# Association schemes obtained from the action of the general unitary group on
isotropic vectors

Nathaniel<EMAIL_ADDRESS>Mathematics and Statistics,
Dordt University, Sioux Center, IA 51250, U. S. A. and Sung Y.
<EMAIL_ADDRESS>Department of Mathematics, Iowa State University,
Ames, IA 50011, U. S. A.

###### Abstract

An infinite family of association schemes obtained from the general unitary
groups acting transitively on the sets of isotropic vectors in the finite
unitary spaces are investigated. We compute the parameters and determine the
character tables for all nontrivial commutative association schemes belonging
to this family. 333This work contains a part of the first author’s Ph.D.
dissertation [8].

Keywords: Schurian association scheme, character table.

MSC-2020 Classification: 05E30 (primary); 20G15, 11E39 (secondary).

##  1 Introduction and preliminaries

An association scheme of class $d$ is a pair
$\mathcal{X}=\left(X,\\{R_{i}\\}_{i\in[d]}\right)$ of a finite set $X$ and a
set of non-empty $d+1$ relations $\\{R_{0},R_{1},\dots,R_{d}\\}$ of $X$ such
that

  1. (1)

$R_{0}=\\{(x,x):x\in X\\}$ is the identity relation;

  2. (2)

$R_{0}\cup R_{1}\cup\cdots\cup R_{d}=X\times X$ and $R_{i}\cap
R_{j}=\emptyset$ for $i\neq j$ in $[d]:=\\{0,1,2,\dots,d\\}$;

  3. (3)

for each $i\in[d]$, $R_{i}^{{}^{\prime}}=R_{i^{\prime}}$ for some
$i^{\prime}\in[d]$ where $R_{i}^{{}^{\prime}}:=\\{(x,y):(y,x)\in R_{i}\\}$;

  4. (4)

for each triple $h,i,j\in[d]$, there exists a non-negative integer
$p_{ij}^{h}$ such that for all $(x,y)\in R_{h}$, the number $|\\{z\in
X:(x,z)\in R_{i},(z,y)\in R_{j}\\}|$ is equal to $p^{h}_{ij}$.

The constants $p_{ij}^{h}$ are called the intersection numbers (parameters) of
the scheme $\mathcal{X}$. If an association scheme
$\mathcal{X}=\left(X,\\{R_{i}\\}_{i\in[d]}\right)$ satisfies that, for all
$h,i,j\in[d]$, $p_{ij}^{h}=p_{ji}^{h}$, then it is said to be commutative. If
for all $i\in[d]$, $R_{i}^{\prime}=R_{i}$, then it is called symmetric.

A major source of association schemes is the set of transitive permutation
groups (cf. [1, 3, 4, BHSWZie]). If a finite group $G$ acts on a finite set
$\Phi$ transitively, the set of the orbitals (also called 2-orbits) of $G$;
that is, the set of the orbits of the action of $G$ on $\Phi\times\Phi$, forms
an association scheme. Such an association scheme
$\mathcal{X}=\left(\Phi,\\{R_{i}\\}_{i\in[d]}\right)$ where
$R_{0},R_{1},\dots,R_{d}$ are the orbitals of the permutation group $G$ on
$\Phi$ (of rank $d+1$) is called a _Schurian association scheme_ (of class
$d$) and denoted by $\mathcal{X}(G,\Phi)$.

In this paper, we study the Schurian association schemes coming from the
transitive action of the finite general unitary groups on the sets of
isotropic vectors of the $n$-dimensional unitary space equipped with a non-
degenerate Hermitian inner product over the finite field $\mathbb{F}_{q^{2}}$
of order $q^{2}$ for all $n\geq 2$ and prime powers $q$. Our aim is to provide
a complete description of these Schurian association schemes in terms of their
character tables. These association schemes help fill the void in the existing
collection of the Schurian association schemes coming from the classical
groups acting on various sets of vectors or subspaces of the corresponding
geometries.

For known interesting examples of Schurian association schemes, we refer the
readers to Brouwer-Cohen-Neumaier [9] for the $P$-polynomial association
schemes (distance-regular graphs) corresponding to the dual polar spaces,
Brouwer-van Maldeghem [10] for known schemes of class 2 (strongly regular
graphs), and Hanaki [13] for all Schurian schemes of small orders (orders up
to 40 or so). For the list of infinite families of Schurian schemes whose
character tables are known, see [19] and the references there.

Given a $d$-class commutative association scheme
$\mathcal{X}=(X,\\{R_{i}\\}_{i\in[d]})$ of order $n$ (i.e., $|X|=n$), let
$A_{i}$ denote the $i$th adjacency matrix representing $R_{i}$; that is,
$\\{0,1\\}$-matrix whose $(x,y)$-entry is defined by

$\left(A_{i}\right)_{xy}=\left\\{\begin{array}[]{ll}1&\mbox{if }(x,y)\in
R_{i}\\\ 0&\mbox{otherwise.}\\\ \end{array}\right.$

By the definition of commutative association scheme, these matrices satisfy
that

  1. (1)

$A_{0}=I$, the identity matrix;

  2. (2)

$A_{0}+A_{1}+\cdots+A_{d}=J$, where $J$ is the all-ones matrix;

  3. (3)

for each $i\in[d]$, $A_{i}^{{}^{\prime}}=A_{i^{\prime}}$ for some
$i^{\prime}\in[d]$ where $A_{i}^{{}^{\prime}}$ denotes the transpose of
$A_{i}$;

  4. (4)

for any $h,i,j\in[d]$, there exists a constant $p_{ij}^{h}$ such that

$A_{i}A_{j}=\sum\limits_{h=0}^{d}p_{ij}^{h}A_{h};$

  5. (5)

$A_{i}A_{j}=A_{j}A_{i}$ for every $i,j\in[d]$.

In the full matrix algebra consisting of all $n\times n$ matrices over the
field of complex numbers, these adjacency matrices of $\mathcal{X}$ generate
the $(d+1)$-dimensional commutative algebra
$\mathcal{A}:=\left<A_{0},A_{1},\dots,A_{d}\right>$ known as the Bose-Mesner
algebra of $\mathcal{X}$. The algebra $\mathcal{A}$ being a semi-simple
algebra admits central primitive idempotents. Let
$E_{0}=\frac{1}{n}J,E_{1},\dots,E_{d}$ denote the primitive idempotents in
$\mathcal{A}$. Then there exist complex numbers $p_{j}(i)$ and $q_{i}(j)$ for
$i,j\in[d]$ such that

$A_{j}=\sum\limits_{i=0}^{d}p_{j}(i)E_{i},\quad
E_{i}=\frac{1}{n}\sum\limits_{j=0}^{h}q_{i}(j)A_{j}.$

The $(d+1)\times(d+1)$ base-change matrices $P$ and $Q$, whose $(i,j)$-entries
are $p_{j}(i)$ and $q_{j}(i)$, respectively, are called the 1st and 2nd
eigenmatrix of $\mathcal{X}$, respectively. The $d+1$ entries
$p_{j}(0),p_{j}(1),\dots,p_{j}(d)$, the entries of the column indexed with $j$
of $P$, are the roots of the minimal polynomial of $A_{j}$ for each $j\in[d]$.
The first eigenmatrix $P$ is also called the _character table_ of the
association scheme. We denote the multiplicities of $\mathcal{X}$ by
$m_{0},m_{1},\dots,m_{d}$, which are the ranks of idempotents
$E_{0},E_{1},\dots,E_{d}$, respectively. Note that $m_{i}$ is the trace of
$E_{i}$ as the eigenvalues of $E_{i}$ are 1 and 0. The $i^{\textnormal{th}}$
valency, i.e., the number of elements of $X$ that are in the
$i^{\textnormal{th}}$ associates with $x$ for any fixed $x\in X$, is denoted
by $k_{i}$. Note that $k_{i}=p_{ii^{\prime}}^{0}$. Then we have the following
formulae for the parameters and eigenvalues of $\mathcal{X}$ (cf. [3, Ch 2]):

###### Proposition 1.1

$\begin{array}[]{llll}&m_{i}=n\big{(}\sum\limits_{j=0}^{d}\frac{|p_{j}(i)|^{2}}{k_{j}}\big{)}^{-1}&&\text{for}\
i\in[d]\\\ &p_{ij}^{h}=\frac{1}{n\cdot
k_{h}}\sum\limits_{l=0}^{d}p_{i}(l)p_{j}(l)\overline{p_{h}(l)}m_{l}&&\text{for}\
h,i,j\in[d].\end{array}$

where $\overline{a}$ denotes the complex conjugate of $a$.

