2 Association schemes, Bose–Mesner algebras, and matrices over subfields

Our notation and terminology generally follow [Reference Brouwer, Cohen and Neumaier6] but see also [Reference Bannai and Ito4, Reference Godsil13]. In this article, all association schemes are commutative. An association scheme (or, more simply, scheme) is an ordered pair $\mathscr {X}=(X,{\mathcal R}),$ where X is a nonempty finite set and ${{\mathcal R}=\{R_0,\ldots , R_d\}}$ is a partition of $X\times X$ satisfying:

1. (i) $R_0= \mathrm {id}_X$ , the identity relation;

2. (ii) for each i, $0\le i\le d$ , there is an $i'$ for which $R_{i'}=R_i^\top = \{ (b,a) \mid (a,b)\in R_i\}$ ;

3. (iii) there exist $p_{ij}^k$ ( $0\le i,j,k\le d$ ) such that $\left | \{ c\in X \mid (a,c)\in R_i, \ (c,b)\in R_j\} \right | =p_{ij}^k$ for any $(a,b)\in R_k$ ;

4. (iv) $p_{ij}^k=p_{ji}^k$ for all i, j, and k.

We may view the pair $(X,R_i)$ as a digraph on the vertex set X. We write $R_i(x) = \{ y\in X | (x,y) \in R_i\}$ . Condition (iii) is then $\left | R_i(a) \cap R_{j'}(b) \right | = p_{ij}^k$ whenever $b\in R_k(a)$ . We typically deal with these directed graphs via their adjacency matrices.

For a nonempty finite set X and a field $\mathbb {K}$ , we denote by $\mathsf {Mat}_X(\mathbb {K})$ the algebra of matrices with rows and columns indexed by the elements of X having entries in $\mathbb {K}$ . For $x,y\in X$ and $A \in \mathsf {Mat}_X(\mathbb {K})$ , the entry in row x, column y of A is denoted $A_{xy}$ . For $[m]=\{1,\ldots ,m\}$ , we will simply write $\mathsf {Mat}_m(\mathbb {K})$ for $\mathsf {Mat}_{[m]}(\mathbb {K})$ . For an association scheme $\mathscr {X}=(X,{\mathcal R})$ as above, the $i^{\mathrm {th}}$ adjacency matrix or $i^{\mathrm {th}}$ Schur idempotent Footnote ² $A_i$ is the matrix in $\mathsf {Mat}_X(\mathbb {Q})$ with $(a,b)$ -entry equal to one if $(a,b)\in R_i$ and a zero otherwise. The fact that the $R_i$ partition $X\times X$ means $\sum _i A_i =J$ , the all ones matrix, $A_i\circ A_j = 0$ for $i\neq j$ , and $A_i\circ A_i =A_i \neq 0$ . Conditions (i)–(iv) are then rephrased as:

1. (i’) $A_0= I$ ;

2. (ii’) for each i, $A_i^\top \in \{A_0,\ldots ,A_d\}$ ;

3. (iii’) $A_i A_j = \sum _{k=0}^d p_{ij}^k A_k$ ;

4. (iv’) $A_i A_j = A_j A_i$ .

We use ${\mathcal R}=\{R_0,\ldots ,R_d\}$ and $\mathcal {A}=\{A_0,\ldots ,A_d\}$ interchangeably. Note that, in this article, a Bose–Mesner algebra $\mathbb {A}= \mathbb {C}[\mathcal {A}] = \operatorname {\mathrm {span}}_{\mathbb {C}}\{A_0,\ldots ,A_d\}$ is always defined over  $\mathbb {C}$ , by extension of scalars, if necessary.

The commuting normal matrices $A_0,\ldots ,A_d$ are simultaneously diagonalizable over $\mathbb {C}$ , and we have a second canonical basis of primitive idempotents $\{ E_0,E_1,\ldots ,E_d\}$ for $\mathbb {A}$ satisfying $E_i E_j = \delta _{ij} E_i$ and $\sum _j E_j = I$ (see, e.g., [Reference Bannai, Bannai, Ito and Tanaka5, Lemma 2.18]). By convention, $E_0=\frac {1}{|X|} J$ . For $0\le j\le d$ , there is a $j^*$ such that $E_{j^*} = E_j^\dagger $ , the Hermitian transpose of $E_j$ . The splitting field of $\mathscr {X}$ (or of $\mathbb {A}$ ) is the subfield of $\mathbb {C}$ obtained from $\mathbb {Q}$ by adjoining all eigenvalues of $A_1,\ldots ,A_d$ .

For a field $\mathbb {K}\subseteq \mathbb {C}$ and a set of matrices ${\mathcal E} \subset \mathsf {Mat}_X(\mathbb {K})$ , we will write $\mathbb {K}[{\mathcal E}]= \operatorname {\mathrm {span}}_{\mathbb {K}}({\mathcal E})$ the $\mathbb {K}$ -span of ${\mathcal E}$ . Assuming $\mathcal {A}$ is the basis of Schur idempotents of some association scheme, $\mathbb {K}[\mathcal {A}]$ is always closed and commutative under both ordinary and Schur multiplication. Under these conditions, $\mathbb {K}[\mathcal {A}]$ is also closed under taking transpose and contains both I and J. While the set $\mathbb {K}[\mathcal {A}]$ is diagonalizable over $\mathbb {C}$ , it is not diagonalizable over $\mathbb {K}$ in general. Among the $2^{d+1}$ idempotents in $\mathbb {C}[\mathcal {A}]$ , we consider those with entries in subfield $\mathbb {K}$ :

(2.1) $$ \begin{align} {\mathcal E}_{\mathbb{K}} = \left\{ F=\sum_{j=0}^d c_j E_j \middle| c_0,\ldots,c_d\in \{0,1\}, \ F\in \mathsf{Mat}_X(\mathbb{K}) \right\}. \end{align} $$

We now introduce terminology for the case where condition (v) holds.

The two standard bases $\{A_0,\ldots ,A_d\}$ and $\{E_0,\ldots ,E_d\}$ for the Bose–Mesner algebra $\mathbb {A}$ of $\mathscr {X}$ are related by the first and second eigenmatricesFootnote ³ P and Q by

(2.2) $$ \begin{align} \hfill A_i = \sum_{j=0}^d P_{ji} E_j, \hspace{1in} E_j = \frac{1}{|X|} \sum_{i=0}^d Q_{ij} A_i. \hfill \end{align} $$

So the matrices $P=\left [ P_{ji} \right ]_{j,i=0}^d$ and $Q=\left [ Q_{ij} \right ]_{i,j=0}^d$ are, up to scalar multiple, inverses of one another. However, each may be obtained from the other in another way. Each digraph $(X,R_i)$ is regular, for $v_i=P_{0i}$ . Set $\Delta _v$ as the $(d+1)\times (d+1)$ diagonal matrix with $(i,i)$ -entry $v_i$ and $\Delta _m$ the $(d+1)\times (d+1)$ diagonal matrix with $(j,j)$ -entry $m_j=\operatorname {\mathrm {rk}} E_j$ . Then, P and Q satisfy both the first and second orthogonality relations [Reference Bannai and Ito4, Theorem II.3.5], [Reference Brouwer, Cohen and Neumaier6, Section 2.2] for association schemes,

(2.3) $$ \begin{align} \hfill PQ = |X| I , \hspace{1in} \Delta_{m} P \Delta_v^{-1} = Q^\dagger. \hfill\end{align} $$

The Krein parameters of X are those scalars $q_{ij}^k$ ( $0\le i,j,k\le d$ ) for which $E_i \circ E_j = \frac {1}{|X|} \sum _{k=0}^d q_{ij}^k E_k$ , and these also satisfy $Q_{hi}Q_{hj} = \sum _{k=0}^d q_{ij}^k Q_{hk}$ for each $0\le h\le d$ . The Krein conditions (e.g., [Reference Brouwer, Cohen and Neumaier6, Theorem 2.3.2]) state that, for each $i,j,k$ , $q_{ij}^k \ge 0$ . We will refer to the extension of $\mathbb {Q}$ by all $q_{ij}^k$ as the Krein field of $\mathscr {X}$ . In some cases, this is a proper subfield of the splitting field; in particular, it is contained in $\mathbb {R}$ even when some $A_i$ is not symmetric. This holds because each $E_i$ is a Hermitian matrix as is $E_i \circ E_j$ .

3 Fusion schemes and the Galois group

Let X be a finite nonempty set and let $\mathscr {X}=(X,\{R_0,R_1,\ldots ,R_d\})$ and $\mathscr {F}=(X,\{R^{\prime }_0, R^{\prime }_1,\ldots $ , $R^{\prime }_e\})$ be association schemes on X. We say that $\mathscr {F}$ is a fusion (scheme) of $\mathscr {X}$ if each $R_i$ is contained in some $R^{\prime }_j$ ; that is, each relation $R^{\prime }_j$ is expressible as a union of basis relations (or by fusing or merging basis relations) in $\{R_0,\ldots ,R_d\}$ . Some authors also refer to $\mathscr {X}$ as a fission scheme with respect to $\mathscr {F}$ while Godsil [Reference Godsil14, Chapter 5], for example, uses the term “subscheme” for what we call a fusion scheme to highlight the relationship between the corresponding Bose–Mesner algebras.

