A symplectic fission scheme for the association scheme of rectangular matrices and its automorphisms

1.   Introduction

The concept of the association scheme together with the partially balanced incomplete block designs was defined in its own right by Bose and Shimamoto in 1952 ^[2]. It was introduced to describe the balance relations among the treatments of partially balanced incomplete block designs. Association schemes have close connections with coding theory, graph theory, and finite group theory and, in particular, provide a framework for studying codes and designs. By the 1980s, association scheme theory had become an important branch of algebraic combinatorics, and the research work on association scheme theory had grown tremendously; see ^[1].

The study of association schemes in China was started by Chang and Hsu in the late 1950s. In the mid 1960s, Wan constructed a family of association schemes on Hermitian matrices and computed the parameters of the lower-dimensional ones and started a new direction of construction of association schemes on matrices. The association scheme theory developed later indicates the association schemes of maximal totally isotropic subspaces and of Hermitian matrices are what is known as primitive P-polynomial and Q-polynomial association schemes. In the late 1970s, Wang continued the study of association schemes of matrices. He derived formulas for the parameters of association schemes of Hermitian matrices and construct association schemes using rectangular matrices and alternate matrices. Later, Wan et al., studied the association schemes of symmetric matrices in odd characteristic. In the 1990s, Wang, with his students, studied the association schemes of symmetric matrices and quadratic forms in even characteristic. Besides the parameters of these association schemes, they discussed the subschemes, quotient schemes, and duality and automorphisms^[3,4,6,7]. So, the study of association schemes of matrices reaches a more complete stage. The results on association schemes of matrices are collected in ^[5].

Let $\mathbb{F}_q$ be the finite field with $q$ elements, and $\mathcal{M}_{mn}(\mathbb{F}_q)$ be the set of $m\times n$ matrices over $\mathbb{F}_q$, where $q$ is a power of a prime number and $m\leq n$. For brevity, we write $\mathcal{M}_{mn}(\mathbb{F}_q)$ by $\mathcal{M}_{mn}$. Let $GL_{n}(\mathbb{F}_{q})$ be the general linear group of degree $n$ over $\mathbb{F}_{q}$ and $G_0 = GL_{m}(\mathbb{F}_{q})\times GL_{n}(\mathbb{F}_{q})$(a direct product). The group $G_0$ acts on $\mathcal{M}_{mn}$:

Let $T_0$ be the group of right translation of $\mathcal{M}_{mn}$, and $G$ be the group generated by $G_0$ and $T_0$. Then $G$ acts transitively on $\mathcal{M}_{mn}$, which determines an association scheme $(\mathcal{M}_{mn}, \{R_{i}\}_{0\leq i \leq m})$, where

It is called the association scheme of rectangular matrices and denoted by ${\rm{Mat}}(m\times n, q)$.

Lemma 1.1. ^[5] (ⅰ) When $m = 1$, ${\rm{Mat}}(m\times n, q)$ is a trivial association scheme, and its automorphism group is $Sym(q^n)$.

(ⅱ) When $1 < m\leq n$, each automorphism of the association scheme ${\rm{Mat}}(m\times n, q)$ must have the following form:

where $P\in GL_{m}(\mathbb{F}_{q}), \ Q\in GL_{n}(\mathbb{F}_{q}), \ A\in \mathcal{M}_{mn}$, and $\sigma$ is an automorphism of $\mathbb{F}_{q}$.

In addition, if $m = n$, the following mapping is also an automorphism

where $^{t}X$ is the transpose of $X$.

Next, let $n = 2\nu$. We replace the group $G_0$ with $\overline{G_0} = GL_{m}(\mathbb{F}_{q})\times Sp_{2\nu}(\mathbb{F}_{q}),$ where $Sp_{2\nu}(\mathbb{F}_{q})$ is the symplectic group of degree $2\nu$ over $\mathbb{F}_q$. Then $\overline{G}$, generated by $\overline{G_0}$ and $T_0$, acts transitively on $\mathcal{M}_{mn}$, which determines a fission scheme of ${\rm{Mat}}(m\times n, q)$. We call it the symplectic fission scheme of ${\rm{Mat}}(m\times n, q)$, denoted by ${\rm{SMat}}(m\times n, q)$. In this paper, we discuss the association classes and inner automorphism group of ${\rm{SMat}}(m\times n, q)$. In particular, we determine the intersection numbers and automorphism group for $m = 1$ and $m = 2$.

