Cartan coherent configurations
Abstract
The Cartan scheme \(\mathcal{X}\) of a finite group G with a (B, N)-pair is defined to be the coherent configuration associated with the action of G on the right cosets of the Cartan subgroup \(B\cap N\) by right multiplication. It is proved that if G is a simple group of Lie type, then asymptotically the coherent configuration \(\mathcal{X}\) is 2-separable, i.e., the array of 2-dimensional intersection numbers determines \(\mathcal{X}\) up to isomorphism. It is also proved that in this case, the base number of \(\mathcal{X}\) equals 2. This enables us to construct a polynomial-time algorithm for recognizing Cartan schemes when the rank of G and the order of the underlying field are sufficiently large. One of the key points in the proof is a new sufficient condition for an arbitrary homogeneous coherent configuration to be 2-separable.
1 Introduction
A well-known general problem in algebraic combinatorics is to characterize an association scheme \(\mathcal{X}\) up to isomorphism by a certain set of parameters [3]. A lot of such characterizations are known when \(\mathcal{X}\) is the association scheme of a classical distance regular graph [4]. In most cases, the parameters can be chosen as a part of the intersection array of \(\mathcal{X}\). However, in general, even the whole array does not determine the scheme \(\mathcal{X}\) up to isomorphism. Therefore, it makes sense to consider the m-dimensional intersection numbers, \(m\ge 1\), introduced in [12] for an arbitrary coherent configuration (for \(m=1\), these numbers are ordinary intersection numbers; the exact definitions can be found in Sect. 2). It was proved in [12] that every Johnson, Hamming or Grassmann scheme is 2-separable, i.e., is determined up to isomorphism by the array of 2-dimensional intersection numbers.
In the present paper, we are interested in coherent configurations (1) in the case where G is a finite group with a (B, N)-pair and \(\mathcal{X}\) is a Cartan scheme of G in the following sense.
Definition 1.1
The Cartan scheme of G with respect to (B, N) is defined to be the coherent configuration (1), where \(\Omega =G/H\) consists of the right cosets of the Cartan subgroup \(H=B\cap N\) and G acts on \(\Omega \) by right multiplication.
Note that the permutation group induced by the action of G is transitive and the stabilizer of the point \(\{H\}\) coincides with H. In a Coxeter scheme of rank at least 3, the point stabilizer equals B. Therefore, such a scheme is a quotient of a suitable Cartan scheme.
The separability problem [13] consists of finding the smallest m for which a coherent configuration is m-separable. The separability problem (in particular, for a Cartan scheme) is easy to solve if the group H is trivial. Indeed, in this case, the permutation group induced by G is regular and the corresponding coherent configuration is 1-separable. The following theorem gives a partial solution to the separability problem for Cartan schemes when G is a finite simple group of Lie type and hence with a (B, N)-pair. In what follows, we denote by \(\mathcal{L}\) the class of all simple groups of Lie type including all exceptional groups and all classical groups \(\Phi (l,q)\), for which \(l\ge l_0\) and \(q\ge al\), where the values of \(l_0\) and a are given in the last two columns of Table 2.
Theorem 1.2
The Cartan scheme \(\mathcal{X}\) of every finite simple group \(G\in \mathcal{L}\) is 2-separable.
Theorem 1.3
Let \(\mathcal{X}\) be the Cartan scheme of a group \(G\in \mathcal{L}\). Then, \(b(\mathcal{X})\le 2\) and \(b(\mathcal{X})=1\) if and only if the group H is trivial.
When the rank of a simple group G of Lie type is small, inequality (3) does not generally hold, but the statements of Theorems 1.2 and 1.3 may still be true. For example, if \(G={{\mathrm{PSL}}}(2,q)\) with even q, then inequality (3) is not true; however, the following assertion holds.
Theorem 1.4
Let \(\mathcal{X}\) be the Cartan scheme of the group \({{\mathrm{PSL}}}(2,q)\), where \(q>3\). Then, \(\mathcal{X}\) is 2-separable and \(b(\mathcal{X})=2\).
We believe that the Cartan scheme of every simple group of Lie type is 2-separable. Moreover, as in the case of classical distance regular graphs, it might be that in most cases such a scheme is 1-separable, i.e., is determined up to isomorphism by the intersection numbers. In this way, one could probably use more subtle results on the structure of finite simple groups and a combinatorial technique in the spirit of [16].