The fusion schemes of a given association scheme form a meet semilattice. We now list this and some other basic facts about fusions.

Given an association scheme $\mathscr {X}=(X,\mathcal {A})$ with Bose–Mesner algebra $\mathbb {A}$ and primitive idempotents $\{E_0,\ldots ,E_d\}$ , any partition $\{{\mathcal O}_0,{\mathcal O}_1,\ldots ,{\mathcal O}_e\}$ of $\{0,1,\ldots ,d\}$ gives us a Schur-closed vector subspace of $\mathbb {A}$ by merging classes, namely, $\mathbb {C}\left [ \left \{ \sum _{i\in {\mathcal O}_j} A_i \middle | 0\le j\le e\right \}\right ]$ . Dually, the subspace $\mathbb {C}\left [ \left \{ \sum _{i\in {\mathcal O}_j} E_i \middle | 0\le j\le e\right \}\right ]$ is always closed under ordinary matrix multiplication. We next discuss how to check whether one of these subspaces is a Bose–Mesner algebra.

The Bannai–Muzychuk criterion is the fundamental computational tool in testing for fusion schemes of an association scheme with known parameters. It comes in two forms.

Suppose $\mathscr {F}=(X,\{A_0',\ldots ,A_e'\})$ is a fusion scheme of $\mathscr {X}=(X,\{A_0,\ldots ,A_d\})$ with $A^{\prime }_j = \sum _{i\in {\mathcal O}_j} A_i$ ( $0\le j\le e$ ) as in the proposition above. The partition $\{{\mathcal O}_0,\ldots ,{\mathcal O}_e\}$ of $\{0,1,\ldots ,d\}$ induces a partition $\{ {\mathcal S}_0,\ldots ,{\mathcal S}_e\}$ of the rows of P as follows [Reference Godsil13, Section 5.2]: the $(e+1)\times (d+1)$ matrix S with $S_{ij}=1$ if $j\in {\mathcal S}_i$ and $S_{ij}=0$ otherwise satisfies $PO=S P_{\mathscr {F}}$ for some $(e+1)\times (e+1)$ matrix $P_{\mathscr {F}}$ which, according to the theorem of Bannai and Muzychuk, is the first eigenmatrix of the corresponding fusion scheme. As $QP=|X|I$ , we then have $O=\frac {1}{|X|}QS P_{\mathscr {F}}$ , or $QS=OQ_{\mathscr {F}}$ , where $Q_{\mathscr {F}}$ is the second eigenmatrix of $\mathscr {F}$ .

Our primary interest in fusion schemes of $\mathscr {X}$ is their connection to subfields of the splitting field of $\mathscr {X}$ .

3.1 The Galois group of an association scheme

To our knowledge, the Galois group of (the splitting field of) an association scheme $\mathscr {X}=(X,\mathcal {A})$ was first studied by Munemasa in [Reference Munemasa19]. We defined this above as the extension of the rationals by all eigenvalues of the scheme; alternatively, if Q is the second eigenmatrix of $\mathscr {X}$ , the splitting field is defined as $\mathbb {Q}(\{Q_{ij} : i,j=1,\ldots ,d\})$ and will be henceforth denoted by $\mathbb {F}$ . Munemasa proved that, for any subfield $\mathbb {K}$ of $\mathbb {F}$ containing all the Krein parameters of $\mathscr {X}$ , ${\mathsf {Gal}}(\mathbb {F}/\mathbb {K})$ is contained in the center of ${\mathsf {Gal}}(\mathbb {F}/\mathbb {Q})$ .

As each $A_i$ is rational, each $\boldsymbol {\sigma }$ in the Galois group ${\mathsf {Gal}}(\mathbb {F}/\mathbb {Q})$ maps each $A_i$ to itself. On the other hand, the Galois group acts faithfully on the set $\{E_0,E_1,\ldots ,E_d\}$ [Reference Martin and Williford18, Theorem 2.1].

Definition 3.1 Let $(X,\mathcal {A})$ be an association scheme with splitting field $\mathbb {F}$ and primitive idempotents $E_0,\ldots ,E_d$ and let $\mathbb {Q} \subseteq \mathbb {K} \subseteq \mathbb {F}$ . Write $M^{\boldsymbol {\sigma }}$ for the matrix obtained by applying $\boldsymbol {\sigma }$ to M entrywise. For each $\boldsymbol {\sigma } \in {\mathsf {Gal}}(\mathbb {F}/\mathbb {K})$ , there is a permutation ${\sigma \in \operatorname {\mathrm {Sym}}(\{0,\ldots ,d\})}$ defined by $E_{j^\sigma }=\left ( E_j\right )^{\boldsymbol {\sigma }}$ . We define the corresponding permutation group

$$\begin{align*}\Sigma_{\mathbb{K}}:=\{ \sigma \mid \boldsymbol{\sigma} \in {\mathsf{Gal}}(\mathbb{F}/\mathbb{K})\}. \end{align*}$$

Since the action of $\Sigma _{\mathbb {Q}}$ is faithful, so too is the action of $\Sigma _{\mathbb {K}}$ , as ${\mathsf {Gal}}(\mathbb {F}/\mathbb {K}) \leq {\mathsf {Gal}}(\mathbb {F}/\mathbb {Q})$ for any $\mathbb {Q} \subseteq \mathbb {K} \subseteq \mathbb {F}$ . One consequence of this is that, for distinct subfields $\mathbb {K}$ and $\mathbb {L}$ of the splitting field $\mathbb {F}$ , ${\mathcal E}_{\mathbb {K}} \neq {\mathcal E}_{\mathbb {L}}$ . In the case where all Krein parameters are rational, we also have an action on the columns of matrix P [Reference Bannai and Ito4, Theorem II.7.3] by the subgroup of ${\mathsf {Gal}}(\mathbb {F}/\mathbb {Q})$ centralizing complex conjugation. Note that, for computational purposes, one uses the fact that $\Sigma _{\mathbb {K}}$ is also the permutation action of ${\mathsf {Gal}}(\mathbb {F}/\mathbb {K})$ on the columns of the second eigenmatrix Q.

Lemma 3.5 Let ${\mathcal Q}_0=\{0\},{\mathcal Q}_1,\ldots ,{\mathcal Q}_e$ be the orbits of the action of $\Sigma _{\mathbb {K}}$ on $\{0,1,\ldots ,d\}$ and, for $0\le \ell \le e$ , define

$$\begin{align*}F_\ell = \sum_{j\in {\mathcal Q}_\ell} E_j. \end{align*}$$

Then, the minimal elements of ${\mathcal E}_{\mathbb {K}}$ are precisely $\{F_0, F_1, \ldots , F_e\}$ ; that is, $E\in {\mathcal E}_{\mathbb {K}}$ if and only if E is expressible as a sum of some subset of $\{F_0, F_1, \ldots , F_e\}$ .

Proof It is clear that each $F_\ell $ has all entries in $\mathbb {K}$ as $F_\ell ^{\boldsymbol {\sigma }}=F_\ell $ for each $\boldsymbol {\sigma }\in {\mathsf {Gal}}(\mathbb {F}/\mathbb {K})$ . Conversely, if $E\in {\mathcal E}_{\mathbb {K}}$ with $E=\sum _{j=0}^d c_j E_j$ with each $c_j\in \{0,1\}$ , then $c_{j^\sigma }=c_j$ for each j and each $\sigma \in \Sigma _{\mathbb {K}}$ so that $EF_\ell = c_j F_\ell $ for any $j\in {\mathcal Q}_\ell $ .

Recall that $\mathscr {X}$ is said to have Property $\Phi _{\mathbb {K}}$ if ${\mathcal E}_{\mathbb {K}}$ is closed under Schur multiplication. That is, Property $\Phi _{\mathbb {K}}$ holds for $\mathscr {X}$ when $\mathbb {C}[{\mathcal E}_{\mathbb {K}}]$ is a Bose–Mesner algebra, denoted by $\mathbb {A}_{\downarrow _{\mathbb {K}}}$ . This occurs precisely when there is a corresponding fusion of $\mathscr {X}$ . In particular, this fusion can be determined by applying the Bannai–Muzychuk criterion (Proposition 3.3) with the partition determined by the orbits ${\mathcal Q}_\ell $ of $\Sigma _{\mathbb {K}}$ on the minimal idempotents defined in Lemma 3.5. (Indeed, if $QO$ has exactly r distinct rows, then the Schur closure of $\mathbb {C}\left [ {\mathcal E}_{\mathbb {K}} \right ]$ has dimension r.)