2.   Preliminaries

2.1.   Definition of association schemes

Definition 2.1. Let $X$ be a nonempty set of cardinality $n$ and $R_0, R_1, \cdots, R_d$ be subsets of $X\times X$ that satisfy the following conditions:

(ⅰ) $R_0 = \{(x, x)|x\in X\}$;

(ⅱ) $X \times X = R_0\cup R_1\cup \cdots \cup R_d, R_i\cap R_j = \emptyset (i\neq j)$;

(ⅲ) for each $i\in \{0, 1, \cdots, d\}$, there exists some $i'\in \{0, 1, \cdots, d\}$ such that $R_i^t = R_{i'}$, where $R_i^t = \{(x, y)|(y, x)\in R_i\}$;

(ⅳ) for any $i, j, k\in \{0, 1, \cdots, d\}$, the number

is constant whenever $(x, y)\in R_k$.

Such a configuration $\mathfrak{X} = (X, \{R_i\}_{0\leq i\leq d})$ is called an association scheme of class $d$ on $X$. $R_0$ is called the trivial or diagonal relation, while the others are called nontrivial relations. Note that $d$ is the number of nontrivial relations. The numbers $p_{ij}^k$ are called the intersection numbers of $\mathfrak{X}$. The association scheme $\mathfrak{X}$ is said to be commutative if

(ⅴ) $p_{ij}^k = p_{ji}^k$ for all $i, j, k\in \{0, 1, ..., d\}$.

Further, $\mathfrak{X}$ is said to be symmetric (or Bose-Mesner type) if

(ⅵ) $i' = i$ for all $i\in \{0, 1, ..., d\}$.

Example 2.1. ^[5] Let $G$ be a finite group acting transitively on a finite set $\Omega$. This induces an action on $\Omega \times \Omega$: for $(x, y)\in \Omega \times \Omega$ and $\sigma \in G$, $(x, y)^{\sigma} = (x^\sigma, y^\sigma)$. Then $G$ no longer acts transitively on $\Omega \times \Omega$ if $|\Omega| = n > 1$. Let $R_0, R_1, \cdots, R_d$ be the orbits of $G$ on $\Omega \times \Omega$, where $R_0 = \{(x, x)|x\in \Omega \}$. Then $\mathfrak{X} = (\Omega, \{R_i\}_{0\leq i\leq d})$ is an association scheme (not necessarily commutative).

Let $\mathfrak{X} = (X, \{R_i\}_{0\leq i\leq d})$ be an association scheme of class $d$ on $X$ and $k_i = p_{ii'}^0$. The number $k_i$ is the number of $y\in X$ such that $(x, y)\in R_i$ for any fixed $x\in X$. It is called the valency of $R_i$. Clearly,

Let $\delta$ be the Kronecker delta: $\delta_{ij} = 0$ for $i\neq j$, and $\delta_{ii} = 1$. Then the following holds:

where $\alpha, \beta, \gamma, \alpha', \beta'\in \{0, 1, \cdots, d\}$, $R_{\alpha'} = \{(x, y)|(y, x)\in R_\alpha\}$, and $R_{\beta'} = \{(x, y)|(y, x)\in R_\beta\}$.

Let $\mathfrak{X} = (X, \{R_i\}_{0\leq i\leq d})$ and $\mathfrak{X'} = (X, \{S_j\}_{0\leq j\leq d'})$ be two association schemes on $X$. If each relation $S_j$ is a union of some $R_i$, then $\mathfrak{X'}$ is said to be a fusion scheme of $\mathfrak{X}$, and $\mathfrak{X}$ is said to be a fission scheme of $\mathfrak{X'}$. Furthermore, let $\mathfrak{Y} = (Y, \{T_i\}_{0\leq k\leq d})$ is an association scheme satisfying $|X| = |Y|$. If a bijection $f: X \rightarrow Y$ induces a permutation $\sigma(f)$ on $\{0, 1, \cdots, d\}$ by $(f(x), f(z))\in T_{i^{\sigma(f)}}$ for $(x, z)\in R_i$, $f$ is called an isomorphism between $\mathfrak{X}$ and $\mathfrak{Y}$. In this case, $\mathfrak{X}$ and $\mathfrak{Y}$ are said to be isomorphic. An isomorphism $f$ from an association scheme $\mathfrak{X}$ to itself is called an automorphism. The set of all automorphisms of $\mathfrak{X}$ becomes a group, called the automorphism group of $\mathfrak{X}$ and denoted by Aut$\mathfrak{X}$. An automorphism $f$ of $\mathfrak{X}$ is called an inner automorphism if it induces the identity permutation on ${0, 1, \cdots, d}$, i.e., $i^{\sigma(f)} = i(i = 0, 1, \cdots, d)$. Clearly, the set of inner automorphisms of $\mathfrak{X}$ becomes a normal subgroup of Aut$\mathfrak{X}$, denoted by Inn$\mathfrak{X}$. The quotient group $\rm{Aut}\mathfrak{X}/\rm{Inn}\mathfrak{X}$ is called the outer automorphism group of $\mathfrak{X}$.