From the computational point of view, Theorems 1.2 and 1.3 can be used for testing isomorphism and recognizing the Cartan schemes satisfying the hypothesis of Theorem 1.2. To this end, it is convenient to represent a coherent configuration \((\Omega ,S)\) as a complete colored graph with vertex set \(\Omega \) such that the color classes of the arcs coincide with the relations of S (the vertex colors match the colors of the loops). It is assumed that the isomorphisms of such colored graphs preserve the colors.
Theorem 1.5
- (1)
given \(D\in \mathcal{G}_n\), test whether \(D\in \mathcal{K}_n\), and (if so) find the corresponding groups G, B, and N;
- (2)
given \(D\in \mathcal{K}_n\) and \(D'\in \mathcal{G}_n\), find the set \({{\mathrm{Iso}}}(D,D')\).
To make the paper possibly self-contained, we cite the basics of coherent configurations in Sect. 2. Theorems 3.1 and 4.1, from which we have deduced Theorems 1.2 and 1.3, are proved in Sects. 3, 4, 5, respectively. Finally, Theorems 1.4 and 1.5 are proved in Sects. 6 and 7, respectively.
Notation. Throughout the paper, \(\Omega \) denotes a finite set.
The diagonal of the Cartesian product \(\Omega \times \Omega \) is denoted by \(1_\Omega \); for any \(\alpha \in \Omega \), we set \(1_\alpha =1_{\{\alpha \}}\).
For a relation \(r\subset \Omega \times \Omega \), we set \(r^*=\{(\beta ,\alpha ):\ (\alpha ,\beta )\in r\}\) and \(\alpha r=\{\beta \in \Omega :\ (\alpha ,\beta )\in r\}\) for all \(\alpha \in \Omega \).
For \(S\in 2^{\Omega ^2}\), we denote by \(S^\cup \) the set of all unions of the elements of S and put \(S^*=\{s^*:\ s\in S\}\) and \(\alpha S=\cup _{s\in S}\alpha s\), where \(\alpha \in \Omega \).
For \(g\in {{\mathrm{Sym}}}(\Omega )\), we set \({{\mathrm{Fix}}}(g)=\{\alpha \in \Omega :\ \alpha ^g=\alpha \}\); in particular, if \(\chi \) is the permutation character of a group \(G\le {{\mathrm{Sym}}}(\Omega )\), then \(\chi (g)=|{{\mathrm{Fix}}}(g)|\) for all \(g\in G\).
The identity of a group G is denoted by e; the set of non-identity elements in G is denoted by \(G^\#\).
2 Coherent configurations
2.1 Main definitions
The point set \(\Omega \) is a disjoint union of fibers, i.e., the sets \(\Gamma \subseteq \Omega \), for which \(1_{\scriptscriptstyle \Gamma }\in S\) For any basis relation \(r\in S\), there exist uniquely determined fibers \(\Gamma \) and \(\Delta \) such that \(r\subseteq \Gamma \times \Delta \). Moreover, the number \(|\gamma r|=c_{rr^*}^t\) with \(t=1_{\scriptscriptstyle \Gamma }\), does not depend on the choice of \(\gamma \in \Gamma \). This number is called the valency of r and denoted by \(n_r\). The maximum of all valencies is denoted by \(k=k(\mathcal{X})\).
2.2 Point extensions and the base number
Definition 2.1
A set \(\{\alpha ,\beta ,\ldots \}\subseteq \Omega \) is a base of the coherent configuration \(\mathcal{X}\) if the extension \(\mathcal{X}_{\alpha ,\beta ,\ldots }\) with respect to the points \(\alpha ,\beta ,\ldots \) is complete; the smallest cardinality of a base is called the base number of \(\mathcal{X}\) and denoted by \(b(\mathcal{X})\).
It is easily seen that \(0\le b(\mathcal{X})\le n-1\), where \(n=|\Omega |\), and the equalities are attained for the complete and trivial coherent configurations on \(\Omega \), respectively. It is also obvious that \(b(\mathcal{X})\le 1\), whenever the coherent configuration \(\mathcal{X}\) is 1-regular.
2.3 Coherent configurations and permutation groups
Lemma 2.2
Let G be a transitive permutation group and \(\mathcal{X}={{\mathrm{Inv}}}(G)\). If \(b(\mathcal{X})\le 2\), then \(G={{\mathrm{Aut}}}(\mathcal{X})\).
Proof
2.4 Indistinguishing number
The following lemma gives a formula for the indistinguishing number of the coherent configuration of a transitive permutation group. Recall that the fixity \({{\mathrm{fix}}}(G)\) of a permutation group G is the maximum number of elements fixed by non-identity permutations [18].
Lemma 2.3
Proof
We complete this subsection by a statement that helps to compute the values of the permutation character of a transitive group.