###### Proposition 1.2

The character table $P=\big{[}p_{j}(i)\big{]}$ of
$\mathcal{X}=(X,\\{R_{i}\\}_{i\in[d]})$ satisfies the orthogonality relations:

$\begin{array}[]{llll}&\sum\limits_{j=0}^{d}\frac{1}{k_{j}}p_{j}(i_{1})\cdot\overline{p_{j}(i_{2})}=\delta_{i_{1}i_{2}}\frac{n}{m_{i_{1}}},&&i_{1},i_{2}\in[d]\\\
&\sum\limits_{i=0}^{d}m_{i}p_{j_{1}}(i)\overline{p_{j_{2}}(i)}=\delta_{j_{1}j_{2}}\cdot
n\cdot k_{j_{1}},&&j_{1},j_{2}\in[d]\end{array}$

where $\delta_{ij}=\left\\{\begin{array}[]{ll}1&\mbox{if }i=j\\\ 0&\mbox{if
}i\neq j\end{array}\right.$.

There is another matrix algebra associated with $\mathcal{X}$. Let $B_{i}$,
$i\in[d]$, be the $i$th intersection matrix defined by
$\left(B_{i}\right)_{jh}=p_{ij}^{h}.$ Then

$B_{i}B_{j}=\sum\limits_{h=0}^{d}p_{ij}^{h}B_{h}.$

It follows that the algebra
$\mathcal{B}:=\left<B_{0},B_{1},\dots,B_{d}\right>$ over $\mathbb{C}$ is
isomorphic to $\mathcal{A}:=\left<A_{0},A_{1},\dots,A_{d}\right>$. In
particular, the minimal polynomial of $B_{i}$ coincides with that of $A_{i}$
for each $i\in[d]$, and thus, the eigenvalues of $B_{i}$s are the entries of
the character table of the scheme $\mathcal{X}$. Therefore, it is possible
that the character table can be determined from the eigenvalues of $B_{i}$
with or without additional information. We will refer to this approach of
constructing the character table of $\mathcal{X}$ as an elementary and ‘direct
approach’.

We note that there is an ‘indirect approach’ that has been used as well. When
$G$ acts transitively on $\Phi$, there is a natural one-to-one correspondence
between $\Phi$ and the set $G_{x}\backslash G$ of the cosets of a point
stabilizer $G_{x}$ for $x\in\Phi$. Thus, the permutation representation of the
action $G$ on $\Phi$ is identical to that of the action $G$ on the set
$H\backslash G$ of cosets of $H$, where $H=G_{x}$ for $x\in\Phi$. The
permutation character of the representation is the same as the induced
character $1^{G}_{H}$ of $H$ in $G$. We also note that there is a one-to-one
correspondence between any two of the following three:

> (i) the set of orbitals of the permutation group $(G,\Phi)$,
> (ii) the set of suborbits of $(G,\Phi)$ and
> (iii) the set of double cosets $Hg_{i}H$ of $H$ in $G$, $g_{i}\in G$.

It follows that there are links between the following three in terms of
associated algebra:

> (i) the Bose-Mesner algebra of the Schurian scheme $\mathcal{X}(G,\Phi)$;
> (ii) the centralizer algebra (Hecke algebra) of $(G,\Phi)$;
> (iii) the double coset algebra, the subalgebra spanned by the set of simple
> quantities $\frac{1}{|H|}\sum\limits_{g\in Hg_{i}H}g$ in group algebra.

Although we take the direct approach to construct the character table of a
Schurian association scheme $\mathcal{X}(G,\Phi)$ with $G=GU(n,q)$ and
$\Phi=\Phi(n,q)$ for suitable $(n,q)$, there is an alternative way to find the
entries of the character table for some cases. Namely, its character table may
be determined by using the relationship between Bose-Mesner algebra and the
associated algebra of the permutation group $(G,\Phi)$ as mentioned above. For
instance, suppose all orbitals are self-paired (thus, the scheme is symmetric)
and the induced character $1^{G}_{H}$ is multiplicity-free, so that

$1^{G}_{H}=\chi_{0}+\chi_{1}+\cdots+\chi_{d},$

where $\chi_{0}=1_{G}$ and $\chi_{1},\chi_{2},\dots,\chi_{d}$ are distinct
irreducible characters of $G$. Then the entries $p_{j}(i)$ of the character
table $P=\left[p_{j}(i)\right]$ of the scheme $\mathcal{X}(G,H\backslash G)$
can be explicitly expressed as:

$p_{j}(i)\ \ =\ \ \frac{1}{|H|}\sum\limits_{a\in HGH}\chi_{i}(a)\ \ =\ \
\frac{1}{|H|}\sum\limits_{l}|Hg_{j}H\cap C_{l}|\cdot\chi_{i}(c_{l})$

where $C_{0},C_{1},\dots,C_{l},\dots,C_{d}$ is the conjugacy classes of $G$
and $c_{l}$ is the class representative of $C_{l}$. However, finding the
entries $p_{j}(i)$ of the character table using the above equation is
sometimes involved if it is not impossible (cf. [19] and references in it).

The highlights of the remaining sections may be described as the following.

  * Sec 2.

The fact that $GU(n,q)$ is acting transitively on $\Phi(n,q)$ for $n,q$, with
$n\geq 2,q\geq 2$ is verified and then, the orbitals of the action are
classified.

  * Sec 3.

The intersection numbers of the association schemes
$\mathcal{X}(GU(n,q),\Phi(n,q))$ for all prime powers $q$ and all valid values
of $n$ ($n\geq 2$) are calculated.

  * Sec 4.

The fact that association scheme $\mathcal{X}(GU(n,q),\Phi(n,q))$ is
commutative if and only if $q=2$ is proved in Theorem 4.1.

  * Sec 5.

The character table of the commutative scheme $\mathcal{X}(GU(n,2),\Phi(n,2))$
(for arbitrary $n$) is constructed.

##  2 The orbitals of permutation group $GU(n,q)$ on $\Phi(n,q)$

In this section, we shall see that for each prime power $q$ and integer $n\geq
2$, the permutation action of finite unitary group $GU(n,q)$ on the set
$\Phi(n,q)$ of isotropic vectors in the finite unitary space
$\mathbb{F}_{q^{2}}^{n}$ is transitive, and thus, its orbitals form an
association scheme. Our aim is to give a complete description of orbitals that
form the relation set of the association scheme. 444For more information on
the facts on finite fields and finite classical geometries that are used in
this discussion, we refer the readers to the books by Wan [21], Grove [12] and
Taylor [20].

Given a prime power $q$, let $V=\mathbb{F}_{q^{2}}^{n}$ be the $n$-dimensional
unitary space over $\mathbb{F}_{q^{2}}$, that is, $V$ is the vector space
equipped with a non-degenerate Hermitian inner product on which $GU(n,q)$ is
acting. Recall that the field $\mathbb{F}_{q^{2}}$ has an involution
$a\mapsto\bar{a}=a^{q}$ whose fixed field is $\mathbb{F}_{q}$. We will
repeatedly use the fact that: (i) For any $\lambda\in\mathbb{F}_{q}^{*}$, the
equation $x\bar{x}=\lambda$ has exactly $q+1$ solutions in
$\mathbb{F}_{q^{2}}^{*}$. (ii) For any $\lambda\in\mathbb{F}_{q}$, the
equation $x+\bar{x}=\lambda$ has exactly $q$ solutions in $\mathbb{F}_{q^{2}}$
(cf. [21]).

