Krein parameters of fiber-commutative coherent configurations

TL;DR

This paper demonstrates that Krein parameters for fiber-commutative coherent configurations are essentially unique, simplifying the Krein condition and the absolute bound, with implications for generalized quadrangles.

Contribution

It establishes the essential uniqueness of Krein parameters in fiber-commutative coherent configurations and simplifies related bounds and conditions.

Findings

Krein parameters are essentially uniquely defined for fiber-commutative coherent configurations.

The Krein condition reduces to checking positive semidefiniteness of finitely many matrices.

Simplified absolute bound using matrices of Krein parameters.

Abstract

For fiber-commutative coherent configurations, we show that Krein parameters can be defined essentially uniquely. As a consequence, the general Krein condition reduces to positive semidefiniteness of finitely many matrices determined by the parameters of a coherent configuration. We mention its implications in the coherent configuration defined by a generalized quadrangle. We also simplify the absolute bound using the matrices of Krein parameters.

Equations

$$\mathfrak{A}=\underset{s\in S}{⨁}{\mathfrak{C}}_s,$$

$${\epsilon }_{i,j}^s{\epsilon }_{k,l}^s$$

$${{\epsilon }_{i,j}^s}^{\ast }$$

$${\epsilon }_{i,j}^s\in \bigcup _{k,l=1}^f{\mathfrak{A}}_{k,l}(i,j\in F_s,s\in S).$$

$${\epsilon }_{i,i}^s\in \bigcup _{k=1}^f{\mathfrak{A}}_k(i\in F_s,s\in S).$$

$$\overline{{\epsilon }_{i,j}^s}\in \bigcup _{k,l=1}^f{\mathfrak{A}}_{k,l}(i,j\in F_s,s\in S).$$

$$\overline{{\epsilon }_{i,j}^s}={\epsilon }_{i,j}^{\hat{s}}(i,j\in F_s,s\in S).$$

$${\epsilon }_1=\sum _{k=1}\frac{1}{|X_k|}{J}_{k,k}.$$

$${\epsilon }_{k,l}^1=\frac{1}{\sqrt{|X_k||X_l|}}{J}_{k,l}$$

$$n_{\lambda }{\epsilon }_{\lambda }\circ n_{\mu }{\epsilon }_{\mu }=\sum _{\nu \in \Lambda }{q}_{\lambda ,\mu }^{\nu }{n}_{\nu }{\epsilon }_{\nu }.$$

$$\sum _{i,j\in F_s}\sum _{k,l\in F_t}{b}_{i,j}{c}_{k,l}{q}_{(i,j,s),(k,l,t)}^{(m,n,u)}.$$

$${\tilde{Q}}_{s,t}^u(B,C)\in {\mathcal{P}}_{F_u}(B\in {\mathcal{P}}_{F_s},C\in {\mathcal{P}}_{F_t}).$$

$$\eta (i,j,s)=(k,l)\text{ if }{\epsilon }_{i,j}^s\in {\mathfrak{A}}_{k,l}.$$

$$F_s^{\prime }=\{k\mid 1\le k\le f,(k,k)\in \{\eta (i,i,s)\mid i\in F_s\}\}.$$

$${\epsilon }_{\lambda }\circ {\epsilon }_{\mu }=\frac{{\delta }_{\eta (\lambda ),\eta (\mu )}}{n_{\lambda }}\sum _{\begin{array}{c}\nu \in \Lambda \\ \eta (\nu )=\eta (\lambda )\end{array}}{q}_{\lambda ,\mu }^{\nu }{\epsilon }_{\nu }.$$

$$F_s=\{i\in \{1,\dots ,f\}\mid \dim ({\mathfrak{C}}_s\cap {\mathfrak{A}}_{i,i})=1\}.$$

$${\epsilon }_{i,j}^s\circ {\epsilon }_{i,j}^t=\frac{1}{\sqrt{|X_i||X_j|}}\sum _{\begin{array}{c}u\in S\\ F_u\ni i,j\end{array}}{q}_{(i,j,s),(i,j,t)}^{(i,j,u)}{\epsilon }_{i,j}^u.$$

