Smith and Critical groups of Polar Graphs
aa r X i v : . [ m a t h . C O ] D ec SMITH AND CRITICAL GROUPS OF POLAR GRAPHS.
VENKATA RAGHU TEJ PANTANGI AND PETER SIN.A bstract . We compute the elementary divisors of the adjacency and Laplacian matricesof families of polar graphs. These graphs have as vertices the isotropic one-dimensionalsubspaces of finite vector spaces with respect to non-degenerate forms, with adjacencygiven by orthogonality.
1. I ntroduction
Let
Γ = ( ˜ V , ˜ E ) be an undirected simple connected graph. Let A be the adjacency matrixof Γ with respect to some arbitrary ordering of ˜ V . Let D be the diagonal matrix with D ii being the degree of the i th vertex of Γ . Then L : = D − A is called the Laplacian matrixof Γ . Multiplication by A and L are endomorphisms of Z ˜ V , the free Z module with ˜ V as a basis set. The cokernel of A is called the Smith group S ( Γ ) of Γ . The finite partof cokernel of L is called the critical group K ( Γ ) of Γ . As a consequence of Kirchho ff ’sMatrix Tree Theorem, the order of K ( Γ ) is equal to the number of spanning trees of Γ (cf.[14]). The critical groups of various graphs arise in combinatorics in the context of chipfiring games (cf. [1]), as the abelian sandpile group in statistical mechanics (cf. [6]), andalso in arithmetic geometry. One may refer to [11] for a discussion on these connections.It is therefore of some interest to compute the Smith groups and critical groups of graphs.In this paper, we calculate the Smith groups and critical groups of families of polargraphs . These graphs are strongly regular graphs (SRGs) associated with finite classicalpolar spaces. Given a finite classical polar space P associated with a quadratic, symplecticor hermitian space V , the polar graph Γ ( V ) is the graph whose vertex set is the set P of points of P (isotropic 1-dimensional subspaces of V ) and whose adjacency is definedby orthogonality with respect to the underlying form. We have six infinite families ofpolar graphs arising from the six families of finite classical polar spaces. Throughout thepaper the Smith and critical groups of Γ ( V ) will be denoted by S and K respectively. Thecomputation of S and K is equivalent to finding the elementary divisors of A and L , so canbe carried out one prime at a time. The group G ( V ) of form-preserving isomorphisms on V acts a group of automorphisms of Γ ( V ). We use properties of strongly regular graphsand modular representation theory of G ( V ) to compute S and K . This methodology is anextension of the methodologies used for other computations in the literature, as found in[2], [5], [12] and [7].Our results giving the elementary divisors of S and K for every polar graph are stated in §
3. The structures of K and S depend on the relationship of primes to various combinatorialparameters of Γ ( V ) which we treat first in §
2. Then in § S and K for each of the six familiesof polar spaces. Key words and phrases. invariant factors, elementary divisors, Smith normal form, critical group, sandpilegroup, adjacency matrix, Laplacian, Polar graph.This work was partially supported by a grant from the Simons Foundation (
We now describe our approach in more detail. Let us consider a matrix X ∈ M n × n ( Z ), aprime ℓ , and a positive integer a ∈ Z ≥ . By e we denote the ℓ -rank of X and for a , e a be the multiplicity of ℓ a as an elementary divisor of the finite part of Z n / X ( Z n ). Let Z ℓ be the ring of ℓ -adic integers and F ℓ = Z ℓ /ℓ Z ℓ the field of ℓ -elements. In §
4, we establishsome general results on elementary divisors by relating them to the Z ℓ -modules M a : = { y ∈ Z n ℓ | Xy ∈ ℓ a Z n ℓ } and the F ℓ -modules M a : = ( M a + ℓ Z n ℓ ) /ℓ Z n ℓ . By the structure theorem forfinitely generated modules over a principal ideal domain, we have e a = dim( M a / M a + ).Suppose X is now either the adjacency matrix A or the Laplacian matrix L of Γ ( V ).As Γ ( V ) is a strongly regular graph, X has three distinct eigenvalues. When a prime ℓ divides at most two of the three eigenvalues, the reduction X ℓ of X modulo ℓ has non-zero eigenvalues. If ℓ divides all three eigenvalues, then X ℓ is nilpotent. The division intothe nilpotent and non-nilpotent cases is an important one from a technical point of view,and is also reflected in structure of S and K . The nilpotence of X ℓ is characterized by afew arithmetic conditions together with the geometry on V . Table 13 of § ffi ce. In § ℓ -elementary divisors of S and K when X ℓ is not nilpotent. In this case,we show that each M i is either im( X ℓ − α I ) or ker( X − α I ) for some eigenvalue α of X .When X ℓ is nilpotent, we use more representation theory. As G ( V ) preserves adjacency, X may be viewed as a Z ℓ G ( V )-endomorphism of the Z ℓ G ( V )-permutation module Z ℓ P ,and the modules M a introduced above are F ℓ G ( V )-submodules of the F ℓ G ( V )-permutationmodule F ℓ P . The action of G ( V ) on P has permutation rank 3. The submodule structureof the permutation module F ℓ P has been determined in [10], [9], [8], and [13] in cross-characteristics , that is, when ℓ is not equal to the characteristic of the underlying fieldof the polar space. From the submodule structures determined in [10], [9], [8], and [13]it can be seen that the length of F ℓ P is at most 6. Thus we expect at most five distinct ℓ -elementary divisors for S and K . Using the information about submodules we locatethe submodules M i in the submodule lattice and thereby determine the dimensions of thesubmodules M i . This detailed analysis, which forms the bulk of the paper, is carried out in § §
13. The identification of the submodules M i of F ℓ P is a further refinement of thedetailed picture of the submodule structure of permutation modules for classical groups,and may be of independent interest.2. D efinitions and N otation Definition 1.
A strongly regular graph (SRG) with parameters (˜ v , ˜ k , ˜ λ, ˜ µ ) is a ˜ k-regulargraph on ˜ v vertices such that:(1) Any two adjacent vertices have ˜ λ neighbours;(2) Any two non-adjacent vertices have ˜ µ neighbours. Let q = p t be a power of a prime, and let F q , F q be finite fields of order q , and q respec-tively. Let V be either a vector space over F q endowed with a non-degenerate symplecticform, a quadratic form, or a vector space over F q carrying a non-degenerate Hermitianform. By ˜ q , we denote the size of the underlying field associated with V . We note that˜ q = q in the Hermitian case and is q in the other cases.Let P be the set of all singular 1-spaces in V . Given two distinct v , u ∈ P , we say v ∼ u if and only if v and u are orthogonal. Let Γ ( V ) be the graph on P , whose adjacencyis defined by the relation ∼ . Based on the parity of dim( V ) and the geometry on V , we MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 3 classify Γ ( V ) into six families. We associate a parameter h ∈ { , , , , } with each family.Another parameter associated with each family is the dimension of the maximal totallyisotropic subspace of V , denoted by z . The graph Γ ( V ) is a graph on ( ˜ q z − + h +
1) ˜ q z − q − q is the size of the underlying field. Any strongly regular Γ ( V ) is isomorphicto one of the following graphs.(1) When V = F mq is a symplectic space (with m ≥ Γ ( V ) by Γ s ( q , m ). Inthis case h = z = m , and Sp(2 m , q ) ≤ Aut ( Γ s ( q , m )).(2) When V = F m + q is an endowed with a non-degenerate quadratic form (with m ≥ Γ ( V ) by Γ o ( q , m ). In this case h = z = m , and O(2 m , q ) ≤ Aut ( Γ o ( q , m )).(3) When V = F mq (with m ≥
3) is endowed with a non-degenerate elliptic quadraticform, we denote Γ ( V ) by Γ o − ( q , m ). In this case h = z = m −
1, and O − (2 m , q ) ≤ Aut ( Γ o − ( q , m )).(4) When V = F mq (with m ≥
3) is endowed with a non-degenerate hyperbolicquadratic form, we denote Γ ( V ) by Γ o + ( q , m ). In this case h = z = m , andO + (2 m , q ) ≤ Aut ( Γ o + ( q , m )).(5) When V = F mq (with m ≥
2) is endowed with a non-degenerate Hermitian qua-dratic form, we denote Γ ( V ) by Γ ue ( q , m ). In this case h = , z = m , andU(2 m , q ) ≤ Aut ( Γ ue ( q , m )).(6) When V = F m + q (with m ≥
2) is endowed with a non-degenerate Hermitianquadratic form, we denote Γ ( V ) by Γ uo ( q , m ). In this case h = , z = m , andU(2 m + , q ) ≤ Aut ( Γ ue ( q , m )).From now on, we assume that Γ ( V ) is one of the six graphs described above. A polargraph is a graph of the form Γ ( V ). By G ( V ), we denote the group of form-preservingautomorphisms of V . For example when Γ ( V ) = Γ s ( q , m ), we have G ( V ) = Sp(2 m , q ).We denote the number of j -dimensional subspaces of F dq , by h dj i q . By standard countingarguments we have h dj i q = Q ji = q d − i + − q − . Note that q may be replaced by a variable z todefine h dj i z : = Q ji = z d − i + − z − .By standard arguments (cf. [3] § Lemma 2.
The graph Γ ( V ) is a strongly regular graph with parameters v : = ( ˜ q z − + h + " z ˜ q , k : = ˜ q " z − ˜ q ( ˜ q z − + h + ,λ : = ( ˜ q − + ˜ q ( ˜ q z − + h + " z − ˜ q , µ : = k ˜ q . Here ˜ q = q when Γ ( V ) is either Γ ue ( q , m ) or Γ uo ( q , m ) ; and ˜ q = q in other cases. Given a matrix X ∈ M n × m ( Z ), let Ab( X ) be the finite part of Z n / X ( Z m ). Now if weare given a prime ℓ and a positive integer a , by e a we denote the multiplicity of ℓ a as anelementary divisor of the matrix X . Then e is the ℓ -rank of X and for a >
0, the multiplicityof ℓ a as an elementary divisor of Ab( X ) is e a , by the structure theorem for finitely generatedabelian groups.Throughout the paper, the adjacency and Laplacian matrices of Γ ( V ) will be denoted by A and L . The Smith group of Γ ( V ) is Ab( A ) and the critical group is Ab( L ). Throughoutthe paper, S and K will denote the Smith and critical groups of Γ ( V ) respectively. VENKATA RAGHU TEJ PANTANGI AND PETER SIN.
It follows from the definition of strongly regular graphs (cf. [4] Theorem 8 . .
2) that A satisfies A − ( µ − λ ) A + ( µ − k ) I = µ J . Here J is the matrix of all ones. By observing that : = P y ∈ P y is an eigenvector for A , corresponding to the eigenvalue k , we have ( A − kI )( A − ( µ − λ ) A + ( µ − k ) I ) = . Using this relation, the following Lemma can be derived usingelementary linear algebra.
Lemma 3.
Let A be an adjacency matrix of Γ ( V ) , then L = kI − A is the Laplacian. Let rbe the positive root of z − ( µ − λ ) z + ( µ − k ) , and s the negative root. Let t = k − r, andu = k − s. Then the following hold.(1) We have r = ˜ q z − − , s = − ( ˜ q z − + h + ,t = h z − i ˜ q ( ˜ q z − + h + , and u = ( ˜ q z − + h + h z i ˜ q .(2) A has three eigenvalues, ( k , r , s ) with multiplicities (1 , f , g ) . Where,f = ˜ q h ( ˜ q z − + h + q z − q − q h − + , and g = ˜ q ( ˜ q z − + h + q z − − q − q h − + .(3) L has eigenvalues (0 , t , u ) : = (0 , k − r , k − s ) with multiplicities (1 , f , g ) .(4) ( z − k )( z − r )( z − s ) is the minimal polynomial of A and ( z )( z − t )( z − u ) is the minimalpolynomial of L.(5) ( A − rI )( A − sI ) = µ J.(6) ( L − tI )( L − uI ) = − µ J.(7) ( z − kI )( z − rI ) f ( z − sI ) g is the characteristic polynomial of A.(8) z ( z − tI ) f ( z − uI ) g is the characteristic polynomial of L. As A is a non-singular matrix, the order of the Smith group S = | det ( A ) | and thus, | S | = kr f s g . As a consequence of Kirchho ff ’s Matrix Tree Theorem (cf. [14]), we have | K | = t f u g v . 3. M ain R esults Our main results are presented in Theorems 4 and 5 below. A Polar graph as defined inthe previous section is isomorphic to one of Γ s ( q , m ) (with m ≥ Γ o ( q , m ) (with m ≥ Γ o − ( q , m ) (with m ≥ Γ o + ( q , m ) (with m ≥ Γ ue ( q , m ) ( with m ≥ Γ uo ( q , m )(with m ≥ ℓ , the multiplicity e i of ℓ i as an elementary divisor of A (respectively L )is given in terms of parameters defined in the first two rows of the Tables 1 to 6 (respec-tively Tables 7 to 12). The parameters x , f , g defined in the first rows of the tables aredimensions of certain G ( V ) representations. In particular, f and g are the multiplicities ofthe eigenvalues r and s (of A ) respectively. The second rows define parameter a , d , w (re-spectively a , b , c , d in the case of L ) as ℓ -adic valuations of certain divisors of eigenvalues k , r , s of A (respectively t , u of K ). We note that v ℓ ( r ) = a + w , v ℓ ( s ) = d , v ℓ ( k ) = a + d , v ℓ ( t ) = a + c , and v ℓ ( u ) = b + d . Example . The 4th and 5th rows of table 7 show that the 2-elementary divisors L when Γ ( V ) = Γ s ( q , m ) (with q odd)i) are 2 , 2 , 2 d + , 2 d + b + with multiplicities g + f − g −
1, 1, and g − m is even; MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 5 ii) and are 2 , 2 , 2 a + c , 2 a + c + with multiplicities g , 1, f − g −
1, and g respectively, when m is odd.Parameters a , b , c , d , f and g are as defined in the first two rows of table 7. Theorem 4.
