Difference families, skew Hadamard matrices, and Critical groups of doubly regular tournaments

In this paper we investigate the structure of the critical groups of doubly regular tournaments (DRTs) associated with skew Hadamard difference families (SDFs) with one, two, or four blocks. Brown and Reid found the existence of a skew Hadamard matrix of order $n+1$ to be equivalent to the existence of a DRT on $n$ vertices. A well known construction of a skew Hadamard matrix order $n$ is by constructing skew Hadamard difference sets in abelian groups of order $n-1$. The Paley skew Hadamard matrix is an example of one such construction. Szekeres and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with two blocks. Wallis and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with four blocks. In this paper we consider the critical groups of DRTs associated with skew Hadamard matrices constructed from skew Hadamard difference families with one, two or four blocks. We compute the critical groups of DRTs associated with skew Hadamard difference families with two or four blocks. We also compute the critical group of the Paley tournament and show that this tournament is inequivalent to the other DRTs we considered. Consequently we prove that the associated skew Hadamard matrices are not equivalent.


Introduction.
A Hadamard matrix H of order n is an n × n matrix of +1's and −1's such that HH ⊺ = nI. It is well known that if n is the order of a Hadamard matrix, then n = 1, 2 or n ≡ 0 (mod 4). It is conjectured that Hadamard matrices of order n exist for all n ≡ 0 (mod 4). The smallest n for which there is no known Hadamard matrix is n = 668 (c.f. [11]). In this paper we deal with skew Hadamard matrices. A Hadamard matrix H is said to be skew if H + H ⊺ = 2I.
Two Hadamard matrices are considered equivalent if one can be obtained from the other by negating rows or columns, or by interchanging rows or columns. It is of interest to determine the equivalence of Hadamard matrices of the same order. Every Hadamard matrix of order n is equivalent to a Hadamard matrix of the form 1 1 ⊺ n−1 −1 n−1 H 0 . All the Hadamard matrices we consider in this paper are assumed to be in this form.
In this paper, we are interested in the inequivalence of skew Hadamard matrices. If H 1 and H 2 are two Hadamard matrices of same order but with different Smith normal forms, then they are inequivalent. In [14], it was found that the Smith normal form of any skew Hadamard matrix of order 4m is diag [1, 2, . . . , 2 2m−1 , 2m, . . . , 2m 2m−1 , 4m]. So Smith normal form fails to distinguish inequvivalent skew Hadamard matrices of the same order. In this article we consider a different invariant associated with skew Hadamard matrices.
A tournament T n of order n is a directed graph obtained by assigning directions to every edge of a complete graph on n vertices. Given vertices v, w of T n , by d(v) we denote the outdegree of v and by d(v, w) we denote the number of vertices dominated by v and w. A DRT with parameters (n, k, λ) is a tournament of order n such that for every pair of distinct vertices v, w, we have d(v) = k and d(v, w) = λ. It is easy to see that n = 4λ + 3 and k = 2λ + 1. Theorem 2 of [20] shows that a skew Hadamard matrix of order n + 1 exists if and only if there is a DRT on n vertices. Given a 1 formal sum of vertices dominated by v, encodes adjacency of the graph. The map ν Q : is called the Laplacian map. The critical group K of Γ is the finite part of the cokernal of ν Q . The critical group is an invariant of the graph. Now let β be an ordered basis of Z V that is obtained by fixing an order on V. The adjacency matrix M of Γ is the matrix representation of ν M with respect to β. We define the Laplacian matrix Q to be the matrix representation of ν Q with respect to β. Let ∆ be the diagonal matrix whose vth diagonal entry is ∆ v . Then we have Q = ∆ − M. If Q v is the matrix obtained by deleting the vth row and vth column of Q, then the Matrix-tree theorem (eg. [22, 5.64 and 5.68 | is the number of spanning trees of Γ. For a nice survey on critical groups of graphs, we refer to [23, §3]. Some papers with computations of critical groups of families of graphs include [27], [13], [12], [5], [2], [10], [8], [4], [19], and [18]. In [12], Lorenzini examined the proportion of graphs with cyclic critical groups among graphs with critical groups of particular order. One effective way of constructing skew Hadamard matrices/DRTs is by using skew difference families. Let (G, +) be an additive finite abelian group of order n. A skew difference family (SDF) on l blocks with parameters (n, k, λ) is An SDF with one block in G is called a skew Hadamard difference set.
We will now describe a few SDFs found in literature. The earliest construction is that of the Paley difference set by Paley [17]. It was conjectured that Paley difference set was the only(upto equvivalence) SDF with one block. Ding and Yuan [7] disproved the conjecture by constructing other SDFs with one block. Szekeres [25,26], Whiteman [29] found an SDF with two blocks in (F q , +), where either q ≡ 5 (mod 8); or q = p e with p ≡ 5 (mod 8) a prime and e ≡ 2 (mod 4). Wallis and Whiteman [28] constructed an SDF with four blocks in (F q , +), where q ≡ 9 (mod 16). Momihara and Xiang [15] generalised the constructions by Szekres, Wallis and Whiteman to obtain the following result.
Proposition 1. [15, Theorem 1.5] Let u ≥ 2 be an integer and q be a prime power such that q ≡ 2 u + 1 (mod 2 u+1 ). Then for any positive integer e, there exists a skew Hadamard difference family with 2 u−1 blocks in (F q e , +).
Szekeres [25] also proved the following result. In this paper, we compute the critical groups of the three families of DRTs described below. Given X ⊂ G, by δ X we denote the characteristic function of X in G. (i) Let (G, +) be an additive abelian group of order 2λ + 1 and (A, B) be an SDF with two blocks in G, with parameters (2λ + 1, λ, λ − 1). Then SZ(G, A, B) is the graph with vertex set V = {v 0 } ∪ {a g | g ∈ G} ∪ {b g | g ∈ G}, whose adjacency map ν M : for all g ∈ G. Theorem 2 of [25] shows that SZ(G, A, B) is a DRT with parameters (4λ + 3, 2λ + 1, λ). Setting u = 2 in Proposition 1 provides us with a famiy of SDFs with two blocks. Proposition 2 provides another such family. Theorem 4 describes the critical group of SZ(G, A, B). We utilize the natural action of group G on the vertex set of SZ(G, A, B) to compute the Smith normal form of its Laplacian.