Definition 3.2 Let $\mathscr {X}=(X,{\mathcal R})$ be an association scheme satisfying Property $\Phi _{\mathbb {K}}$ . We denote by $\mathscr {X}_{\downarrow _{{\mathbb {K}}}}$ the fusion scheme which has Bose–Mesner algebra $\mathbb {A}_{\downarrow _{{\mathbb {K}}}}$ . We shall call $\mathscr {X}_{\downarrow _{{\mathbb {K}}}}$ the Galois fusion of $\mathscr {X}$ with respect to $\mathbb {K}$ in light of Lemma 3.5 and the Bannai–Muzychuk criterion. We shall refer to $\mathscr {X}_{\downarrow _{{\mathbb {Q}}}}$ simply as the Galois fusion of $\mathscr {X}$ . (Note that we use the notation $\mathscr {X}_{\downarrow _{{\mathbb {K}}}}$ only when $\mathscr {X}$ has Property $\Phi _{\mathbb {K}}$ .)

The following is also well known.

Proposition 3.6 A commutative association scheme has only real eigenvalues if and only if it is symmetric. Given any non-symmetric commutative scheme $(X,{\mathcal R})$ with splitting field $\mathbb {F}$ , the Galois group ${\mathsf {Gal}}\left (\mathbb {F}/(\mathbb {F} \cap \mathbb {R}) \right )$ has complex conjugation as its sole non-identity element and the relative Galois fusion of our non-symmetric scheme with respect to this subgroup of its Galois group is simply the symmetrization of $(X,{\mathcal R})$ .

Remark 3.7 This is a special case of a much more general phenomenon: stratifiability of a homogeneous coherent configuration.

Proposition 3.8 Let $\mathbb {K}_1$ and $\mathbb {K}_2$ be subfields of the splitting field of $\mathscr {X}$ . If $\mathscr {X}$ satisfies Property $\Phi _{\mathbb {K}_1}$ and Property $\Phi _{\mathbb {K}_2}$ , then $\mathscr {X}$ satisfies Property $\Phi _{{\mathbb {K}_1 \cap \mathbb {K}_2}}$ .

Proof By Lemma 2.2(iii), $\mathbb {C} \left [ {\mathcal E}_{\mathbb {K}_1 \cap \mathbb {K}_2} \right ] =\mathbb {C} \left [ {\mathcal E}_{\mathbb {K}_1} \cap {\mathcal E}_{\mathbb {K}_2} \right ] =\mathbb {C} \left [ {\mathcal E}_{\mathbb {K}_1} \right ] \cap \mathbb {C} \left [ {\mathcal E}_{\mathbb {K}_2} \right ]$ is a Bose–Mesner algebra.

We now introduce some additional notation which will be useful in discussing Galois fusions and will also be used in the following sections.

Definition 3.3 Let ${\mathcal Q}_0=\{0\},{\mathcal Q}_1,\ldots ,{\mathcal Q}_e$ be the orbits of the action of $\Sigma _{\mathbb {K}}$ on $\{0,1,\ldots ,d\}$ . We define a map $\iota :\{0, \ldots , d\} \to \{0, \ldots , e\}$ by $\iota (i) = j$ for $i \in \mathcal {Q}_j$ . Given this partition of $\{0,\ldots ,d\}$ , we shall set $\overline {Q}=QO,$ where O is the $(d+1)\times (e+1)$ matrix with $(i,j)$ -entry equal to one if $i\in {\mathcal Q}_j$ and zero otherwise. Note that we will use $\iota _{\mathscr {X}}$ and $\overline {Q}_{\mathscr {X}}$ when such a clarification is needed.

The map $\iota $ is clearly surjective with $\iota ^{-1}(j) = \mathcal {Q}_j$ . It is injective precisely when $\mathbb {F}=\mathbb {K}$ . Henceforth, we shall restrict $F_j$ to not simply any sums of the minimal idempotents, but to those arising from the Galois orbits with respect to some subfield $\mathbb {K}$ ; henceforth,

$$\begin{align*}F_j = \sum_{i \in \iota^{-1}(j)}E_i. \end{align*}$$

Example 3.9 Consider the scheme $\mathscr {X}$ built from Cayley graphs on $\mathbb {Z}_4 \times \mathbb {Z}_2$ specified by its relation matrix $\sum i A_i$ and the relation matrix of its Galois fusion $\mathscr {X}_{\downarrow _{\mathbb {Q}}}$ together with their corresponding second eigenmatrices $Q_{\mathscr {X}}$ and $Q_{\mathscr {X}_{\downarrow _{\mathbb {Q}}}}$ . Here, we see this last matrix is obtained from $\overline {Q}_{\mathscr {X}}$ by the removal of repeated rows:

$$\begin{align*}\mathscr{X}: \begin{bmatrix} 0 & 1 & 2 & 3 & 4 & 4 & 4 & 4 \\ 1 & 0 & 3 & 2 & 4 & 4 & 4 & 4 \\ 3 & 2 & 0 & 1 & 4 & 4 & 4 & 4 \\ 2 & 3 & 1 & 0 & 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 & 0 & 1 & 2 & 3 \\ 4 & 4 & 4 & 4 & 1 & 0 & 3 & 2 \\ 4 & 4 & 4 & 4 & 3 & 2 & 0 & 1 \\ 4 & 4 & 4 & 4 & 2 & 3 & 1 & 0 \end{bmatrix}, \quad \mathscr{X}_{\downarrow\mathbb{Q}}: \begin{bmatrix} 0 & 1 & 2 & 2 & 3 & 3 & 3 & 3 \\ 1 & 0 & 2 & 2 & 3 & 3 & 3 & 3 \\ 2 & 2 & 0 & 1 & 3 & 3 & 3 & 3 \\ 2 & 2 & 1 & 0 & 3 & 3 & 3 & 3 \\ 3 & 3 & 3 & 3 & 0 & 1 & 2 & 2 \\ 3 & 3 & 3 & 3 & 1 & 0 & 2 & 2 \\ 3 & 3 & 3 & 3 & 2 & 2 & 0 & 1 \\ 3 & 3 & 3 & 3 & 2 & 2 & 1 & 0 \end{bmatrix} \end{align*}$$

$$\begin{align*}Q_{\mathscr{X}}= \left[ \begin{array}{@{}crrrr@{}} 1 & 1 & 2 & 2 & 2 \\ 1 & 1 & -2 & -2 & 2 \\ 1 & 1 & -2i & 2i & -2 \\ 1 & 1 & 2i & -2i & -2 \\ 1 & \!\!-1 & 0 & 0 & 0 \end{array} \right], \ \overline{Q}_{\mathscr{X}}= \left[ \begin{array}{@{}rrrr@{}} 1 & 1 & 4 & 2 \\ 1 & 1 & \!-4 & 2 \\ 1 & 1 & 0 & \!-2 \\ 1 & 1 & 0 & \!-2 \\ 1 & \!-1 & 0 & 0 \end{array} \right], \ Q_{\mathscr{X}_{\downarrow_{\mathbb{Q}}}}= \left[ \begin{array}{@{}rrrr@{}} 1& 1& 2& 4 \\ 1& \!\!-1& 0& 0 \\ 1& 1& \!\!-2& 0 \\ 1& 1& 2& \!\!-4 \end{array}\right]. \end{align*}$$

The splitting field of $\mathscr {X}$ is $\mathbb {Q}(i)$ and the permutation action of ${\mathsf {Gal}}(\mathbb {Q}(i)/\mathbb {Q})$ on the idempotents gives $\langle (2,3) \rangle $ . Hence, $F_0 = E_0$ , $F_1 = E_1$ , $F_2 = E_2 + E_3$ , $F_3 = E_4$ , and $\iota : \{0, 1, 2, 3, 4\} \to \{0, 1, 2, 3\}$ by $\iota (0) = 0$ , $\iota (1) = 1$ , $\iota (2) = 2$ , $\iota (3) = 2,$ and $\iota (4) = 3$ . This is described by $\overline {Q}$ , where columns 2 and 3 of Q have been replaced by their sum. Repeated rows indicate that relations 2 and 3 fuse, and by the Bannai–Muzychuk criterion, we have a valid fusion $\mathscr {X}_{\downarrow _{\mathbb {Q}}}$ with second eigenmatrix $Q_{\mathscr {X}_{\downarrow _{\mathbb {Q}}}}$ .

Note that not every scheme has property $\Phi _{\mathbb {Q}}$ as the following example shows.