2.2.   Symplectic geometry over the finite field

Let

where $I^{(\nu)}$ is the $\nu\times \nu$ identity matrix. The set of all $2\nu \times 2\nu$ matrices $T$ over $\mathbb{F}_q$ satisfying $TK{^{t}T} = K$ forms a group with respect to the matrix multiplication, called the symplectic group of degree $2\nu$ over $\mathbb{F}_q$, and is denoted by $Sp_{2\nu}(\mathbb{F}_{q})$. A $2\nu\times 2\nu$ matrix $T$ is called a generalized symplectic matrix of degree $2\nu$ over $\mathbb{F}_{q}$ if $TK{^{t}T} = kK$ for some $k\in \mathbb{F}_{q}^{*}$. The set of generalized symplectic matrices of degree $2\nu$ over $\mathbb{F}_{q}$ forms a group with respect to the matrix multiplication, which is called the generalized symplectic group of degree $2\nu$ over $\mathbb{F}_{q}$ and denoted by $GSp_{2\nu}(\mathbb{F}_{q}).$

Let $\mathbb{F}_{q}^{(2\nu)}$ be the $2\nu$-dimensional row vector space over $\mathbb{F}_{q}$. There is a natural action of $Sp_{2\nu}(\mathbb{F}_{q})$ on $\mathbb{F}_{q}^{(2\nu)}$ by the vector matrix multiplication as follows:

The space $\mathbb{F}_{q}^{(2\nu)}$ together with this action is called the $2\nu$-dimensional symplectic space over $\mathbb{F}_{q}$. Suppose that $P$ is an $m$-dimensional vector subspace of $\mathbb{F}_{q}^{(2\nu)}$. We use the same letter $P$ to denote a matrix representation of $P$, i.e., $P$ is an $m\times 2\nu$ matrix whose rows form a basis of $P$. It is clear that a matrix representation of a subspace is not unique. Two $m\times 2\nu$ matrices $P_1$ and $P_2$ of rank $m$ represent the same subspace if and only if there is an $m\times m$ nonsingular matrix $Q$ such that $P_1 = QP_2$. A subspace $P$ is said to be of type $(m, s)$ if the dimension of $P$ is $m$ and rank$(PK{^tP}) = 2s$.

Lemma 2.1. ^[5] Subspaces of type $(m, s)$ exist in $\mathbb{F}_{q}^{(2\nu)}$ if and only if $2s\le m\le \nu+s$.

Lemma 2.2. ^[5] Let $P_1$ and $P_2$ be two $m$-dimensional subspaces of $\mathbb{F}_{q}^{(2\nu)}$. Then there is a $T\in Sp_{2\nu}(\mathbb{F}_{q})$ such that $P_1 = AP_2T$, where $A\in GL_{m}(\mathbb{F}_{q})$, if and only if $P_1$ and $P_2$ are of the same type. In other words, $Sp_{2\nu}(\mathbb{F}_{q})$ acts transitively on each set of subspaces of the same type.

Corallary 2.1. Let $P$ be a subspace of type $(m, s)$ in $\mathbb{F}_{q}^{(2\nu)}$, where $2s\le m\le \nu+s$. Then there are $A\in GL_{m}(\mathbb{F}_{q})$ and $T\in Sp_{2\nu}(\mathbb{F}_{q})$ such that

Lemma 2.3. ^[5] Let $2s\leq m\leq \nu+s$. Then the number of subspaces of type $(m, s)$ in $\mathbb{F}_{q}^{(2\nu)}$ is given by

In this paper, we define $\Pi_{i\in \phi}f(i) = 1$, where $\phi$ is empty set and $f(i)$ is a function about $i$. For example, $\Pi_{i = 2}^{1}(q^{i}-1) = 1$.