Then by W(G, A, B, C, D) we denote a graph with vertex set
We require the adjacency map µ M : for all g ∈ G.
Let (g 1 , g 2 , . . . , g 2λ+1 ) be an ordering on G. Consider the ordered basis Let M be the matrix representation of µ M with respect to β. Theorem 12 of [28] states that is a DRT with parameters (8λ + 7, 4λ + 3, 2λ + 1). Setting u = 3 in Proposition 1 provides us with a famiy of SDFs with four blocks. Theorem 5 describes the critical group of W(G, A, B, C, D). We utilize the natural action of group G on the vertex set of W(G, A, B, C, D) to compute the Smith normal form of its Laplacian.
(iii) The third family we consider is the family of Paley tournaments. Let A be a skew Hadamard difference set in an abelian group G of order 4λ + 3. By DRT (G, A) we denote the graph with vertex set {[g]| g ∈ G} and arc set . Let p t be a power of a prime p with q ≡ 3 (mod 4) and let F q be the finite field of order q. Let H be the set of non-zero squares in F q . It is well known that H is a skew Hadamard difference set in the additive group (F q , +) of the field. The Paley tournament graph P(q) is DRT (G, H), that is, it is the Cayley graph on (F q , +) with "connection" set being the multiplicative subgroup of squares in F q . Theorem 7 describes the critical group of P(q). This was essentially computed in [5], in which the authors describe the critical group of the Paley graph. This computation involves some Jacobi sums involving the quadratic character ψ. The only difference between our computation here and that in [5] is that ψ(−1) = −1 in our case.