Example 3.10 The Coxeter graph (considered as a metric association scheme) has second eigenmatrix Q, below, and corresponding matrix $\overline {Q}$ :

$$\begin{align*}Q = \begin{bmatrix} 1 & 8 & 6 & 7 & 6 \\ 1 & 16/3 & -2+2\sqrt{2} & -7/3 & -2-2\sqrt{2} \\ 1 & 4/3 & -2\sqrt{2} & -7/3 & 2\sqrt{2} \\ 1 & -4/3 & -1 & 7/3 & -1\\ 1 & -8/3 & 2+\sqrt{2} & -7/3 & 2-\sqrt{2} \end{bmatrix}, \quad \overline{Q} = \begin{bmatrix} 1 & 8 & 12 & 7 \\ 1 & 16/3 & -4 & -7/3 \\ 1 & \phantom{-}4/3 & \phantom{-}0 & -7/3 \\ 1 & -4/3 & -2 & \phantom{-}7/3 \\ 1 & -8/3 & \phantom{-}4 & -7/3 \end{bmatrix}. \end{align*}$$

That is, we have summed the columns of Q according to the Galois orbits to obtain $\overline {Q}$ , which clearly fails the Bannai–Muzychuk criterion. Hence, the Coxeter graph does not satisfy property $\Phi _{\mathbb {Q}}$ . This means that the Coxeter graph does not have a Galois fusion over any proper subfield.

Problem 3.11 For any association scheme $\mathscr {X}$ , when does Property $\Phi _{\mathbb {K}}$ hold for $\mathbb {Q} \subseteq \mathbb {K} \subseteq \mathbb {F}$ ? In particular, when does Property $\Phi _{\mathbb {Q}}$ hold? That is, what are the Galois fusions of $\mathscr {X}$ with respect to subfields of $\mathbb {F}$ ?

4 Delsarte designs and irrational eigenvalues

We continue with the notation used in Sections 2 and 3.

For $T\subseteq \{1,\ldots ,d\},$ a subset $\emptyset \subset C \subseteq X$ is a Delsarte T-design if the characteristic vector ${\mathbf x}={\mathbf x}_C$ of C (with ${\mathbf x}_a=1$ if $a\in C$ and ${\mathbf x}_a=0$ otherwise) satisfies $E_j {\mathbf x} = 0$ for all $j\in T$ . Given C, the largest set of indices for which C is a design is $T(C)=\{ j \mid E_j{\mathbf x}=0\}$ . The inner distribution $a=a(C)=[a_0,\ldots ,a_d]$ and dual distribution (or MacWilliams transform) $b=b(C)=[b_0,\ldots ,b_d]$ defined by

$$ \begin{align*}\hfill a_i = \frac{1}{|C|} {\mathbf x}^\top A_i {\mathbf x} \hspace{1in} b_j = \frac{|X|}{|C|} {\mathbf x}^\top E_j {\mathbf x} \hfill\end{align*} $$

are related by $b=aQ$ , and we know that $a\ge 0$ and $b\ge 0$ (see, e.g., [Reference Brouwer, Cohen and Neumaier6, Section 2.5]). Evidently, $j\in T(C) \Leftrightarrow b_j=0$ .

We may also consider weighted Delsarte T-designs where each vertex in C is assigned a weight in $\mathbb {Q}$ ; we call such a function on the vertex set a rational Delsarte design. If the weights are in $\mathbb {Z}_+$ then they may be considered as “repeated points” or as a multiset. Any rationally weighted Delsarte T-design corresponds to a vector ${\mathbf x}$ with rational entries. As for non-weighted Delsarte T-designs, we have the same properties for $T(C)$ and the MacWilliams transform, after suitably adjusting the inner distribution to

$$\begin{align*}a_i = \frac{{\mathbf x}^\top A_i {\mathbf x}}{{\mathbf x}^\top {\mathbf x}}. \end{align*}$$

We are typically interested in 01-characteristic vectors. However, even when considering weighted Delsarte designs, the characteristic vector has rational entries. This means that we have ${\mathbf x}_C = \sum _{j \not \in T(C)} E_j {\mathbf x}_C$ in $\mathbb {Q}^{X}$ . Hence, if $F_0, \ldots , F_e$ are the minimal elements of $\mathcal {E}_{\mathbb {Q}}$ , then ${\mathbf x}_C$ must be expressible as ${\mathbf x}_C = \sum _{j \in S} F_j{\mathbf x}_C$ for some $S \subseteq \{0, \ldots , e\}$ . We will develop this idea in what follows, so we restrict ourselves to $\mathbb {K}=\mathbb {Q}$ . We note, however, that the results generalize to other choices of $\mathbb {K}$ .

Proof (i) For $\boldsymbol {\sigma } \in {\mathsf {Gal}}(\mathbb {F}/\mathbb {Q})$ and corresponding $\sigma \in \Sigma _{\mathbb {Q}} \subseteq \operatorname {\mathrm {Sym}}(\{0,\ldots ,d\})$ , we have

$$ \begin{align*}(aQ)^{\boldsymbol{\sigma}}= a(Q)^{\boldsymbol{\sigma}}= [b_0,b_{1^\sigma},\ldots,b_{d^\sigma}]\end{align*} $$

so that $b_j=0$ if and only if $b_{j'}=0$ for all $j'$ in the orbit ${\mathcal Q}_\ell $ containing j. Parts (ii) and (iii) follow immediately.

Theorem 4.1 has important implications in practice, since it means that the study of Delsarte designs in an association scheme with irrational eigenvalues is essentially equivalent to the study of subsets satisfying $F_j {\mathbf x}=0$ for all j in some reduced set ${T' \subseteq \{1,\ldots ,e\}}$ . In many cases, this greatly reduces the complexity of a search, for example, when using linear programming.

We have a stronger result than Theorem 4.1 when property $\Phi _{\mathbb {Q}}$ holds.

This says in essence that it is sufficient to consider the T-designs of the $\mathscr {X}_{\downarrow _{\mathbb {Q}}}$ when studying T-designs of $\mathscr {X}$ . This is helpful in particular when showing non-existence or when classifying T-designs. Note, however, that they may have different combinatorial properties. This is essentially used in an ad hoc manner in [Reference Bamberg, Giudici, Lansdown and Royle2].

Theorem 4.3 establishes a correspondence between the designs of $\mathscr {X}$ and those of $\mathscr {X}_{\downarrow _{\mathbb {Q}}}$ , which has curious implications for pairs of non-isomorphic schemes having isomorphic Galois fusions. We call $\mathscr {X} =(X, \mathcal {R})$ isomorphic to $\mathscr {Y}=(Y, \mathcal {S})$ , ${\mathcal S}=\{S_0,\ldots ,S_e\}$ , if there is a bijection $\sigma _1$ from X to Y and a bijection $\sigma _2$ from $\{0, \ldots , d\}$ to $\{0, \ldots , e\}$ such that $\{\left (\sigma _1(x), \sigma _1(y)\right ) \mid (x,y) \in R_i\} = S_{\sigma _2(i)}$ in which case we write $\mathscr {X\kern3pt}^{(\sigma _1, \sigma _2)} = \mathscr {Y}$ . For some $T \subseteq \{1, \ldots , d\},$ we define $\mathfrak{D}_T(\mathscr {X\kern3pt}) = \{ C \subset X : C \text { is a} T\text {-Design of } \mathscr {X\kern3pt}\}$ , that is, $\mathfrak{D}_T(\mathscr {X\kern3pt})$ is the collection of all T-designs of $\mathscr {X}$ .

Example 4.6 Let $\mathscr {X}$ be as in Example 3.9 and let $\mathscr {Y\kern3pt}$ be the association scheme on the same vertex set $\mathbb {Z}_4\times \mathbb {Z}_2$ defined by the following relation matrix $\sum i A_i$ and having second eigenmatrix $Q_{\mathscr {Y}}$ as follows:

$$\begin{align*}\mathscr{Y}: \begin{bmatrix} 0 & 1 & 2 & 2 & 3 & 3 & 4 & 4 \\ 1 & 0 & 2 & 2 & 3 & 3 & 4 & 4 \\ 2 & 2 & 0 & 1 & 4 & 4 & 3 & 3 \\ 2 & 2 & 1 & 0 & 4 & 4 & 3 & 3 \\ 4 & 4 & 3 & 3 & 0 & 1 & 2 & 2 \\ 4 & 4 & 3 & 3 & 1 & 0 & 2 & 2 \\ 3 & 3 & 4 & 4 & 2 & 2 & 0 & 1 \\ 3 & 3 & 4 & 4 & 2 & 2 & 1 & 0 \end{bmatrix}, \quad Q_{\mathscr{Y}}= \left[ \begin{array}{rrrrr} 1 & 1 & 1 & 1 & 4 \\ 1 & 1 & 1 & 1 & \!\!-4 \\ 1 & \!\!-1 & \!\!-1 & 1 & 0 \\ 1 & \!\!-i & i & \!\!-1 & 0 \\ 1 & i & \!\!-i & \!\!-1 & 0 \end{array} \right]. \end{align*}$$