$$(Q_{s,t}^u{)}_{i,j}=q_{(i,j,s),(i,j,t)}^{(i,j,u)}.$$

$$Q_{s,t}^1={\delta }_{\hat{s},t}\mathrm{tr}({\epsilon }_{j,j}^t)J_{F_{s,t,1}}.$$

$$\left({\epsilon }_{i,j}^s\circ {\epsilon }_{i,j}^t\right)\sum _{k,l\in F_1}{\epsilon }_{k,l}^1$$

$$=\frac{1}{\sqrt{|X_i||X_j|}}(Q_{s,t}^1{)}_{i,j}\sum _{l\in F_1}{\epsilon }_{i,l}^1.$$

$$\mathrm{tr}\left(({\epsilon }_{i,j}^s\circ {\epsilon }_{i,j}^t)\sum _{k,l\in F_1}{\epsilon }_{k,l}^1\right)$$

$$=\frac{1}{\sqrt{|X_i||X_j|}}\sum _{x,y\in X}({\epsilon }_{i,j}^s\circ {\epsilon }_{i,j}^t{)}_{x,y}(\text{by eqrefmatunit})$$

$$=\frac{1}{\sqrt{|X_i||X_j|}}\mathrm{tr}({{\epsilon }_{i,j}^s}^T{\epsilon }_{i,j}^t)$$

$$=\frac{1}{\sqrt{|X_i||X_j|}}\mathrm{tr}(\overline{{\epsilon }_{j,i}^s}{\epsilon }_{i,j}^t)(\text{by }{{\epsilon }_{i,j}^s}^{\ast }={\epsilon }_{j,i}^s)$$

$$=\frac{1}{\sqrt{|X_i||X_j|}}\mathrm{tr}({\epsilon }_{j,i}^{\hat{s}}{\epsilon }_{i,j}^t)$$

$$=\frac{1}{\sqrt{|X_i||X_j|}}{\delta }_{\hat{s},t}\mathrm{tr}({\epsilon }_{j,j}^t).$$

$$(\mathbf{z}{)}_k=\frac{({\mathbf{z}}_s{)}_k({\mathbf{z}}_t{)}_k}{({\mathbf{z}}_u{)}_k}(k\in F_{s,t,u}).$$

$$({\mathbf{z}}_s^{\ast }{\mathbf{z}}_s{)}_{i,j}{\epsilon }_{i,j}^s\circ ({\mathbf{z}}_t^{\ast }{\mathbf{z}}_t{)}_{i,j}{\epsilon }_{i,j}^t$$

$$=({\mathbf{z}}_s^{\ast }{\mathbf{z}}_s{)}_{i,j}({\mathbf{z}}_t^{\ast }{\mathbf{z}}_t{)}_{i,j}({\epsilon }_{i,j}^s\circ {\epsilon }_{i,j}^t)$$

$$=\frac{({\mathbf{z}}_s^{\ast }{\mathbf{z}}_s{)}_{i,j}({\mathbf{z}}_t^{\ast }{\mathbf{z}}_t{)}_{i,j}}{\sqrt{|X_i||X_j|}}\sum _{\begin{array}{c}u\in S\\ F_u\ni i,j\end{array}}(Q_{s,t}^u{)}_{i,j}{\epsilon }_{i,j}^u$$

$$=\frac{1}{\sqrt{|X_i||X_j|}}\sum _{\begin{array}{c}u\in S\\ F_u\ni i,j\end{array}}\frac{({\mathbf{z}}_s^{\ast }{\mathbf{z}}_s{)}_{i,j}({\mathbf{z}}_t^{\ast }{\mathbf{z}}_t{)}_{i,j}}{({\mathbf{z}}_u^{\ast }{\mathbf{z}}_u{)}_{i,j}}(Q_{s,t}^u{)}_{i,j}({\mathbf{z}}_u^{\ast }{\mathbf{z}}_u{)}_{i,j}{\epsilon }_{i,j}^u.$$