Lemma 2.4
Proof
Clearly, \(\alpha ^g\in {{\mathrm{Fix}}}(x)\) if and only if \(Hgx=Hg\), which holds if and only if there is \(h\in H\) satisfying \(x=h^{g}\). In particular, this yields (11).
To prove that the left-hand side of (12) is contained in the right-hand side, let \(x=h_0^{g_0}\), that is the set \({{\mathrm{Fix}}}(x)\) is non-empty. Suppose that g is an arbitrary element of G with \(\alpha ^g\in {{\mathrm{Fix}}}(x)\). Then, there is \(h\in H\) such that \(h^g=x=h_0^{g_0}\). Put \(y=gg_0^{-1}\). Since the elements \(h_0\) and \(h=h_0^{y^{-1}}\) are conjugate in G, they are also conjugate in N, so there is \(n\in N\) with \(h_0^{{y}^{-1}}=h_0^{{n}^{-1}}\). It follows that \(y=nc\), where \(c\in C\). Therefore, \(g=ncg_0\), so \(\alpha ^g\in {{\mathrm{Fix}}}(x)\) implies that \(g\in NCg_0\). To establish the reverse inclusion, for every \(n\in N\), set \(h=h_0^{n^{-1}}\). Then, \(h^{ncg_0}=h_0^{cg_0}=x\) for every \(c\in C\). By the argument of the first paragraph, this proves \(\alpha ^{NCg_0}\subseteq {{\mathrm{Fix}}}(x)\).
Obviously, \(|NCg_0|=|N:(C\cap N)||C|\). Now, the first equality in (13) is the direct consequence of (12), because \(\alpha ^g=\alpha ^{g'}\) if and only if \(g'g^{-1}\in H\). Since \(|C|=|G|/|x^G|\) and \(|G|/|H|=|\Omega |\), the second equality follows. \(\square \)
2.5 Algebraic isomorphisms and m-dimensional intersection numbers
Saying that coherent configurations \(\mathcal{X}\) and \(\mathcal{X}'\) have the same intersection numbers, we mean that formula (14) holds for a certain algebraic isomorphism. Thus, the exact meaning of the phrase “the coherent configuration \(\mathcal{X}\) is determined up to isomorphism by the intersection numbers” is that \(\mathcal{X}\) is separable.
Let \(m\ge 1\) be an integer. According to [13], the m-extension of a coherent configuration \(\mathcal{X}\) with point set \(\Omega \) is defined to be the smallest coherent configuration on \(\Omega ^m\), which contains the Cartesian m-power of \(\mathcal{X}\) and for which the set \({{\mathrm{Diag}}}(\Omega ^m)\) is the union of fibers. The intersection numbers of the m-extension are called the m-dimensional intersection numbers of the configuration \(\mathcal{X}\). Now, m-separable coherent configurations for \(m>1\) are defined essentially in the same way as for \(m=1\). The exact definition can be found in the survey [13], whereas in the present paper, we need only the following result, which immediately follows from [13, Theorems 3.3 and 5.10].
Theorem 2.5
Let \(\mathcal{X}\) be a coherent configuration admitting a 1-regular extension with respect to \(m-1\) points, \(m\ge 1\). Then,\(\mathcal{X}\) is m-separable.\(\square \)
Corollary 2.6
Let \(\mathcal{X}\) be a coherent configuration admitting a 1-regular one-point extension. Then, \(\mathcal{X}\) is 2-separable and \(b(\mathcal{X})\le 2\).\(\square \)
3 A sufficient condition for 1-regularity of a point extension
3.1 Main theorem
The aim of this section is to prove the following statement underlying the combinatorial part in the proof of the main results of this paper.
Theorem 3.1
Let \(\mathcal{X}\) be a homogeneous coherent configuration on n points with indistinguishing number c and maximum valency k. Suppose that \(2c(k-1)<n\), i.e., inequality (3) holds. Then, every one-point extension of \(\mathcal{X}\) is 1-regular.
The proof of Theorem 3.1 will be given in the end of this section. The idea is to deduce the 1-regularity of the point extension \(\mathcal{X}_\alpha \) from Lemma 3.6 stating that inequality (3) implies the connectedness of the binary relations \(s_{\mathrm{max}}\) and \(s_\alpha \) defined in Sect. 3.2. Note that this condition itself implies that any pair from \(s_{\mathrm{max}}\) forms a base of \(\mathcal{X}\) (Lemma 3.3).