Let V be a either a vector space over F q endowed with a non-degeneratesymplectic form, quadratic form, or a vector space over F q carrying a non-degenerateHermitian form. Further assume dim( V ) ≥ when V is carrying a symplectic / Hermitianform, and dim( V ) ≥ when V is endowed with a non-degenerate quadratic form. Considerthe graph Γ ( V ) , its Smith group S and a prime ℓ | | S | . If ℓ = p, the ℓ -part of S is Z / q Z when Γ ( V ) is either Γ ue ( q , m ) or Γ uo ( q , m ) , and Z / q Z in other cases. If ℓ , p, the elementarydivisors of S are as described in Tables 1, 2, 3, 4, 5, and 6. In these, δ i j is if i = j and otherwise. ( f , g ) : = (cid:18) q ( q m − q m − + q − , q ( q m + q m − − q − (cid:19) ( a , d , w ) : = (cid:18) v ℓ ( h m − i q ) , v ℓ ( q m − + , v ℓ ( q − (cid:19) Prime Arithmetic conditions Non-zero divisor multiplicities ℓ = m is even and q is odd e = g + e w = f − g −
1, and e d + w = g + m is odd and q is odd e = g , e a = e a + w = f − g −
1, and e a + w + = g + ℓ , d = e = g + δ a , , e a = δ w , ( f ) + + δ a , ( g ), and e a + w = f + δ w , . a = w = e = f , and e d = g + T able
1. Smith group of Γ s ( q , m ).See § h = z = m ) and § Γ s ( q , m ). ( x , f , g ) : = q m − q q − , q ( q m − q m − + q − , q ( q m + q m − − q − ! ( a , d , w ) : = (cid:18) v ℓ ( h m − i q ) , v ℓ ( q m − + , v ℓ ( q − (cid:19) Prime Arithmetic conditions Non-zero divisor multiplicities ℓ = m is even and q is odd e = x + e d = g − x , e w = f − x −
1, and e d + w = x + m is odd and q is odd e = x , e = g − x + δ a , , e a = + δ a , ( g − x ), e a + w = f − x −
1, and e a + w + = x + ℓ , d = e = g + δ a , , e a = δ w , ( f ) + + δ a , ( g ), and e a + w = f + δ w , . a = w = e = f and e d = g + T able
2. Smith group of Γ o ( q , m ).See § h = z = m ) and § Γ o ( q , m ). ( f , g ) : = (cid:18) q ( q m − − q m − + q − , q ( q m + q m − − q − (cid:19) ( a , d , w ) : = (cid:18) v ℓ ( h m − i q ) , v ℓ ( q m − + , v ℓ ( q − (cid:19) Prime Arithmetic conditions Non-zero divisor multiplicities ℓ = m is even and q odd. e = g , e a = e a + w = f − g −
1, and e a + d + w = g + m is odd and q is odd. e = g + e w = f − g − e w + = g + ℓ , q + ≡ ℓ ) and m is even e = g , e a = f − g , and e a + d = g + q . − ℓ ) and d = e = g + δ a , , e a = δ w , ( f ) + + δ a , ( g ), and e a + w = f + δ w , . q . − ℓ ) and a = w = e = f , and e d = g + T able
3. Smith group of Γ o − ( q , m ). VENKATA RAGHU TEJ PANTANGI AND PETER SIN.
See § h = z = m −
1) and §
10 for computation of the Smith group of Γ o − ( q , m ). ( f , g ) : = (cid:18) q ( q m − q m − + q − , q ( q m − + q m − − q − (cid:19) ( a , d , w ) : = (cid:18) v ℓ ( h m − i q ) , v ℓ ( q m − + , v ℓ ( q − (cid:19) Prime Arithmetic conditions Non-zero divisor multiplicities ℓ = m is even and q odd. e = f , e = g + − f , and e w + = f . m is odd and q is odd. e = f − e d = g − f + + δ a , d , e a = δ a , d ( g − f + +
1, and e a + w + d = f . ℓ , q + ≡ ℓ ) and m is odd e = f − e a = + δ a , d ( g − f + e d = δ a , d + ( g − f + e a + b = f . q . − ℓ ) and d = e = g + δ a , , e a = δ w , ( f ) + + δ a , ( g ), and e a + w = f + δ w , . q . − ℓ ) and a = w = e = f and e d = g + T able
4. Smith group of Γ o + ( q , m ).See § h = z = m ) and §
11 for the computation of the Smith group of Γ o + ( q , m ). ( x , f , g ) : = ( q m − q m − + q − q − , q h m i q ( q m − + q + , q h m − i q ( q m − + q − ( a , d , w ) : = (cid:18) v ℓ ( h m − i q ) , v ℓ ( q m − + , v ℓ ( q − (cid:19) Prime Arithmetic conditions Non-zero divisor multiplicities ℓ | q + ℓ ∤ m and ℓ ∤ m − e = x + e w = f − x − + δ w , d ( g − x ), e d = δ w , d ( f − x − + g − x ,and e w + d = x + ℓ | m − e = x , e a = + δ a , d ( g − x ), e d = g − x + δ a , d , e w + a = f − x −
1, and e w + a + d = x + ℓ | m e = x + e d = g − x , e w = f − x −
1, and e w + d = x + ℓ ∤ q + d = e = g + δ a , , e a = δ w , ( f ) + + δ a , ( g ), and e a + w = f + δ w , . a = w = e = f and e d = g + T able
5. Smith group of Γ ue ( q , m ).See § h = / z = m ) and §
12 for the computation of the Smith group of Γ ue ( q , m ). ( f , g ) : = q h m i q ( q m − + q + , q h m − i q ( q m − − q − ( a , d , w ) : = (cid:18) v ℓ ( h m − i q ) , v ℓ ( q m + + , v ℓ ( q − (cid:19) Prime Arithmetic conditions Non-zero divisor multiplicities ℓ | q + ℓ ∤ m e = g , e a = e w + a = f − g −
1, and e w + a + d = g + ℓ | m e = g + e w = f − g −
1, and e w + d = g + ℓ ∤ q + d = e = g + δ a , , e a = δ w , ( f ) + + δ a , ( g ), and e a + w = f + δ w , . a = w = e = f and e d = g + T able
6. Smith group of Γ uo ( q , m ).See § h = / z = m ) and §
13 for the computation of the Smith group of Γ uo ( q , m ). MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 7
Theorem 5.
Let V be a either a vector space over F q endowed with a non-degeneratesymplectic form, quadratic form, or a vector space over F q carrying a non-degenerateHermitian form. Further assume dim( V ) ≥ when V is carrying a symplectic / Hermitianform, and dim( V ) ≥ when V is endowed with a non-degenerate quadratic form. Considerthe graph Γ ( V ) , its critical group K and a prime ℓ | | K | . The ℓ -elementary divisors of K areas described in Tables 7, 8, 9, 10, 11, and 12. In these, δ i j is if i = j and otherwise. ( f , g ) : = (cid:18) q ( q m − q m − + q − , q ( q m + q m − − q − (cid:19) ( a , b , c , d ) : = (cid:18) v ℓ ( h m − i q ) , v ℓ ( h m i q ) , v ℓ ( q m + , v ℓ ( q m − + (cid:19) Prime Arithmetic conditions Non-zero divisor multiplicities ℓ = m is even and q is odd e = g + e = f − g − e d + =
1, and e d + b + = g − m is odd and q is odd e = g , e a = e a + c = f − g −
1, and e a + c + = g . ℓ , b = d = e = g + δ a , , e a = δ c , ( f − + + δ a , ( g ), and e a + c = f − + δ c , . a = c = e = f + δ d , , e d = δ b , ( g ) + + δ d , ( f ), and e b + d = g − + δ b , T able
7. Critical group of Γ s ( q , m ).See § h = z = m ) and § Γ s ( q , m ).( x , f , g ) : = q m − q q − , q ( q m − q m − + q − , q ( q m + q m − − q − ! ( a , b , c , d ) : = (cid:18) v ℓ ( h m − i q ) , v ℓ ( h m i q ) , v ℓ ( q m + , v ℓ ( q m − + (cid:19) Prime Arithmetic conditions Non-zero divisor multiplicities ℓ = m is even and q is odd e = x + e = f − x − e d + = + δ b , ( g − x ), e b + d = g − x + δ b , ,and e d + b + = x − m is odd and q is odd e = x , e = g − x + δ a , , e a = δ a , ( g − x ) + e a + c = f − x − e a + c + = x . ℓ , b = d = e = g + δ a , , e a = δ c , ( f − + + δ a , ( g ), and e a + c = f − + δ c , . a = c = e = f + δ d , , e d = δ b , ( g ) + + δ d , ( f ), and e b + d = g − + δ b , T able
8. Critical group of Γ o ( q , m ).See § h = z = m ) and § Γ o ( q , m ). ( f , g ) : = (cid:18) q ( q m − − q m − + q − , q ( q m + q m − − q − (cid:19) ( a , b , c , d ) : = (cid:18) v ℓ ( h m − i q ) , v ℓ ( h m − i q ) , v ℓ ( q m + , v ℓ ( q m − + (cid:19) Prime Arithmetic conditions Non-zero divisor multiplicities ℓ = m is even and q odd. e = g , e a = e a + = f − g −
1, and e a + d + = g . m is odd and q is odd. e = g + e c = f − g − e c + =
1, and e b + c + = g − ℓ , q + ≡ ℓ ) and m is even e = g , e a = f − g , and e a + d = gq + ≡ ℓ ) and m is odd e = g + e c = f − g , and e b + c = g − q . − ℓ ) and b = d = e = g + δ a , , e a = δ c , ( f − + + δ a , ( g ), and e a + c = f − + δ c , . q . − ℓ ) and a = c = e = f + δ d , , e d = δ b , ( g ) + + δ d , ( f ), and e b + d = g − + δ b , T able
9. Critical group of Γ o − ( q , m ).See § h = z = m −
1) and §
10 for computation of the critcal group of Γ o − ( q , m ). VENKATA RAGHU TEJ PANTANGI AND PETER SIN. ( f , g ) : = (cid:18) q ( q m − q m − + q − , q ( q m − + q m − − q − (cid:19) ( a , b , c , d ) : = (cid:18) v ℓ ( h m − i q ) , v ℓ ( h m i q ) , v ℓ ( q m − + , v ℓ ( q m − + (cid:19) Prime Arithmetic conditions Non-zero divisor multiplicities ℓ = m is even and q odd. e = f , e c + = δ b , c ( g − f + + e b + = g − f + + δ b , c , and e b + c + = f − m is odd and q is odd. e = f − e a = + δ a , d ( g + − f ), e d = g + − f + δ a , d , and e a + d + = f − ℓ , q + ≡ ℓ ) and m is odd e = f − e a = + δ a , d ( g − f + e d = δ a , d + ( g − f +
1) and e a + d = f − q + ≡ ℓ ) and m is even e = f , e c = + δ b , c ( g − f + e b = g − f + + δ b , c , and e b + c = f − q . − ℓ ) and b = d = e = g + δ a , , e a = δ c , ( f − + + δ a , ( g ), and e a + c = f − + δ c , . q . − ℓ ) and a = c = e = f + δ d , , e d = δ b , ( g ) + + δ d , ( f ), and e b + d = g − + δ b , T able
10. Critical group of Γ o + ( q , m ).See § h = z = m ) and §
11 for the computation of the critical group of Γ o + ( q , m ). ( x , f , g ) : = ( q m − q m − + q − q − , q h m i q ( q m − + q + , q h m − i q ( q m − + q − ( a , b , c , d ) : = (cid:18) v ℓ ( h m − i q ) , v ℓ ( h m i q ) , v ℓ ( q m − + , v ℓ ( q m − + (cid:19) Prime Arithmetic conditions Non-zero divisor multiplicities ℓ | q + ℓ ∤ m and ℓ ∤ m − e = x + e d = f − x − + δ c , d ( g − x ), e c = δ c , d ( f − x − + g − x , and e c + d = x . ℓ | m − e = x , e a = + δ a , c ( g − x ), e c = δ a , c + ( g − x ), e a + d = f − x −
1, and e c + d = x . ℓ | m e = x + e d = f − x − e b + d = g − x + δ b , d , e d = + δ b , d ( g − x ),and e b + d = x − ℓ ∤ q + b = d = e = g + δ a , , e a = δ c , ( f − + + δ a , ( g ), and e a + c = f − + δ c , . a = c = e = f + δ d , , e d = δ b , ( g ) + + δ d , ( f ), and e b + d = g − + δ b , T able
11. Critical group of Γ ue ( q , m ).See § h = / z = m ) and §
12 for the computation of the critical group of Γ ue ( q , m ). ( f , g ) : = q h m i q ( q m − + q + , q h m − i q ( q m − − q − ( a , b , c , d ) : = (cid:18) v ℓ ( h m − i q ) , v ℓ ( h m i q ) , v ℓ ( q m − + , v ℓ ( q m + + (cid:19) Prime Arithmetic conditions Non-zero divisor multiplicities ℓ | q + ℓ ∤ m e = g , e a = e a + d = f − g −
1, and e a + c + d = g . ℓ | m e = g + e d = f − g − e d = e b + d = g − ℓ ∤ q + b = d = e = g + δ a , , e a = δ c , ( f − + + δ a , ( g ), and e a + c = f − + δ c , . a = c = e = f + δ d , , e d = δ b , ( g ) + + δ d , ( f ), and e b + d = g − + δ b , T able
12. Critical group of Γ uo ( q , m ).See § h = / z = m ) and §
13 for the computation of the critical group of Γ uo ( q , m ). MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 9
Remark.