Given row vectors $x=(x_{1},x_{2},\dots,x_{n})$ and
$y=(y_{1},y_{2},\dots,y_{n})$ in $V$, denote the Hermitian inner product of
$x$ and $y$ by

$\langle
x,y\rangle=xy^{*}=x_{1}y_{1}^{q}+x_{2}y_{2}^{q}+\cdots+x_{n}y_{n}^{q}.$

Recall that the set of isotropic vectors in $V$ is

$\Phi(n,q):=\\{x\in V-\\{0\\}:\langle x,x\rangle=0\\}$

and the number of isotropic vectors in $V$ is:

$|\Phi(n,q)|=(q^{n}-(-1)^{n})(q^{n-1}-(-1)^{n-1})$

In what follows, $GU(n,q)$ and $\Phi(n,q)$ will be often denoted by $G$ and
$\Phi$, respectively, for short, once $n$ and $q$ are fixed. The group action
here is described for $x\in\Phi$ and $U\in G$ by $xU$ using typical matrix
multiplication. We extend the group action of $G$ on $\Phi$ to $G$ acting on
$\Phi\times\Phi$ in the obvious way described by $(x,y)U=(xU,yU)$.

In order to verify that the action of $G$ on $\Phi$ is indeed transitive, we
need to recall the following well-known results. Note that a hyperbolic pair
in $V$ is a pair of vectors $(u,v)$ such that $u$ and $v$ are isotropic, and
the Hermitian inner product $\langle u,v\rangle=1$. The subspace spanned by
$\\{u,v\\}$ is called a hyperbolic line.

###### Proposition 2.1

Suppose that $L$ is a two-dimensional subspace of $V$ which contains an
isotropic vector $u$ with respect to a non-degenerate Hermitian inner product.
Then there exists a vector $v$, such that $L={Span}\\{u,v\\}$, and $(u,v)$ is
a hyperbolic pair.

It is known that $V$ can be decomposed into

$V=L_{1}\bot L_{2}\bot\cdots\bot L_{m}\bot W$

where each $L_{i}$ is a hyperbolic line, with $m$ as the Witt index, and such
that $W$ does not contain any isotropic vectors. For our case, we can ignore
$W$, since we are concerned with isotropic vectors only, though in general for
finite fields, dim$(W)$ = 0 or 1 because of the previous proposition, which
ensures that we can find more isotropic vectors while the dimension of
remaining subspace is at least 2. (cf. [20, pp.116-117]).

###### Proposition 2.2

For a vector space $V$ that is decomposed as $V=L_{1}\bot L_{2}\bot\cdots\bot
L_{m}\bot W,$ let $(u_{i},v_{i})$ be hyperbolic pairs that form a basis for
each corresponding $L_{i}$, and let $w$ be a basis for $W$ with $\langle
w,w\rangle=1$. Then $G$ acts regularly on the basis vectors $u_{1},\dots
u_{m},v_{1},\dots,v_{m},w$. That is, the action of $G$ on $\Phi$ is transitive
and semiregular.

From these propositions we can deduce the following.

###### Theorem 2.3

$GU(n,q)$ acts transitively on $\Phi(n,q)$, for all $n\geq 2$ and all prime
powers $q$.

Proof: Let $u,v\in\Phi(n,q)$, and let $(u,u^{\prime})$ be a hyperbolic pair of
$V=\mathbb{F}_{q^{2}}^{n}$, which exists by Proposition 2.1. Define $L_{1}$ as
the hyperbolic line spanned by $(u,u^{\prime})$, and decompose $V$ into
hyperbolic lines

$V=L_{1}\bot L_{1}^{\bot}=L_{1}\bot L_{2}\bot\cdots\bot L_{m}.$

Now it is shown that there exists a unitary matrix that takes $u$ to $v$ in
case by case.

  1. (i)

Suppose $v\in L_{1}^{\bot}$. Then again by the above proposition there is a
hyperbolic pair $(v,v^{\prime})$ that spans some $L_{i}$ with $i\neq 1$. We
can choose hyperbolic pairs arbitrarily to form a basis for each of the
remaining hyperbolic lines. Then we have a basis for $V$ including $u$ and $v$
as basis elements. By Proposition 2.2, $GU(n,q)$ acts regularly on these
vectors, and thus transitively. Hence, there is a unitary matrix that takes
$u$ to $v$.

  2. (ii)

Suppose $v\in L_{1}$ and $(u,v)$ forms a basis for $L_{1}$. Then again we can
arbitrarily choose hyperbolic pairs to extend to a basis for $V$ and apply
Proposition 2.2 to obtain a unitary matrix that takes $u$ to $v$.

  3. (iii)

Suppose $v\in L_{1}$ and $(u,v)$ does not form a basis for $L_{1}$. Then $u$
and $v$ are linearly dependent, i.e. $v=\beta u$. Recall that $(u,u^{\prime})$
does form a basis for $L_{1}$, so $(v,u^{\prime})$ also forms a basis for
$L_{1}$. By (ii), there is a unitary matrix $U\in GU(n,q)$ such that
$vU=u^{\prime}$. Similarly, there is a unitary matrix $U^{\prime}\in GU(n,q)$
such that $uU^{\prime}=u^{\prime}$. Hence, there is a unitary matrix
$U^{\prime}U^{-1}\in GU(n,q)$ that takes $u$ to $v$, since

$uU^{\prime}U^{-1}=u^{\prime}U^{-1}=v.$

Therefore, $GU(n,q)$ indeed acts transitively on the set of isotropic vectors.

In order to classify the orbitals of the transitive permutation group
$(GU(n,q),\Phi(n,q))$, we will first define the desired partition of
$\Phi(n,q)\times\Phi(n,q)$, and then proceed to show that these are indeed the
orbitals of the action of $GU(n,q)$ on $\Phi(n,q)$; and so the desired
association relations.

Let $\alpha$ be an arbitrary but fixed primitive element of
$\mathbb{F}_{q^{2}}$. Consider the following sets where $i,j\in[q^{2}-2]$ and
$\Phi=\Phi(n,q)$.

$\begin{array}[]{c}S_{\alpha^{i}}(n,q)=\\{(x,y)\in\Phi\times\Phi:y=\alpha^{i}x\\}\\\
R_{\alpha^{j}}(n,q)=\\{(x,y)\in\Phi\times\Phi:\langle
x,y\rangle=\alpha^{j}\\}\\\ T(n,q)=\\{(x,y)\in\Phi\times\Phi:\langle
x,y\rangle=0,y\notin{Span}\\{x\\}\\}.\end{array}\qquad\qquad(*)$

###### Lemma 2.4

For any $n$ and $q$ and for each $i,j\in[q^{2}-2]$, the sets $S_{\alpha^{i}}$,
$R_{\alpha^{j}}$’s, and $T$ defined above partition $\Phi\times\Phi$, i.e.

$\Phi\times\Phi\ \ =\ \ T\ \cup\ \bigcup_{i=0}^{q^{2}-2}S_{\alpha^{i}}\ \cup\
\bigcup_{j=0}^{q^{2}-2}R_{\alpha^{j}}.$

Proof: This follows from the fact that if $\langle x,y\rangle\neq 0$, then
$(x,y)$ lies in exactly one of the $R_{\alpha^{i}}$’s. If $\langle
x,y\rangle=0$ then either $(x,y)\in T$ or $y\in{Span}\\{x\\}$. In the latter
case, $y=\alpha^{i}x$ for some $i$, implying that $(x,y)\in S_{\alpha^{i}}$.
Hence, each element of $\Phi\times\Phi$ lies in exactly one of the sets above
$(*)$.

Notice that the three types of orbitals: $S_{\alpha}$’s, $R_{\alpha}$’s, and
$T$, correspond directly with Cases (iii), (ii), and (i) respectively,
addressed in the proof of Theorem 2.3. The next task is to show that these
sets indeed form the orbitals of the group action, that is, showing that
$\Phi(n,q)\times\Phi(n,q)$ is not partitioned any finer or courser by the
action. This is displayed by showing each set closed under the group action so
that none of them need to be combined to form the relations, and that
$GU(n,q)$ does not refine any further. We will use the following result, known
as Witt’s Extension Theorem, repeatedly.