3.2 Relations \(s_{\mathrm{max}}\) and \(s_\alpha \)
Lemma 3.2
Proof
Lemma 3.3
If \(s_{\mathrm{max}}\) and all \(s_\alpha \), \(\alpha \in \Omega \), are connected relations, then \(\{\alpha ,\beta \}\) is a base of the coherent configuration \(\mathcal{X}\) for each \(\beta \in \Omega \) such that \((\alpha ,\beta )\in s_{\mathrm{max}}\).
Proof
Denote by \(\Gamma _0\) the set of all points \(\gamma \in \Gamma \) with \(\gamma s_{\mathrm{max}}\subseteq \Gamma \). Then, \(\alpha \in \Gamma _0\), because in view of (17), the set \(\alpha s_{\mathrm{max}}\) contains \(\beta \in \Gamma \). Therefore, \(\Gamma _0\) contains the connected component of \(s_{\mathrm{max}}\) that contains \(\alpha \). Since \(s_{\mathrm{max}}\) is connected, this implies that \(\Gamma _0=\Omega \) and hence \(\Gamma =\Omega \). By the definition of \(\Gamma \), this means that the fibers of \(\mathcal{X}_{\alpha ,\beta }\) are singletons. Thus, \(\{\alpha ,\beta \}\) is a base of \(\mathcal{X}\).\(\square \)
3.3 Connected components of \(s_\alpha \)
One can treat \(s_\alpha \) also as a graph with vertex set \(\alpha s_{\mathrm{max}}\) and edge set \(s_\alpha \). The set of all connected components of this graph which contain a vertex in \(\alpha u\) for a fixed \(u\in S_{\mathrm{max}}\) is denoted by \(C_\alpha (u)=C(u)\).
Lemma 3.4
Let \(u,v\in S_{\mathrm{max}}\). Suppose that \(C(u)\cap C(v)\ne \varnothing \). Then, \(C(u)=C(v)\) and \(|\alpha u\cap C|=|\alpha v\cap C|\) for all \(C\in C(u)\).
Proof
Lemma 3.5
- (1)
if either \(n_u>n_v\), or \(n_u=n_v\) and \(C(u)\ne C(v)\), then \(p_u(\delta )\ge k\),
- (2)
if \(n_u=n_v\), \(C(u)=C(v)\), and \(|C(u)|>1\), then \(p_u(\delta )\ge k/2\).
Proof
3.4 The connectedness of \(s_{\mathrm{max}}\) and \(s_\alpha \)
Using Lemmas 3.4 and 3.5, we will prove that the hypothesis of Theorem 3.1 gives a sufficient condition for the graphs \(s_\alpha \) and \(s_{\mathrm{max}}\) to be connected.
Lemma 3.6
Suppose that \(2c(k-1)<n\) and \(k\ge 2\). Then, the graphs \(s_\alpha \) and \(s_{\mathrm{max}}\) are connected. Moreover, \(|\alpha s_{\mathrm{max}}|>n/2\).
Proof
To prove that the graph \(s_{\mathrm{max}}\) is also connected, assume to the contrary that one of its components, say C, has at most n / 2 points. Let \(\alpha \in C\) and \(u\in S_{\mathrm{max}}\). Then, \(\alpha u\subseteq C\) and \(n_u>n_v\) for all \(v=r(\alpha ,\delta )\) with \(\delta \in \Omega {\setminus } C\). By statement (1) of Lemma 3.5, this implies that \(p_u(\delta )\ge k\) for all such \(\delta \). Thus, inequality (25) holds again, which contradicts the hypothesis of the lemma.
3.5 Proof of Theorem 3.1
4 Inequality (3) in simple groups of Lie type
The main purpose of the following two sections is to prove Theorem 4.1 below, from which Theorems 1.2 and 1.3 were deduced in Introduction. In this section, we reduce the proof to Lemma 4.3, which will be proved in the next section.
Theorem 4.1
For the Cartan scheme \(\mathcal{X}\) of every group \(G\in \mathcal{L}\), inequality (3) holds.