We observe that the two families of polar graphs Γ s ( q , m ) and Γ o ( q , m ) are SRGswith the same parameters but di ff erent Smith and critical groups. This is an examplewhere Smith and critical groups are distinguishing invariants for two families of isospectralgraphs. 4. S mith normal form Let R be any PID and T : R m → R n be a linear transformation. By the structure theoremfor finitely generated modules over PIDs, we have { α i } si = ⊂ R \ { } such that α i | α i + andcoker( T ) (cid:27) R n − s ⊕ s M i = R /α i R . By [ T ] we denote the matrix representation of T with respect to standard bases. Then theabove equation tells us that we can find P ∈ GL( R n ), and Q ∈ GL( R m ) such that P [ T ] Q = " Y ( s × n − s ) ( m −× s ) ( n − s × n − s ) , where Y = diag( α , . . . , α s ). The matrix P [ T ] Q is called the Smith normal form of T .Let ℓ ∈ R be a prime dividing α s . Given j ∈ Z ≥ , we define e j ( ℓ ) : = |{ α i | v ℓ ( α i ) = j }| .Now e j ( ℓ ) is the multiplicities of ℓ j as an ℓ -elementary divisors of coker( T ). If R = Z , then e j ( ℓ ) is the multiplicity of ℓ j as an elementary divisor of the abelian group coker( T ).Let R ℓ be the ℓ -adic completion of R . We have R n ℓ / T ( R m ℓ ) (cid:27) R n − s ℓ ⊕ M j > (cid:16) R ℓ /ℓ j R ℓ (cid:17) e j ( ℓ ) . Define M j ( T ) : = { z ∈ R m ℓ | T ( z ) ∈ ℓ j R n ℓ } . For ease of notation, we denote M j ( T ) by M j and e j ( ℓ ) by e j . We have R m ℓ = M ( T ) ⊃ M ( T ) ⊃ . . . ⊃ M n ( T ) ⊃ · · · .Let F = R ℓ /ℓ R ℓ . If M ⊂ R m ℓ is a submodule, define M = ( M + ℓ R m ℓ ) /ℓ R m ℓ . Then M is an F -vector space.The following Lemma follows from the structure theorem. Lemma 6. e j : = dim( M j ( T ) / M j + ( T )) . So we have, dim( M j ( T )) − dim(ker( T )) = X t ≥ j e t . (1)Following the notation in §
2, given a matrix C ∈ M n × m ( Z ), the finite part of Z n / C ( Z m ) isdenoted by Ab( C ). The following lemma, which will be applied frequently, is Lemma 3.1of [7]. We include a short proof for the convenience of the reader. Lemma 7.
Let C be an n × m integer matrix. Fix a prime ℓ and let d = v ℓ ( | Ab( C ) | ) . LetM i : = M i ( C ) be as defined above and e i : = e i ( ℓ ) be the ℓ -elementary divisors of C. Supposethat we have two sequences of integers < t < t . . . < t j and s > s . . . > s j > s j + = dim(ker( C )) satisfying the following conditions.(A) dim( M t i ) ≥ s i , for all ≤ i ≤ j.(B) d = j P i = ( s i − s i + ) t i .Then the following hold.(a) e = m − s .(b) e t i = s i − s i + . (c) e a = for a < { t . . . t i , . . . t j } .Proof. We have d = X i ≥ ie i ≥ j − X k = X t k ≤ i < t k + ie i + X i ≥ t j ie i ≥ j − X k = t k X t k ≤ i < t k + e i + t j X i ≥ t j e i . (2)Application of equation (1) given above the lemma yields j − X k = t k X t k ≤ i < t k + e i + t j X i ≥ t j e i = j − X k = (cid:16) t k (dim( M t k ) − dim( M t k + )) (cid:17) + t j (cid:16) dim( M t j ) − dim(ker( C )) (cid:17) . Now application of conditions (A) and (B) in the statement gives us j − X k = (cid:16) t k (dim( M t k ) − dim( M t k + )) (cid:17) + t j (cid:16) dim( M t j ) − dim(ker( C )) (cid:17) ≥ j X i = ( s i − s i + ) t i = d . (3)So the inequalities (2) and (3) are in fact equations and thus the lemma follows. (cid:3) The following result is 12 . . Lemma 8.
Let C be an n × n integer matrix with an integer eigenvalue φ with geomet-ric multiplicity c. Fix a prime ℓ dividing both | Ab ( C ) | and φ , with v ℓ ( φ ) = d. Then dim( M d ( C )) ≥ c.Proof. Let V φ be the eigenspace of Q n ℓ . Then V φ ∩ Z n ℓ is a pure Z ℓ -submodule ( Z ℓ -directsummand) of Z n ℓ of rank c . It is clear that V φ ∩ Z n ℓ ⊂ M d ( C ). As V φ ∩ Z n ℓ is pure, we have V φ ∩ Z n ℓ ⊂ M d ( C ). (cid:3)
5. N ilpotence of A and K modulo ℓ .We recall from § Γ ( V ) is an SRG with parameters ( v , k , λ, µ ) specified in Lemma 2.Following notations fixed in § A will denote the adjacency matrix of Γ ( V ) and L = kI − A will denote the Laplacian matrix. By J , we denote the all-one matrix of same size as A .We also recall from Lemma 3 that A has eigenvalues k , r , s , with multiplicities 1, f , and g respectively; and that L has eigenvalues 0, t = k − r , u = k − s , with multiplicities 1, f , and g respectively. The values of r , s , t , u , f , and g are specified in Lemma 3. We alsoobserved that | S | = kr f s g and that | K | = t f u g v .Deducing from Lemma 3 that k = − ˜ qs r ˜ q − ℓ | | S | if and only if ℓ | ˜ qrs .Since tuv is an integer, we see that ℓ | | K | if and only if ℓ | tu . In the context of Lemma 8and Lemma 7, it is useful to investigate the ℓ -adic valuations of eigenvalues r , s of A ; andthose of eigenvalues t and u of L .Given X ∈ M n × n ( Z ), by X ℓ we denote the reduction of X modulo ℓ . The matrix X ℓ isnilpotent if and only if all eigenvalues of X are divisible by ℓ . Now the discussion in theabove paragraph and enables us to make the following observations.1) Since the ˜ q is coprime to both r and s , we see that A ℓ is nilpotent if and only if ℓ | r and ℓ | s .2) L ℓ is nilpotent if and only if ℓ | t and ℓ | u .The following Lemma completely classifies all the pairs ( Γ ( V ) , ℓ ) for which A ℓ or L ℓ isnilpotent. MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 11
Lemma 9.
Consider the graph Γ ( V ) and let X be either the adjacency matrix or the Lapla-cian matrix of Γ ( V ) . Let ℓ be a prime and X ℓ be the reduction of X (mod ℓ ) . Then condi-tions for nilpotence of X ℓ are encoded in Table 13. Γ ( V ) ℓ Arithmetic conditions Nilpotence of X ℓ Γ s ( q , m ) or Γ o ( q , m ) ℓ = q is odd True ℓ , Γ o − ( q , m ) ℓ = q is odd True ℓ , ℓ | q + m is even True ℓ , ℓ | q + m is odd True for L ℓ and False for A ℓ ℓ , ℓ ∤ q + Γ o + ( q , m ) ℓ = q is odd True ℓ , ℓ | q + m is odd True ℓ , ℓ | q + m is even True for L ℓ and False for A ℓ ℓ , ℓ ∤ q + Γ ue ( q , m ) or Γ uo ( q , m ) ℓ ℓ | q + ℓ ∤ q + ℓ = q is even False. T able
13. Conditions on ℓ .The proof follows by observing that X ℓ is nilpotent if and only if all three eigenvaluesof X are divisible by ℓ .Finding ℓ -elementary divisors of S and K in the “non-nilpotent” cases is a bit easier.Lemma 3 gives us ( A − rI )( A − sI ) = µ J and ( L − tI )( L − uI ) = − µ J . In this case Lemma 8and the equations above help us construct two integer sequences satisfying the hypothesisof Lemma 7. We will do these computations in § G ( V ), the group of form pre-serving linear isomorphisms of V . Let us consider the case when ℓ is a “nilpotent” prime.We may treat A and L as elements of End Z ℓ ( Z ℓ P ), where Z ℓ P is the free Z ℓ module with P (vertex set of Γ ( V )) as a basis. The action on P by elements of the group G ( V ) pre-serves adjacency and thus commutes with the actions of A and L . This implies that the F ℓ P subspaces M i ( A ) and M i ( L ) constructed as in § F ℓ G ( V )-submodules of thepermutation module F ℓ P . The action of G ( V ) on P has permutation rank 3. The submod-ule structure of the permutation module F ℓ P has been determined in [10], [9], [8], and[13] in cross-characteristics , that is, when ℓ ∤ ˜ q . We use these results along with Lemma 7to finish our computations.6. W hen A ℓ and L ℓ are not nilpotent .In this section we deal with Γ ( V ) and a prime ℓ such that A ℓ and L ℓ are not nilpotent.Table 13 can be used to look up all possible pairs ( V , ℓ ) such that A ℓ (equivalently L ℓ ) arenot nilpotent.6.1. Elementary divisors of S . The graph Γ ( V ) is one of Γ s ( q , m ), Γ o ( q , m ), Γ o − ( q , m ), Γ o + ( q , m ), Γ uo ( q , m ) and Γ ue ( q , m ). Following the notation in Lemma 3, we have r = h z − i ˜ q ( ˜ q − s = − ( ˜ q z − + h + µ = h z − i ˜ q ( ˜ q z − + h + k = ˜ q µ . Here ˜ q = q for Γ uo ( q , m ) and Γ uo ( q , m ); and ˜ q = q for other graphs. If ℓ | | S | , we saw in § A ℓ is not nilpotent if and only if ℓ does not divide r and s simultaneously. Assume that ℓ does not divide r and s simultaneously, and that ℓ | | S | . As | S | = kr f s g and k = ˜ qs r ˜ q − ℓ divides exactly one of ˜ q , r , and s .In this subsection, we identity A ℓ with A and M i ( A ) with M i .6.1.0.1. Case 1: ℓ | r and ℓ ∤ s ˜ q . We set v ℓ ( h z − i ˜ q ) = a and v ℓ ( ˜ q − = w . Then v ℓ ( s ) = v ℓ ( k ) = v ℓ ( µ ) = a , v ℓ ( r ) = a + w , and v ℓ ( | S | ) = ( a + w ) f + a . As ℓ | r , one of a and w isnecessarily non-zero.By Lemma 3, ( z − k )( z − s ) g ( z − r ) f is the characteristic equation of A . Reducing modulo ℓ ,we see that z f ( z − k )( z − s ) g is the characteristic polynomial of A . By Lemma 3, we observethat minimal polynomial of L divides ( z − k )( z − s )( z ), and thus all the Jordan blocks of L associated with s have size 1. Therefore, the geometric multiplicity of s as an eigenvalueof A is g . We can now conclude that dim(im( A − sI )) = f + A ( A − sI ) = − r ( A − sI ) + µ J . Since a = v ℓ ( µ ) ≤ v ℓ ( r ), we see thatim( A − sI ) ⊂ M a . Thus dim( M a ) ≥ f + r is an eigenvalue of valuation a + w , Lemma 8 implies that dim( M a + w ) ≥ f .We apply Lemma 7 to conclude the following.(1) Assume that w =
0, then a ,
0. As A is non-singular, ker( A ) = { } . Now byLemma 7, setting j = t = a , s = f + ≤ dim( M a ), s = = dim(ker( A )), wehave e a = f + e = g , and e i = i .(2) Assume a =
0, then w ,
0. As A is non-singular, ker( A ) = { } . Now by Lemma 7,setting j = t = w , s = f ≤ dim( M w ), s =
0, we have e w = f , e = g + e i = i .(3) Assume aw ,
0. As A is non-singular, ker( A ) = { } . Now by Lemma 7, setting j = t = a + w , t = w s = f + ≤ dim( M w ), s = f ≤ dim( M w ), s =
0, wehave e a + w = e a = f , e = g , and e i = i .6.1.0.2. Case 2: ℓ | s and ℓ ∤ r ˜ q . Set v ℓ ( s ) = d . As ℓ ∤ r , we have v ℓ ( r ) =
0. Then v ℓ ( k ) = v ℓ ( µ ) = d , and v ℓ ( | S | ) = dg + d .Lemma 3 gives us A ( A − rI ) = − s ( A − rI ) + µ J . This shows that im( A − rI ) ⊂ M d .By Lemma 3, z ( z − s ) g ( z − r ) f is the characteristic polynomial of A . Reducing mod ℓ , wesee that z g + ( z − r ) f is the characteristic polynomial of A . Also Lemma 3, we can deducethat z ( z − r ) is the minimal polynomial of A , and thus the geometric multiplicity of r asan eigenvalue of A is f . We can now conclude that dim(im( A − rI )) = g +
1. Thereforedim( M d ) ≥ g +
1. So by Lemma 7, setting j = s = g + s =
0, and t = d , we have e d = g + e = f and e i = i .6.1.0.3. Case 3: ℓ | ˜ q and ℓ ∤ rs . Set v ℓ ( ˜ q ) = ǫ . Then v ℓ ( k ) = v ℓ ( | S | ) = a . As k isan eigenvalue of valuation ǫ , Lemma 8 shows that dim( M ǫ ) ≥
1. Thus by Lemma 7, wededuce that e ǫ = e = f + g , and e i = i .6.2. Elementary divisors of K . The graph Γ ( V ) is one of Γ s ( q , m ), Γ o ( q , m ), Γ o − ( q , m ), Γ o + ( q , m ), Γ uo ( q , m ) and Γ ue ( q , m ). Following the notation in Lemma 3, we have µ = h z − i ˜ q ( ˜ q z − + h + t = h z − i ˜ q ( ˜ q z − + h + u = ( ˜ q z − + h + h z i ˜ q . and v = h z i ˜ q ( ˜ q z − + h + q = q for Γ uo ( q , m ) and Γ uo ( q , m ); and ˜ q = q for other graphs.If ℓ | | K | , we saw in 5 that L ℓ is not nilpotent if and only if ℓ does not divide t and u simultaneously. Assume that ℓ does not divide t and u simultaneously, and that ℓ | | K | . Werecall that | K | = t f u g / v .In this subsection, we identity L ℓ with L and M i ( L ) with M i . MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 13
Case 1: ℓ | t and ℓ ∤ u . In this case, v ℓ ( t ) > v ℓ ( u ) =
0. We set v ℓ ( h z − i ˜ q ) = a and v ℓ (( ˜ q z − + h + = c . Now, we have v ℓ ( t ) = a + c , v ℓ ( µ ) = a , and v ℓ ( v ) = c . Since | K | = t f u g / v , we have v ℓ ( | K | ) = ( a + c ) f − c . As L is a matrix of nullity 1, we havedim(ker( L )) =
1. As t is an eigenvalue of ℓ -valuation a + c and geometric multiplicity f .So Lemma 8 implies that dim( M a + c ) ≥ f .Lemma 3 gives us L ( L − uI ) = − t ( L − uI ) − µ J . So im( L − uI ) ⊂ M a . Again by Lemma3, z ( z − u ) g ( z − t ) f is the characteristic polynomial of L . Reducing mod ℓ , we see that z f + ( z − u ) g is the characteristic polynomial of L . From Lemma 3 we deduce that minimalpolynomial of L divides z ( z − u ). Thus all the Jordan blocks of L associated with u are ofsize 1. Therefore the geometric multiplicity of u as an eigenvalue of L is g . We can nowconclude that dim(im( L − uI )) = f +
1, and thus dim( M c ) ≥ f + a =
0, then c ,
0. So by Lemma 7, setting j = s = f , s = dim(ker( L )) = t = c , we have e c = f − e = g +
1, and e i = i .(2) Assume c =
0, then a ,
0. So by Lemma 7, setting j = s = f + s = dim(ker( L )) = t = a , we have e a = f , e = g , and e i = i .(3) Assume ac ,
0. By Lemma 7, setting j = s = f + s = f , s = dim(ker( L )) = t = c , t = a + c we have e = g , e a = e a + c = f − e i = i .6.2.0.2. Case 2: ℓ | u and ℓ ∤ t . In this case, v ℓ ( u ) >
0, and v ℓ ( t ) =
0. We set v ℓ ( h z i ˜ q ) = b and v ℓ (( ˜ q z − + h + = d . We have v ℓ ( u ) = b + d , v ℓ ( µ ) = d , and v ℓ ( v ) = b . Since | K | = t f u g / v ,we have v ℓ ( | K | ) = ( b + d ) g − b . As L is a matrix of nullity 1, we have dim(ker( L )) = u is an eigenvalue of valuation d + b and geometric multiplicity g , Lemma 8 impliesdim( M d + b ) ≥ g .By Lemma 3, we have L ( L − tI ) = − u ( L − tI ) − µ J . So im( L − tI ) ⊂ M d . Lemma 3 tellsthat z ( z − u ) g ( z − t ) f is the characteristic polynomial of L . Reducing modulo ℓ , we see that z f + ( z − t ) g is the characteristic polynomial of L . From Lemma 3 we deduce that minimalpolynomial of L divides z ( z − t ). Thus all the Jordan blocks of L associated with t are ofsize 1. Therefore the geometric multiplicity of t as an eigenvalue of L is f . We can nowconclude dim(im( L − tI )) = g + M d ) ≥ g + b =
0, then d ,
0. So by Lemma 7, setting j = s = g + s = dim(ker( L )) = t = d , we have e = f , e d = g , and e i = i .(2) Assume d =
0, then b ,
0. So by Lemma 7, setting j = s = g , s = dim(ker( L )) = t = b , we have e = f + e b = g −
1, and e i = i .(3) Assume bd ,
0. Then by Lemma 7, setting j = s = g + s = g , s = dim(ker( L )) = t = d , t = d + b we have e = f , e d = e d + b = g − e i = i .7. W hen A ℓ and L ℓ are nilpotent .Let ℓ be a prime and Γ ( V ) be a polar graph such that A ℓ or L ℓ is nilpotent. In this case,we use representation theory of G ( V ) to compute the ℓ -elementary divisors of S and K . The action of G ( V ) on Γ ( V ) commutes with A and L . Thus the vector spaces M i ( A ) and M i ( L ) are in fact G ( V )-submodules of F ℓ P . We recall that the set P which is the set of allsingular 1-spaces in V is the vertex set of Γ V .The action of G ( V ) on P is a rank 3 permutation action. When ℓ is not the characte-teristic of the field associated with the underlying vector space V , the submodule structureof F ℓ P is given in [8], [13], [10] and [9]. We use the submodule structures present inliterature to determine M i ( A ) and M i ( L ) and consequently find the elementary divisors of S and K .We now define some submodules of F ℓ P . These are some important submodules of F ℓ P defined in [8], [13], [10] and [9].1) Given any subspace Z of V , we denote [ Z ] to be the sum of all isotropic one- dimen-sional subspace of Z . We denote [ V ] by , henceforth known as the all-one vector.2) Consider A and L to be elements of End( Q ℓ P ). Define V r = ker( A − rI ) = ker( L − tI ),and V s = ker( A − sI ) = ker( L − uI ). Then define V r to be the subspace V r ∩ Z ℓ P of F ℓ P , and V s to be the subspace V s ∩ Z ℓ P of F ℓ P . As V r ∩ Z ℓ P and V s ∩ Z ℓ P are pure submodulesof Z ℓ P , we have dim( V r ) = dim( V r ) = f and dim( V s ) = dim( V s ) = g .3) We define C to be the linear subspace of F ℓ P spanned by { [ W ] | W is a maximal totally isotropic subspace of V } .