######  Proposition 2.5 (Witt’s Extension Theorem [22])

Suppose $V_{1},V_{2}\subseteq\mathbb{F}_{q^{2}}^{n}$ are subspaces and that
there is a linear isomorphism $U^{\prime}:V_{1}\rightarrow V_{2}$ such that
$\langle xU^{\prime},yU^{\prime}\rangle=\langle x,y\rangle$ for any $x,y\in
V_{1}$. Then there exists $U\in GU(n,q)$ such that $U|_{V_{1}}=U^{\prime}$.

###### Lemma 2.6

$GU(n,q)$ acts transitively on $S_{\alpha^{i}}$, for each $i\in[q^{2}-2]$.

Proof: Take $(x,y)\in S_{\alpha^{i}}$, then $y=\alpha^{i}x$, so for any $U\in
GU(n,q)$, we have $yU=\alpha^{i}xU$. Thus, $(xU,yU)=(x,y)U\in S_{\alpha^{i}}$.

To show transitivity, take $(x,y),(w,z)\in S_{\alpha^{i}}$ we want to show
that some $U\in GU(n,q)$ takes $(x,y)$ to $(w,z)$. Since $GU(n,q)$ acts
transitively on $\Phi(n,q)$ by Theorem 2.3, there exists a $U\in GU(n,q)$ such
that $w=xU$. Then $yU=\alpha^{i}xU=\alpha^{i}w=z$. Hence, $(x,y)U=(w,z)$, as
desired.

###### Lemma 2.7

$GU(n,q)$ acts transitively on $R_{\alpha^{j}}$, for each $j\in[q^{2}-2]$.

Proof: Take $(x,y)\in R_{\alpha^{j}}$, then $\langle
x,y\rangle=xy^{*}=\alpha^{j}$, so for any $U\in GU(n,q)$,

$\langle xU,yU\rangle=xU(yU)^{*}=xUU^{*}y^{*}=xy^{*}=\alpha^{j}.$

Thus, $(xU,yU)=(x,y)U\in R_{\alpha^{j}}$.

To show transitivity, take $(x,y),(w,z)\in R_{\alpha^{j}}$. There exists a
linear isomorphism

$U^{\prime}:{Span}\\{x,y\\}\to{Span}\\{w,z\\}$

such that $xU^{\prime}=w$ and $yU^{\prime}=z$, as they are both 2-dimensional
subspaces. Since

$\langle x,x\rangle=\langle y,y\rangle=\langle w,w\rangle=\langle
z,z\rangle=0\;\mbox{ and }\;\langle x,y\rangle=\langle w,z\rangle=\alpha^{j},$

$U^{\prime}$ is an isometry. Therefore, by Witt’s Extension Theorem
$U^{\prime}$ extends to a $U\in GU(n,q)$ mapping $(x,y)\mapsto(w,z)$.

###### Lemma 2.8

$GU(n,q)$ acts transitively on $T$.

Proof: Take $(x,y)\in T$, then $\langle x,y\rangle=0$, but $y\neq\alpha x$ for
any $\alpha\in\mathbb{F}_{q^{2}}$. So for any $U\in GU(n,q)$,

$\langle xU,yU\rangle=xU(yU)^{*}=xUU^{*}y^{*}=xy^{*}=0.$

Now, by way of contradiction, suppose $yU=\alpha xU$ for some
$\alpha\in\mathbb{F}_{q^{2}}$, then $yUU^{-1}=\alpha xUU^{-1}$, implying that
$y=\alpha x$, a contradiction. Hence, $(xU,yU)\in T$ as desired.

To show transitivity, take $(x,y),(w,z)\in T$, we have that
$y\notin{Span}\\{x\\}$ and $z\notin{Span}\\{w\\}$, so $Span\\{x,y\\}$ and
$Span\\{w,z\\}$ are 2-dimensional subspaces. We proceed similarly to before by
taking a linear isomorphism $U^{\prime}:{Span}\\{x,y\\}\to{Span}\\{w,z\\}$
such that $xU^{\prime}=w$ and $yU^{\prime}=z$. Again, this is an isometry
since all of the vectors are isotropic, and $\langle x,y\rangle=\langle
w,z\rangle=0.$ Applying Witt’s Extension Theorem gives a map $U\in GU(n,q)$
with $(x,y)\mapsto(w,z)$.

This shows that the aforementioned sets have the desired structure, but it
remains to show that these are indeed orbitals by showing that they are
nonempty. We show that regardless of $n$ and $q$ each $R_{\alpha}$ and
$S_{\alpha}$ is nonempty; however, the set $T$ is, in fact, empty for $n=2$
and $n=3$, which will be addressed separately.

###### Lemma 2.9

$S_{\alpha^{i}}(n,q)\neq\emptyset$ for all $n$ and $q$, and all
$i\in[q^{2}-2]$.

Proof: Consider $x=(1,a,0,\dots,0)$, where $a\in\mathbb{F}_{q^{2}}$ such that
$a\bar{a}=-1$. It is clear that $x\in\Phi(n,q)$.

Let $y=\alpha^{i}x$, then

$\langle
y,y\rangle=\langle\alpha^{i}x,\alpha^{i}x\rangle=\alpha^{i}\alpha^{iq}\langle
x,x\rangle=0.$

So $y\in\Phi(n,q)$ as well, and $(x,y)\in S_{\alpha^{i}}$, hence
$S_{\alpha^{i}}\neq\emptyset.$

###### Lemma 2.10

$R_{\alpha^{j}}(n,q)\neq\emptyset$ for all $n$ and $q$, and all
$j\in[q^{2}-2]$.

Proof: Let $x\in\Phi(n,q)$. Then by Proposition 2.1, we can choose an
isotropic vector $y\in\Phi(n,q)$ such that $\langle x,y\rangle=1$. Consider
$\alpha^{j}x$. Clearly,
$\langle\alpha^{j}x,\alpha^{j}x\rangle=\alpha^{j}(\alpha^{j})^{q}\langle
x,x\rangle=0$, so $\alpha^{j}x\in\Phi(n,q)$. Moreover,
$\langle\alpha^{j}x,y\rangle=\alpha^{j}\langle x,y\rangle=\alpha^{j}$. Hence,
$(\alpha^{j}x,y)\in R_{\alpha^{j}}.$

Now we address the set $T$.

###### Lemma 2.11

If $n=2$ or $n=3$, then $T(n,q)=\emptyset$ for all $q$. If $n\geq 4$, then
$T(n,q)\neq\emptyset.$

Proof: Notice that an element of $T$ consists of a pair of linearly
independent isotropic vectors, meaning they span a two-dimensional totally
isotropic subspace. By Theorem 5.7 or [21], we have that $n\geq 4$, so
$T=\emptyset$ when $n\in\\{2,3\\}$.

$T$ can be seen to be non-empty for $n\geq 4$ by considering
$x=(1,a,0,0,\dots,0)$ and $y=(0,0,1,a,0,\dots,0)$. Clearly these are isotropic
and $\langle x,y\rangle=0$, but $x\notin{Span}\\{y\\}$.

We are finally in a position to identify the association relations of
$\mathcal{X}(GU(n,q),\Phi(n,q))$:

###### Theorem 2.12

For $n\in\\{2,3\\}$, the relations of $\mathcal{X}(GU(n,q),\Phi(n,q))$ are the
$R_{\alpha^{i}}(n,q)$’s together with the $S_{\alpha^{i}}(n,q)$’s for
$i\in[q^{2}-2]$. For $n\geq 4$ the relations are these sets together with $T$.

Moreover,

$\textnormal{rank}(\mathcal{X}(GU(n,q),\Phi(n,q)))=\begin{cases}2q^{2}-2&n\in\\{2,3\\}\\\
2q^{2}-1&n\geq 4.\end{cases}$

Proof: This follows directly from the preceding lemmata.

##  3 Intersection numbers of $\mathcal{X}(GU(n,q),\Phi(n,q))$

We now compute the intersection numbers of $\mathcal{X}(GU(n,q),\Phi(n,q))$.
Theorem 2.12 provides enough information to determine the valencies of
$\mathcal{X}$ for the $n=2$ and $n=3$ cases.