We generally follow the notation of well-known Carter’s book [6] with some exceptions that we explain below inside parentheses. If \(\Phi _l\) is a simple Lie algebra of rank l, then \(\Phi _l(q)\) is the simple Chevalley group of rank l over a field of order q. Let B, N, and \(H=B\cap N\) be a Borel, monomial, and Cartan subgroups of a simple Chevalley group \(\Phi _l(q)\) as in [6], while \(W=N/H\) be the corresponding Weyl group. Then, [6, Proposition 8.2.1] implies that the subgroups B and N form a (B, N)-pair of \(\Phi _l(q)\). If \(\tau \) is a symmetry of the Dynkin diagram of \(\Phi _l\) of order t, then \({}^t\Phi _l(q)\) is the simple twisted group of Lie type (in [6] such a group is denoted as \({}^t\Phi _l(q^t)\)).^{2} Again B, N, and \(H=B\cap N\) stand for Borel, monomial, and Cartan subgroups of a simple twisted group of Lie type, and \(W=N/H\) is the Weyl group (in [6], they are denoted by \(B^1\), \(N^1\), and so on). It follows from [6, Theorem 13.5.4] that in this case, B and N form a (B, N)-pair of \({}^t\Phi _l(q)\) again. For the sake of brevity, we will use notation \({}^t\Phi _l(q)\) for all simple groups of Lie type, assuming that t is the empty symbol in the case of untwisted groups. Recall also that the order w of the Weyl group W does not depend on the order of the underlying field.
Let G be a finite simple group of Lie type, and let \(\mathcal{X}=(\Omega ,S)\) be the Cartan scheme of G, where the corresponding (B, N)-pair is as in the previous paragraph (see Definition 1.1). In particular, \(\Omega =G/H\) and \(S={{\mathrm{Orb}}}(G,\Omega ^2)\). Put \(n=|\Omega |\), \(k=k(\mathcal{X})\), and \(c=c(\mathcal{X})\).
Lemma 4.2
Proof
Observe that G satisfies the hypotheses of Lemmas 2.3 and 2.4. Indeed, the transitivity of the action of G on \(\Omega \) is evident, while the monomial subgroup N of G satisfies the additional condition from Lemma 2.4 due to [6, Proposition 8.4.5]. This enables us to estimate the indistinguishing number c and get the required inequality (3) with the help of Lemma 4.3 below, which is proved in the next section.
Lemma 4.3
Lemma 4.4
Let G be a simple group of Lie type. Suppose that there exists an integer m such that inequality (28) holds. Then, for the Cartan scheme \(\mathcal{X}\) of G, inequality (3) is satisfied.
Proof
5 Proof of Lemma 4.3
Exceptional groups
\({}^t\Phi _l\) | \(m_0\) | |H| | |W| |
---|---|---|---|
\(E_8\) | \(q^{112}\) | \((q-1)^8\) | \(2^{14}\cdot 3^5\cdot 5^2\cdot 7\) |
\(E_7\) | \((1/2)q^{64}\) | \((q-1)^7/(2,q-1)\) | \(2^{10}\cdot 3^4\cdot 5\cdot 7\) |
\(E_6\) | \((1/3)q^{30}\) | \((q-1)^6/(3,q-1)\) | \(2^{7}\cdot 3^4\cdot 5\) |
\({}^2E_6\) | \((1/3)q^{30}\) | \((q-1)^4(q+1)^2/(3,q+1)\) | \(2^{7}\cdot 3^2\) |
\(F_4\) | \(q^{16}\) | \((q-1)^4\) | \(2^{7}\cdot 3^2\) |
\(G_2\) | \(q^{3}(q^3-1)\) | \((q-1)^2\) | \(2^{2}\cdot 3\) |
\({}^3D_4\) | \(q^{16}\) | \((q-1)(q^3-1)\) | \(2^2\cdot 3\) |
\({}^2F_4\) | \(q^{6}(q-1)(q^3+1)\) | \((q-1)^2\) | \(2^4\) |
\({}^2G_2\) | \(q^2(q^2+q+1)\) | \(q-1\) | 2 |
\({}^2B_2\) | \(q^2(q-1)\) | \(q-1\) | 2 |
Classical groups I
\({}^t\Phi _l\) | Conditions | \(m_0\) | |H| | |W| | \(l_0\) | a |
---|---|---|---|---|---|---|
\(A_l\) | \(\frac{q^{2l}}{2}\) | \(\frac{(q-1)^l}{(l+1,q-1)}\) | \((l+1)!\) | 7 | 4 | |