Then, $\mathscr {X}$ and $\mathscr {Y}$ are non-isomorphic but $\mathscr {X}_{\downarrow _{\mathbb {Q}}} = \mathscr {Y}_{\downarrow _{\mathbb {Q}}}$ . Now, $\iota _{\mathscr {Y}}:\{0,1,2,3,4\} \to \{0,1,2,3\}$ by $\iota _{\mathscr {Y\kern1.5pt}}(0)=0$ , $\iota _{\mathscr {Y\kern1.5pt}}(1)=1$ , $\iota _{\mathscr {Y\kern1.5pt}}(2)=1$ , $\iota _{\mathscr {Y\kern1.5pt}}(3)=2$ , and $\iota _{\mathscr {Y\kern1.5pt}}(4)=3$ . Consider C from Example 4.4, which is a $\{1,2,3\}$ -design in $\mathscr {X}$ and a $\{1, 2\}$ -design in $\mathscr {X}_{\downarrow _{\mathbb {Q}}} = \mathscr {Y}_{\downarrow _{\mathbb {Q}}}$ . So C is a $\iota _{\mathscr {Y}}^{-1}(1) \cup \iota _{\mathscr {Y}}^{-1}(2) = \{3, 4\}$ -design of $\mathscr {Y}$ . Note that $\mathscr {X}_{\downarrow _{\mathbb {Q}}}$ and $\mathscr {Y}_{\downarrow _{\mathbb {Q}}}$ are equal and not just isomorphic, so no permutation of the vertices needs to be applied to C for it to be a design of $\mathscr {Y}$ . It is not hard for the interested reader to completely enumerate all T-designs for both $\mathscr {X}$ and $\mathscr {Y\kern3pt}$ by computer. Indeed, in light of Theorem 4.5, one need only enumerate them for $\mathscr {X}_{\downarrow _{\mathbb {Q}}}$ .

5 Conjugacy class association schemes

We begin this section with a brisk review of the general theory of conjugacy class association schemes. These are well-studied and serve as fundamental examples in some standard texts; see [Reference Bannai and Ito4, p. 54], [Reference Godsil13, pp. 230–231, p. 264], [Reference Bannai, Bannai, Ito and Tanaka5, p. 51], but also [Reference Bannai and Ito4, pp. 19–21] and [Reference Brouwer, Cohen and Neumaier6, Section 2.9].

Let G be a finite group with decomposition $G = \mathcal {C}_0\cup \mathcal {C}_1 \cup \cdots \cup \mathcal {C}_d$ into conjugacy classes where $\mathcal {C}_0=\{1\}$ . We build matrices in $\mathsf {Mat}_G(\mathbb {C})$ as follows. For $0\le i\le d$ , define $A_i$ , the $i^{\mathrm {th}}$ adjacency matrix, as the 01-matrix in $\mathsf {Mat}_G(\mathbb {C})$ having a one in position $(g,h)$ if $g^{-1}h\in \mathcal {C}_i$ . This is the adjacency matrix of the digraph with vertex set G and arc set $R_i = \left \{ (g,ga) \middle | g\in G, \ a\in \mathcal {C}_i \right \}$ . The Bose–Mesner algebra $\mathbb {A}$ , defined above as the vector space span of $\{A_0,\ldots ,A_d\}$ , is also the center of the group algebra $\mathbb {C} [\hat {G}],$ where $\hat {G}$ is the group of $v=|G|$ permutation matrices in $\mathsf {Mat}_G(\mathbb {C})$ given by the right regular representation $g\cdot x=xg^{-1}$ ; the permutation matrix $P_g \in \hat {G}$ has a one in position $(x,xg^{-1})$ for $x\in G$ and $P_g {\mathbf e}_x = {\mathbf e}_{xg}$ , where ${\mathbf e}_{x} \in \mathbb {C}^G$ is the standard basis vector corresponding to $x\in G$ .

We have a partition ${\mathcal R}=\{R_0,\ldots ,R_d\}$ of $G\times G$ into binary relations. It is well-known that $(G,{\mathcal R})$ is a commutative association scheme [Reference Bannai and Ito4, p. 20]; the scheme is symmetric precisely when all conjugacy classes are inverse-closed. Each normal subgroup of G determines an imprimitivity system of this association scheme; if ${N \unlhd G}$ , then N is expressible as a union of conjugacy classes, $\mathcal {C}_0\cup \cdots \cup \mathcal {C}_s$ say, and $(X,R_1\cup \cdots \cup R_s)$ is a disjoint union of $|G:N|$ complete graphs of size $|N|$ . Moreover, every imprimitivity system of a conjugacy class scheme arises in this way: the component of the disconnected graph containing the identity is a union of conjugacy classes closed under multiplication.

There are exactly $d+1$ irreducible characters of $G,$ including the trivial character $\chi _0:G \rightarrow \{1\}$ . The character table ${\mathsf {T}}$ of G has one row for each irreducible character and one column for each conjugacy class; in row j, column i, the value is $\chi _j(g)$ for $g\in \mathcal {C}_i$ . We denote the basis of primitive idempotents of the Bose–Mesner algebra $\mathbb {A}$ by $\{E_0,\ldots ,E_d\}$ and define the first and second eigenmatrices P and Q by the equations (2.2) as usual.

Though it is well-known, we will show below using eigenvectors of the $A_i$ that for the conjugacy class scheme of a finite group G, we have $Q_{ij} = \chi _j(1) \, \overline {\chi _j(g)}$ and ${P_{ji} = |\mathcal {C}_i|\chi _j(g)/\chi _j(1)}$ ( $g\in \mathcal {C}_i$ ).Footnote ⁴ Denoting the corresponding irreducible representations by $\rho _0,\ldots ,\rho _d$ , we have $\chi _j(g)=\mathsf {tr} \rho _j(g)$ and the right regular representation $\rho :G\rightarrow \hat {G}$ can be block diagonalized as a direct sum of irreducible representations [Reference Dummit and Foote9, Theorem 18.10]; in the notation of Bannai and Ito [Reference Bannai and Ito4, p. 17], we have

$$ \begin{align*}\rho \sim \rho_0 + f_1 \rho_1 + \cdots + f_d \rho_d,\end{align*} $$

where, in the block diagonalization of the right regular representation $\rho $ , the $j^{\mathrm {th}}$ irreducible representation $\rho _j$ appears with multiplicity equal to its degree $f_j$ .

Our final goal in this section is to diagonalize the Bose–Mesner algebra of a conjugacy class association scheme.

For $0\le j\le d$ , fix a choice of representation $\rho _j:G\rightarrow \mathsf {Mat}_{f_j}(\mathbb {C})$ with $\mathsf {tr} \rho _j(g)=\chi _j(g)$ for each $g\in G$ . With $f=f_j$ , consider the bijection $\upsilon :[f]\times [f] \rightarrow [f^2]$ , vectorizing each matrix $R=[r_{ij}] \in \mathsf {Mat}_f(\mathbb {C})$ into $[r_{11},r_{12},\ldots ,r_{ff}]$ . Construct a $|G|\times f^2$ matrix $U_j$ with row g equal to $u_g:=\upsilon \left ( \rho (g)\right )$ .

Proof Let j be given, $0\le j\le d$ , and write simply $U=U_j$ as constructed above and write $\rho=\rho_j$ . We will show that, for each i, $0\le i\le d$ ,

$$ \begin{align*}A_i U = \frac{ |\mathcal{C}_i| \chi_j(g_i)}{f_j} U,\end{align*} $$

where $g_i\in \mathcal {C}_i$ . Let $C = \sum _{x\in \mathcal {C}_i} \rho (x)$ . Then, for any $g\in G$ , $\rho (g)^{-1} C\rho (g) = C$ so that $C\rho (g) = \rho (g) C$ . By Schur’s Lemma [Reference Serre23, Proposition 4, 2.2], [Reference Ledermann16, Theorem 1.4], $C=\theta I_{f_j}$ for some $\theta $ . Thus, $\sum _{x\in \mathcal {C}_i} \rho (gx)= \rho (g) C = \theta \rho (g)$ , and the row g of $A_iU$  is

$$\begin{align*}\sum_{x\in \mathcal{C}_i} u_{gx} =\upsilon\left( \sum_{x\in \mathcal{C}_i} \rho({gx}) \right) = \theta u_g. \end{align*}$$

In other words, $A_iU=\theta U$ for some $\theta \in \mathbb {C}$ . Taking the trace of both sides of the equation $\sum _{x\in \mathcal {C}_i} \rho (x) = \theta I$ , we find $ |\mathcal {C}_i| \chi _j(g_i) = \theta f_j$ .