Main results.
Let K be the critical group of a DRT with parameters (4λ + 3, 2λ + 1, λ), by K 1 we denote the subgroup of order (λ + 1) 2λ+1 . Let K 2 be the subgroup of K of order (4λ + 3) 2λ . We observe that K = K 1 ⊕ K 2 . In §4 we show that K 1 depends only on the parameter λ.
Theorem 3. Let λ be a positive integer and let K denote the critical group of a DRT with parameters (4λ+3, 2λ+1, λ).
The result below describes the critical group of SZ(G, A, B). We prove this in §5.  2λ . Let p be a prime and let rk p (Q) denote the p-rank of Q. If p | λ + 1, then rk p (Q) = 2λ + 1; and if p | 4λ + 3, then rk p (Q) = 2λ + 2.
The result below describes the critical group of W(G, A, B, C, D). We prove this in §6. 3 Theorem 5. Let λ be a positive integer and let (A, B, C, D) be an SDF in an additive abelian G with |G| = 2λ+1. Let Q denote the Laplacian matrix of W(G, A, B, C, D) and by K we denote its critical group. Then K = (Z/(2λ + 2)Z) 4λ+3 ⊕ (Z/(8λ + 7)Z) 4λ+2 . Let p be a prime and let rk p (Q) denote the p-rank of Q. If p | λ + 1, then rk p (Q) = 4λ + 3; and if p | 4λ + 3, then rk p (Q) = 4λ + 4.
The following describes the critical group of P(q). We prove this in §7.
Theorem 6. Let p be a prime and t be a positive integer such that q := p t ≡ 3 (mod 4). Let Q denote the Laplacian matrix of P(q) and by K we denote its critical group. Then the p-rank of Q is p + 1 2 (3) and for 1 ≤ i < t, Remark 1. Let q be a prime power satisfying q ≡ 3 (mod 4). Proposition 2 provides us with an SDF with two blocks Both P(q) and SZ(G, A, B) are DRT's with parameters (q, (q − 1)/2, (q − 3)/4). Theorems 4 and 6 show that these graphs have non isomorphic critical groups. Therefore these graphs are not isomorphic and thus the associated Hadamard matrices are not equivalent.
Remark 2. Letq be a prime power such thatq ≡ 5 (mod 8). Proposition 1 guarantees the existence of an SDF with two blocks in (F q , +). Let (A, B) be an SDF in (Fq, +). Let's also assume that q = 2q + 1 is also a power of a prime. Theorems 4 and 6 show that that SZ(Fq, A, B) and P(q) are not isomorphic and thus the associated Hadamard matrices are not equivalent.
Remark 3. Letq be a prime power such thatq ≡ 9 (mod 16). Proposition 1 guarantees the existence of an SDF with four blocks in (F q , +). Let (A, B, C, D) be an SDF in (Fq, +). Let's also assume that q = 4q + 3 is also a power of a prime. Theorems 5 and 6 show that that W (Fq, A, B, C, D) and P(q) are not isomorphic and thus the associated Hadamard matrices are not equivalent.
Remark 4. We found that the critical groups of SZ (G, A, B) and W (G, A, B, C, D) depend only on the order of G. However this is not the case for DRTs constructed from skew Hadamard difference sets. Let q ≡ 3 (mod 4) be a prime power. To construct P(q), the set H of quadratic residues in (F q , +) was used. Another example of skew Hadamard difference set is the set DY(1) = {x 10 − x 6 − x 2 | x ∈ F × 3 n } in the additive group (F 3 n , +), with n odd. This was constructed by Ding and Yuan [7]. By DRT (3 n , DY(1)), we denote the DRT with vertex set {[x]| x ∈ F 3 n } and arc set With the help of a computer, we can find that the SNFs of the Laplacians of DRT (3 5 , DY(1)) and P(243) are different. It was conjectured in [6] that there are at least five inequvivalent difference sets in (F 3 n , +) for all odd n > 3.