4) We define C ′ to be the linear subspace of F ℓ P spanned by { [ W ] − [ W ′ ] | W , W ′ are maximal totally isotropic subspaces of V } .
5) We define U to be ( J − A ℓ )( F ℓ P ), where J is the matrix of all 1 ′ s.6) We define U ′ to be the subspace spanned by { ( J − A ℓ )( v ) − ( J − A ℓ )( u ) | v , u ∈ P } .7) Let ( , ) be the symmetric bilinear form on F ℓ P with P as an orthonormal basis. If Z is a subspace, then Z ⊥ denotes the orthogonal complement of Z with respect to ( , ).In the following Lemma, we collect some inclusion relations involving the modulesdefined above, M i ( A )’s, and M i ( L )’s. Lemma 10.
Let ℓ be a prime and Γ ( V ) be one of { Γ s ( q , m ) , Γ o ( q , m ) , Γ o − ( q , m ) , Γ o + ( q , m ) , Γ uo ( q , m ) , Γ ue ( q , m ) } such that A ℓ or L ℓ is nilpotent. Also let ˜ q be the size of the fieldassociated with V. Then the following hold.(1) We have V s ⊂ M v ℓ ( s ) ( A ) , V r ⊂ M v ℓ ( r ) ( A ) , V s ⊂ M v ℓ ( u ) ( L ) , and V t ⊂ M v ℓ ( t ) ( L ) .(2) Given α = v ℓ ( h z − i ˜ q ) , we have C ⊂ M α ( A ) and C ⊂ M α ( L ) .(3) Given β = v ℓ ( rs ) , we have U ⊂ M β ( A ) .(4) Given γ = v ℓ ( tu ) , we have U ′ ⊂ M γ ( L ) (5) If ℓ | s, then we have U ⊂ M δ ( L ) . Here δ = v ℓ ( ts ) .Proof.
1. The eigenspace associated with an eigenvalue α of A is the same as the eigenspaceassociated with eigenvalue k − α of L . The proof of (1) now follows from the proof ofLemma 8.2. Let W be a maximal totally isotropic subspace of V and let v ∈ P . As W is anisotropic subspace, if v ⊂ W , then v is adjacent to every other 1-space of W , a total of h z i ˜ q −
1. Assume that v W . Let u ∈ P and u ⊂ W , then v is adjacent to u if and onlyif u is one of the h z − i ˜ q v ⊥ ∩ W . Here v ⊥ is the orthogonalcomplement of v with respect to the form on V . Thus we have A ([ W ]) = h z − i ˜ q + r [ W ].Since L = kI − A , we also have L ([ W ]) = − h z − i ˜ q + t [ W ]. Using h z − i ˜ q | t , we arrive at 2. MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 15
3. From Table 13, we observe that nilpotence of A ℓ or L ℓ implies ℓ | q +
1. Therefore wehave A − / qJ ≡ A + J (mod ℓ ). We note that im( J ) = F ℓ , and thus im( A + J ) = im( J − A ) = U . Using AJ = kJ , and µ = k / q and Lemma 3, we have A ( A − / qJ − ( r + s ) I ) = − rsI . As ℓ | r and ℓ | s , we can conclude 3.4. This follows by using LJ =
0, Lemma 3, and calculations similar to those above.5. Let v ∈ P and let W be any maximal totally isotropic subspace of V . From Lemma3, we have L ( L − ( t + u ) I ) = − tuI − µ J . From the computations above, we have L ([ W ]) = − h z − i ˜ q + t [ W ]. These two observations together with LJ = L L ( v ) − ( t + u ) v + J ( v ) − µ h z − i ˜ q [ W ] = − tuL ( v ) − t µ h z − i ˜ q [ W ] . (4)Lemma 2 and Lemma 3 show that µ h z − i ˜ q = − s . Since ℓ | s , we have L ( v ) − ( t + u ) v + J ( v ) − µ h z − i ˜ q [ W ] ≡ L ( v ) + J ( v ) (mod ℓ ) ≡ J ( v ) − A ( v ) (mod ℓ ) . (5)Now (4) and (5) yield 5. (cid:3)
8. 2- elementary divisors of S and K when Γ ( V ) = Γ s ( q , m ).Given the graph Γ s ( q , m ) and a prime ℓ , table 13 shows that A ℓ (equivalently L ℓ ) isnilpotent if and only if ℓ = q is odd. In this section we will compute the 2-elementarydivisors of S and K when Γ ( V ) = Γ s ( q , m ) and q is odd. We set h =
1, and z = m in Lemma3 and Lemma 2 to get the parameters for this graph. The graph Γ s ( q , m ) is an SRG withparameters v = h m i q , k = q h m − i q (1 + q m − ), λ = h m − i q −
2, and µ = h m − i q .The adjacency matrix A has eigenvalues ( k , r , s ) = ( k , q m − − , − (1 + q m − )) with mul-tiplicities (1 , f , g ) = (cid:18) , q ( q m − q m − + q − , q ( q m + q m − − q − (cid:19) . So the Laplacian L has eigenvalues(0 , t , u ) = (0 , k − r , k − s ) = (cid:18) , h m − i q (1 + q m ) , h m i q (1 + q m − ) (cid:19) with multiplicities (1 , f , g ).From now on in this section, we denote Γ s ( q , m ) by Γ s .8.1. Submodule Structure.
We now recall from § C , C ′ U , U ′ , V r and V s in the context of the graph Γ s . In this case G ( V ) = Sp(2 m , q ). From Theorem 2.13 andRemark 2.15 of [8], we have the following result. Theorem 11.
The F Sp(2 m , q ) submodule structure for F P is given by the followingHasse diagrams. m is even F P < > ⊥ CC ′ ( C ′ ) ⊥ C ⊥ < > { } m is odd F P C rrrrr ▲▲▲▲▲▲ < > ⊥ ▼▼▼▼qqqqqq C ′ ( C ′ ) ⊥ ▼▼▼▼▼ssss < > ▲▲▲▲▲ C ⊥ qqqqqq { } We have U = ( C ′ ) ⊥ , U ′ = C ⊥ , dim( C ) = f + , dim( C ′ ) = f , dim(( C ′ ) ⊥ ) = g + , and dim( C ⊥ ) = g. -elementary divisors when m is even. Elementary divisors of S . We identify M i ( A ) with M i and A with A . Since m is even, h m − i q is odd. Thus have v ( q − = v ( q m − − = v ( r ). We set v ( q − = w and v ( s ) = v (1 + q m − ) = d . So we have v ( k ) = d = v ( µ ). As | S | = kr f s g , we obtain v ( | S | ) = d + w f + dg . Lemma 10 implies that U ⊂ M d + w . By Theorem 11, we concludethat dim( M d + w ) ≥ dim( U ) = g +
1. As r is an eigenvalue of 2-valuation w and geometricmultiplicity f , Lemma 8 implies that dim( M w ) ≥ f . So by Lemma 7, setting j = s = f , s = g + s = dim(ker( A )) = t = w , and t = d + w , we obtain e = g + e w = f − g − e d + w = g +
1, and e i = i .8.2.0.2. Elementary divisors of K . We identify M i ( L ) with M i and L with L . In this case h m i q is even and thus v ( q m − >
1. This implies that v ( q m + =
1. Set v ( h m i q ) = b and v ( s ) = v ( q m − + = d . Then v ( t ) = v ( u ) = b + d , v ( k ) = v ( µ ) = b , and v ( | K | ) = f + ( b + d ) g − ( b +
1) (as | K | = t f u g / v ).Lemma 10 gives us U ′ ⊂ M b + d + and U ⊂ M d + . Now by Theorem 11, we can seethat dim( M b + d + ) ≥ g and dim( M d + ) ≥ g +
1. As t is an eigenvalue of valuation 1 andgeometric multiplicity f , Lemma 8 gives us dim( M ) ≥ f .So by Lemma 7, setting j = s = f , s = g + s = g , s = dim(ker( L )) =
1, and t = t = d +
1, and t = d + b +
1, we conclude that e = g + e = f − g − e d + = e b + d + = g −
1, and e i = i .8.3. 2 -elementary divisors when m is odd. Elementary divisors of S . We identify M i ( A ) with M i and A with A . In this case m is odd and thus h m − i q is even, and therefore v ( q m − − > v ( q m − + = v ( h m − i q ) = a and v ( q − = w . Now v ( r ) = a + w , v ( s ) = v ( q m − + = v ( k ) = v ( µ ) = a +
1. As | S | = kr f s g , we have v ( | S | ) = ( a + w ) f + g + a +
1. ByLemma 10, we have C ⊂ M a and U ⊂ M a + w + . Theorem 11 implies dim( M a ) ≥ f + M a + w + ) ≥ g +
1. As r is an eigenvalue of valuation a + w and geometric multiplicity f , Lemma 8 implies that dim( M a + w ) ≥ f . MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 17
So by Lemma 7, setting j = s = f + s = f , s = g + s = dim(ker( A )) = t = a , t = a + w , and t = a + w +
1, we have e = g , e a = e a + w = f − g − e a + w + = g +
1, and e i = i .8.3.0.2. Elementary divisors of K . We identify M i ( L ) with M i and L with L . As h m − i q is even we have v ( − + q m − ) >
1, and thus v ( s ) = v (1 + q m − ) =
1. Set v ( q m + = c and v ( h m − i q ) = a . We then have v ( t ) = a + c , v ( u ) = v ( v ) = c , v ( k ) = v ( µ ) = a + v ( | K | ) = ( a + c ) f + g − c .By Lemma 10, we have C ⊂ M a and U ⊂ M a + c + . Now application of Theorem 11gives us dim( M a ) ≥ f + M a + c + ) ≥ g +
1. As t is an eigenvalue of valuation a + c and geometric multiplicity f , Lemma 8 implies that dim( M a + c ) ≥ f .So by Lemma 7, setting j = s = f + s = f , s = g + s = dim(ker( L )) = t = a , t = a + c , and t = a + c +
1, we may conclude that e = g , e a = e a + c = f − g − e a + c + = g , and e i = i .9. 2- elementary divisors of S and K when Γ ( V ) = Γ o ( q , m ).Given the graph Γ o ( q , m ) and a prime ℓ , table 13 shows that A ℓ (equivalently L ℓ ) isnilpotent if and only if ℓ = q is odd. In this section we compute the 2-elementarydivisors of S and K when Γ ( V ) = Γ o ( q , m ). We set h =
1, and z = m in Lemma 3 andLemma 2 to get parameters for this graph. The graph Γ o ( q , m ) is an SRG with parameters v = h m i q , k = q h m − i q (1 + q m − ), λ = h m − i q −
2, and µ = h m − i q .The Adjacency matrix A has eigenvalues ( k , r , s ) = ( k , q m − − , − (1 + q m − )) with mul-tiplicities (1 , f , g ) = (cid:18) , q ( q m − q m − + q − , q ( q m + q m − − q − (cid:19) . So the Laplacian L has eigenvalues(0 , t , u ) = (0 , k − r , k − s ) = (cid:18) , h m − i q (1 + q m ) , h m i q (1 + q m − ) (cid:19) with multiplicities (1 , f , g ).From now on in this section, we denote Γ o ( q , m ) by Γ o .9.1. Submodule structure.