$$b_{m,n}{c}_{m,n}(Q_{s,t}^u{)}_{m,n}=(B^{\prime }\circ C^{\prime }\circ Q_{s,t}^u{)}_{m,n},$$

$$R_{1,1,1}$$

$$R_{1,1,2}$$

$$R_{1,1,3}$$

Full text

1. Introduction

Coherent configurations are defined by D. G. Higman in [4]. A special class of coherent configurations, called homogeneous coherent configurations, are also known as association schemes. The Krein condition asserts that Krein parameters of a commutative association scheme are non-negative real numbers (see [1, Theorem 3.8]), and it can rule out the existence of some putative association schemes. A generalization of this property was formulated by Hobart [5], who proved that certain matrices determined by the parameters are positive semidefinite. However, some complication arises due to the fact that an analogue of Krein parameters cannot be defined uniquely. This seems to be an obstacle to develop a nice theory for coherent configurations parallel to commutative association schemes. In this paper, we restrict ourselves to fiber-commutative coherent configurations. This restriction enables us to define a basis of matrix units almost uniquely for each simple two-sided ideals of the adjacency algebra $\mathfrak{A}$ of the coherent configuration. Since the algebra $\mathfrak{A}$ is closed with respect to the entry-wise product, we can define Krein parameters as the expansion coefficients of the entry-wise product of two basis elements. These Krein parameters can then be collected in a matrix form, which we call the matrix of Krein parameters, and we show that this is a positive semidefinite hermitian matrix. This is a generalization of the Krein condition for commutative association schemes to fiber-commutative coherent configurations. Our main theorem asserts that, the general Krein condition formulated by Hobart [5] which in general consists of infinitely many inequalities, is equivalent to the positive semidefiniteness of the finitely many matrices of Krein parameters. As an illustration, we consider the fiber-commutative coherent configurations defined by generalized quadrangles. We write down all matrices of Krein parameters, and show that one cannot derive any consequence other than the well-known inequalities established in [2, 3]. Moreover, we also simplify the absolute bounds for fiber-commutative coherent configurations due to [6].

This paper is organized as follows. In Section 2, we prepare notation and formulate the Krein condition for coherent configurations. In Section 3, we define matrices of Krein parameters for fiber-commutative coherent configurations, and give our main theorem. In Section 4, we compute the matrices of Krein parameters for the coherent configuration obtained from a generalized quadrangle. In Section 5, we apply our result to simplify the absolute bounds for fiber-commutative coherent configurations.

2. Preliminaries

For a finite set $X$, we denote by $\mathrm{M}_{X}(\mathbb{C})$ the algebra of square matrices with entries in $\mathbb{C}$ whose rows and columns are indexed by $X$. We also denote by $J_{X}$ the all-ones matrix in $\mathrm{M}_{X}(\mathbb{C})$.

Definition 2.1**.**

Let $X=\coprod_{i=1}^{f}X_{i}$ be a partition of a finite set $X$. For all pairs $i,j\in\{1,\dots,f\}$, let $X_{i}\times X_{j}=\coprod_{k=1}^{r_{i,j}}R_{i,j,k}$ be a partition, and let $\mathcal{I}=\{(i,j,k)\mid 1\leq i,j\leq f,1\leq k\leq r_{i,j}\}$. For $I\in\mathcal{I}$, let $A_{I}\in\mathrm{M}_{X}(\mathbb{C})$ denote the adjacency matrix of the relation $R_{i,j,k}\subset X\times X$. The pair $\mathcal{C}=(\mathcal{X},\{R_{I}\}_{I\in\mathcal{I}})$ is called a coherent configuration if the following conditions hold:

1. (i)

For each $i\in\{1,\dots,f\}$, there exists $k\in\{1,\dots,r_{i,i}\}$ such that $A_{i,i,k}=I_{X_{i}}$, where $I_{X_{i}}$ is the $\{0,1\}$-matrix indexed by $X\times X$ with $1$ on $(x,x)$-entry for $x\in X_{i}$ and [math] otherwise. 2. (ii)

$\sum_{I\in\mathcal{I}}A_{I}=J_{X}$. 3. (iii)

For any $I\in\mathcal{I}$, there exists $I^{\prime}\in\mathcal{I}$ such that $A_{I}^{T}=A_{I^{\prime}}$. 4. (iv)

For any $I,J\in\mathcal{I}$, $A_{I}A_{J}=\sum_{K\in\mathcal{I}}p_{I,J}^{K}A_{K}$ for some scalars $p_{I,J}^{K}$.