Preliminaries
3.1. Smith Normal Forms. Let R be a Principal Ideal Domain and Z : R m → R n be a linear transformation. By the structure theorem for finitely generated modules over PIDs, we have Let [Z] denote the matrix representation of Z with respect to the standard bases. Then the above equation tells us that we can find P ∈ GL n (R), and Q ∈ GL m (R) such that where Y = diag(s 1 (Z), . . . , s r (Z)).
The following result which is Theorem 1 of [16] gives a relation between SNF of the product of two matrices and the SNFs of the individual matrices. Consider a prime p ∈ R and a square matrix N with entries in R, whose SNF over R is diag (s 1 (N), . . . , s i N, . . . s n (N)). Let S p be any unramified extension of the local ring R p . If diag(p j 1 , . . . , p j i , . . . , p j n ) is the SNF of N considered as a matrix over S p , then p j i ||s i (N). So while finding Smith normal forms, we can focus on one prime at a time.
The following is a well know result about adjacency matrices of DRTs.

Permutation action and characters.
We will now collect some useful results from character theory. Each of P(q), SZ (G, A, B), W(G, A, B, C, D) is constructed using a finite abelian group (G, +). We use the natural action of G on the vertex set to compute the critical groups. These actions are closely related to the regular action of G on itself.
We define the action of G on Y = {y g | g ∈ G} by h.y g = y g+h . This is the regular action of G. Let p ∤ |G| be a prime and let S be an extension of Q p containing the |G|-th roots of unity. By R we denote the ring of integers of S, and by R Y we denote the free R-module generated by Y as a basis set. Let Irr(G) be the group of R-valued characters of G.
We extend the arguments used in [14] to determine the structure of K 1 . Let p | λ + 1 be a prime integer. Then the Smith normal form of Q over the p-adic numbers Z p gives us the p-part of K 1 . Let s 1 (Q), . . . , s 4λ+3 (Q) be the invariant factors of Q considered as a matrix over Z p From (4) we see that QQ ⊺ ≡ 0 (mod p). Therefore we have rk p (Q) ≤ 4λ + 3 − rk p (Q ⊺ ) and thus rk p (Q) ≤ 2λ + 1. (4), and Lemma 8 we may conclude that (i) As s i (Q) | λ + 1, we can now conclude that s i (Q) = λ + 1 for 2λ + 2 ≤ i ≤ 4λ + 2. It now follows that K 1 = (Z/(λ + 1)Z) 2λ+1 .