###### Theorem 3.1

For $n\in\\{2,3\\}$ the valencies $k_{i}=p^{0}_{ii^{\prime}}$ of
$\mathcal{X}(GU(n,q),\Phi(n,q))$ are

$\displaystyle\overbrace{1,1,\dots,1}^{q^{2}-1},\;\;\overbrace{\frac{|\Phi(n,q)|}{q^{2}-1}-1,\
\frac{|\Phi(n,q)|}{q^{2}-1}-1,\ \dots,\
\frac{|\Phi(n,q)|}{q^{2}-1}-1}^{q^{2}-1}$

for $0\leq i\leq D-1$ with $D=2q^{2}-2$.

Proof: Note that $|R_{\alpha}|$ and $|S_{\alpha}|$ do not depend on $\alpha$.
$|S_{\alpha}|=|\Phi|$ since each vector $x\in\Phi$ uniquely determines
$y=\alpha x\in S_{\alpha}$ for each $\alpha$, so

$k_{0}=k_{1}=\cdots=k_{q^{2}-2}=\frac{|S_{\alpha}|}{|\Phi|}=1.$

The remaining $k_{i}$’s are equal, say $k$, so $|R_{\alpha}|=k|\Phi|$. Hence,
because $\Phi\times\Phi$ is partitioned by only the $R_{\alpha}$’s and the
$S_{\alpha}$’s, we can count the number of pairs of isotropic vectors as

$|\Phi|(q^{2}-1)+k|\Phi|(q^{2}-1)=|\Phi|^{2}$

so it follows that

$\displaystyle k\ \ =\ \ \frac{|\Phi|}{q^{2}-1}-1\ \ =\ \
\frac{(q^{n}-(-1)^{n})(q^{n-1}-(-1)^{n-1})}{q^{2}-1}-1.$

The presence of $T$ as an additional orbital makes the computation for the
valencies of $\mathcal{X}$ much more involved for $n\geq 4$. We start by
relabeling the association relations with sequential indexing using $R_{l}$,
$l\in[D]$ with $D=2q^{2}-2$. Namely, by the cyclic nature of the field, we can
more concisely articulate the relations as follows.

$R_{l}=\begin{cases}S_{\alpha^{l}}&\textnormal{if }0\leq l\leq q^{2}-2\\\
R_{\alpha^{l}}&\textnormal{if }q^{2}-1\leq l\leq 2q^{2}-3\\\ T&\textnormal{if
}l=D=2q^{2}-2.\end{cases}$

Now we want to compute the intersection numbers $p_{ij}^{h}$ based on the
range in which $h,i,$ and $j$ lie. For the sake of simplicity, we will
henceforth refer to the ranges in the following way

$I_{1}:=[q^{2}-2],\qquad\qquad I_{2}:=[2(q^{2}-1)-1]\setminus I_{1}.$

The intersection numbers can be computed based only on whether each of $h,i,j$
is in $I_{1},I_{2}$, or equal to $D$. This results in 27 different cases,
though many of them are immediate or found similarly to one another. We
include one sample calculation of importance below.

###### Lemma 3.2

If $h\in I_{1}$ and $i=j=D$, then $p_{ij}^{h}=q^{2}|\Phi(n-2,q)|$.

Proof: Since the intersection numbers do not depend on the choice of $x$ and
$y$, we let $x=(1,a,0,\dots,0)$, where again $a$ is an element of the field
such that $a\bar{a}=-1$, and let $y=\alpha^{h}x$.

$\displaystyle p_{ij}^{h}$ $\displaystyle=$
$\displaystyle\left|\left\\{z\in\Phi(n,q):\langle x,z\rangle=0,z\notin
Span\\{x\\},\langle z,y\rangle=0,y\notin Span\\{z\\}\right\\}\right|$
$\displaystyle=$ $\displaystyle\left|\left\\{z=\beta
x+z^{\prime}:\beta\in\mathbb{F}_{q^{2}},z_{1}^{\prime}=z_{2}^{\prime}=0,\langle
z^{\prime},z^{\prime}\rangle=0,z\notin Span\\{x\\},y\notin
Span\\{z\\}\right\\}\right|$ $\displaystyle=$
$\displaystyle\left|\left\\{z=\beta
x+z^{\prime}:\beta\in\mathbb{F}_{q^{2}},z_{1}^{\prime}=z_{2}^{\prime}=0,z^{\prime}\in\Phi(n,q)\right\\}\right|$
$\displaystyle=$ $\displaystyle q^{2}|\Phi(n-2,q)|.$

Notice that $z$ satisfying the conditions for $x$ here also ensures that it
satisfies the conditions for $y$. There are $q^{2}$ choices for $\beta$ and
$|\Phi(n-2,q)|$ choices for $z^{\prime}$, since it cannot be the zero vector,
otherwise it would be in the span of $x$.

The following result also assisted in several of these calculations.

###### Lemma 3.3

In $\mathcal{X}(GU(n,q),\Phi(n,q))$, for each $l\in[D]$, the symmetric
conjugate relation $R_{l^{\prime}}$ of $R_{l}$ lies in the same range as
$R_{l}$; that is, $l,l^{\prime}\in I_{1}$, or $l,l^{\prime}\in I_{2}$, or
$l=l^{\prime}=D$.

###### Theorem 3.4

For $n\geq 4$ the valencies of $\mathcal{X}(GU(n,q),\Phi(n,q))$ are

$\overbrace{1,\ 1,\ \dots,\ 1}^{q^{2}-1},\;\;\;\overbrace{q^{2n-3},\
q^{2n-3},\ \dots,\ q^{2n-3}}^{q^{2}-1},\;\;\;q^{2}|\Phi(n-2,q)|.$

Proof: As before, $k_{0}=k_{1}=\cdots=k_{q^{2}-1}=1.$ Observe that
$k_{D}=p_{DD}^{0}$ falls into the case where $i=j=D$ and $h\in I_{1}$, so by
Lemma 3.2, $k_{D}=q^{2}|\Phi(n-2,q)|.$

Now the remaining valencies are all the same, say $k$, so
$|R_{\alpha}|=k|\Phi(n,q)|$. Hence, again by double counting the number of
pairs of isotropic vectors using the partitioning of
$\Phi(n,q)\times\Phi(n,q)$ it follows that

$|\Phi(n,q)|(q^{2}-1)+k|\Phi(n,q)|(q^{2}-1)+k_{D}|\Phi(n,q)|=|\Phi(n,q)|^{2}.$

This gives us

$\displaystyle k(q^{2}-1)\ \ =\ \ |\Phi(n,q)|-(q^{2}-1)-q^{2}|\Phi(n-2,q)|\ \
=\ \ q^{2n-3}(q^{2}-1).$

Therefore, $k=q^{2n-3}$ as desired.

Now using the fact that the valencies are as above, we complete the
calculation of the intersection numbers that are summarized in the table
below.

Table 1: Intersection Numbers for $\mathcal{X}(GU(n,q),\Phi(n,q))$.

| $h\in I_{1}$ | $h\in I_{2}$ | $h=D$
---|---|---|---
$i\in I_{1}$ | $j\in I_{1}$ | $\begin{cases}1&\mbox{if}\ h\equiv i+j\\\ 0&\mbox{else}\end{cases}$ | 0 | 0
| $j\in I_{2}$ | 0 | $\begin{cases}1&\mbox{if }h+i\equiv j\\\ 0&\mbox{else}\end{cases}$ | 0
| $j=D$ | 0 | 0 | 1
$i\in I_{2}$ | $j\in I_{1}$ | 0 | $\begin{cases}1&\mbox{if }h\equiv i+jq\\\ 0&\mbox{else}\end{cases}$ | 0
| $j\in I_{2}$ | ${\begin{cases}q^{2n-3}&\mbox{if }q(h+i)\equiv j\\\ 0&\mbox{else}\end{cases}}$ | ${\begin{cases}|\Phi(n-2,q)|+1&\mbox{if}^{*}\ i+j\equiv h+t\\\ q^{2n-5}+(-q)^{n-3}&\mbox{else}\end{cases}}$ | $q^{2n-5}$
| $j=D$ | 0 | $|\Phi(n-2,q)|$ | $q^{2n-5}$
$i=D$ | $j\in I_{1}$ | 0 | 0 | 1
| $j\in I_{2}$ | 0 | $|\Phi(n-2,q)|$ | $q^{2n-5}$
| $j=D$ | $q^{2}|\Phi(n-2,q)|$ | $|\Phi(n-2,q)|$ | $(q^{2}-1)^{2}+q^{4}|\Phi(n-4,q)|$

All the congruences are modulo $q^{2}-1$, (i.e., should read
$\pmod{q^{2}-1}$), except for the case when $i,j,h\in I_{2}$ which is
congruent modulo $q+1$, that is, ${}^{*}\ i+j\equiv h+t\pmod{q+1}$ where

$t=\begin{cases}0&\textnormal{ if $q$ is even,}\\\ \frac{q+1}{2}&\textnormal{
if $q$ is odd.}\end{cases}$

##  4 Commutativity of $\mathcal{X}(GU(n,q),\Phi(n,q))$

Now that the intersection numbers have been completely determined for all
values of $n$ and $q$, the next step is to construct character tables for
those association schemes that are commutative, i.e. $p_{ij}^{h}=p_{ji}^{h}$
for all $h,i,j\in[D]$. In fact, this is the case only when $q=2$.