\({}^2A_l\) | l odd | \(\frac{q^{4l-3}}{2(q+1)}\) | \(\frac{(q-1)^{\frac{l+1}{2}}(q+1)^{\frac{l-1}{2}}}{(l+1,q+1)}\) | \(2^{\frac{l+1}{2}}\frac{l+1}{2}!\) | 6 | 4 |
\({}^2A_l\) | l even | \(\frac{q^{2l+1}}{2(q+1)}\) | \(\frac{(q-1)^{\frac{l}{2}}(q+1)^{\frac{l}{2}}}{(l+1,q+1)}\) | \(2^{\frac{l}{2}}\frac{l}{2}!\) | 6 | 4 |
\(B_l\) | \(\frac{l(q-1)}{2}\) odd | \(\frac{q^{4l-1}}{4(q+1)}\) | \(\frac{(q-1)^l}{2}\) | \(2^ll!\) | 4 | 4 |
\(B_l\) | \(\frac{l(q-1)}{2}\) even | \(\frac{q^{2l+1}}{4(q+1)}\) | \(\frac{(q-1)^l}{2}\) | \(2^ll!\) | 4 | 4 |
\(C_l\) | \(\frac{q^{4l-4}}{2}\) | \(\frac{(q-1)^l}{(2,q-1)}\) | \(2^ll!\) | 3 | 4 | |
\(D_l\) | \(\frac{q^{4l-3}}{4(q+1)}\) | \(\frac{(q-1)^l}{(4,q^l-1)}\) | \(2^{l-1}l!\) | 4 | 2 | |
\({}^2D_l\) | \(\frac{q^{4l-3}}{4(q+1)}\) | \(\frac{(q-1)^{l-1}(q+1)}{(4,q^l+1)}\) | \(2^{l-1}l!\) | 4 | 2 |
Lemma 5.1
- (1)
If G is one of the groups \(A_l\), \({}^2A_l\) with l even, and \(B_l\) with \(l(q-1)/2\) even, then the number \(r_m\) does not exceed the number in the fourth column of the corresponding row of Table 3.
- (2)
If G is one of the other simple classical groups, then \(\nu (h)\ge 2\) for every \(h\in H^\#\).
Classical groups II
\({}^t\Phi _l\) | Conditions | \(m_1\) | \(r_m\) |
---|---|---|---|
\(A_l\) | \(\frac{q^{4(l-1)}}{2}\) | \(\frac{l(l+1)(q-1)^2}{2}-1\) | |
\({}^2A_l\) | l even | \(\frac{q^{4l-3}}{2(q+1)}\) | \(\frac{(l+1)(q+1)^2}{2}+q\) |
\(B_l\) | \(\frac{l(q-1)}{2}\) even | \(\frac{q^{4l-1}}{4(q+1)}\) | \(\frac{l(q-3)}{2}+1\) |
Proof
It is well known and easily verified that the diagonal subgroup of a perfect classical matrix group contains an element h with \(\nu (h)=1\) only if G is one of the groups in statement (1). Therefore, we need only to estimate \(r_m\) in these cases.
Let \(G=A_l\). Then, the required statement immediately follows from (33) with the help of direct estimation of the number u from above by the number of diagonal matrices in \({{\mathrm{SL}}}(l+1,q)\) with at least \(l-1\) equal diagonal entries.
To complete the proof, we verify inequality (28) for the number m defined by (32). Observe that, due to (32) and Lemma 5.1, the number \(r_m\) equals 0 in all cases when \(m=m_0\). In the latter case, it suffices to verify inequality (31). We proceed further case by case.
For each of the remaining two series of classical groups, the expression on the left-hand side of (28) for \(m=m_0\le m_1\) does not exceed the same expression for \(m=m_1\) (see Tables 2 and 3). Since the expression on the right-hand side in both cases does not depend on whether \(m=m_0\) or not, it suffices to verify (28) for \(G={}^2A_{l}(q)\) (resp., \(G=B_{l}(q)\)) independently of the parity of l (resp., \(l(q-1)/2\)), where \(m_0\) and m are taken as in the case of even l (resp. even \(l(q-1)/2\)).
6 Proof of Theorem 1.4
Lemma 6.1
Proof
It is easy to verify that \(H^x\cap H=1\) for all \(x\in G{\setminus } N\) and \(N=H\cup Hi\). Thus, the required statements follow from formula (6).\(\square \)
Lemma 6.2
- (1)
\(c_{s_u s_{}}^{s_v}=0\) if \(s=s_1\) or \(s_i\), and \(c_{s_u s_{}}^{s_v}=1\) otherwise,
- (2)
if \(s\not \in \{s_1,s_i,s_u,s_v\}\), then \(c_{s_u^{}s^{}_v}^s=1\) or \(c_{s_v^{}s^{}_u}^s=1\).
Proof
Let us verify that the coherent configuration \(\mathcal{X}_\alpha \) is 1-regular for \(\alpha =H\). Indeed, in this case, Theorem 1.4 follows from Corollary 2.6.