Crtical group of SZ(G, A, B).
Let us turn our attention to DRTs of the form SZ (G, A, B). Let (A, B) be an SDF in an abelian group (G, +) of order 2λ + 1. By SZ(G, A, B) we denote the graph with vertex set V = {v 0 } ∪ {a g | g ∈ G} ∪ {b g | g ∈ G} and whose adjacency operator ν M satisfies (1).
We recall that SZ(G, A, B) is a DRT with parameters (4λ + 3, 2λ + 1, λ). Let Q be the Laplacian matrix of SZ(G, A, B) and K be its critical group. By Theorem 3, we have K = (Z/(λ + 1)Z) 2λ+1 ⊕ K 2 , where K 2 is the subgroup of order (4λ + 3) 2λ . Let p | 4λ + 3 be a prime. Determining the SNF of Q over an unramified extension of Z p will give us the p-part of K 2 .
Let t ∈ N such that |G| | (p t − 1), by θ we denote a primitive (p t − 1)st root of unity in Q p . We denote by R, the ring of integers in Q(θ). As p is unramified in R, the p-part of K can be found by determining the SNF of Q over R. By R V , we denote the free module over R with V as a basis set. The matrix Q defines a map ν Q : We now consider the action of (G, +) on V that satisfies (i)h.v 0 = v 0 , (ii) h.a g = a g+h , and (iii)h.b g = b g+h for all g, h ∈ G. This permutation action preserves adjacency. The action of G on V makes R V a permutation module for G. Given χ ∈ Irr(G), we define N χ := {x ∈ R V |g.x = χ(g)x for all g ∈ G}. In other words, N χ is the direct sum of all irreducible submodules of R V affording χ. As G preserves adjacency, ν Q is an RG map. Now by applying Schur's Lemma, we have ν(N χ ) ⊂ N χ .
The action of G decomposes N χ , with ν Q (N χ ) ⊂ N χ . We will now look at ν Q | N χ .
Using Lemma 10 and the relations in (1) yields the following lemma. For χ χ 0 , let Q χ be the matrix representation of ν Q | N χ with respect to the ordered basis (e (χ,a) , e (χ,b) ). let Q χ 0 be the matrix representation of ν Q | N χ 0 with respect to the ordered basis (e (χ 0 ,a) , e (χ 0 ,b) , v). So Q is similar to the block diagonal matrix We see from Lemma 11 that for χ χ 0 , we have T r(Q χ ) = 4λ + 2 − χ(A) − χ(−A). Now as χ is not trivial, we have 0 = χ(G). Using A ∪ −A = G \ {0 G }, we may conclude that T r(Q χ ) = 4λ + 3. The eigenvalues of Q χ are elements of the set and det(Q χ ) = (4λ + 3)(λ + 1). We also observe from Lemma 11 that the difference of the off-diagonal entries of Q χ is 1 and thus one of them is coprime to p. Applying Lemma 7 and det(Q χ ) = (4λ + 3)(λ + 1) we can conclude that diag(1, 4λ + 3) is the SNF of Q χ over R. Similar computations can be used to show that diag(1, 1, 0) is the SNF of Q χ 0 over R. This proves Theorem 4.

Crtical group of W(G, A, B, C, D).
Given an SDF (A, B, C, D) in an additive abelian group (G, +) of order 2λ + 1, we recall that W (G, A, B and whose adjacency operator ν M is defined by (2).
We recall that W (G, A, B, C, D) is a DRT with parameters (8λ + 7, 4λ + 3, 2λ + 2). Let Q be the Laplacian matrix of W(G, A, B, C, D) and K be its critical group. By Theorem 3, we have K = (Z/(2λ + 2)Z) 4λ+3 ⊕ K 2 , where K 2 is the subgroup of order (8λ + 7) 4λ+2 . Let p | 8λ + 7 be a prime Determining the SNF of Q over an unramified extension of Z p will give us the p-part of K 2 .
Let t ∈ N such that |G| | (p t − 1), by θ we denote a primitive (p t − 1)st root of unity in Q p . We denote by R, the ring of integers in Q(θ). As p is unramified in R, the p-part of K can be found by determining the SNF of Q over R. By R V , we denote the free module over R with V as a basis set. The matrix Q defines a map ν Q : Unlike in the case of SZ (G, A, B), the natural G action on W(G, A, B, C, D) does not preserve adjacency, but provides a useful integral basis for R V . We consider the action of G on V that satisfies The result above follows by straightforward applications of the relations in (2) and Lemma 10. Let χ 0 denote the trivial character of G. For χ χ 0 ∈ Irr(G), define N χ to be the R V submodule generated by {e (µ, f ) | µ = a, b, c, d and f = χ, χ −1 }. Let N χ 0 be the submodule generated by {v 1 , v 2 , v 3 , e (µ,χ 0 ) | µ = a, b, c, d}. Lemma 12 shows that ν Q (N χ ) ⊂ N χ for χ χ 0 . Lemma 12 implies ν Q (N χ 0 ) ⊂ N χ 0 . By ν χ we denote ν Q | N χ .

Critical group of P(q).
We now turn our attention to Paley tournament graph P(q). The computation of critical group of P(q) is essentially the same as that of the Paley graph done in [5]. The proofs of results in this section are similar to those in [5].
Let q = p t be a power of a prime p with q ≡ 3 (mod 4). Let K be the group field with q elements and let H be the subgroup of squared in K × . We recall that the Paley tournament graph P(q) is the Cayley graph of (K, +) with "connection" set being H.