3.   Association classes of ${{SMat}}(m\times n, q)$

Let $n = 2\nu$ and $\overline{G_0} = GL_{m}(\mathbb{F}_{q})\times Sp_{2\nu}(\mathbb{F}_{q})$. By the introduction, $\overline{G}$, generated by $\overline{G_0}$ and $T_0$, acts transitively on $\mathcal{M}_{mn}$, which determines the symplectic fission scheme of ${\rm{Mat}}(m\times n, q)$, denoted by ${\rm{SMat}}(m\times n, q)$. Let $R_0, R_1, \cdots, R_d$ be the orbits of $\overline{G}$ on $\mathcal{M}_{mn}\times \mathcal{M}_{mn}$, where $R_0 = \{(X, X)\mid X\in \mathcal{M}_{mn}\}$. Then ${\rm{SMat}}(m\times n, q) = (\mathcal{M}_{mn}, \{R_i\}_{0\leq i\leq d})$.

Definition 3.1. A matrix in $\mathcal{M}_{mn}$ is said to be of type $(t, s)$, if the subspace generated by its row vectors is of type $(t, s)$ in $\mathbb{F}_{q}^{(2\nu)}$. Two matrices $P$ and $Q$ in $\mathcal{M}_{mn}$ are said to be S-equivalent, denoted by $P\sim Q$, if there exist $A\in GL_{m}(\mathbb{F}_{q})$ and $T\in Sp_{2\nu}(\mathbb{F}_{q})$ such that $P = AQT$.

Obviously, the S-equivalence between matrices is an equivalent relationship, and the equivalent classes are the orbits of $\overline{G_0}$ acting on $\mathcal{M}_{mn}$.

Theorem 3.1. Let $P\in \mathcal{M}_{mn}$ be of type $(t, s)$, then $2s\le t\le \mathit{\text{min}}\{m, \nu+s\}$, and

Proof. Obviously, $0\leq t \leq m$. By Lemma 2.2, we have $2s\le t\le \text{min}\{m, \nu+s\}$.

Since $\dim(P) = t$, there is $A_1\in GL_{m}(\mathbb{F}_{q})$ such that

where $P_1$ is the matrix representation of a subspace of type $(t, s)$ in $\mathbb{F}_{q}^{(2\nu)}$. Then, by Corollary 2.1, there are $A_2\in GL_{t}(\mathbb{F}_{q})$ and $T\in Sp_{2\nu}(\mathbb{F}_{q})$ such that

Let

then $A_3A_1PT = M(t, s).$ The theorem holds. □

By the above theorem and Lemma 2.2, we obtain the necessary and sufficient conditions for two matrices to be S-equivalent immediately.

Theorem 3.2. Let $P, \ Q\in \mathcal{M}_{mn}$, then $P$ and $Q$ are S-equivalent if and only if they are of the same type.

Theorem 3.3. Let $n = 2\nu$, then ${\rm{SMat}}(m\times n, q) = (\mathcal{M}_{mn}, \{R_{(t, s)}\})$, where

$X, \ Y\in \mathcal{M}_{mn}$ and $2s\le t\le \mathit{\text{min}}\{m, \nu+s\}$.

The class number $d$ of ${\rm{SMat}}(m\times n, q)$ satisfies

The valency of $R_{(t, s)}$ is given by

where $N(t, s;2\nu)$ is defined in Lemma 2.3.

Proof. We discuss the orbits of $\overline{G}$ on $\mathcal{M}_{mn}\times \mathcal{M}_{mn}$ first. Let $P, \ Q\in \mathcal{M}_{mn}$ and $\tau_1: X\mapsto X-P$ for each $X\in \mathcal{M}_{mn}$. Then $\tau_1\in \overline{G}$ and under this transformation, $(P, Q)$ could be carried into $(0, Q-P)$. Suppose $Q-P$ is of type $(t, s)$, then $2s\le t\le \text{min}\{m, \nu+s\}$ and there is $\tau_2\in \overline{G_0}$ such that $(\tau_2(0), \tau_2(Q-P) = (0, M(t, s))$ by Theorem 3.1. By Theorem 3.2, different $(0, M(t, s))$ represent different orbits. Thus ${\rm{SMat}}(m\times n, q) = (\mathcal{M}_{mn}, \{R_{(t, s)}\})$.