###### Theorem 4.1

The association scheme $\mathcal{X}(GU(n,q),\Phi(n,q))$ is commutative if and
only if $q=2$.

Proof: For the sufficiency of the statement, take $q=2$ and compute the
intersection numbers using the table above in terms of $n$. The size of the
intersection matrices does not depend on $n$ when $n\geq 4$, and when
$n\in\\{2,3\\}$ we simply ignore the last row and last column of each matrix,
as well as the entire last matrix.555This does not, however, affect the check
for commutativity and the intersection matrices for these two cases of $n$ are
shown explicitly in the next section. The intersection matrices of
$\mathcal{X}(GU(n,2),\Phi(n,2))$ are as follows, where
$s=|\Phi(n-2,2)|=2^{2n-5}-(-2)^{n-3}-1$.

$B_{0}=I,\qquad B_{1}=\left(\begin{array}[]{rrrrrrr}0&1&0&0&0&0&0\\\
0&0&1&0&0&0&0\\\ 1&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0\\\ 0&0&0&1&0&0&0\\\
0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&1\end{array}\right),\qquad
B_{2}=\left(\begin{array}[]{rrrrrrr}0&0&1&0&0&0&0\\\ 1&0&0&0&0&0&0\\\
0&1&0&0&0&0&0\\\ 0&0&0&0&1&0&0\\\ 0&0&0&0&0&1&0\\\ 0&0&0&1&0&0&0\\\
0&0&0&0&0&0&1\end{array}\right)$

$B_{3}=\left(\begin{array}[]{ccccccc}0&0&0&1&0&0&0\\\ 0&0&0&0&0&1&0\\\
0&0&0&0&1&0&0\\\ 2^{2n-3}&0&0&s+1&s-(-2)^{n-2}+1&s-(-2)^{n-2}+1&2^{2n-5}\\\
0&0&2^{2n-3}&s-(-2)^{n-2}+1&s+1&s-(-2)^{n-2}+1&2^{2n-5}\\\
0&2^{2n-3}&0&s-(-2)^{n-2}+1&s-(-2)^{n-2}+1&s+1&2^{2n-5}\\\
0&0&0&s&s&s&2^{2n-5}\end{array}\right)$

$B_{4}=\left(\begin{array}[]{lllllll}0&0&0&0&1&0&0\\\ 0&0&0&1&0&0&0\\\
0&0&0&0&0&1&0\\\ 0&0&2^{2n-3}&s-(-2)^{n-2}+1&s+1&s-(-2)^{n-2}+1&2^{2n-5}\\\
0&2^{2n-3}&0&s-(-2)^{n-2}+1&s-(-2)^{n-2}+1&s+1&2^{2n-5}\\\
2^{2n-3}&0&0&s+1&s-(-2)^{n-2}+1&s-(-2)^{n-2}+1&2^{2n-5}\\\
0&0&0&s&s&s&2^{2n-5}\end{array}\right)$

$B_{5}=\left(\begin{array}[]{lllllll}0&0&0&0&0&1&0\\\ 0&0&0&0&1&0&0\\\
0&0&0&1&0&0&0\\\ 0&2^{2n-3}&0&s-(-2)^{n-2}+1&s-(-2)^{n-2}+1&s+1&2^{2n-5}\\\
2^{2n-3}&0&0&s+1&s-(-2)^{n-2}+1&s-(-2)^{n-2}+1&2^{2n-5}\\\
0&0&2^{2n-3}&s-(-2)^{n-2}+1&s+1&s-(-2)^{n-2}+1&2^{2n-5}\\\
0&0&0&s&s&s&2^{2n-5}\end{array}\right)$

$B_{6}=\left(\begin{array}[]{rrrrrrc}0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&1\\\
0&0&0&0&0&0&1\\\ 0&0&0&s&s&s&2^{2n-5}\\\ 0&0&0&s&s&s&2^{2n-5}\\\
0&0&0&s&s&s&2^{2n-5}\\\
4s&4s&4s&s&s&s&2^{2n-5}-(-2)^{n-1}-7\end{array}\right)\;\;\;\;$

It can be verified by inspection that $p_{ij}^{h}=p_{ji}^{h}$ for all values
of $h,i,$ and $j$.

For the necessity, we show that for each $q\geq 3$, there exist values of
$h,i,j$ such that $p_{ij}^{h}\neq p_{ji}^{h}$. Set $h=q$, $i=q^{2}-1$ and
$j=q^{2}$, which falls into the scenario of $h\in I_{1}$, and $i,j\in I_{2}$,
so

$p_{ij}^{h}=\begin{cases}q^{2n-3}&\mbox{if }q(h+i)\equiv j\pmod{q^{2}-1}\\\
0&\mbox{else}.\end{cases}$

We have $j=q^{2}\equiv 1\pmod{q^{2}-1}$, and

$q(h+i)\equiv q(q+(q^{2}-1))\equiv 1\pmod{q^{2}-1}$

so $p_{ij}^{h}=q^{2n-3}$. On the other hand, for $p_{ji}^{h}$ we have $i\equiv
0\pmod{q^{2}-1}$, but

$q(h+j)\equiv q(q+q^{2})\equiv q^{2}+q\equiv q+1\pmod{q^{2}-1},$

which is only congruent to $0\pmod{q^{2}-1}$ when $q=2$. Hence, $p_{ji}^{h}=0$
for all $q\geq 3$. Therefore, $p_{ij}^{h}\neq p_{ji}^{h}$, so the association
scheme is not commutative.

From this theorem, character tables for the association scheme can be
investigated only when $q=2$, which we do in the subsequent section for
generalized $n$.

##  5 Character tables of $\mathcal{X}(GU(n,2),\Phi(n,2))$

We now proceed with the intention of calculating the character table of
$\mathcal{X}(GU(n,q),\Phi(n,q))$ for $q=2$, which requires obtaining the
eigenvalues of the intersection matrices. We seek to construct the generalized
character table for arbitrary $n$. First, the $n=2$ and $n=3$ cases are dealt
with individually, since they produce a character table missing a row and a
column of those obtained from larger $n$.

###  5.1 Character tables of $\mathcal{X}(GU(2,2),\Phi(2,2))$ and
$\mathcal{X}(GU(3,2),\Phi(3,2))$

First, we compute the intersection matrices for
$\mathcal{X}=\mathcal{X}(GU(2,2),\Phi(2,2))$ and
$\mathcal{X}=\mathcal{X}(GU(3,2),\Phi(3,2))$ using the table of intersection
numbers produced in the previous section. We then compute the eigenvalues of
each of the intersection matrices for each case of $n$. These are arranged to
construct character tables satisfying the orthogonality conditions in
Proposition 1.2. The multiplicities $m_{i}$ are calculated using the formula

$\sum_{h=0}^{D}\frac{1}{k_{h}}p_{h}(i)\overline{p_{h}(j)}=\frac{|\Phi(n,2)|}{m_{j}}\delta_{ij}$

For each character table, $\omega=\frac{1}{2}(-1+\sqrt{3}\;i)$, i.e. the
primitive third root of unity, and the rightmost column entries are the
multiplicities, $m_{i}$.

