1 CUBO 12, 2 (2010) Generalized quadrangles and subconstituent algebra 73 where dR = r(r−2)(k−1)2 (k−2+r) , dS = (r−1)(k−2)((k−1)2−(r−1)) (k−2+r) and rk(M00) = 1, rk(P11) = (r − 1), rk(R22) = dR , rk(S22) = dS Proof. The proof is analogous to the one given for the decomposition of N CX . The number of irreducible left T -submodules that appear on each decomposition depends on the rank of the projections to corresponding isotypic leftT -submodule: M : CX → MCX M = M00 + M11 + M22 =      1 0 0 0 J11√ |Ω1||Ω1| 0 0 0 J22√ |Ω2||Ω2|      . P : CX → PCX P := P11 R : CX → RCX R := R22 S : CX → SCX S := S22 From the definition of such matrices, and computing its trace, we get the corresponding ranks. Corollary 5.7. k (1 + (k − 1)(r − 1)) = 3 + 2r(k − 2) + r − 1 + r(r−2)(k−1) 2 (k−2+r) + (r−1)(k−2)((k−1)2−(r−1)) (k−2+r) Proof. One can get the equation by computing the dimensions of the decomposition given in Theorem 5.6. Remark 5.8. In subsections 5.1 , 5.2 we can exchange ”left” by ”right” considering the action of the T -algebra C X × T −→ CX that gives C X T = CX Acknowledgement. This work was finished during a visiting research position at the Depart- ment of Mathematics of UFPE, Recife-Brasil (september-november 2008) supported by FACEPE and CIEM-CONICET. 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