In addition, let ${\rm{SMat}}(m\times n, q)$ is an association scheme of class $d$, then $d+1$ is the number of S-equivalent classes, which is the number of $(t, s)$. Clearly, $0\leq t \leq m$. From $2s\le t\le \text{min}\{m, \nu+s\}$, we deduce $t-\nu \leq s\le [t/2].$ If $t\leq \nu$, then $s$ can take $[t/2]+1$ values. If $t > \nu$, then $s$ can take $[t/2]-(t-\nu)+1$ values. This means that

(ⅰ) If $0\leq m\leq \nu$, then

(ⅱ) If $\nu < m\leq 2\nu$, then the number of ${\rm{SMat}}(m\times n, q)$ is

The results in the theorem can be obtained through simple calculations.

Finally, let's calculate the valency of $R_{(t, s)}$. By ^[5], the number of matrices of rank $t$ in $\mathcal{M}_{mn}$ is

For different $t$-dimensional subspaces, there are the same number of representation matrices of rank $t$ in $\mathcal{M}_{mn}$. The number of $t$-dimensional subspaces in $\mathbb{F}_{q}^{(n)}$ is

Thus, there are $n_t/N(t, n)$ matrices in $\mathcal{M}_{mn}$ that represent the same $t$-dimensional subspace. By Lemma 2.3, there are $n_tN(t, s;2\nu)/N(t, n)$ matrices in $\mathcal{M}_{mn}$ that represent the same subspace of type $(t, s)$. The theorem holds. □

Theorem 3.4. Let $m = 1$, then ${\rm{SMat}}(m\times n, q) = (\mathcal{M}_{mn}, \{R_{(t, s)}\})$ be a trivial association scheme, where $(t, s) = (0, 0), (1, 0)$. The valencies are $k_{(0, 0)} = 1, k_{(1, 0)} = q^{2\nu}-1.$ For the intersection numbers,

(ⅰ) when $i = (0, 0)$, $p_{ij}^k = \delta_{jk}$.

(ⅱ) when $i = (1, 0)$, $p_{ij}^k$ could be obtained by the following table, whose rows are indexed by the value of $j$ and columns indexed by the value of $k$.

Proof. The values of $(t, s)$, $k_{(0, 0)}$ and $k_{(1, 0)}$ could be obtained by Theorem 3.3 immediately. For the intersection numbers,

(ⅰ) when $i = (0, 0)$, $p_{ij}^k = \delta_{jk}$ by (2.1).

(ⅱ) when $i = (1, 0)$, $p_{(1, 0)(0, 0)}^{(0, 0)} = 0$, $p_{(1, 0)(0, 0)}^{(1, 0)} = 1$, $p_{(1, 0)(1, 0)}^{(0, 0)} = k_{(1, 0)}$ and $p_{(1, 0)(0, 0)}^{(1, 0)}+p_{(1, 0)(1, 0)}^{(1, 0)} = k_{(1, 0)}$ by Eq (2.1). Thus, $p_{(1, 0)(1, 0)}^{(1, 0)} = k_{(1, 0)}-1.$ □

Example 3.1 Let $m = 1, n = 2\nu = 2$ and $q = 2$, then ${\rm{SMat}}(m\times n, q) = (\mathcal{M}_{mn}, \{R_{(t, s)}\})$ be a trivial association scheme, where $(t, s) = (0, 0), (1, 0)$. The valencies are $k_{(0, 0)} = 1, k_{(1, 0)} = 3.$ For the intersection numbers,

(ⅰ) $p_{(0, 0)(0, 0)}^{(0, 0)} = p_{(0, 0)(1, 0)}^{(1, 0)} = 1$ and $p_{(0, 0)(0, 0)}^{(1, 0)} = p_{(0, 0)(1, 0)}^{(0, 0)} = 0$.

(ⅱ) $p_{(1, 0)(0, 0)}^{(0, 0)} = 0, \ = p_{(1, 0)(0, 0)}^{(1, 0)} = 1, \ p_{(1, 0)(1, 0)}^{(0, 0)} = 3,$ and $p_{(1, 0)(1, 0)}^{(1, 0)} = 2.$

Example 3.2. Let $m = 1, n = 2\nu = 4$ and $q = 3$, then ${\rm{SMat}}(m\times n, q) = (\mathcal{M}_{mn}, \{R_{(t, s)}\})$ be a trivial association scheme, where $(t, s) = (0, 0), (1, 0)$. The valencies are $k_{(0, 0)} = 1, k_{(1, 0)} = 80.$ For the intersection numbers,

(i) $p_{(0, 0)(0, 0)}^{(0, 0)} = p_{(0, 0)(1, 0)}^{(1, 0)} = 1$ and $p_{(0, 0)(0, 0)}^{(1, 0)} = p_{(0, 0)(1, 0)}^{(0, 0)} = 0$.