We now recall from § C , C ′ U , U ′ , V r and V s in the context of the graph Γ o . In this case G ( V ) = O(2 m + , q ). From Theorem 1 . .
5, and Lemma 7 . Theorem 12.
The module U ′⊥ has a submodule M containing V s such that dim( M / V s ) = . The relative positions of M, V s , C, V r , U ⊥ , U ′ , and U in the F O(2 m + , q ) submodulelattice of ( U ′ ) ⊥ are given by the following diagrams. m is odd. U ′⊥ ❉❉❉❉❉❉❉❉④④④④④④④④④ C U ⊥ ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ V r M ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ U ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ V s h i U ′ m is even. U ′⊥ ❉❉❉❉❉❉❉❉④④④④④④④④ C U ⊥ ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ V r M ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ U ❇❇❇❇❇❇❇❇ V s ④④④④④④④④ U ′ h i Here y : = dim( V s / U ′ ) = ( q m − q m ) − q q + , d : = dim( V r / U ) = ( q m + q m + q )2( q + − ,and x : = dim( U ′ ) = q m − q q − , dim( C / V r ) = , dim( M / V s ) = , and dim( U / U ′ ) = .Remark. [13] proves the above for m ≥
3. For m =
2, we refer to Theorem 3 . C of k L should be changed from h M | M ∈ L i k to h η , ( M ) | M ∈ L i k , andthe related definition of C + should be similarly corrected.9.2. 2 -elementary divisors when m is even. Elementary divisors of S . We identify M i ( A ) with M i and A with A . Since h m − i q is odd, we have v ( q − = v ( q m − − v ( q − = w , v (1 + q m − ) = d ,we have v ( k ) = d = v ( µ ), v ( r ) = w , v ( s ) = d , and v ( | S | ) = d + dg + w f .Case 1: Assume that w =
1. In this case, as v ( q m − − = w =
1, we have d = v ( q m − + = v ( q m − − + >
1. As r , s are integer eigenvalues with non-zero 2-valuations, Lemma10 gives us V r ⊂ M and V s ⊂ M . Now by Theorem 12, we have V r + V s = U ⊥ ⊂ M andhence dim( M ) ≥ dim( U ⊥ ) = f + g + − dim( U ) = f + g − x .By Lemma 10, U ⊂ M d + . By Theorem 12 we have dim U = x +
1, and thusdim( M d + ) ≥ x + s is an integer eigenvalue of geometric multiplicity g and 2-valuation d . Lemma 10gives us V s ⊂ M d . Also U ⊂ M d + ⊂ M d . So by Theorem 12, we have V s + U = M ⊂ M d and hence dim( M d ) ≥ dim( M ) = g +
1. We have,1( f + g − x − g − + d ( g + − ( x + + ( d + x + = f + d ( g + = v ( | S | ) . So by Lemma 7, setting j = s = f + g − x , s = g + s = x + s = dim(ker( A )) = t = t = d , and t = d +
1, we may conclude that e = x + e = f − x − e d = g − x , e d + = x +
1, and e i = i > MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 19
Case 2: Assume that w >
1. In this case, d =
1. As r , s are integer eigenvalues with non-zero 2-valuations, we have by Lemma 10, V r ⊂ M and V s ⊂ M . Thus by Theorem 12, U ⊥ ⊂ M . Hence dim( M ) ≥ f + g + − x − U ⊂ M w + . Thus dim( M w + ) ≥ x + f + g − x − f ) + w ( f − ( x + + ( w + x + = w f + g + = v ( | S | ) . So by Lemma 7, setting j = s = f + g − x , s = f , s = x + s = dim(ker( A )) = t = t = w , and t = w +
1, we may conclude that e = x + e = g − x , e w = f − x − e w + = x +
1, and e i = i > Elementary divisors of K . We identify M i ( L ) with M i and L with L . In this case, m is even. Since h m i q is even, we have v ( q m − > v ( q m + =
1. We set v ( h m i q ) = b and v ( s ) = v ( q m − + = d . So we have v ( t ) = v ( u ) = d + b , v ( k ) = v ( µ ) = d , and v ( | K | ) = f + ( b + d ) g − ( b + A ≡ L (mod 2), from Smith group computation above we conclude that dim( M ) ≥ f + g − x . As u is an eigenvalue of valuation d + b , Lemma 10 implies V s ⊂ M d + b , andthus dim( M d + b ) ≥ g . By Lemma 10, we have U ′ ⊂ M d + b + and U ⊂ M d + . Thereforedim( M d + b + ) ≥ x , by Theorem 12.Since V s ⊂ M d + b ⊂ M d + and U ⊂ M d + , by Theorem 12 we have M d + ⊂ M . We maynow conclude that dim( M d + ) ≥ g + f + g − x − g − + ( d + g + − g ) + ( d + b )( g − x ) + ( d + b + x − = f + ( d + b ) g − ( b + = v ( | K | ) . We apply Lemma 7 to conclude the following. • If b >
1, set j = s = f + g − x , s = g + s = g , s = x s = dim(ker( L )) = t = t = d + t = d + b , t = d + b +
1, then by Lemma 7, we have e = x + e = f − x − e d + = e d + b = g − x , e d + b + = x −
1, and e i = i . • If b =
1, set j = s = f + g − x , s = g + s = x , s = dim(ker( L )) = t = t = d + t = d + b +
1, then by Lemma 7, we have e = x + e = f − x − e + d = g + − x , e d + b + = x −
1, and e i = i .9.3. 2 -elementary divisors of S and K , when m is odd. Elementary divisors of S . We identify M i ( A ) with M i and A with A . In thiscase m is odd. As m is odd, h m − i q is even and thus v ( h m − i q ) >
0. Set v ( h m − i q ) = a and v ( q − = w . So we have v ( r ) = a + w , v ( s ) = v ( k ) = v ( µ ) = a +
1, and v ( | S | ) = ( a + w ) f + g + ( a + M a ) ≥ f + C ) = f + s is an integer eigenvalue of multiplicity g with 2-valuation 1, Lemma 10implies V s ⊂ M . Since C ⊂ M a ⊂ M aswell , Theorem 12 implies M ⊃ V s + C = ( U ′ ) ⊥ .So dim( M ) ≥ dim( U ′⊥ ) = f + g + − dim( U ′ ) = f + g + − x . As r is an integer eigenvalueof multiplicity f with 2-valuation a + w , Lemma 10 implies dim( M a + w ) ≥ f . By Lemma10, we also have U ⊂ M a + w + . From Theorem 12, we conclude dim( M a + w + ) ≥ x + f + g + − x − ( f + + a ( f + − f ) + ( a + w )( f − ( x + + ( a + b + x + = g + a + (( a + w ) f ) + = v ( | S | ) . We may conclude the following from Lemma 7. • If a >
1, setting j = s = f + g + − x , s = f + s = f , s = x + s = dim(ker( A )) = t = t = a , t = a + w , and t = a + w +
1, by Lemma 7we get e = x , e = g − x , e a = e a + w = f − x − e a + w + = x + e i = i > • If a =
1, setting j = s = f + g + − x , s = f , s = x + s = dim(ker( A )) = t = t = a + w , and t = a + w +
1, by Lemma 7, e = x , e = g + − x , e a + w = f − x − e a + w + = x +
1, and e i = i > Elementary divisors of K . We identify M i ( L ) with M i and L with L . In this case, m is odd. As h m − i q is even, we have v ( − + q m − ) > v ( s ) = v (1 + q m − ) = v ( q m + = c and v ( h m − i q ) = a . We have v ( t ) = a + c , v ( u ) = v ( v ) = c , v ( k ) = v ( µ ) = a +
1, and v ( | K | ) = ( a + c ) f + g − c .By Lemma 10, C ⊂ M a and U ⊂ M a + c + . Thus we have dim( M a ) ≥ f + M a + c + ) ≥ x +
1, by Theorem 12. As u is an integer eigenvalue of geometric multi-plicity g with 2-valuation 1, Lemma 8 implies M ⊃ V s . Since C ⊂ M a ⊂ M , Theorem12 implies M ⊃ ( U ′ ) ⊥ . Thus dim( M ) ≥ f + g + − x . As t is an integer eigenvalue ofmultiplicity g with 2-valuation a + b , Lemma 10 implies M a + c ⊃ V r . So dim( M a + c ) ≥ f .We have 1( f + g + − x − ( f + + a ( f + − f ) + ( a + c )( f − ( x + + ( a + c + x + − = g + ( a + c ) f − b = v ( | K | ) . Using Lemma 7, we conclude the following. • If a >
1, setting j = s = f + g + − x , s = f + s = f , s = x + s = dim(ker( L )) = t = t = t = a + c , and t = a + c +
1, by Lemma 7we have e = x , e = g − x , e a = e a + c = f − x − e a + c + = x , and e i = i . • If a =
1, by similar arguments we can deduce that e = x , e = g − x + e a + c = f − x − e a + c + = x , and e i = i .10. ℓ - elementary divisors of S and K when Γ ( V ) = Γ o − ( q , m ), and ℓ | q + Γ o − ( q , m ) and a prime ℓ , table 13 shows that A ℓ is nilpotent if and onlyif either i) ℓ = q is odd; or ii) ℓ is odd with ℓ | q + m is even. We also have L ℓ is nilpotent if and only if ℓ | q + ℓ -elementary divisors of S and K when Γ ( V ) = Γ o − ( q , m )and ( ℓ, q , m ) satisfy the arithmetic conditions given above.From now on in this section, we denote Γ o − ( q , m ) by Γ o − , and ℓ is a prime that meetsthe description in the previous paragraph. We set h = z = m − Γ o − ( V ) is an SRG withparameters v = h m − i q ( q m + k = q h m − i q ( q m − + λ = q h m − i q ( q m − + − − q m − ,and µ = h m − i q ( q m − + A are ( k , r , s ) = ( k , q m − − , − (1 + q m − )) , withmultiplicities (1 , f , g ) = (cid:18) , q ( q m − − q m − + q − , q ( q m + q m − − q − (cid:19) .So the Laplacian L has eigenvalues (0 , t , u ) = (cid:18) , h m − i q (1 + q m ) , h m − i q (1 + q m − ) (cid:19) withmultiplicities (1 , f , g ). MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 21
Submodule structure.
We now recall from § C , C ′ U , U ′ , V r and V s in the context of the graph Γ o − . In this case G ( V ) = O − (2 m , q ). By Corollary 8 . .
7, Lemma 8 .
8, Lemma 8 .
9, Corollary 8 .
10, and Corollary 8 .
11 of [13], we havethe following result.
Theorem 13.