Each subset $X_{i}\subset X$ is called a fiber, $|\mathcal{I}|$ is called the rank, and $p_{I,J}^{K}$ are called the parameters of $\mathcal{C}$.

For the remainder of this section, we fix a coherent configuration $\mathcal{C}$ as in Definition 2.1. Let $\mathfrak{A}$ be the subalgebra of $\mathrm{M}_{X}(\mathbb{C})$ spanned by $\{A_{I}\mid I\in\mathcal{I}\}$. This algebra is called the adjacency algebra of $\mathcal{C}$. The subspace $\mathfrak{A}_{k,l}$ is defined as the subspace consisting of matrices whose entries are zero except those indexed by $X_{k}\times X_{l}$. For brevity, we write $\mathfrak{A}_{k}=\mathfrak{A}_{k,k}$. Each $\mathfrak{A}_{k}$ forms the adjacency algebra of a coherent configuration with single fiber, that is, an association scheme on $X_{k}$.

Definition 2.2**.**

The coherent configuration $\mathcal{C}$ is said to be fiber-commutative if the algebra $\mathfrak{A}_{k}$ is commutative for all $k\in\{1,\dots,f\}$. Similarly, $\mathcal{C}$ is said to be fiber-symmetric if the algebra $\mathfrak{A}_{k}$ consists only of symmetric matrices for all $k\in\{1,\dots,f\}$.

Let $\{\Delta_{s}\mid s\in S\}$ be a set of representatives of all irreducible matrix representations of $\mathfrak{A}$ over $\mathbb{C}$ satisfying $\Delta_{s}(A)^{*}=\Delta_{s}(A^{*})$ for any $A\in\mathfrak{A}$, where $*$ denotes the transpose-conjugate. Since $\mathfrak{A}$ is semisimple, $\mathfrak{A}$ is completely reducible. In other words, $\mathfrak{A}$ is decomposed into

[TABLE]

where $\mathfrak{C}_{s}$ is a simple two-sided ideal affording $\Delta_{s}$. Moreover, for each $s\in S$, $\Delta_{s}|_{\mathfrak{C}_{s}}$ is an isomorphism from $\mathfrak{C}_{s}$ to $\mathrm{M}_{e_{s}}(\mathbb{C})$. This implies that there exists a basis $\{\varepsilon_{i,j}^{s}\in\mathfrak{A}\mid i,j\in F_{s}\}$ of $\mathfrak{C}_{s}$ satisfying

[TABLE]

where $|F_{s}|=e_{s}$. Note that there is a good reason not to take $F_{s}=\{1,\dots,e_{s}\}$. This will become clear after Lemma 3.1. By [6, Theorem 8], we can choose $\varepsilon_{i,j}^{s}$ in such a way that

[TABLE]

Note that, since $\mathfrak{A}_{k,l}\mathfrak{A}_{k^{\prime},l^{\prime}}=0$ if $l\neq k^{\prime}$, (3) implies

[TABLE]

This is also mentioned in the proof of [6, Theorem 8].

Since $\overline{\mathfrak{C}_{s}}$ is also a simple two-sided ideal, there exists $\hat{s}\in S$ such that $\mathfrak{C}_{\hat{s}}=\overline{\mathfrak{C}_{s}}$. If $\mathcal{C}$ is fiber-symmetric, then $s=\hat{s}$ for all $s\in S$ by (4). Note that $\{\overline{\varepsilon_{i,j}^{s}}\mid i,j\in F_{s}\}$ is a basis of $\mathfrak{C}_{\hat{s}}$ satisfying (1). Since $\overline{\mathfrak{A}_{k,l}}=\mathfrak{A}_{k,l}$ for all $k,l\in\{1,\dots,f\}$,

[TABLE]

This implies that we can choose $\{\varepsilon_{i,j}^{s}\mid i,j\in F_{s}\}$ and $\{\varepsilon_{i,j}^{\hat{s}}\mid i,j\in F_{s}\}$ in a manner compatible with complex conjugation.