(ⅱ) $p_{(1, 0)(0, 0)}^{(0, 0)} = 0, \ = p_{(1, 0)(0, 0)}^{(1, 0)} = 1, \ p_{(1, 0)(1, 0)}^{(0, 0)} = 80,$ and $p_{(1, 0)(1, 0)}^{(1, 0)} = 79.$

Theorem 3.5. Let $m = 2$, then the association classes of ${\rm{SMat}}(m\times n, q)$ are $R_{(0, 0)}, R_{(1, 0)},$ $R_{(2, 0)}($vanishes when $n = 2)$, and $R_{(2, 1)}$. Their valencies are $k_{(0, 0)} = 1, k_{(1, 0)} = (q+1)(q^{2\nu}-1),$ $k_{(2, 0)} = q(q^{2\nu}-1)(q^{2(\nu-1)}-1)($vanishes when $n = 2)$, $k_{(2, 1)} = q^{2\nu-1}(q-1)(q^{2\nu}-1).$ For the intersection numbers,

(ⅰ) when $i = (0, 0)$, $p_{ij}^k = \delta_{jk}$.

(ⅱ) when $i\neq (0, 0)$, the intersection numbers $p_{ij}^k$ could be obtained by Tables 1–3. The rows are indexed by the value of $j$ and columns indexed by the value of $k$. When $n = 2$, there are no matrices of type $(2, 0)$, thus related intersection numbers $p_{ij}^k$ disappear.

Proof. The values of $(t, s)$, $k_{(0, 0)}$ and $k_{(1, 0)}$ could be obtained by Theorem 3.3 immediately. For the intersection numbers,

(ⅰ) when $i = (0, 0)$, $p_{ij}^k = \delta_{jk}$ by (2.1).

(ⅱ) when $i\neq (0, 0)$, we only take the 1st table as an example to give the proof in the case of $i = (1, 0)$; others' are similar.

Let

(1) By (2.1), $p_{i0}^k = \delta_{ik}, \ p_{ij}^0 = k_i\delta_{ij'}.$ Thus,

and

(2) We will compute the value of $p_{(1, 0)(1, 0)}^{(1, 0)}$.

By definition, $p_{(1, 0)(1, 0)}^{(1, 0)} = |\{ X\in \mathcal{M}_{2n}|X\sim M(1, 0), X-M(1, 0)\sim M(1, 0)\}|$.

If $(x_{1}, x_{2}, \cdots, x_{2\nu}) = 0$, then $(y_{2}, \cdots, y_{2\nu}) = 0$ and $y_1\neq 0.$ In this case, $X$ has $q-1$ choices.

If $(x_{1}, x_{2}, \cdots, x_{2\nu})\neq 0$, then by $X\sim M(1, 0)$, we have $(y_1, y_{2}, \cdots, y_{2\nu}) = k(x_{1}, x_{2}, \cdots, x_{2\nu})$, where $k\in \mathbb{F}_{q}$. At this time, if $(x_{2}, \cdots, x_{2\nu}) = 0$, $X$ has $q^2-q-1$ choices. If $(x_{2}, \cdots, x_{2\nu})\neq 0$, then $k = 0$, and $X$ has $q(q^{2\nu-1}-1)$ choices.

Above all, $p_{(1, 0)(1, 0)}^{(1, 0)} = q-1+q^2-q-1+q(q^{2\nu-1}-1) = q^{2v}+q^{2}-q-2$.

(3) When $n > 2$, we will compute the value of $p_{(1, 0)(1, 0)}^{(2, 0)}$.

By definition, $p_{(1, 0)(1, 0)}^{(2, 0)} = |\{ X\in \mathcal{M}_{2n}|X\sim M(1, 0), X-M(2, 0)\sim M(1, 0)\}|$.

If $(x_{1}, x_{2}, \cdots, x_{2\nu}) = 0$, then $y_{2} = 1$ and $(y_3, \cdots, y_{2\nu}) = 0$. In this case, $X$ has $q$ choices.

If $(x_{1}, x_{2}, \cdots, x_{2\nu})\neq 0$, then by $X\sim M(1, 0)$, we have $(y_1, y_{2}, \cdots, y_{2\nu}) = k(x_{1}, x_{2}, \cdots, x_{2\nu})$, where $k\in \mathbb{F}_{q}$. At this time, $(x_{3}, \cdots, x_{2\nu}) = 0$, then $X$ has $q^2$ choices.

Above all, $p_{(1, 0)(1, 0)}^{(2, 0)} = q^2+q$.