By Wedderburn’s structure theorem for semisimple algebras (see also [Reference Serre23, p. 48]), the image $\{ \rho (g) \mid g\in G\}$ is full-dimensional in $\mathsf {Mat}_f(\mathbb {C})$ . In fact, extending $\rho $ linearly to the full group algebra, we have that $\rho : \mathbb {C}[G] \rightarrow \mathsf {Mat}_{f_j}(\mathbb {C})$ is surjective. So the $f_j^2$ columns of U are linearly independent. Distinct irreducible representations $\rho _j,\rho _k$ of G correspond to orthogonal columns of the matrix Q, so these $f_j^2$ vectors are orthogonal to the eigenvectors obtained in this same manner from the remaining d irreducible representations of G. So we have $\sum _{j=0}^d f_j^2$ linearly independent eigenvectors for the Bose–Mesner algebra. We then use the standard identity $\sum _{j=0}^d f_j^2=|G|$ (see, e.g., [Reference Dummit and Foote9, Theorem 18.10(3)]) to finish the proof.

The first and second eigenmatrices of the conjugacy class association scheme $P=[P_{ji}]$ and $Q=[Q_{ij}]$ defined by (2.2) above satisfy the standard first and second orthogonality relations for association schemes (2.3), where $v_i=|\mathcal {C}_i|$ , but in this case, of course, these reduce to the classical orthogonality relations for group characters [Reference Bannai and Ito4, Theorems I.4.4 and I.4.9], [Reference Serre23, Section 2.3]. Our construction of the matrix U above shows $m_j=f_j^2$ . So the second orthogonality relation gives us $Q_{ij} = f_j \overline {\chi _j(x_i)}$ . In summary (cf. [Reference Bannai and Ito4, Theorem II.7.2], with $g_i$ a representative of $\mathcal {C}_i$ ,

(5.1) $$ \begin{align} P_{ji} = \frac{1}{f_j} |\mathcal{C}_i| \chi_j(g_i), \hspace{1in} Q_{ij} = f_j \ \overline{\chi_j(g_i)}. \end{align} $$

The Krein parameters of the conjugacy class scheme are closely related to the decomposition of Kronecker products of irreducible representations into irreducibles. If $f_i$ denotes the degree of $\rho _i$ and

$$\begin{align*}\rho_i \otimes \rho_j \sim r_{ij}^0 \rho_0 + \cdots + r_{ij}^d \rho_d , \end{align*}$$

then $\chi _i \circ \chi _j = \sum _{k=0}^d r_{ij}^k \chi _k$ and, after scaling each column by its degree and using [Reference Brouwer, Cohen and Neumaier6, Lemma 2.3.1(vii)], we find $q_{ij}^k = \frac {f_if_j}{f_k} r_{ij}^k$ .

The conjugacy classes of size one are precisely $\left \{ \{a\} \middle | a\in Z(G) \right \}$ and the eigenspaces of dimension one correspond to the elements of the abelianization ${G'=G/[G,G]}$  of G.

The intersection numbers $p_{ij}^k$ are given by

$$ \begin{align*}p_{ij}^k = \left| \mathcal{C}_i \cap z\mathcal{C}_j^{-1} \right|,\end{align*} $$

where $z \in \mathcal {C}_k$ and $\mathcal {C}_j^{-1} = \{ h^{-1} \mid h\in \mathcal {C}_j\}$ . To see this, consider any $x,y\in G$ with ${x^{-1}y\in \mathcal {C}_k}$ ; find $g\in G$ with $g^{-1} x^{-1} y g = z$ so that $xg^{-1} \left ( \mathcal {C}_i \cap R_{j'}(z) \right ) g = R_i(x) \cap R_{j'}(y)$ .

5.1 Galois group and rational fusions

From here forward, let $\zeta _\ell $ denote a primitive (complex) $\ell ^{\mathrm {th}}$ root of unity.

The splitting field of the finite group G is defined to be the extension of $\mathbb {Q}$ by the entries of its character table. This then agrees with the concept of the splitting field [Reference Munemasa19] of the conjugacy class association scheme by (5.1). Let $\mathbb {F}$ denote the splitting field of the conjugacy class scheme of G. As discussed in Section 3.1, each $\boldsymbol {\sigma }\in {\mathsf {Gal}}(\mathbb {F}/\mathbb {Q})$ acts on the set of primitive idempotents $\{E_0,\ldots ,E_d\}$ ; to $\boldsymbol {\sigma }$ , associate the permutation $\sigma \in \Sigma _{\mathbb {Q}}$ defined by $E_{j^\sigma }=\left ( E_j\right )^{\boldsymbol {\sigma }}$ , where the action of $\boldsymbol {\sigma }$ on matrices is entrywise.

Recall that $x,y\in G$ are rationally conjugate if $g\langle x\rangle g^{-1} = \langle y \rangle $ for some $g\in G$ . Clearly, if x and y are conjugate in G, then x and y are rationally conjugate; any equivalence class $\mathcal {C}$ of this “rationally conjugate” relation is a union of conjugacy classes of G and so $R_{\mathcal {C}}=\{ (x,y) \mid x^{-1}y\in \mathcal {C} \}$ is expressible as a union of basis relations of the conjugacy class scheme.

The exponent of finite group G, i.e., the smallest positive integer k such that $g^k=1$ for all $g\in G$ , divides $n=|G|$ . If G has exponent k, then, in any representation $\rho :G\rightarrow {\mathsf {GL}}_m(\mathbb {C})$ , $\rho (g)^k=I$ for each $g\in G$ , and so the eigenvalues of $\rho (g)$ are $k^{\mathrm {th}}$ roots of unity. Hence, the trace $\chi (g) =\mathsf {tr} \rho (g)$ belongs to the cyclotomic extension $\mathbb {Q}( \zeta _k)$ , and this is contained in $\mathbb {Q}( \zeta _n)$ . In some cases (e.g., whenever all conjugacy classes are inverse-closed and the eigenvalues are all real), the splitting field of G can be a proper subfield of $\mathbb {Q}( \zeta _k)$ . Motivated by this important class of examples, Simon Norton asked whether the splitting field of any commutative association scheme is contained in some cyclotomic extension of the rational numbers [Reference Bannai and Ito4, p. 183].

While we choose here to cite a reference dealing directly with Schur rings, the connection between rational characters and rational conjugacy classes of G is well studied (indeed, this connection is the origin of the term “rational conjugacy class”). For instance, the fact that there are equally many rational conjugacy classes of G as orbits of the full Galois group can be found in Serre [Reference Serre23, Chapter 13].

Theorem 5.2 (Muzychuk et al. [Reference Muzychuk, Ponomarenko and Chen22])

The conjugacy class association scheme $(G,{\mathcal R})$ of any finite group G satisfies Property $\Phi _{\mathbb {Q}}$ . The basis relations of the corresponding fusion scheme are the relations $R_{\mathcal {C}}$ as $\mathcal {C}$ ranges over the rational conjugacy classes of G as defined above.

Proof An idempotent $\sum _{j=0}^d c_jE_j$ belongs to ${\mathcal E}_{\mathbb {Q}}$ if and only if the corresponding character $\psi (g) = \sum _{j=0}^d c_j \chi _j(g)$ is rational for each $g\in G$ .

As shown above in our discussion of the exponent, the splitting field $\mathbb {F}$ is contained in a cyclotomic extension $\mathbb {Q}(\zeta _n)$ and, for each m with $\gcd (m,n)=1$ , there is a Galois automorphism $\boldsymbol {\sigma }\in {\mathsf {Gal}}(\mathbb {Q}(\zeta _n)/\mathbb {Q})$ with $\zeta _n^{\boldsymbol {\sigma }}=\zeta _n^m$ .

Now suppose $\langle x\rangle = \langle y\rangle $ . Then, $y=x^m$ for some m with $\gcd (m,n)=1$ . If $\psi $ is a rational character of G, say $\psi (g) = \mathsf {tr} \rho (g)$ for some complex representation $\rho :G \rightarrow {\mathsf {GL}}_f(\mathbb {C})$ , then $\psi (x) = \mathsf {tr} \rho (x) = \sum _{j=1}^f \omega _j$ for some $n^{\mathrm {th}}$ roots of unity $\omega _1,\ldots ,\omega _f$ . We then have

$$ \begin{align*}\psi(y) = \mathsf{tr} \, \rho\left( x^m \right) = \mathsf{tr} \, \rho\left( x\right)^m = \sum_{j=1}^f \omega_j^m =\sum_{j=1}^f \omega_j^{\boldsymbol{\sigma}} =\left( \sum_{j=1}^f \omega_j \right)^{\boldsymbol{\sigma}} = \psi(x)^{\boldsymbol{\sigma}} = \psi(x),\end{align*} $$

since $\psi $ is rational. Now, if y is instead conjugate to some $x^m$ with $\langle x^m\rangle = \langle x \rangle $ , then $\psi (y) = \psi \left (x^m\right )=\psi (x)$ . This tells us that the adjacency matrix of the relation $R_{\mathcal {C}}$ , $A=\displaystyle {\sum _{\mathcal {C}_i\subseteq \mathcal {C}}} A_i$ , satisfies $A\circ E= \alpha _E A$ (some $\alpha _E$ ) for each $E\in {\mathcal E}_{\mathbb {Q}}$ , so A belongs to the Schur closure of $\mathbb {C}[{\mathcal E}_{\mathbb {Q}}]$ . Equivalently, collapsing over orbits of ${\mathsf {Gal}}(\mathbb {F}/\mathbb {Q})$ , we find that the rows i and j of $\overline {Q} =QO$ are equal whenever $\mathcal {C}_i$ and $\mathcal {C}_j$ are contained in the same rational conjugacy class. So the number $e+1$ of Galois orbits ${\mathcal Q}_j$ is at most the number of rational conjugacy classes. Going in the other direction, the adjacency matrix A of any relation $R_{\mathcal {C}}$ belongs to the full Bose–Mesner algebra and has only rational eigenvalues, so A belongs to $\mathbb {Q}[{\mathcal E}_{\mathbb {Q}}]$ .