Definition 2.3**.**

For each $s\in S$, a basis $\{\varepsilon_{i,j}^{s}\mid i,j\in F_{s}\}$ of $\mathfrak{C}_{s}$ is called a basis of matrix units for $\mathfrak{C}_{s}$ if (1) and (3) hold. If $\{\varepsilon_{i,j}^{s}\mid i,j\in F_{s}\}$ is a basis of matrix units for $\mathfrak{C}_{s}$ for each $s\in S$, then their union is called bases of matrix units for $\mathfrak{A}$ provided that $F_{s}=F_{\hat{s}}$ and

[TABLE]

Note that bases of matrix units are not determined uniquely (see [4]), but we will see later that they are essentially unique for the fiber-commutative case.

Lemma 2.4**.**

*The center of $\mathfrak{A}$ is contained in $\bigoplus_{k=1}^{f}\mathfrak{A}_{k}$. *

Proof.

This is immediate from (4), since $\sum_{i\in F_{s}}\varepsilon_{i,i}^{s}$ is the central idempotent corresponding to $\mathfrak{C}_{s}$. ∎

Let $J_{k,l}$ be the matrix in $\mathfrak{A}$ with $1$ in all entries indexed by $X_{k}\times X_{l}$ and [math] otherwise. Without loss of generality, we may assume that $\mathfrak{C}_{1}=\mathfrak{A}\varepsilon_{1}\mathfrak{A}$, where

[TABLE]

This implies that we may also assume that

[TABLE]

for any $k,l\in F_{1}$, where $F_{1}=\{1,\dots,f\}$.

For the reminder of this section, we fix bases of matrix units $\{\varepsilon_{i,j}^{s}\mid s\in S,\;i,j\in F_{s}\}$ for $\mathfrak{A}$. Let $\Lambda_{s}=F_{s}^{2}\times\{s\}$ for each $s\in S$ and $\Lambda=\coprod_{s\in S}\Lambda_{s}.$ Moreover, we denote $\varepsilon_{\lambda}=\varepsilon_{i,j}^{s}$ for $\lambda=(i,j,s)\in\Lambda$. Define $n_{\lambda}=\sqrt{|X_{k}||X_{l}|}$, where $\lambda\in\Lambda$ and $\varepsilon_{\lambda}\in\mathfrak{A}_{k,l}$. Let $\circ$ denote the Hadamard (entry-wise) product of matrices. Since $\mathfrak{A}$ is closed with respect to $\circ$, there exist $q_{\lambda,\mu}^{\nu}\in\mathbb{C}$ such that

[TABLE]

Definition 2.5**.**

The complex numbers $q_{\lambda,\mu}^{\nu}$ appearing in (7) are called Krein parameters with respect to bases of matrix units $\{\varepsilon_{\lambda}\mid\lambda\in\Lambda\}$.

Let $\mathcal{P}_{F}$ denote the set of the all positive semidefinite hermitian matrices in $\mathrm{M}_{F}(\mathbb{C})$.

Theorem 2.6** (Krein conditions [5, Lemma 1]).**

For any $s,t,u\in S$, $B=(b_{i,j})\in\mathrm{M}_{F_{s}}(\mathbb{C})$ and $C=(c_{i,j})\in\mathrm{M}_{F_{t}}(\mathbb{C})$, let $\tilde{Q}_{s,t}^{u}(B,C)$ denote the matrix in $\mathrm{M}_{F_{u}}(\mathbb{C})$ whose $(m,n)$-entry is

[TABLE]

Then

[TABLE]

Let $\eta$ be the mapping from $\Lambda$ to $\{1,\dots,f\}^{2}$ defined by $\varepsilon_{\lambda}\in\mathfrak{A}_{\eta(\lambda)}$ for $\lambda\in\Lambda$, or equivalently,

[TABLE]

Lemma 2.7**.**

For each $s\in S$, define

[TABLE]

Then $\eta(\Lambda_{s})={F^{\prime}_{s}}^{2}$.