(4) When $n > 2$, we will compute the value of $p_{(1, 0)(2, 0)}^{(2, 0)}$.

By definition, $p_{(1, 0)(2, 0)}^{(2, 0)} = |\{ X\in \mathcal{M}_{2n}|X\sim M(1, 0), X-M(2, 0)\sim M(2, 0)\}|$.

If $(x_{1}, x_{2}, \cdots, x_{2\nu}) = 0$, then $(y_1, y_2, \cdots, y_{2\nu})\neq 0$, $(y_2-1, y_3, \cdots, y_{2\nu})\neq 0$ and $y_{\nu+1} = 0$. In this case, $X$ has $q^{2\nu-1}-q-1$ choices.

If $(x_{1}, x_{2}, \cdots, x_{2\nu})\neq 0$, then by $X\sim M(1, 0)$, we have $(y_1, y_{2}, \cdots, y_{2\nu}) = k(x_{1}, x_{2}, \cdots, x_{2\nu})$, where $k\in \mathbb{F}_{q}$. At this time, if $(x_{3}, \cdots, x_{2\nu}) = 0$, then $X$ has $q^3-q^2-q$ choices. If $(x_{3}, \cdots, x_{2\nu}) = 0$, then $X$ has $q^{2\nu}-q^3$ choices.

Above all, $p_{(1, 0)(2, 0)}^{(2, 0)} = q^{2v}+q^{2v-1}-q^{2}-2q-1$.

(5) Other values of $p_{ij}^k$ could be obtained by $\sum_{j = 0}^d{p_{ij}^k} = k_i, \ k_\gamma p_{\alpha\beta}^\gamma = k_\beta p_{\alpha'\gamma}^\beta = k_\alpha p_{\gamma\beta'}^\alpha$ in (2.1).

□

Example 3.3. Let $m = 2, n = 2\nu = 4$ and $q = 2$, then the association classes of ${\rm{SMat}}(m\times n, q)$ are $R_{(0, 0)}, R_{(1, 0)},$ $R_{(2, 0)}$, and $R_{(2, 1)}$. Their valencies are $k_{(0, 0)} = 1, k_{(1, 0)} = 45,$ $k_{(2, 0)} = 90$, $k_{(2, 1)} = 120.$ For the intersection numbers,

(ⅰ) when $i = (0, 0)$, $p_{(0, 0)(0, 0)}^{(0, 0)} = p_{(0, 0)(1, 0)}^{(1, 0)} = p_{(0, 0)(2, 0)}^{(2, 0)} = p_{(0, 0)(2, 1)}^{(2, 1)} = 1$, and $p_{ij}^k = 0$ for other cases.

(ⅱ) when $i\neq (0, 0)$, the intersection numbers $p_{ij}^k$ are as follows (Tables 4–6).

4.   Automorphism group of ${{SMat}}(m\times n, q)$

In this section, we give the inner automorphism group of ${\rm{SMat}}(m\times n, q)$ as follows.

Theorem 4.1. Let $n = 2\nu$ and $q$ is a power of a prime number.

(ⅰ) When $m = 1$, the automorphism group of ${\rm{SMat}}(m\times n, q)$ is $Sym(q^n)$.

(ⅱ) When $1 < m\leq n$, each inner automorphism of the association scheme ${\rm{SMat}}(m\times n, q)$ must have the following form:

where $P\in GL_{m}(\mathbb{F}_{q}), \ Q\in GSp_{2\nu}(\mathbb{F}_{q}), \ A\in \mathcal{M}_{mn}$, and $\sigma$ is an automorphism of $\mathbb{F}_{q}$.

In addition, if $m = n$, the following mapping is also an inner automorphism

Proof. (ⅰ) When $m = 1$, ${\rm{SMat}}(m\times n, q)$ is a trivial association scheme, thus its automorphism group is $Sym(q^n)$.

(ⅱ) When $1 < m\leq n$, let $P\in GL_{m}(\mathbb{F}_{q}), \ Q\in GSp_{2\nu}(\mathbb{F}_{q}), \ A\in \mathcal{M}_{mn}$, and $\sigma$ be an automorphism of $\mathbb{F}_{q}$. For $(X, Y)\in R_{(s, t)}$,

Thus, every $\tau_{P, Q, A, \sigma}$ is an inner automorphism of ${\rm{SMat}}(m\times n, q)$.