7 Proof of Theorem 1.5
Theorem 7.1
Let \(\mathcal{X}\) and \(\mathcal{X}'\) be coherent configurations on n points. Then given an algebraic isomorphism \(\varphi :\mathcal{X}\rightarrow \mathcal{X}'\), all the elements of the set \({{\mathrm{Iso}}}(\mathcal{X},\mathcal{X}',\varphi )\) can be listed in time \((bn)^{O(b)}\), where \(b=b(\mathcal{X})\).
Step 1. Find the coherent configuration \(\mathcal{X}={{\mathrm{WL}}}(\mathcal{S})\).
Step 2. If there are no distinct points \(\alpha ,\beta \) such that the coherent configuration \(\mathcal{X}_{\alpha ,\beta }={{\mathrm{WL}}}(\mathcal{S}_{\alpha ,\beta })\) is complete, then \(b(\mathcal{X})>2\) and \(D\not \in \mathcal{K}_n\).
Step 3. Find all the elements of the group \(G={{\mathrm{Iso}}}(\mathcal{X},\mathcal{X},{{\mathrm{id}}})\) by the algorithm of Theorem 7.1. If G is not simple, then \(D\not \in \mathcal{K}_n\).
Step 4. Analyzing the number |G|, check that \(G\in \mathcal{L}\). If not, then \(D\not \in \mathcal{K}_n\); otherwise, set p to be the characteristic of the ground field associated with G.
Step 5. Fix a point stabilizer H of G and find \(P\in {{\mathrm{Syl}}}_p(G)\), for which relations (42) hold. If there is no such P, then \(D\not \in \mathcal{K}_n\).
Step 6. Now, \(D\in \mathcal{K}_n\) and \(\mathcal{X}\) is the Cartan scheme of G with respect to (B, N), where \(B=N_G(P)\) and \(N=N_G(H)\).\(\square \)
The first four steps of the algorithm remain the same as before if we do not assume that the rank of (B, N) is at least 2. But in this case, one can find a 2-transitive representation of the group G; here, a complete classification of all 2-transitive groups is useful (see, e.g., [11, Sec. 7.7]). This enables us to find the groups B and N.
To solve the isomorphism problem, let \(D\in \mathcal{K}_n\) and \(D'\in \mathcal{G}_n\). Denote by \(\mathcal{S}\) and \(\mathcal{S}'\) the sets of color classes of D and \(D'\), respectively. Without loss of generality, we may assume that there is a color preserving bijection \(\psi :\mathcal{S}\rightarrow \mathcal{S}'\). Then, one can apply the canonical version of the Weisfeiler-Leman algorithm presented in [21, Section M], where, in fact, the following statement was proved.
Theorem 7.2
Let \(\mathcal{S}\) and \(\mathcal{S}'\) be m-sets of binary relations on an n-element set. Then, given a bijection \(\psi :\mathcal{S}\rightarrow \mathcal{S}'\) one can check in time \(mn^{O(1)}\) whether or not there exists an algebraic isomorphism \(\varphi :{{\mathrm{WL}}}(\mathcal{S})\rightarrow {{\mathrm{WL}}}(\mathcal{S}')\) such that \(\varphi |_\mathcal{S}=\psi \). Moreover, if \(\varphi \) does exist, then it can be found within the same time.\(\square \)
Footnotes
- 1.
In the complete colored graph representing \(\mathcal{X}\), k is the maximum number of the monochrome arcs incident to a vertex, and c is the maximum number of triangles with fixed base; the other two sides of which are monochrome arcs.
- 2.
In the case of Suzuki and Ree groups, \(q=2^{2\alpha +1}\) for \({}^2B_2(q)\) and \({}^2F_4(q)\), and \(q=3^{2\alpha +1}\) for \({}^2G_2(q)\), where \(\alpha >1\) is an integer.
- 3.
The alternative way to establish the same is to apply Zenkov’s theorem [22]. It yields that since H is abelian, there is an element \(g_0\in G\) such that \(H\cap H^{g_0}\) lies in the Fitting subgroup of G, which is trivial if the group G is simple.
- 4.
- 5.
It is worth mentioning that despite [5, Tables 3.7–3.9] contain the bounds on the sizes of conjugacy classes in the group \({\text {Inndiag}}(G)\) rather than G itself, the bounds for \(m_0\) in Table 5 are correct, because \(|G:C_G(h)|=|{\text {Inndiag}}(G):C_{{\text {Inndiag}}(G)}(h)|\) for every \(h\in H\) (see, e.g., the definition of the diagonal automorphism in [6, Sec. 12.2]).