Proof.

First, we prove $\eta(\Lambda_{s})\subset{F^{\prime}_{s}}^{2}$. For $(i,j,s)\in\Lambda_{s}$, suppose $\eta(i,j,s)=(k,l)$. Namely, $\varepsilon_{i,j}^{s}\in\mathfrak{A}_{k,l}$. By (1) and (4), $\varepsilon_{i,i}^{s}\in\mathfrak{A}_{k}$ and $\varepsilon_{j,j}^{s}\in\mathfrak{A}_{l}$ hold. Thus $\eta(i,i,s)=(k,k)$ and $\eta(j,j,s)=(l,l)$ and these mean $k,l\in F^{\prime}_{s}$.

Conversely, suppose $\eta(i,i,s)=(k,k)$ and $\eta(j,j,s)=(l,l)$, where $i,j\in F_{s}$. Then $\varepsilon_{i,i}^{s}\in\mathfrak{A}_{k}$ and $\varepsilon_{j,j}^{s}\in\mathfrak{A}_{l}$. By (1), we obtain $\varepsilon_{i,j}^{s}\in\mathfrak{A}_{k,l}$. Thus $(k,l)=\eta(i,j,s)\in\eta(\Lambda_{s})$. ∎

Lemma 2.8**.**

Let $\lambda,\mu,\nu\in\Lambda$. If $q_{\lambda,\mu}^{\nu}\neq 0$, then $\eta(\lambda)=\eta(\mu)=\eta(\nu)$.

Proof.

By the definition of $\eta$, $\varepsilon_{\lambda}\in\mathfrak{A}_{\eta(\lambda)}$, $\varepsilon_{\mu}\in\mathfrak{A}_{\eta(\mu)}$, and $\varepsilon_{\nu}\in\mathfrak{A}_{\eta(\nu)}$ hold. If $\eta(\lambda)\neq\eta(\mu)$, then $\varepsilon_{\lambda}\circ\varepsilon_{\mu}=0$, and this means $q_{\lambda,\mu}^{\nu}=0$ for any $\nu\in\Lambda$. If $\eta(\lambda)=\eta(\mu)\neq\eta(\nu)$, then $\varepsilon_{\lambda}\circ\varepsilon_{\mu}\in\mathfrak{A}_{\eta(\lambda)}$ and this means that $q_{\lambda,\mu}^{\nu}=0$. ∎

By Lemma 2.8, the expansion (7) is simplified to

[TABLE]

For brevity, we write a basis of matrix units $\{\varepsilon_{i,j}^{s}\mid i,j\in F_{s}\}$ as $\{\varepsilon_{i,j}^{s}\}$ and we define $Z\circ\{\varepsilon_{i,j}^{s}\}=\{\zeta_{i,j}\varepsilon_{i,j}^{s}\mid i,j\in F_{s}\}$ for a matrix $Z=(\zeta_{i,j})\in\mathrm{M}_{F_{s}}(\mathbb{C})$.

Lemma 2.9**.**

*Fix $s\in S$. Let $Z=(\zeta_{i,j})\in\mathrm{M}_{F_{s}}(\mathbb{C})$. If $Z\circ\{\varepsilon_{i,j}^{s}\}$ is a basis of matrix units for $\mathfrak{C}_{s}$, then $Z$ is a positive semidefinite matrix with rank one and $|\zeta_{i,j}|=1$ for all $i,j\in F_{s}$. *

Proof.