Conversely, let $\tau$ be an inner automorphism of ${\rm{SMat}}(m\times n, q)$, i.e., $\tau$ induces the identity permutation on ${(t, s)}$. It is easy to verify that $\tau$ is also an automorphism of ${\rm{Mat}}(m\times n, q)$. Thus, we can assume $\tau = \tau_{P, Q, A, \sigma}$, where $P\in GL_{m}(\mathbb{F}_{q}), \ Q\in GL_{n}(\mathbb{F}_{q}), \ A\in \mathcal{M}_{mn}$, and $\sigma$ is an automorphism of $\mathbb{F}_{q}$. In the following, we will prove $Q\in GSp_{2\nu}(\mathbb{F}_{q})$, that is, $QK{^{t}Q} = kK$, where $k\in \mathbb{F}_{q}^*$. Denote the $i$-th unit row vector by $e_i$.

(1) Let $1\leq i\leq n$ and

Since $(0, M)\in R_{(1, 0)},$ $(\tau(0), \tau(M)) = (A, PM^{\sigma}Q+A)\in R_{(1, 0)},$ i.e., $PM^{\sigma}Q\sim M(1, 0)$. Thus,

where $\alpha_{i}$ be the $i$-th row vector of $Q$. This means that $(QK^{t}Q)_{ii} = 0$ when $1\leq i\leq n$.

(2) Let $1\leq i\neq j\leq\nu$ and

Since $(0, M)\in R_{(2, 0)},$ $(\tau(0), \tau(M)) = (A, PM^{\sigma}Q+A)\in R_{(2, 0)},$ i.e., $PM^{\sigma}Q\sim M(2, 0)$. Thus,

This means that $(QK^{t}Q)_{ij} = 0$ when $1\leq i\neq j\leq\nu$. Similarly, it can be proven that $(QK^{t}Q)_{ij} = 0$ when $\nu\leq i\neq j\leq2\nu$ or $1\leq i\leq \nu, \nu < j\leq 2\nu(j-i\neq \nu)$.

(3) Let $1\leq i\leq \nu, j = i+\nu$, and $M$ be shown as the case (2). Since $(0, M)\in R_{(2, 1)},$ $(\tau(0), \tau(M)) = (A, PM^{\sigma}Q+A)\in R_{(2, 1)},$ i.e., $PM^{\sigma}Q\sim M(2, 1)$. Thus, the rank of

is 2. Let $\alpha_{i}K^{t}{\alpha_{j}} = k_{i}$, then $k_{i}\neq 0$, and

where $J = diag\{k_1, k_2, \cdots, k_\nu\}$.

(4) Let $1\leq i < j\leq \nu$, and

then $(0, M)\in R_{(2, 0)}$. Through similar discussions, we have

Thus $k_{i} = k_{j}$. If we assume $k_{i} = k_{j} = k$, then $QK^{t}Q = kK$. Thus $Q\in GSp_{2\nu}(\mathbb{F}_{q})$. The theorem now follows from Lemma 1.1.

It should be pointed out the cases (2) and (4) don't appear when $n = 2$. □

When $m = 2$, by Theorem 3.5, all valencies $k_{(s, t)}$ of ${\rm{SMat}}(m\times n, q)$ are distinct. Thus, each automorphism of ${\rm{SMat}}(m\times n, q)$ must be an inner automorphism. We have the following theorem.

Theorem 4.2. When $m = 2$, each automorphism of the association scheme ${\rm{SMat}}(m\times n, q)$ must have the following form:

where $P\in GL_{m}(\mathbb{F}_{q}), \ Q\in GSp_{2\nu}(\mathbb{F}_{q}), \ A\in \mathcal{M}_{mn}$, and $\sigma$ is an automorphism of the finite field $\mathbb{F}_{q}$.

In addition, if $n = 2$, the following mapping is also an automorphism

5.   Conclusions

In this paper, we construct a symplectic fission scheme for the association scheme of $m\times n$ rectangular matrices over the finite field $\mathbb{F}_q$, denoted by ${\rm{SMat}}(m\times n, q)$. Its association classes and inner automorphism group are discussed. In particular, we determine the intersection numbers and automorphism group of ${\rm{SMat}}(m\times n, q)$ for $m = 1$ and $m = 2$.

Author contributions

Yang Zhang, Shuxia Liu, and Liwei Zeng: Conceptualization, Methodology, Validation, Writing-original draft, Writing-review and editing. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The work was supported by National Natural Science Foundation of China under Grant Numbers 12171139.

Conflict of interest

All authors declare no conflicts of interest in this paper.