Table 2: Character Table of $\mathcal{X}(GU(2,2),\Phi(2,2))$

$P\ =$ 1 1 1 2 2 2 1 1 $\omega$ $\overline{\omega}$ 2 $2\overline{\omega}$
$2{\omega}$ 1 1 $\overline{\omega}$ $\omega$ 2 $2{\omega}$
$2\overline{\omega}$ 1 1 1 1 $-1$ $-1$ $-1$ 2 1 $\omega$ $\overline{\omega}$
$-1$ $-\overline{\omega}$ $-{\omega}$ 2 1 $\overline{\omega}$ $\omega$ $-1$
$-{\omega}$ $-\overline{\omega}$ 2

Table 3: Character Table of $\mathcal{X}(GU(3,2),\Phi(3,2))$

$P\ =\ $ 1 1 1 8 8 8 1 1 $\omega$ $\overline{\omega}$ $-4$
$-4\overline{\omega}$ $-4{\omega}$ 3 1 $\overline{\omega}$ $\omega$ $-4$
$-4{\omega}$ $-4\overline{\omega}$ 3 1 1 1 $-1$ $-1$ $-1$ 8 1 $\omega$
$\overline{\omega}$ $2$ $2\overline{\omega}$ $2{\omega}$ 6 1
$\overline{\omega}$ $\omega$ $2$ $2{\omega}$ $2\overline{\omega}$ 6

Notice that

$|\Phi(n,2)|=\sum_{i=0}^{D}m_{i}=\begin{cases}9&\mbox{if }n=2\\\ 27&\mbox{if
}n=3,\end{cases}$

and the zeroth row gives the valencies, as desired. It is straightforward to
verify that the row sums are 0 other than the zeroth row, and that for all
$h,i,j\in[D]$,

$p_{ij}^{h}=\frac{1}{|\Phi|k_{h}}\sum_{l=0}^{D}p_{i}(l)p_{j}(l)\overline{p_{h}(l)}m_{l}.$

We note that the character tables of $\mathcal{X}(GU(2,2),\Phi(2,2))$ and
$\mathcal{X}(GU(3,2),\Phi(3,2))$ are counted in Hanaki’s classification of
small association schemes with character tables listed in “as09[10]” and
“as27[403]” in [13].

###  5.2 Character table of $\mathcal{X}(GU(n,2),\Phi(n,2))$ for $n\geq 4$

###### Theorem 5.1

For $n\geq 4$, the character table of $\mathcal{X}(GU(n,2),\Phi(n,2))$ is
given by

$P\ =\ $ $1$ $1$ $1$ $2^{2n-3}$ $2^{2n-3}$ $2^{2n-3}$ $2^{2n-3}$
$-(-2)^{n-1}-4$ $1$ $1$ $\omega$ $\overline{\omega}$ $-(-2)^{n-1}$
$-(-2)^{n-1}\overline{\omega}$ $-(-2)^{n-1}{\omega}$ $0$ $m_{1}$ $1$
$\overline{\omega}$ $\omega$ $-(-2)^{n-1}$ $-(-2)^{n-1}{\omega}$
$-(-2)^{n-1}\overline{\omega}$ $0$ $m_{2}$ $1$ $1$ $1$ $-(-2)^{n-2}$
$-(-2)^{n-2}$ $-(-2)^{n-2}$ $3(-2)^{n-2}-3$ $m_{3}$ $1$ $\omega$
$\overline{\omega}$ $-(-2)^{n-2}$ $-(-2)^{n-2}\overline{\omega}$
$-(-2)^{n-2}{\omega}$ $0$ $m_{4}$ $1$ $\overline{\omega}$ $\omega$
$-(-2)^{n-2}$ $-(-2)^{n-2}{\omega}$ $-(-2)^{n-2}\overline{\omega}$ $0$ $m_{5}$
$1$ $1$ $1$ $-(-2)^{n-3}$ $-(-2)^{n-3}$ $-(-2)^{n-3}$ $3(-2)^{n-3}-3$ $m_{6}$

where $\omega$ is the primitive third root of unity.

Proof: First we find the intersection matrices of
$\mathcal{X}(GU(n,2),\Omega(n,2))$ for arbitrary $n$, as shown in the proof of
Theorem 4.1. We use the help of MATLAB to compute the eigenvalues of each in
terms of $n$. The eigenvalues are arranged in such a way that they satisfy the
orthogonality conditions of Proposition 1.2. It is routine to also verify the
following formulae for each $h,i,j\in[D]$ using these intersection numbers and
the proposed character table as another verification

$p_{ij}^{h}=\frac{1}{|\Phi|\cdot
k_{h}}\sum\limits_{l=0}^{D}p_{i}(l)p_{j}(l)\overline{p_{h}(l)}m_{l}\qquad\qquad
p_{i}(h)p_{j}(h)=\sum_{l=0}^{D}p_{ij}^{l}p_{l}(h).$

From this we can derive the multiplicities as well.

###### Corollary 5.2

For $n\geq 4$, the multiplicities of the character table of
$\mathcal{X}(GU(n,2),\Phi(n,2))$ are

$m_{0}=1$ $m_{1}=m_{2}=(2^{n}-(-1)^{n})(2^{n-1}-(-1)^{n-1})/9$
$m_{3}=4(2^{n}-(-1)^{n})(2^{n-3}-(-1)^{n-3})/9$
$m_{4}=m_{5}=2(2^{n}-(-1)^{n})(2^{n-1}-(-1)^{n-1})/9$
$m_{6}=8(2^{n-1}-(-1)^{n-1})(2^{n-2}-(-1)^{n-2})/9$

Proof: These multiplicities are all computed again using the formula

$\sum_{h=0}^{D}\frac{1}{k_{h}}p_{h}(i)\overline{p_{h}(j)}=\frac{|\Phi(n,2)|}{m_{j}}\delta_{ij}.$

This finishes our investigation of the association scheme
$\mathcal{X}(GU(n,q),\Phi(n,q))$ for all values of $n$ and $q$, since all
intersection numbers were found, as well as all character tables that exist.

###  5.3 Fusion schemes of $\mathcal{X}(GU(n,2),\Phi(n,2))$

In the classification of association schemes, often knowledge of the character
table of its fusion or fission scheme is useful. A fusion scheme of a Schurian
association scheme is not necessarily Schurian; however, we observe that
$\mathcal{X}(GU(n,2),\Phi(n,2))$ has two fusion schemes, both of which are
Schurian. One is its symmetrization and the other is a 2-class scheme if $n=2$
or $3$, and a 3-class scheme if $n\geq 4$. Note that each of these fusion
schemes can be obtained by taking the semidirect product of $GU(n,q)$ and a
suitable cyclic group and acting on the set of isotropic vectors, $\Phi(n,q)$.

Before we discuss the character tables of interest, we first recall some basic
facts about the fusion and fission of a commutative schemes (cf. [2, 7, 14,
15, 11, 16, 17]).

###### Definition 5.3

Let $\mathcal{X}=\big{(}X,\\{R_{i}\\}_{0\leq i\leq d}\big{)}$ and
$\widehat{\mathcal{X}}=\big{(}X,\\{\widehat{R}_{\alpha}\\}_{0\leq\alpha\leq
e}\big{)}$ be commutative association schemes defined on $X$. If for every
$i\in[d]$, $R_{i}\subseteq\widehat{R}_{\alpha}$ for some $\alpha\in[e]$, then
we say that $\widehat{\mathcal{X}}$ is a fusion scheme of $\mathcal{X}$, and
$\mathcal{X}$ is a fission scheme of $\widehat{\mathcal{X}}$. For the
notation, we will denote all the symbols belonging to $\widehat{\mathcal{X}}$
by a $\widehat{}$ placed over the symbols (such as,
$\widehat{p}_{\alpha\beta}^{\gamma}$, $\widehat{m}_{i}$, $\widehat{k}_{i}$,
$\widehat{P}$, $\widehat{p}_{j}(i)$, etc.) whenever we need to distinguish
them from those belonging to $\mathcal{X}$.

The following two criteria for fusion will be used repeatedly in this section.