References
- 1.Abramenko, P., Parkinson, J., Van Maldeghem, H.: Distance regularity in buildings and structure constants in Hecke algebras. arXiv:1508.03912 [math.CO] 1–23 (2015)
- 2.Bailey, R.F., Cameron, P.J.: Base size, metric dimension, and other invariants of groups and graphs. Bull. Lond. Math. Soc. 43, 209–242 (2011)MathSciNetCrossRefMATHGoogle Scholar
- 3.Bannai, E., Ito, T.: Algebraic Combinatorics. I Benjamin/Cummings, Menlo Park, CA (1984)MATHGoogle Scholar
- 4.Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3, 18th edn. Springer, Berlin (1989)Google Scholar
- 5.Burness, T.C.: Fixed point ratios in actions of finite classical groups, II. J. Algebra 309, 80–138 (2007)MathSciNetCrossRefMATHGoogle Scholar
- 6.Carter, R.W.: Simple Groups of Lie Type. Wiley, London (1972)MATHGoogle Scholar
- 7.Carter, R.W.: Finite Groups of Lie Type. Conjugacy Classes and Complex Characters. Wiley, London (1985)MATHGoogle Scholar
- 8.De Medts, T., Haot, F., Tent, K., Van Maldeghem, H.: Split BN-pairs of rank at least 2 and the uniqueness of splittings. J. Group Theory 8(1), 1–10 (2005)MathSciNetCrossRefMATHGoogle Scholar
- 9.Deriziotis, D.: The centralizers of semisimple elements of the Chevalley groups \(E_7\) and \(E_8\). Tokyo J. Math. 6(1), 191–216 (1983)MathSciNetCrossRefMATHGoogle Scholar
- 10.Deriziotis, D.: Conjugacy classes and centralizers of semisimple elements in finite groups of Lie type, Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, Heft 11 (1984)Google Scholar
- 11.Dixon, J.D., Mortimer, B.: Permutation Groups, Graduate Texts in Mathematics. Springer, New York (1996)Google Scholar
- 12.Evdokimov, S., Ponomarenko, I.: Separability number and schurity number of coherent configurations. Electron. J. Comb. 7, R31 (2000)MathSciNetMATHGoogle Scholar
- 13.Evdokimov, S., Ponomarenko, I.: Permutation group approach to association schemes. Eur. J. Comb. 30, 1456–1476 (2009)MathSciNetCrossRefMATHGoogle Scholar
- 14.Kimmerle, W., Lyons, R., Sandling, R., Teague, D.N.: Composition factors from the group ring and Artin’s theorem on orders of simple groups. Proc. Lond. Math. Soc. Ser. 60(1), 89–122 (1990)MathSciNetCrossRefMATHGoogle Scholar
- 15.Liebeck, M.W., Shalev, A.: Simple groups, permutation groups, and probability. J. Am. Math. Soc. 12(2), 497–520 (1999)MathSciNetCrossRefMATHGoogle Scholar
- 16.Muzychuk, M., Ponomarenko, I.: On Pseudocyclic association schemes. Ars Math. Contemp. 5(1), 1–25 (2012)MathSciNetMATHGoogle Scholar
- 17.Ponomarenko, I.: Bases of schurian antisymmetric coherent configurations and isomorphism test for schurian tournaments. J. Math. Sci. 192(3), 316–338 (2013)MathSciNetCrossRefMATHGoogle Scholar
- 18.Saxl, J., Shalev, A.: The fixity of permutation groups. J. Algebra 174, 1122–1140 (1995)MathSciNetCrossRefMATHGoogle Scholar
- 19.Veldkamp, F.D.: Roots and maximal tori in finite forms of semisimple algebraic groups. Math. Ann. 207, 301–314 (1974)MathSciNetCrossRefMATHGoogle Scholar
- 20.Vdovin, E.P.: On intersections of solvable Hall subgroups in finite simple exceptional groups of Lie type. Proc. Steklov Institute Math. 285(1), S1–S8 (2014)MathSciNetCrossRefMATHGoogle Scholar
- 21.Weisfeiler, B. (ed.): On Construction and Identification of Graphs, Springer Lecture Notes in Mathematics, 558 (1976)Google Scholar
- 22.Zenkov, V.I.: Intersection of abelian subgroups in finite groups. Math. Notes 56(2), 869–871 (1994)MathSciNetCrossRefMATHGoogle Scholar
- 23.Zieschang, P.-H.: Theory of Association Schemes. Springer, Berlin (2005)MATHGoogle Scholar
- 24.Zieschang, P.-H.: Trends and lines of development in scheme theory. Eur. J. Comb. 30, 1540–1563 (2009)MathSciNetCrossRefMATHGoogle Scholar