Example 5.3 The “gcd scheme” for $\mathbb {Z}_n$ described in Example 3.4 is precisely the rational fusion of the conjugacy class scheme of $\mathbb {Z}_n$ .

Example 5.4 The alternating group $A_4$ has character table and eigenmatrices

$$ \begin{align*}{\mathsf{T}}=\left[ \begin{array}{crcc} 1 & 1 & 1 & 1 \\ 1 & 1 & \zeta\ & \zeta^2 \\ 1 & 1 & \zeta^2 & \zeta\ \\ 3 &\!\!-1 & 0 & 0 \end{array} \right], \quad Q =\left[ \begin{array}{cccr} 1 & 1 & 1 & 9 \\ 1 & 1 & 1 & \!\!-3 \\ 1 & \zeta & \zeta^2 & 0 \\ 1 & \zeta^2 & \zeta & 0 \end{array} \right], \quad P =\left[ \begin{array}{crcc} 1 & 3 & 4 & 4 \\ 1 & 3 & 4 \zeta & 4\zeta^2 \\ 1 & 3 & 4 \zeta^2 & 4\zeta \\ 1 &\!\!-1 & 0 & 0 \end{array} \right],\end{align*} $$

where $\zeta $ is a primitive cube root of unity. We fuse the conjugate pair of eigenspaces to find $\bar {Q} = \left [ \begin {array}{crr} 1 & 2 & 9 \\ 1 & 2 & \!\!-3 \\ 1 &\!\!-1& 0 \end {array} \right ]$ and $\bar {P} = \left [ \begin {array}{crr} 1 & 3 & 8 \\ 1 & 3 & \!\!-4 \\ 1 &\!\!-1& 0 \end {array} \right ]$ ; the Galois fusion is the association scheme of the complete multipartite graph $\overline {3K_4}$ .

Observe that, as a consequence, Theorems 4.3 and 4.5 are always applicable for conjugacy class schemes.

Problem 5.5 Does there exist a finite group G whose rational fusion is a primitive 2-class association scheme? Note that some graphs with Paley parameters, e.g., $\mathsf {Paley}(q^2)$ but also $\mathrm {srg}(225,112,55,56)$ , have rational eigenvalues.

Problem 5.6 Suppose $\mathbb {E}$ and $\mathbb {K}$ are subfields of the splitting field of the conjugacy class scheme $(G,{\mathcal R})$ . Is there a simple relationship between $\mathbb {E}$ and $\mathbb {K}$ such that, whenever $(G,{\mathcal R})$ satisfies Property $\Phi _{\mathbb {E}}$ , it must also satisfy Property $\Phi _{\mathbb {K}}$ ?

6 Case study: Dicyclic groups

The dicyclic group ${\mathsf {Dic}}_n$ is the group of order $4n$ with presentation

$$ \begin{align*}{\mathsf{Dic}}_n = \left\langle \ x,y \ \middle| \ x^{2n}=1, \ y^2=x^n, \ y^{-1}xy=x^{-1} \ \right\rangle.\end{align*} $$

For n even, this group has the same character table as the dihedral group $D_{2n}$ of the same order. In fact, the two association schemes are isomorphic. Since the dihedral case is studied elsewhere, we will focus on the case where n is odd.

Multiplication and conjugation (using $y^{-1}=yx^n$ ) are given by

$$ \begin{align*} \begin{array}{l|ll} g\cdot h & \phantom{y}x^\ell & yx^\ell \\ \hline \phantom{y}x^k & \phantom{y}x^{k+\ell} & yx^{\ell-k} \\ {y}x^k & {y}x^{k+\ell} & \phantom{y}x^{n+\ell-k} \end{array}\qquad\qquad \begin{array}{l|ll} h^g & \phantom{y}x^\ell & yx^\ell \\ \hline \phantom{y}x^k & \phantom{y}x^{\ell} & yx^{\ell+2k} \\ {y}x^k & \phantom{y}x^{-\ell} & {y}x^{2k-\ell} \end{array}. \hfill \phantom{.} \end{align*} $$

The $n+2$ nontrivial conjugacy classes are $\mathcal {C}_k=\{x^k,x^{2n-k} \}$ , $1\le k< n$ , $\mathcal {C}_n=\{x^n\}$ ,

$$ \begin{align*}\mathcal{C}_{n+1} = \{ yx^{2k} : k=0,\ldots,n-1 \}, \quad \mathcal{C}_{n+2} = \{ yx^{2k+1} : k=0,\ldots,n-1 \},\end{align*} $$

where $\mathcal {C}_{n+2} = \mathcal {C}_{n+1}^{(-1)}$ since n is assumed odd.

The conjugacy class association scheme contains two vertex-disjoint subschemes (or induced schemes) isomorphic to the association scheme of the $2n$ -gon: one induced on the cyclic subgroup $\langle x\rangle $ and the other on $y\langle x\rangle $ . In both the association scheme of the dihedral group $D_{2n}$ and this association scheme, the remaining pairs are partitioned into two relations $R_{n+1}\cup R_{n+2} = \left ( \langle x\rangle \times y\langle x\rangle \right ) \cup \left ( y\langle x\rangle \times \langle x\rangle \right )$ . Whereas, in the dihedral case, these two relations are both isomorphic to two copies of the complete bipartite graph $K_{n,n}$ , in the dicyclic case, the relations $R_{n+1}$ and $R_{n+2}$ are orientations of $K_{2n,2n}$ . More precisely, pairs in $R_{n+1}$ incident to $x^k \in {\mathsf {Dic}}_n$ are $(x^k,yx^{k+2\ell })$ and $(yx^{n-k+2\ell },x^k)$ ( $0\le \ell < n$ ).

The (transpose of the) eigenmatrix for the association scheme of the cycle $C_{2n}$ appears as a submatrix of our character table. The splitting field of $C_{2n}$ is well-known to be the real part of a cyclotomic extension, $\mathbb {Q}(\zeta _{2n}) \cap \mathbb {R}$ , and we extend this by the square root of $-1$ to get our splitting field $\mathbb {F} = \left ( \mathbb {Q}(\zeta _{2n}) \cap \mathbb {R} \right ) (i) = \mathbb {Q}(\zeta _{2n}+\zeta _{2n}^{-1}, i)$ .

The cyclotomic extension $\mathbb {Q}(\zeta _m)$ has Galois group $\left ( \mathbb {Z}_m\right )^\times $ [Reference Dummit and Foote9, Section 14.6]; the Galois automorphism corresponding to $k\in \mathbb {Z}_m$ with $\gcd (k,m)=1$ is given by ${\zeta _m\mapsto \zeta _m^k}$ . (See also [Reference Conrad7] for a nice exposition.) We have Galois group ${\mathsf {Gal}}(\mathbb {F}/\mathbb {Q})$ where our splitting field $\mathbb {F}$ is a subfield of $\mathbb {Q}(\zeta _{4n})$ whose Galois group is $\left ( \mathbb {Z}_{4n}\right )^\times $ . Since all subgroups are normal, all intermediate fields $\mathbb {Q} \subset \mathbb {K} \subseteq \mathbb {Q}(\zeta _{4n})$ are Galois extensions. So $[\mathbb {F}:\mathbb {Q}]=[\mathbb {Q}(\zeta _{2n}):\mathbb {Q}]= \phi (2n)$ , the Euler totient function. Since n is assumed to be odd, $\mathbb {F}$ is a degree $\phi (n)$ extension of the rationals. By [Reference Dummit and Foote9, Proposition 14.19], ${\mathsf {Gal}}(\mathbb {F}/\mathbb {Q}(i))\cong {\mathsf {Gal}}( \mathbb {Q}(\zeta _{2n}+\zeta _{2n}^{-1})/\mathbb {Q})$ , the Galois group of the splitting field of the $2n$ -cycle, and ${\mathsf {Gal}}(\mathbb {F}/\mathbb {Q}(\zeta _{2n}+\zeta _{2n}^{-1})) \cong \mathbb {Z}_2$ . Moreover, writing $\zeta =\zeta _{4n}$ now, the subgroup of ${\mathsf {Gal}}\left ( \mathbb {Q}(\zeta )/\mathbb {Q} \right )$ that stabilizes $\mathbb {F}$ is $N=\langle \zeta \mapsto \zeta ^{2n-1} \rangle $ , since $k=1$ and ${k=2n-1}$ are the only exponents with $\zeta ^{nk}=\zeta ^n$ and $\zeta ^{2k}+\zeta ^{-2k}=\zeta ^2+\zeta ^{-2}$ . We now have ${\mathsf {Gal}}(\mathbb {F}/\mathbb {Q}) \cong \left ( \mathbb {Z}_{4n}\right )^\times / \langle 2n-1\rangle $ . So, for $1\le k\le 4n-1$ with $\gcd (k,4n)=1$ , $\zeta \mapsto \zeta ^k$ and $\zeta \mapsto \zeta ^{2n-k}$ act identically on $\mathbb {F}$ , and we need only consider $0<k<n$ and $3n<k<4n$ or $-n<k<n$ with $\gcd (k,2n)=1$ .