Since $Z\circ\{\varepsilon_{i,j}^{s}\}$ is a basis of matrix units for $\mathfrak{C}_{s}$, $Z\circ\{\varepsilon_{i,j}^{s}\}$ satisfies (1). This means that $\zeta_{i,j}^{s}\zeta_{j,k}^{s}=\zeta_{i,k}^{s}$ and $\overline{\zeta_{j,i}^{s}}=\zeta_{i,j}^{s}$ for any $i,j,k\in F_{s}$. Thus $|\zeta_{i,j}|=1$ holds. Moreover, Since $\zeta_{i,j}^{s}=\overline{\zeta_{1,i}^{s}}\zeta_{1,j}^{s}$ holds, $Z$ is expressed as $Z=\mathbf{z}^{*}\mathbf{z},$ where $\mathbf{z}=(\zeta_{1,j})_{j\in F_{s}}.$ Thus $Z$ is a positive semidefinite matrix with rank one. ∎

3. Fiber-commutative coherent configurations

In this section, we also use the same notation as the previous section. In other words, $\mathfrak{A}$ is the adjacency algebra of a coherent configuration $\mathcal{C}$, $\mathfrak{A}$ is decomposed into the direct sum of simple ideals as $\mathfrak{A}=\bigoplus_{s\in S}\mathfrak{C}_{s}$. Moreover $\{\varepsilon_{i,j}^{s}\}$ is a basis of matrix units for $\mathfrak{C}_{s}$, and their union over $s\in S$ is bases of matrix units for $\mathfrak{A}$. In this section, we assume that the coherent configuration $\mathcal{C}$ is fiber-commutative.

Lemma 3.1**.**

For any $s\in S$ and $k,l\in\{1,\dots,f\}$, $\mathrm{dim}(\mathfrak{C}_{s}\cap\mathfrak{A}_{k,l})\leq 1$. In other words, the number of pairs $(i,j)$ satisfying $\varepsilon_{i,j}^{s}\in\mathfrak{A}_{k,l}$ is at most $1$.

Proof.

By (3), for each $i,j\in F_{s}$, there exist $k,l$ such that $\varepsilon_{i,j}^{s}\in\mathfrak{A}_{k,l}$. Thus it suffices to show $\#\{(i,j,s)\in\Lambda_{s}\mid\varepsilon_{i,j}^{s}\in\mathfrak{A}_{k,l}\}\leq 1$. Suppose $\varepsilon_{i,j}^{s},\varepsilon_{i^{\prime}j^{\prime}}^{s}\in\mathfrak{A}_{k,l}$ and $i\neq i^{\prime}$. By (4), we have $\varepsilon_{i,i}^{s},\varepsilon_{i^{\prime},i^{\prime}}^{s}\in\mathfrak{A}_{k}$. Thus $\varepsilon_{i,i^{\prime}}^{s}=\varepsilon_{i,i}\varepsilon_{i,i^{\prime}}\varepsilon_{i^{\prime},i^{\prime}}\in\mathfrak{A}_{k}$ holds. Since $\mathfrak{A}_{k}$ is commutative, $\varepsilon_{i,i^{\prime}}^{s}=\varepsilon_{i,i}^{s}\varepsilon_{i,i^{\prime}}^{s}=\varepsilon_{i,i^{\prime}}^{s}\varepsilon_{i,i}^{s}=0$, and this is a contradiction. Therefore, we obtain $i=i^{\prime}$ and similarly, $j=j^{\prime}$. ∎

Note that Lemma 3.1 is stated implicitly by Hobart and Williford in the proof of [6, Corollary 10]. Since $\eta|_{\Lambda_{s}}:F_{s}^{2}\times\{s\}\rightarrow\{1,\dots,f\}^{2}$ is injective by Lemma 3.1, the set $F_{s}$ can be taken to be the subset $F^{\prime}_{s}$ of $\{1,\dots,f\}$ defined in Lemma 2.7.

Definition 3.2**.**

For $s\in S$, we define the support for $\mathfrak{C}_{s}$ to be the subset

[TABLE]

By the definition of $F_{s}$, we can take $\eta$ as $\eta(i,j,a)=(i,j)$ for $i,j\in F_{s}$. Indeed, by (4), we may suppose $\varepsilon_{i,i}^{s}\in\mathfrak{A}_{i,i}$ for all $i\in F_{s}$. Then by $\varepsilon_{i,j}^{s}=\varepsilon_{i,i}^{s}\varepsilon_{i,j}^{s}\varepsilon_{j,j}^{s}\in\mathfrak{A}_{i,j},$ we have $\eta(i,j,s)=(i,j)$.