###### Proposition 5.4

For a given scheme $\mathcal{X}=\big{(}X,\\{R_{i}\\}_{0\leq i\leq d}\big{)}$
and a partition $\Lambda=\\{\Lambda_{\alpha}\\}_{0\leq\alpha\leq e}$ of $[d]$
with $\Lambda_{0}=\\{0\\}$,
$\widehat{\mathcal{X}}=\big{(}X,\\{\widehat{R}_{\alpha}\\}_{0\leq\alpha\leq
e}\big{)}$ becomes a scheme with the relations defined by
$\widehat{R}_{\alpha}=\bigcup\limits_{i\in\Lambda_{\alpha}}R_{i}$, for
$\alpha\in[e]$, if and only if

  * $(i)$

$\widehat{R}^{\prime}_{\alpha}\ \ =\ \
\bigcup\limits_{i\in\Lambda_{\alpha}}R^{\prime}_{i}\ \ =\ \
\bigcup\limits_{j\in\Lambda_{\alpha^{\prime}}}R_{j}\ \ =\ \
\widehat{R}_{\alpha^{\prime}}$ for some $\alpha^{\prime}\in[e]$, and

  * $(ii)$

for any $\alpha,\beta,\gamma\in[e]$, and any $h,k\in\Lambda_{\gamma}$,

$\sum\limits_{i\in\Lambda_{\alpha}}\
\sum\limits_{j\in\Lambda_{\beta}}p_{ij}^{h}=\sum\limits_{i\in\Lambda_{\alpha}}\
\sum\limits_{j\in\Lambda_{\beta}}\
p_{ij}^{k}\equiv\widehat{p}_{\alpha\beta}^{\gamma}.$

###### Proposition 5.5

([2, 15, 17]) Let $\mathcal{X}=\big{(}X,\\{R_{i}\\}_{0\leq i\leq d}\big{)}$ be
a scheme, and $\Lambda=\\{\Lambda_{\alpha}\\}_{0\leq\alpha\leq e}$ be a
partition of $[d]$ such that $\Lambda_{0}=\\{0\\}$. Suppose for every
$\alpha\in[e]$,
$\bigcup\limits_{i\in\Lambda_{\alpha}}R_{i^{\prime}}=\bigcup\limits_{j\in\Lambda_{\alpha^{\prime}}}R_{j}$
for some $\alpha^{\prime}\in[e]$. Then $\Lambda$ gives rise to a fusion scheme
$\widehat{\mathcal{X}}=\big{(}X,\\{\widehat{R}_{\alpha}\\}_{0\leq\alpha\leq
e}\big{)}$ with
$\widehat{R}_{\alpha}=\bigcup\limits_{i\in\Lambda_{\alpha}}R_{i}$ if and only
if there exists a partition
$\Lambda^{*}=\\{\Lambda_{\alpha}^{*}\\}_{0\leq\alpha\leq e}$ of $[d]$ with
$\Lambda_{0}^{*}=\\{0\\}$ such that each
$\big{(}\Lambda_{\beta}^{*},\Lambda_{\alpha}\big{)}$-block of the character
table $P$ of $\mathcal{X}$ has a constant row sum. In this case, the constant
row sum $\sum\limits_{j\in\Lambda_{\alpha}}p_{j}(i)$ for
$i\in\Lambda_{\beta}^{*}$ of the block
$(\Lambda_{\beta}^{*},\Lambda_{\alpha})$ is the $(\beta,\alpha)$-entry
$\widehat{p}_{\alpha}(\beta)$ of the fusion character table $\widehat{P}$.

That is to say, character tables of fusion schemes are simply the sum of the
characters of the relations that are fused together, removing the duplicate
rows that are created.

We note that the fusion scheme obtained by fusing each non-symmetric relation
of a scheme $\mathcal{X}$ with its symmetric conjugate is called the
_symmetrization_ of $\mathcal{X}$. It is clear that every non-symmetric
commutative association scheme has at least one symmetric fusion scheme,
namely, its symmetrization.

###### Corollary 5.6

Each of $\mathcal{X}(GU(2,2),\Phi(2,2))$ and $\mathcal{X}(GU(3,2),\Phi(3,2))$
has its symmetrization. The character tables, denoted by $\bar{P}$, of their
symmetrizations are given by

$\bar{P}(GU(2,2),\Phi(2,2))=\begin{array}[]{|cccc|r|}\hline\cr 1&2&2&4&1\\\
1&-1&2&-2&2\\\ 1&2&-1&-2&2\\\ 1&-1&-1&1&4\\\
\hline\cr\end{array};\qquad\bar{P}(GU(3,2),\Phi(3,2))=\begin{array}[]{|cccc|r|}\hline\cr
1&2&8&16&1\\\ 1&-1&-4&4&6\\\ 1&2&-1&-2&8\\\ 1&-1&2&-2&12\\\
\hline\cr\end{array}.$

Each of $\mathcal{X}(GU(2,2),\Phi(2,2))$ and $\mathcal{X}(GU(3,2),\Phi(3,2))$
also has a 2-class fusion scheme whose relations $\widehat{R}_{1}$ and
$\widehat{R}_{2}$ are obtained as $\widehat{R}_{1}=R_{1}\cup R_{2}$ and
$\widehat{R}_{2}=R_{3}\cup R_{4}\cup R_{5}$. The character tables for these
fusion schemes, all of which are shown to be Schurian association schemes, are
given by

$\widehat{P}=\begin{array}[]{|ccc|r|}\hline\cr 1&2&6&1\\\ 1&2&-3&2\\\
1&-1&0&6\\\
\hline\cr\end{array};\qquad\widehat{P}=\begin{array}[]{|ccc|r|}\hline\cr
1&2&24&1\\\ 1&2&-3&8\\\ 1&-1&0&18\\\ \hline\cr\end{array}.$

Proof: It is clear from Proposition 5.5 and the character tables from Section
5.1.

The character tables for these fusion schemes are listed in Hanaki’s
classification of small association schemes with character tables listed in
“as09[5], as27[383], as09[2]” and “as27[2]” in [13], respectively.

###### Corollary 5.7

For $n\geq 4$, the character table of the symmetrization of
$\mathcal{X}(GU(n,2),\Phi(n,2))$ is given by

$\bar{P}=\begin{array}[]{|ccccc|r|}\hline\cr
1&2&2^{2n-3}&2^{2n-2}&2^{2n-3}-(-2)^{n-1}-4&1\\\
1&2&-(-2)^{n-2}&(-2)^{n-1}&3(-2)^{n-2}-3&4(2^{n}-(-1)^{n})(2^{n-3}-(-1)^{n-3})/9\\\
1&2&-(-2)^{n-3}&(-2)^{n-2}&3(-2)^{n-3}-3&8(2^{n-1}-(-1)^{n-1})(2^{n-2}-(-1)^{n-2})/9\\\
1&-1&-(-2)^{n-1}&(-2)^{n-1}&0&(2^{2n}+(-2)^{n}-2)/9\\\
1&-1&-(-2)^{n-2}&(-2)^{n-2}&0&(2^{2n+1}+2(-2)^{n}-4)/9\\\
\hline\cr\end{array}.$

The character table of the 3-class fusion scheme whose relations
$\widehat{R}_{1}$, $\widehat{R}_{2}$ and $\widehat{R}_{3}$ are defined by
$\widehat{R}_{1}=R_{1}\cup R_{2}$, $\widehat{R}_{2}=R_{3}\cup R_{4}\cup R_{5}$
and $\widehat{R}_{3}=R_{6}$ is given by

$\widehat{P}\ =\ $ $1$ $2$ $3\cdot 2^{2n-3}$ $2^{2n-3}$ $-(-2)^{n-1}-4$ $1$
$1$ $2$ $-3(-2)^{n-2}$ $3(-2)^{n-2}-3$
$4(2^{n}-(-1)^{n})(2^{n-3}-(-1)^{n-3})/9$ $1$ $2$ $-3(-2)^{n-3}$
$3(-2)^{n-3}-3$ $8(2^{n-1}-(-1)^{n-1})(2^{n-2}-(-1)^{n-2})/9$ $1$ $-1$ $0$ $0$
$(2^{2n}+(-2)^{n}-2)/3$ .

Proof: Omitted.

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