With this notation for elements of ${\mathsf {Gal}}(\mathbb {F}/\mathbb {Q})$ , let us see how $\boldsymbol {\theta }_k:\zeta \mapsto \zeta ^k$ acts on the columns of matrix Q, i.e., on the rows of the character table. The character table for ${\mathsf {Dic}}_n$ takes the following general form. Let $\kappa ^{(r)} = \zeta ^{2r}+\zeta ^{-2r}=2\cos \left ( \frac {\pi r}{n} \right )$ to save space.

$$ \begin{align*} \begin{array}{c|ccccccc|rr} \mathcal{C} & \mathcal{C}_0 & \mathcal{C}_1 & \mathcal{C}_2 & \cdots & \mathcal{C}_k & \cdots & \mathcal{C}_n & \mathcal{C}_{n+1} & \mathcal{C}_{n+2} \\ |\mathcal{C}| & 1 & 2 & 2 & \cdots & 2 & \cdots & 1 & n & n \\ \hline \chi_{0} & 1 & 1 & 1 & \cdots & 1 & \cdots & 1 & 1 & 1 \\ \chi_{1} & 1 & 1 & 1 & \cdots & 1 & \cdots & 1 & -1 & -1 \\ \chi_{2} & 1 & \!\!\!\!-1 & 1 & \cdots & (-1)^k & \cdots & (-1)^n & i & -i \\ \chi_{3} & 1 & \!\!\!\!-1 & 1 & \cdots & (-1)^k & \cdots & (-1)^n & -i & i \\ \hline \psi_{1} & 2 & \kappa^{(1)} & \kappa^{(2)} & \cdots & \kappa^{(k)} & \cdots & \!\!\!\!-2 & 0 & 0 \\ \psi_{2} & 2 & \kappa^{(2)} & \kappa^{(4)} & \cdots & \kappa^{(2k)} & \cdots & 2 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots & \vdots \\ \psi_{n-1} & 2 & \kappa^{(n-1)} & \kappa^{(2n-2)} & \cdots & \kappa^{(nk-k)} & \cdots & (-1)^{n-1}2 & 0 & 0 \\ \end{array} \end{align*} $$

We compute $\boldsymbol {\theta }_k\left ( \kappa ^{(r)}\right ) = \boldsymbol {\theta }_k \left ( \zeta ^{2r}+\zeta ^{-2r}\right )= \zeta ^{2kr}+\zeta ^{-2kr}= \kappa ^{(kr)}$ when $kr<n$ . As this suggests, $\boldsymbol {\theta }_k$ and $\boldsymbol {\theta }_{-k}$ have the same action on $\kappa ^{(1)},\ldots ,\kappa ^{(n-1)}$ , but one fixes $\pm i,$ while the other swaps them, namely, the one among $\pm k$ that is three modulo four.

If, for integers $k,m$ with $m>1$ , we write $k \ {\scriptscriptstyle \mathrm {{MOD}}} \ m = k'$ for that unique $k' \equiv k \pmod {m}$ in the range $-\frac {m}{2} < k' \le \frac {m}{2}$ , then we have

$$ \begin{align*}\boldsymbol{\theta}_k : \psi_\ell \mapsto \psi_{\ell^{\theta_k}} = \psi_{|k\ell \ {\scriptscriptstyle\mathrm{{MOD}}} \ 2n|}~.\end{align*} $$

So each permutation $\theta _k \in \Sigma _{\mathbb {F}}$ defined by $E_\ell ^{\boldsymbol {\theta }_k} = E_{\ell ^{\theta _k}}$ fixes $0$ and $1$ , fixes (resp., swaps) $2$ and $3$ if $k\equiv 1\ \pmod {4}$ (resp., $k\equiv 3\ \pmod {4}$ ), and for $E_{j+3}$ corresponding to $\psi _{j}$ , $\theta _k(j+3)= |kj \ {\scriptscriptstyle \mathrm {{MOD}}} \ 2n|+3$ .

We can simplify things significantly if we merely wish to work out the orbits of $\Sigma _{\mathbb {Q}}$ on $\{0,1,\ldots ,d\}$ . We have orbits ${\mathcal Q}_0=\{0\}$ , ${\mathcal Q}_1=\{1\}$ , and ${\mathcal Q}_2=\{2,3\}$ and the remaining orbits are precisely the orbits of the Galois group of the $2n$ -cycle acting on $\{ \psi _1,\ldots , \psi _{n-1}\}$ as the transpose of the eigenmatrix of this (self-dual) scheme appears among the first $n+1$ columns of ${\mathsf {T}}$ after deleting repeated rows.

So our rational decomposition tool gives the following result in this case.

Note that, since $\gcd ( \ell ^{\theta _k},2n)$ is divisible by $\gcd (\ell ,2n)$ , the set $\psi _\ell , \psi _{2\ell },\psi _{3\ell },\ldots $ is a union of Galois orbits.

Every subgroup of ${\mathsf {Dic}}_n$ is either cyclic or dicyclic. Since we are restricting to the case where n is odd, k is even if and only if $2n/k$ is odd. For ${H=\langle x^k\rangle }$ , we have $|H|= 2n/k$ and $H \unlhd {\mathsf {Dic}}_n$ with inner distribution of the form ${a = [1,0,\ldots ,0,2,0,\ldots ,0,2,0,\ldots \ldots , 0,0],}$ where $a_0=1$ , $a_i=2$ if $k|i$ and $i<n$ , and $a_n=0$ or $1$ depending on whether k is even or odd, respectively. (The remaining entries of a are zero.) The dual distribution $b=aQ$ has entries naturally indexed by $\chi _0,\chi _1,\chi _2,\chi _3,\psi _1,\ldots ,\psi _{n-1}$ with values

$$ \begin{align*} \begin{array}{|c|cc|cc|c|l|} \hline \ell=2n/k & \chi_0,& \chi_1 & \chi_2,&\chi_3 & \psi_{\ell i} (1\le i \le \lfloor (k-1)/2 \rfloor) & \psi_j (\text{otherwise}) \\ \hline |H|=\ell \text{ even} & \ell & \ell & 0 & 0 & 4\ell & 0 \\ |H|=\ell \ \text{ odd} & \ell & \ell & \ell & \ell & 4\ell & 0 \\ \hline \end{array} \end{align*} $$

So the idempotents outside $T(H)$ are either $\{E_{\chi _0}, E_{\chi _1},E_{\psi _\ell },E_{\psi _{2\ell }},E_{\psi _{3\ell }},\ldots \}$ or $\{E_{\chi _0}, E_{\chi _1}$ , $E_{\chi _2}$ , $E_{\chi _3},E_{\psi _\ell },E_{\psi _{2\ell }},E_{\psi _{3\ell }},\ldots \}$ .

The case where H is dicyclic is almost identical. Here, the cyclic subgroup $H \cap \langle x\rangle $ behaves as above and $H \setminus \langle x\rangle $ contributes either $a_{d-1}=|H|/2=2n/k$ when k is even or $a_{d-1}=a_d=|H|/4=n/k$ when k is odd. In the first case, the first few entries of the MacWilliams transform are $aQ = \left [ \begin {array}{@{}ccccc@{}} \frac {4n}{k} & 0 & 0 & 0 & \cdots \end {array} \right ]$ and, in the second case, $aQ = \left [ \begin {array}{@{}ccccc@{}} \frac {4n}{k} & 0 & \frac {4n}{k} & \frac {4n}{k} & \cdots \end {array} \right ]$ ; in both cases, the entries of $aQ$ indexed by $\psi _1,\ldots ,\psi _{n-1}$ are exactly as in the table above for the cyclic subgroups.