Given a prime ℓ with ℓ | q + . The relative positions of C, V r , V s , U ′ , U,and h i in the F ℓ O − (2 m , q ) submodule structure of C are in the following diagrams. Wehave dim( C ) = f + . ℓ = and m is evenCV r U ●●●●●●●●⑥⑥⑥⑥⑥⑥⑥⑥ h i U ′ = V s ℓ = and m is oddCV r UU ′ = V s h i ℓ , , ℓ | q + , and m is even.C ●●●●●●●●●● U ❋❋❋❋❋❋❋❋❋ V r h i U ′ = V s ℓ , , ℓ | q + , and m is odd.C ①①①①①①①①①① ●●●●●●●●● U ❊❊❊❊❊❊❊❊ V r ②②②②②②②② U ′ = V s h i -elementary divisors of S and K when m is odd. Elementary divisors of S . We identify M i ( A ) with M i and A with A . Since m is odd, h m − i q is an odd number. So we have v ( r ) = v ( q m − − = v ( q − h m − i q is an even number, v ( q m − − >
1, and thus v ( q m − + = = v ( s ). We also have v ( k ) = v ( µ ) =
1. Setting v ( q − = w , we have v ( r ) = w and v ( | S | ) = w f + g + U ⊂ M w + . So by Theorem 13, we get dim( M w + ) ≥ g + r is an integer eigenvalue with valuation w , Lemma 8 implies dim( M w ) ≥ f .Now we have ( w )( f − ( g + + ( w + g + = w f + g + = v ( | S | ). So by Lemma7, setting j = t = w , t = w + s = f , s = g + s = dim(ker( A )) =
0, we concludethat e = g + e w = f − g − e w + = g +
1, and e i = i .10.2.0.2. Elementary divisors of K . We identify M i ( L ) with M i and L with L . Set v ( q m + = c and v ( h m − i q ) = b . Since m is odd, we have v ( s ) = v ( q m − + =
1. So v ( t ) = c , v ( u ) = b + v ( v ) = c + b , v ( µ ) =
1, and v ( | K | ) = c f + ( b + g − ( c + b ). As t is an eigenvalue of valuation f , by Lemma 8, we have dim( M c ) ≥ f .Lemma 10 implies that U ⊂ M c + and U ′ ⊂ M b + c + . Therefore dim( M b + c + ) ≥ g anddim( M c + ) ≥ g +
1, by Theorem 13.Now, c ( f − ( g + + ( c + g + − g ) + ( b + c + g − = c f + ( b + g − b − c .So by Lemma 7, setting j = s = f , s = g + s = g , s = dim(ker( L )) = t = c , t = c +
1, and t = b + c +
1, we may conclude that e = g + e c = f − g − e c + = e b + c + = g − e i = i .10.3. 2 -elementary divisors of S and K when m is even. Elementary divisors of S . We identify M i ( A ) with M i and A with A . As m iseven, h m − i q is even and h m − i q is odd. Set v ( h m − i q ) = a , v ( q m − − = v ( q − = w and v ( q m − + = d . Then v ( r ) = a + w , v ( s ) = d , v ( k ) = v ( µ ) = a + d , and v ( | S | ) = ( a + w ) f + dg + a + d .Lemma 10, implies that C ⊂ M a . Thus by Theorem 13 we have dim( M a ) ≥ f + r is an eigenvalue of valuation a + w , Lemma 8 implies dim( M a + w ) ≥ f . By Lemma10, U ⊂ M a + d + w . Thus by Theorem 13, dim( M a + d + w ) ≥ g + a ( f + − f ) + ( a + w )( f − ( g + + ( a + d + w )( g + = ( a + w ) f + dg + a + d .So by Lemma 7, setting j = s = f + s = f , s = g + s = dim(ker( A )) = t = a , t = a + w , and t = a + d + w , we have e = g , e a = , e a + w = f − g − , e a + d + w = g + e i = i .10.3.0.2. Elementary divisors of K . We identify M i ( L ) with M i and L with L . As m iseven, h m − i q is even and h m − i q is odd. Set v ( h m − i q ) = a and v ( s ) = v ( q m − + = d .As h m i q is even, we have v ( q m − > v ( q m + =
1. Since h m − i q is odd, we have v ( v ) =
1. We have v ( t ) = a + v ( u ) = d , and v ( µ ) = a + d , and v ( | K | ) = ( a + f + dg − C ⊂ M a and U ⊂ M a + d + . Therefore by Theorem 13, we havedim( M a ) ≥ f + M a + d + ) ≥ g +
1. As t is an integer eigenvalue of L with valuation a +
1, Lemma 8 implies dim( M a + ) ≥ f .We have a ( f + − f ) + ( a + f − ( g + + ( a + d + g + − = ( a + f + dg − = v ( | K | ).Therefore by Lemma 7, setting j = s = f + s = f , s = g + s = dim(ker( L )) = t = a , t = a +
1, and t = a + d +
1, we have e = g , e a = e a + = f − g − e a + d + = g ,and e i = i , ℓ -elementary divisors of S and K when m is even, ℓ , and ℓ | q + . Elementary divisors of S . We identify M i ( A ) with M i and A ℓ with A . In thiscase ℓ is an odd prime dividing q + m is even. Thus v ℓ ( h m − i q ) = v ( r ). Set v ℓ ( h m − i q ) = a and v ℓ ( s ) = v ℓ ( q m − + = d . Then v p ( r ) = a , v ℓ ( k ) = a + d = v ℓ ( µ ), and v ℓ ( | S | ) = a f + dg + a + d .By Lemma 10, we have C ⊂ M a and U ⊂ M a + d . Thus dim( M a ) ≥ f + M a + d ) ≥ g +
1, by Theorem 13.We have a ( f + − ( g + + ( a + d )( g + = v ℓ ( | S | ). So by Lemma 7, setting j = s = f + s = g + s = dim(ker( A )) = t = a , t = a + d , we conclude e = ge a = f − g , e a + d = g +
1, and e i = i .10.4.0.2. Elementary divisors of K . We identify M i ( L ) with M i and L ℓ with L . In thiscase ℓ is an odd prime with m even. We set v ℓ ( h m − i q ) = a , and v ℓ ( q m − + = d . As q ≡ − ℓ ), we have v ℓ ( v ) = v ℓ (cid:18)(cid:18)h m − i q ( q m + (cid:19)(cid:19) =
0. Thus v ℓ ( t ) = a , v ℓ ( u ) = d , and v ℓ ( | K | ) = a f + dg . MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 23
Lemma 10 gives us C ⊂ M a and U ⊂ M a + d . Now Theorem 13 gives us dim( M a ) ≥ f + M a + d ) ≥ g + a ( f + − ( g + + ( a + d )( g + − = a f + dg = . So by Lemma 7, setting j = s = f + s = g + s = dim(ker( A )) = t = a , t = a + d , we have e = g , e a = f − g , e a + d = g , and e i = i , ℓ -elementary divisors of K when m is odd, ℓ , and ℓ | q + . In this case we have q ≡ − ℓ ) and thus r ≡ s ≡ − ℓ ) and thus ℓ ∤ | S | .However we see that ℓ | t and ℓ | u and thus ℓ | | K | . In this section we compute the ℓ -elementary divisors of K .10.5.0.1. Elementary divisors of K . We identify M i ( L ) with M i and L ℓ with L . Set v ℓ ( q m + = c and v ℓ ( h m − i q ) = b . Since m is odd, we have v ℓ ( s ) = v ℓ ( q m − + =
0. So v ℓ ( t ) = c , v ℓ ( u ) = b , v ℓ ( v ) = c + b , v ℓ ( µ ) =
0, and v ℓ ( | K | ) = c f + bg − ( c + b ).Lemma 10 gives us U ′ ⊂ M b + c and V r ⊂ M c . Therefore dim( M b + c ) ≥ g and dim( M c ) ≥ f , by Theorem 13. Now we have v ℓ ( | K | ) = c f + bg − c − b = c ( f − g ) + ( b + c )( g − j = s = f , s = g + s = dim(ker( L )) = t = c , t = b + c , we may conclude that e = g + e c = f − g , e b + c = g −
1, and e i = i .11. ℓ - elementary divisors of S and K when Γ ( V ) = Γ o + ( q , m ), and ℓ | q + Γ o + ( q , m ) and a prime ℓ , table 13 shows that A ℓ is nilpotent if and onlyif either i) ℓ = q is odd; ii) or ℓ is odd with ℓ | q + m is odd. Also L ℓ is nilpotentif and only if ℓ | q + ℓ -elementary divisors of S and K when Γ ( V ) = Γ o + ( q , m ) and ( ℓ, q , m ) satisfy the arithmetic conditions given above.From now on in this section, we denote Γ o + ( q , m ) by Γ o + , and ℓ is a prime that meets thedescription in the previous paragraph. We set h =
0, and z = m in Lemma 3 and Lemma 2to get parameters for this graph. Thus Γ o + ( V ) is an SRG with parameters v = h m i q ( q m − + k = q h m − i q ( q m − + λ = q h m − i q ( q m − + − − q m − , and µ = h m − i q ( q m − + A are ( k , r , s ) = ( k , q m − − , − (1 + q m − )),with multiplicities (1 , f , g ) = (cid:18) , q ( q m − q m − + q − , q ( q m − + q m − − q − (cid:19) .So L has eigenvalues (0 , t , u ) = (0 , k − r , k − s ) = (cid:18) , h m − i q (1 + q m − ) , h m i q (1 + q m − ) (cid:19) with multiplicities (1 , f , g ).11.1. Submodule Structure.
We now recall from § C , C ′ U , U ′ , V r and V s in the context of the graph Γ o + . In this case G ( V ) = O + (2 m , q ). By Corollary 2.10,6.5, Lemma 6.6 of [13] we have the following result. Theorem 14.
Let ℓ be a prime with ℓ | q + . Then the relative positions of U ⊥ , C, V s , V r ,U, U ′ , and h i in the F ℓ O + (2 m , q ) submodule structure of ( U ′ ) ⊥ are given in the followingdiagrams. We have dim(( U ′ ) ⊥ ) = g + . ℓ = and m is even ( U ′ ) ⊥ ●●●●●●●● C U ⊥ ①①①①①①①① U = V r ❋❋❋❋❋❋❋❋❋ V s U ′ h i ℓ = and m is odd ( U ′ ) ⊥ ●●●●●●●● C U ⊥ ①①①①①①①① U = V r ❋❋❋❋❋❋❋❋❋ V s h i U ′ ℓ , , m is odd and ℓ | q + U ′ ) ⊥ U ⊥ ③③③③③③③③③ ❉❉❉❉❉❉❉❉❉ C ❈❈❈❈❈❈❈❈❈ V s ✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌ U ❉❉❉❉❉❉❉❉❉ V r h i U ′ ℓ , , m is odd and ℓ | q + U ′ ) ⊥ U ⊥ ④④④④④④④④④ ❉❉❉❉❉❉❉❉❉ C ❇❇❇❇❇❇❇❇❇ V s ✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌ U ❈❈❈❈❈❈❈❈ V r U ′ h i Here, dim( U ′ ) = f − , dim( U ) = f , dim( C ) = f + , dim( U ⊥ ) = f + g + − dim( U ) = g + . -elementary divisors of S and K when m is even. Elementary divisors of S . We identify M i ( A ) with M i and A with A . In thiscase m is even. Therefore h m − i q is odd. So we have v ( r ) = v ( q m − − = v ( q − v ( s ) = v ( q m − + =
1, and v ( k ) = v ( µ ) =
1. Setting v ( q − = w , we have v ( | S | ) = w f + g + V r ⊂ M and V s ⊂ M . Thus by Theorem 14, we see that M ⊃ U ⊥ ,and hence dim( M ) ≥ g +
1. Lemma 10 gives us U ⊂ M w + . Thus by Theorem 14, we getdim( M w + ) ≥ f .We have 1( g + − f ) + ( w + f = v ( | S | ). So by Lemma 7, setting j = s = g + s = f , s = dim(ker( A )) = t =
1, and t = w +
1, we conclude e = f , e = g + − f , e w + = f , and e i = i .11.2.0.2. Elementary divisors of K . We identify M i ( L ) with M i and L with L . In this case m is even, so we have v ( s ) = v ( q m − + = v ( k ) = v ( µ ) =
1. Set v ( q m − + = c MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 25 and v ( h m i q ) = b . We now have v ( t ) = c , v ( u ) = b + v ( v ) = c + b , and v ( | K | ) = c f + ( b + g − b − c .Lemma 10 gives us U ′ ⊂ M b + c + , U ⊂ M c + , and V s ⊂ M b + . We use Lemma 7 toconclude the following.(1) If c < b , we have M c + ⊃ M b + and thus V s ⊂ M c + . Also since U ⊂ M c + ,Theorem 14 implies U ⊥ ⊂ M c + . Hence dim( M c + ) ≥ g +
1. Again by Theorem14 dim( M b + ) ≥ dim( V s ) ≥ g , and dim( M b + c + ) ≥ dim( U ′ ) ≥ f − c + g + − g ) + ( b + g − ( f − + ( b + c + f − − = v ( | K | ). Nowby Lemma 7, setting j = s = g + s = g , s = f − s = dim(ker( L )) = t = c + t = b +
1, and t = b + c +
1, we have e = f , e c + = e b + = g − f + e b + c + = f −
2, and e i = i .(2) If c > b , By arguments similar to those above we can show dim( M b + ) ≥ g + M c + ) ≥ f , and dim( M b + c + ) ≥ f − b + g + − f ) + ( c + f − ( f − + ( b + c + f − − = v ( | K | ). ApplyingLemma 7 as above, we get e = f , e c + = e b + = g − f + e b + c + = f −
2, and e i = i , b = c , by similar arguments, we can show e = f , e c + = g − f + e c + = f − e i = i .11.3. 2 -elementary divisors of S and K when m is odd. Elementary divisors of S . We identify M i ( A ) with M i and A with A . As m isodd, we have v ( q m − + = v ( h m − i q ) >
1. We set v ( h m − i q ) = a , v ( q − = w and v ( q m − + = d . So v ( r ) = a + w , v ( s ) = d , v ( k ) = v ( µ ) = a + d , and v ( | S | ) = ( a + w ) f + dg + a + d .By Lemma 10, we have C ⊂ M a , U ⊂ M a + w + d , and V s ⊂ M d . Using Lemma 7 wearrive at the following conclusions.(1) Assume that d < a , then M d ⊃ M a ⊃ C . Now since V s and C are subsets of M d ,Theorem 14 gives us ( U ) ′⊥ ⊂ M d , and thus dim( M d ) ≥ g +
2. Since C ⊂ M a ,we have dim( M a ) ≥ f +
1. Again by Theorem 14 we get dim( M a + w + d ) ≥ f = dim( U ).Now, d ( g + − ( f + + a ( f + − f ) + ( a + w + d )( f ) = v ( | S | ). So by Lemma7, setting j = s = g + s = f + s = f , s = dim(ker( A )), t = d , t = a ,and t = a + w + d , we have e = f − e d = g − f + e a = e a + w + d = f , and e i = i .(2) If a < d , by arguments similar to the ones above, we can show dim( M a ) ≥ g + U ⊂ M a + w + d ⊂ M d and V s ⊂ M d , Theorem 14 implies dim( M d ) ≥ g +
1, anddim( M a + w + d ) ≥ f .Now, a ( g + − ( g + + d ( g + − f ) + ( a + w + d )( f ) = v ( | S | ). ApplyingLemma 7 as above, we have e = f − e d = g − f + e a = e a + w + d = f , and e i = i .(3) If a = d , by similar arguments we can show that e = f − e a = g + − f , e a + d + w = f , and e i = i .11.3.0.2. Elementary divisors of K . We identify M i ( L ) with M i and L with L . As m is odd, we have v ( q m − + = v ( h m − i q ) >
1. Set v ( h m − i q ) = a and v ( s ) = v ( q m − + = d . So v ( t ) = a + v ( u ) = d , v ( v ) =
1, and v ( | K | ) = ( a + f + dg − C ⊂ M a and U ⊂ M a + d + . Again by Lemma 10 we have V s ⊂ M d . Using Lemma 7 we arrive at the following conclusions. (1) Assume d < a , then C ⊂ M a ⊂ M d . As V s ⊂ M d , Theorem 14 implies ( U ′ ) ⊥ ⊂ M d , and hence dim( M d ) ≥ dim( U ′⊥ ) = g +
2. Also dim( M a ) ≥ dim( C ) = f + M a + d + ) ≥ dim( U ) = f .Now, d ( g + − ( f + + a ( f + − f ) + ( a + d + f − = dg + ( a + f − a − = v ( | K | ). So by Lemma 7, setting j = s = g + s = g + s = f , s = dim(ker( L )) = t = a , t = d , and t = a + d +
1, we have e = f − e a = e d = g + − f , e a + d + = f −
1, and e i = i .(2) If a < d , by arguments similar to those above, dim( M a ) ≥ g +
2. As ∈ M d ,and V s ⊂ M d , by Theorem 14, it follows that dim( M d ) ≥ g +
1. We also havedim( M a + d + ) ≥ f .Now, a ( g + − ( g + + d ( g + − f ) + ( a + d + f − = dg + ( a + f − a − = v ( | K | ).By applying Lemma 7 as above, we have e = f − e a = e d = g + − f , e a + d + = f −
1, and e i = i .(3) If a = d , by arguments similar to those above we may show that e = f − e a = g + − f , e a + = f −
1, and e i = i .11.4. ℓ -elementary divisors of S and K when m is odd, and ℓ | q + . Elementary divisors of S . We identify M i ( A ) with M i and A ℓ with A . In thiscase ℓ | q + m odd. So we have v ℓ ( h m − i q ) = v ℓ ( r ). We set v ℓ ( h m − i q ) = v ℓ ( r ) = a , and v ℓ ( s ) = v ℓ ( q m − + = d . Then v ℓ ( k ) = a + d = v ℓ ( µ ) and v ℓ ( | S | ) = a f + dg + a + d .By Lemma 10, we have U ⊂ M a + d , V s ⊂ M d , and V r ⊂ M a . We now apply Theorem14 and Lemma 7 to conclude the following.(1) Assume a < d , then V s ⊂ M a ⊂ M d . Since V r ⊂ M a , Theorem 14 impliesdim( M a ) ≥ dim( U ⊥ ) = g +
1. As U ⊂ M a + d ⊂ M a , Theorem 14 impliesdim( M a ) ≥ dim( U ⊥ ) ≥ g +
1. From above, we have dim( M a + d ) ≥ dim( U ) = f .Now, a ( g + − ( g + + d ( g + − f ) + ( a + d ) f = v ℓ ( | S | ). By Lemma 7, setting j = s = g + s = g + s = f , s = dim(ker( A )) = t = a , t = d , and t = a + d , we have e = f − e a = e d = g − f + e a + d = f , and e i = i .(2) If d < a , by arguments similar to those above, we can show that dim( M d ) ≥ g + U ⊂ M a + d ⊂ M a and V r ⊂ M a , Theorem 14 implies dim( M a ) ≥ dim( C ) ≥ f +
1. From above, we have dim( M a + d ) ≥ dim( U ) = f .Now, b ( g + − ( f + + d ( f + − f ) + ( a + d ) f = v ℓ ( | S | ). By applying Lemma7, we get e = f − e a = e d = g − f + e a + d = f , and e i = i .(3) If a = d , by arguments similar to those above, we get e = f − e a = g − f + e a + d = f , and e i = i .11.4.0.2. Elementary divisors of K . We identify M i ( L ) with M i and L ℓ with L . In thiscase ℓ | q + m odd, we have v ℓ ( h m − i q ) > v ℓ ( q m − + =
0, and v ℓ ( h m i q ) =
0. Setting v ℓ ( h m − i q ) = a v ( s ) = v ℓ ( q m − + = d , we have v ℓ ( t ) = a , v ℓ ( u ) = c , v ℓ ( v ) =
0, and v ℓ ( | K | ) = a f + dg . As L ≡ − A (mod 2 a + d ), we have M i ( L ) = M i ( A ) for all i ≤ a + c . So we have U ⊂ M a + d , V s ⊂ M d , and V r ⊂ M a Using this fact and Lemma 7 we conclude the following.(1) If a < d , then M a ⊃ M d ⊃ V s . Since M a ⊃ V r as well, by Theorem 14 we havedim( M a ) ≥ g +
2, dim( M d ) ≥ g +
1, and dim( M a + d ) ≥ f . MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 27
Now a ( g + − ( g + + d ( g + − f ) + ( a + b )( f − = v ℓ ( | K | ). So by Lemma 7,setting j = s = g + s = g + s = f , s = dim(ker( L )) = t = a , t = d ,and t = a + d , we have e = f − e a = e d = g − f +
1, and e a + d = f −
1, and e i = i .(2) If a > d , we have M d ⊃ M a . So by similar arguments dim( M d ) ≥ g +
2. And bythe above we have dim( M a ) ≥ g +
1, and dim( M a + d ) ≥ f .Now a ( g + − ( f + + d ( f + − f ) + ( a + d )( f − = v ℓ ( | K | ). By Lemma7, we have e = f − e a = e d = g − f +
1, and e a + d = f −
1, and e i = i .(3) If a = d , by arguments similar to those above, we have e = f − e a = g − f + e a = f −
1, and e i = i .11.5. ℓ -elementary divisors of K when m is even and ℓ | q + . In this case we have q ≡ − ℓ ) and thus r ≡ s ≡ − ℓ ) and thus ℓ ∤ | S | .However we see that ℓ | t and ℓ | u and thus ℓ | | K | . In this section we compute the ℓ -elementary divisors of K .11.5.0.1. Elementary divisors of K . We identify M i ( L ) with M i and L ℓ with L . Set v ℓ ( q m − + = c and v ℓ ( h m i q ) = b . Since m is odd, we have v ℓ ( s ) = v ℓ ( q m − + = v ℓ ( t ) = c , v ℓ ( u ) = b , v ℓ ( v ) = c + b , v ℓ ( µ ) =
0, and v ℓ ( | K | ) = c f + bg − ( c + b ).Lemma 10 gives us V r ⊂ M c , V s ⊂ M b and U ′ ⊂ M b + c . We now use Lemma 7 toconclude the following.(1) If c < b , we have M c ⊃ M b and thus V s ⊂ M c . Also since V r ⊂ M c , Theorem 14implies U ⊥ ⊂ M c . Hence dim( M c ) ≥ g +
1. Again by Theorem 14 dim( M b ) ≥ dim( V s ) ≥ g , and dim( M b + c ) ≥ dim( U ′ ) ≥ f − c )( g + − g ) + ( b )( g − ( f − + ( b + c )( f − − = v ( | K | ). Now by Lemma7, setting j = s = g + s = g , s = f − s = dim(ker( L )) = t = c , t = b , and t = b + c , we have e = f , e c = e b = g − f + e b + c = f −
2, and e i = i .(2) If c > b , By arguments similar to those above we can show dim( M b ) ≥ g +
1. Wealso have dim( M c ) ≥ f , and dim( M b + c ) ≥ f − b )( g + − f ) + ( c )( f − ( f − + ( b + c )( f − − = v ( | K | ). ApplyingLemma 7 as above, we get e = f , e c = e b = g − f + e b + c = f −
2, and e i = i , b = c , by similar arguments, we can show e = f , e c = g − f + e c = f − e i = i .12. ℓ - elementary divisors of S and K when Γ ( V ) = Γ ue ( q , m ), and ℓ | q + Γ ue ( q , m ) and a prime ℓ , table 13 shows that A ℓ (equivalently L ℓ ) isnilpotent if and only if ℓ | q +
1. In this section we compute the ℓ -elementary divisors of S and K when Γ ( V ) = Γ ue ( q , m ) and ℓ | q + h = , and z = m in Lemma 3 and Lemma 2 to get parameters for this graph.Thus Γ ue ( q , m ) is an SRG with parameters v = h m i q ( q m − + k = q h m − i q ( q m − + λ = ( q − + q (( q ) m − + h m − i q , and µ = h m − i q ( q m − + The adjacency matrix A has eigenvalues ( k , r , s ) = ( k , q m − − , − (1 + q m − )), with mul-tiplicities (1 , f , g ) = , q h m i q ( q m − + q + , q h m − i q ( q m − + q − . So L has eigenvalues(0 , t , u ) = (cid:18) , h m − i q (1 + q m − ) , h m i q (1 + q m − ) (cid:19) with multiplicities (1 , f , g ).12.1. Submodule Structure.
We now recall from § C , C ′ U , U ′ , V r and V s in the context of the graph Γ ue . In this case G ( V ) = U(2 m , q ). By Corollary 2 . .
5, and Lemma 4.6 of [13], we have the following result.
Theorem 15.
When ℓ | q + , the module U ′⊥ has a submodule M containing V s suchthat dim( M / V s ) = . The relative positions of M,U ⊥ , C, V s , V r , U, U ′ , and h i in the F ℓ U(2 m , q ) submodule structure of ( U ′ ) ⊥ are given in the following diagrams. ℓ ∤ m ( U ′ ) ⊥ ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ ✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸ U ⊥ ☛☛☛☛☛☛☛☛☛☛☛☛☛☛ ✷✷✷✷✷✷✷✷✷✷✷✷✷✷ C M ③③③③③③③③③③③③③③③③③③③③③ V r V s ☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞ U ❊❊❊❊❊❊❊❊❊ h i U ′ ℓ | m U ′⊥ ✌✌✌✌✌✌✌✌✌✌✌✌✌✌ U ⊥ ❈❈❈❈❈❈❈❈✌✌✌✌✌✌✌✌✌✌✌✌✌✌ C M ⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤ V r V s ✌✌✌✌✌✌✌✌✌✌✌✌✌✌ U ❈❈❈❈❈❈❈❈ U ′ h i Here δ : = dim( V r / U ) = q m − q + , y : = dim( V s / U ′ ) = ( q m − q m − − q )( q + , x : = dim( U ′ ) = ( q m − q m − + q − q − , and dim( M / V s ) = dim( U / U ′ ) = dim( C / V r ) = . ℓ -elementary divisors of S and K when ℓ | q + and ℓ ∤ m . Elementary divisors of S . We identify M i ( A ) with M i and A ℓ with A . In thiscase, ℓ | q + ℓ ∤ m . Set v ℓ ( q − = w , v ℓ ( h m − i q ) = a , and v ℓ ( q m − + = d . Then v ℓ ( r ) = w + a , v ℓ ( s ) = d , v ℓ ( k ) = v ℓ ( µ ) = a + d , and v ℓ ( | S | ) = ( w + a ) f + dg + a + d .Now s is an eigenvalue of valuation d and k is an eigenvalue of valuation a + d . SoLemma 8 implies V s ⊂ M d and h i ⊂ M a + d ⊂ M d . So by Theorem 15, we have M d ⊃ V s ⊕ h i .Lemma 10 implies C ⊂ M a , U ⊂ M w + a + d , and V r ⊂ M w + a . MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 29
For any positive integer n , we have q n + + = ( q + (cid:18)h n + i − q (cid:19) . Therefore if ℓ | q + v ℓ ( q n + + = v ℓ ( q +
1) if and only if ℓ | n + v ℓ ( h m − i q ) = ℓ | m − v ℓ ( q − = v ℓ ( q +
1) if and only if ℓ , ℓ ∤ m −
1. In this case, a =
0, since v ℓ ( h m − i q ) =
0. We have v ℓ ( | S | ) = w f + dg + d . We apply Lemma 7 and Theorem 15 to arrive at the following results.(1) Assume w < d , then we have as M d ⊃ V s ⊕ h i , and V r ⊂ M d ⊂ M w ⊃ V r . So byTheorem 15, U ⊥ = V r + V s ⊃ M w . We saw that M w + d ⊃ U . Again by Theorem 15,we have dim( M w ) ≥ dim( U ⊥ ) = f + g + − dim( U ) = f + g − x , dim( M d ) ≥ g + M w + d ) ≥ dim( U ) = x + a ( f + g − x − ( g + + d (( g + − ( x + + ( w + d )( x + = w f + dg + d = v ℓ ( | S | ). So by Lemma 7, setting j = s = f + g − x , s = g + s = x + s = dim(ker( A )) = t = w , t = d , and t = w + d , we have e = x + e w = f − x − e d = g − x , e w + d = x +
1, and e i = i .(2) If w > d , by arguments similar to those above we can show that M d ⊃ U ⊥ , M w ⊃ V r , and M w + d ⊃ U . Applying Lemma 7 as above, we can conclude that e = x + e w = f − x − e d = g − x , e w + d = x +
1, and e i = i .(3) If w = d , again by arguments similar to those above, we can show that e = x + e w = f + g − x − e w + d = x +
1, and e i = i .Subcase 2: When ℓ | m −
1. In this case, a ,
0, but ℓ ∤ m −
3. So d = v ℓ ( q m − + = v ℓ ( q + ≤ v ℓ ( q − = w , with the equality holding if and only if ℓ ,
2. So we have either a ≤ d < w + a < w + a + d , or d < a < w + a < w + a + d .(1) If a < d < w + a < w + a + d , we have M d ⊂ M a . So by Theorem 15 M a ⊃ M d ⊃ C + V s = ( U ′ ) ⊥ . Since M w + a ⊃ V r and d < w + a , Theorem 15 implies M d ⊃ V s + V r = U ⊥ . We also have M w + a ⊃ V r and M w + a + d ⊃ U . Thus wehave dim( M a ) ≥ dim( U ′⊥ ) = f + g + − x , dim( M w ) ≥ dim( U ⊥ ) = f + g − x ,dim( M w + a ) ≥ f , and dim( M w + a + d ) ≥ x + a ( f + g + − x − ( f + g − x )) + d ( f + g − x − f ) + ( w + d )( f − x − + ( w + a + d )( x + = ( w + a ) f + dg + d + a = v ℓ ( | S | ) . Thus by Lemma 7, setting j = s = f + g + − x , s = f + g − x , s = f , s = x + s = dim(ker( A )) = t = a , t = d , t = w + a , and t = w + a + d ,we conclude that e = x , e a = e d = g − x , e w + a = f − x − e w + a + d = x +
1, and e i = i .(2) If d < a < w + a < w + a + d , by arguments similar to those above, we can show M d ⊃ ( U ′ ) ⊥ , M a ⊃ C M w + d ⊃ V r , and M w + a + d ⊃ U . Now by applying Lemma 7like above, we have e = x , e a = e d = g − x , e w + a = f − x − e w + a + d = x + e i = i .(3) If d = a < w + a < w + a + d , by similar arguments, M a ⊃ ( U ′ ) ⊥ , M w + a ⊃ V r , and M w + a + d ⊃ U . Now by applying Lemma 7 like above, we can show that e = x , e a = e d = g + − x , e w + a = f − x − e w + a + d = x +
1, and e i = i .12.2.0.2. Elementary divisors of K . We identify M i ( L ) with M i and L ℓ with L . We set v ℓ ( h m − i q ) = a , v ℓ ( q m − + = d , and v ℓ ( q m − + = c . Then v ℓ ( k ) = v ℓ ( µ ) = a + d , v ℓ ( v ) = v ℓ (cid:18)h m i q ( q m − + (cid:19) = c . So v ℓ ( t ) = a + c , v ℓ ( u ) = d , and v ℓ ( | K | ) = ( a + c ) f + dg − c .As L ( ) =
0, we have h i ⊂ M i for all i . Since s is an eigenvalue of valuation d ,Lemma 8 implies M d ⊃ V r . So by Theorem 15, we see that M d ⊃ V s ⊕ h i . Since t is aneigenvalue of valuation a + c , Lemma 8 implies V r ⊂ M a + c . Lemma 10 gives us C ⊂ M a and U ′ ⊂ M a + c + d . Thus by Theorem 15, U ′ ⊕ h i = U ⊂ M a + c + d .Subcase 1: When ℓ ∤ m −
1. In this case, a =
0, as v ℓ ( h m − i q ) =
0. Thus v ℓ ( | K | ) = c f + dg − c .We apply Lemma 8, and Theorem 15 to conclude the following.(1) If c < d < d + c , From the information we gathered above, we have M c ⊃ V r and M c ⊃ M d ⊃ V s ⊕ h i . Applying Theorem 15 gives us M c ⊃ U ⊥ and hencedim( M c ) ≥ f + g − x . We also have by Theorem 15, dim( M d ) ≥ g + M d + c ) ≥ x + c ( f + g − x − ( g + + d ( g + − ( x + + ( c + d )( x + − = c f + dg − c = v ℓ ( | K | ). So by Lemma 7, setting j = s = f + g − x , s = g + s = x + s = dim(ker( L )) = t = c , t = d , and t = c + d , we have e = x + e c = f − x − e d = g − x , e c + d = x , and e i = i .(2) If d < c < d + c , By arguments similar to those above, we can show M d ⊃ U ⊥ ,and dim( M d ) ≥ f + g − x ; M c ⊃ V r , and dim( M c ) ≥ f +
1; and M d + c ⊃ U , anddim( M d + c ) ≥ x +
1. By applying Lemma 7 as above, we can show that e = x + e c = f − x − e d = g − x , e c + d = x , and e i = i .(3) If c = d < d + c , by arguments similar to those above we can show e = x + e c = f + g − x − e c + d = x , and e i = i .Subcase 2: When ℓ | m −
1. As q ≡ − ℓ ) and ℓ | m −
1, by the observations atthe beginning of the subsection, we have c = d . We apply Lemma 7 and Theorem 15 toconclude the following.(1) Assume that a < c = d < a + d < a + c + d . As C ⊂ M a , and V s ⊂ M c ⊂ M d , byTheorem 15 M d ⊃ ( U ′ ) ⊥ , and thus dim( M d ) ≥ f + g + − x . Also by Theorem 15,since V r ⊂ M a + d ⊂ M c , M c ⊃ U ⊥ and thus dim( M c ) ≥ f + g − x . We also have U ⊂ M a + c + d , V r ⊂ M a + d , and thus dim( M d + a ) ≥ f , and dim( M a + c + d ) ≥ x + a ( f + g + − x − ( f + g − x )) + c ( f + g − x − ( f )) + ( a + d )( f − ( x + + ( a + c + d )( x + − = ( a + d ) f + cg − ( a + d ) = v ℓ ( | K | ). So by Lemma 7, setting j = s = f + g + − x , s = f + g − x , s = f , s = x + s = dim(ker( L )) = t = a , t = c , t = a + d , and t = a + c + d , we have e = x , e a = e c = g − x , e a + d = f − x − e a + c + d = x , and e i = i .(2) Assume that c = d < a < a + d < b + c + d . Then by arguments similar to thoseabove, M c ⊃ C + V s = ( U ′ ) ⊥ , M a ⊃ C , M a + d ⊃ V r , and M a + c + d ⊃ U . Nowapplying Lemma 7 as above, we have e = x , e a = e c = g − x , e a + d = f − x − e a + c + d = x , and e i = i .(3) If c = d = a < a + d < a + c + d , then by arguments similar to those above, we canshow e = x , e c = g + − x , e a + d = f − x − e a + c + d = x , and e i = i .12.3. ℓ -elementary divisors of S and K when ℓ | q + , and ℓ | m . Elementary divisors of S . We identify M i ( A ) with M i and A ℓ with A . In thiscase ℓ ∤ m − ℓ ∤ h m − i q and v ℓ ( q m − − = v ℓ ( q − ℓ ∤ m − ℓ ∤ m −
1, we have v ℓ ( s ) = v ℓ ( q m − + = v ℓ ( q m − + = v ℓ ( q + ≤ v ℓ ( q − w = v ℓ ( q − = v ℓ ( r ) , v ℓ ( q m − + = v ℓ ( s ) = d . MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 31
We have v ℓ ( | S | ) = w f + dg + d , and v ℓ ( k ) = v ℓ ( µ ) = d Observe that d ≤ w < d + w . As r is an eigenvalue of valuation w ≥ d and s , k areeigenvalues of valuation d , by Lemma 8 and Theorem 15, M d ⊃ V r + V s = U ⊥ , and M w ⊃ V r . By Lemma 10, U ⊂ M w + d . Thus by Theorem 15,dim( M d ) ≥ f + g − x , dim( M w ) ≥ f and dim( M w + d ) ≥ x + d ( f + g − x − f ) + w ( f − ( x + + ( d + w )( x + = v ℓ ( | S | ). So by Lemma 7,setting j = s = f + g − x , s = f , s = x + s = dim(ker( A )) = t = d , t = w , and t = w + d , we have e = x + e d = g − x , e w = f − x − e w + d = x +
1, and e i = i .12.3.0.2. Elementary divisors of K . We identify M i ( L ) with M i and L ℓ with L . Since ℓ ∤ m −
1, we have ℓ ∤ h m − i q and thus v ℓ ( q m − − = v ℓ ( q − ℓ ∤ m − ℓ ∤ m −
1, we have v ℓ ( s ) = v ℓ ( q m − + = v ℓ ( q m − + = v ℓ ( q + ≤ v ℓ ( q − v ℓ ( q m − + = d and v ℓ (cid:18)h m i q (cid:19) = b .We have v ℓ ( k ) = v ℓ ( µ ) = d , v ℓ ( v ) = b + d , v ℓ ( t ) = d , v ℓ ( u ) = d + b , and v ℓ ( | K | ) = d f + ( d + b ) g − ( b + d ).As L ( ) =
0, we have h i ⊂ M i for all i . Since t is an eigenvalue of valuation d ,and u is an eigenvalue of valuation b + d , by Lemma 8 and Theorem 15, we see that M d ⊃ V r + V s = U ⊥ and M d + b ⊃ V s .By Lemma 10, we have U ⊂ M d and U ′ ⊂ M b + d .We apply Lemma 7 and Theorem 15 we arrive at the following conclusions.(1) If b < d , 2 d > b + d , we have M b + d ⊃ M d + V s ⊃ U + V s . Thus by Theorem 15,we see that M b + d ⊃ M and thus dim( M b + d ) ≥ g +
1. Since M d ⊃ U ⊥ , M d ⊃ U and M b + d ⊃ U ′ , Theorem 15 implies dim( M ) d ≥ f + g − x dim( M d ) ≥ x +
1, anddim( M b + d ) ≥ x .Now, d ( f + g − x − ( g + + ( b + d )( g + − ( x + + d ( x + − x ) + ( b + d )( x − = v ℓ ( | K | ). So by Lemma 7, setting j = s = f + g − x , s = g + s = x + s = x , s = dim(ker( L )) = t = d , t = b + d , t = d , and t = b + d , we have e = x + e d = f − x − e b + d = g − x , e d = e b + d = x −
1, and e i = i .(2) If b > d , by similar arguments, M d ⊃ M , M b + d ⊃ V s , M d ⊃ U ⊥ , M d ⊃ U and M b + d ⊃ U ′ . Applying Lemma 7 like in the above case, we have e = x + e d = f − x − e b + d = g − x , e d = e b + d = x −
1, and e i = i .(3) If b = d , by arguments similar to those above, we have e = x + e b = f − x − e b + d = g − x + e d + b = x −
1, and e i = i .13. ℓ - elementary divisors of S and K when Γ ( V ) = Γ uo ( q , m ), and ℓ | q + Γ uo ( q , m ) and a prime ℓ , table 13 shows that A ℓ (equivalently L ℓ ) isnilpotent if and only if ℓ | q +
1. In this section we compute the ℓ -elementary divisors of S and K when Γ ( V ) = Γ uo ( q , m ) and ℓ | q + Γ uo ( q , m ) by Γ uo , and ℓ is a prime that meetsthe description in the previous paragraph. We set h = , and z = m in Lemma 3 andLemma 2 to get parameters for this graph. Thus Γ uo ( q , m ) is an SRG with parameters v = h m i q ( q m + + k = q h m − i q ( q m − + λ = ( q − + q (( q ) m − + h m − i q , and µ = h m − i q ( q m − + The adjacency matrix A has eigenvalues ( k , r , s ) = ( k , q m − − , − (1 + q m − )) withmultiplicities (1 , f , g ) = , q h m i q ( q m − + q + , q h m − i q ( q m − − q − . So the Laplacian L has eigenvalues (0 , t , u ) = (0 , k − r , k − s ) = (cid:18) , h m − i q (1 + q m + ) , h m i q (1 + q m − ) (cid:19) withmultiplicities (1 , f , g ).13.1. Submodule structure.
We now recall from § C , C ′ U , U ′ , V r and V s in the context of the graph Γ uo . In this case G ( V ) = U(2 m + , q ). By Corollary2 .
10, Corollary 5 .
6, and Proposition 5 .
14 of [13], we have the following result.
Theorem 16. If ℓ | q + , the following are true.(1) If ℓ ∤ m, then C ⊃ V r ⊃ U = h i ⊕ V s ⊃ V s = U ′ .(2) If ℓ | m, then C ⊃ V r ⊃ U ⊃ V s = U ′ ⊃ h i .(3) We have dim( C ) = f + and dim( U ) = g + . Elementary divisors of S and K , when ℓ ∤ m , and ℓ | q + . Elementary divisors of S . We identify M i ( A ) with M i and A ℓ with A . We set v ℓ ( q − = w , v ℓ ( h m − i q ) = a and v ℓ ( q m − + = d . So we have v ℓ ( r ) = w + a , v ℓ ( s ) = d , v ℓ ( k ) = v ℓ ( µ ) = a + d , and v ℓ ( | S | ) = ( w + a ) f + dg + a + d .We have a < w + a < w + a + d . As r is an eigenvalue of valuation w + a , by Lemma8, we have M a + w ⊃ V r . By Lemma 10, M a ⊃ C , and M w + a + d ⊃ U . Thus by Theorem 16,dim( M a ) ≥ f +
1, dim( M w + a ) ≥ f , and dim( M w + a + d ) ≥ g + a ( f + − f ) + ( w + a )( f − ( g + + ( w + a + d )( g + = ( w + a ) f + dg + a + w = v ℓ ( | S | ).So by Lemma 7, setting j = s = f + s = f , s = g + s = dim(ker( A )) = t = a , t = w + a , and t = w + a + d , we have e = g , e a = e w + a = f − g − e w + a + d = g + e i = i .13.2.0.2. Elementary divisors of K . We identify M i ( L ) with M i and L ℓ with L . We set v ℓ ( h m − i q ) = a , v ℓ ( q m − + = c , and v ℓ ( s ) = v ℓ ( q m + + = d So we have v ℓ ( t ) = a + d , v ℓ ( u ) = c , v ℓ ( v ) = d , and v ℓ ( | K | ) = ( a + d ) f + cg − d .By Lemma 10 we have C ⊂ M a and U ⊂ M a + d + c . As t is an eigenvalue of valuation a + d by Lemma 8, we have M a + d ⊃ V t = V r . By Theorem 16, we have dim( M a ) ≥ f + M a + d ) ≥ f , and dim( M a + c + d ) ≥ g + a ( f + − f ) + ( a + d )( f − ( g + + ( a + c + d )( g + − = v ℓ ( | K | ). So by Lemma7, setting j = s = f + s = f , s = g + s = dim(ker( L )) = t = a , t = a + b ,and t = a + b + c , we have e = g , e a = e d + a = f − g − e d + a + c = g , and e i = i .13.3. Elementary divisors of S and K , when ℓ | m , and ℓ | q + . Elementary divisors of S . We identify M i ( A ) with M i and A ℓ with A . As ℓ | m and q ≡ − ℓ ), we have v ℓ ( q − = v ℓ ( r ), and v ℓ ( s ) = v ℓ ( k ) = . Set v ℓ ( q − = w and v ℓ ( q m − + = d . We have v ℓ ( | S | ) = w f + dg + d .As r is an eigenvalue of valuation w , Lemma 8 implies V r ⊂ M w . By Theorem 10, wehave M w + d ⊃ U . By Theorem 16, we have dim( M w ) ≥ f , and dim( M w + d ) ≥ g + w ( f − ( g + + ( w + d )( g + = v ℓ ( | S | ). So by Lemma 7, setting j = s = f , s = g + s = dim(ker( A )) = t = w , and t = w + d , we have e = g + e w = f − g − e w + d = g +
1, and e i = i . MITH AND CRITICAL GROUPS OF POLAR GRAPHS. 33
Elementary divisors of K . We identify M i ( L ) with M i and L ℓ with L . As ℓ | m ,we have v ℓ ( q m + + = v ℓ ( q m − + v ℓ ( h m i q ) = b and v ℓ ( q m + + = d . We have v ℓ ( t ) = d , v ℓ ( u ) = b + d , v ℓ ( v ) = b + d , and v ℓ ( | K | ) = d f + ( b + d ) g − ( b + d ).As t is an eigenvalue of valuation d , we have M d ⊃ V r . Lemma 10 gives us M b + d ⊃ U ′ and U ⊂ M d . By Theorem 16, we have dim( M d ) ≥ f , dim( M d ) ≥ g +
1, anddim( M b + d ) ≥ g .Now, d ( f − ( g + + (2 d )( g + − g ) + ( b + d )( g − = v ℓ ( | K | ). So by Lemma 7, setting j = s = f , s = g + s = g , s = dim(ker( L )) = t = d , t = d , and t = b + d ,we have e = g + e d = f − g − e d = e b + d = g −
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E-mail address : [email protected] E-mail address : [email protected] D epartment of M athematics , U niversity of F lorida , G ainesvilleainesville