For brevity, we write $F_{s,t,u}=F_{s}\cap F_{t}\cap F_{u}$. Note that $F_{1}=\{1,\dots,f\}$ holds by (6). By Lemma 2.8, (11) can be written as follows: for $(i,j)\in F_{s}^{2}\cap F_{t}^{2}$,

[TABLE]

Definition 3.3**.**

For $s,t,u\in S$, let $Q_{s,t}^{u}\in\mathrm{M}_{F_{s,t,u}}(\mathbb{C})$ be the matrix with $(i,j)$-entry

[TABLE]

The matrix $Q_{s,t}^{u}$ is called the matrix of Krein parameters with respect to the bases of matrix units $\{\varepsilon_{i,j}^{s}\},\{\varepsilon_{i,j}^{t}\},\{\varepsilon_{i,j}^{s}\}$ for $\mathfrak{C}_{s},\mathfrak{C}_{t},\mathfrak{C}_{u}$.

Note that, by (12), $Q_{s,t}^{u}=Q_{t,s}^{u}$ holds for any $s,t,u\in S$. Moreover, the matrix $Q_{s,t}^{u}$ is hermitian by (2) and (12).

Proposition 3.4**.**

For any $s,t\in S$, we have $Q_{1,s}^{t}=\delta_{s,t}J_{F_{1,s,t}}$.

Proof.

Immediate from (6), (12) and Definition 3.3. ∎

Proposition 3.5**.**

For any $s,t\in S$,

[TABLE]

In particular, $\mathrm{tr}(\varepsilon_{j,j}^{t})$ is independent of $j\in F_{t}$.

Proof.

By (12),

[TABLE]

We compute the trace of each side of this identity. By (6), the trace of the right-hand side is $(Q_{s,t}^{1})_{i,j}/{\sqrt{|X_{i}||X_{j}|}}$. On the other hand, the trace of the left-hand side is

[TABLE]

By the properties of the trace, $\mathrm{tr}({\varepsilon_{i,j}^{s}}^{T}\varepsilon_{i,j}^{t})=\mathrm{tr}(\varepsilon_{i,j}^{t}{\varepsilon_{i,j}^{s}}^{T})$ and this implies $\mathrm{tr}(\varepsilon_{j,j}^{t})=\mathrm{tr}(\varepsilon_{i,i}^{t})$. Thus we obtain $(Q_{s,t}^{1})_{i,j}=\delta_{\hat{s},t}\mathrm{tr}(\varepsilon_{i,i}^{t})=\delta_{\hat{s},t}\mathrm{tr}(\varepsilon_{j,j}^{t})$, and the result follows. ∎

Proposition 3.6**.**

For $s,t,u\in S,$ let $\mathbf{z}_{s}\in\mathbb{C}^{F_{s}},\mathbf{z}_{t}\in\mathbb{C}^{F_{t}},\mathbf{z}_{u}\in\mathbb{C}^{F_{u}}$ be vectors whose entries consist of complex numbers with absolute value $1$. Define $\mathbf{z}\in\mathbb{C}^{F_{s,t,u}}$ by

[TABLE]

Then $\mathbf{z}^{*}\mathbf{z}\circ Q_{s,t}^{u}$ is the matrix of Krein parameters with respect to $\mathbf{z}_{s}^{*}\mathbf{z}_{s}\circ\{\varepsilon_{i,j}^{s}\},\mathbf{z}_{t}^{*}\mathbf{z}_{t}\circ\{\varepsilon_{i,j}^{t}\},\mathbf{z}_{u}^{*}\mathbf{z}_{u}\circ\{\varepsilon_{i,j}^{u}\}$.

Proof.

By (12) and Definition 3.3, we have

[TABLE]

Thus the result follows. ∎

In particular, if $Q_{s,t}^{u}$ is positive semidefinite, then $Z\circ Q_{s,t}^{u}$ is also positive semidefinite. Thus the positive semidefiniteness of $Q_{s,t}^{u}$ is independent of the choice of bases of matrix units.

Theorem 3.7**.**

For any $s,t,u\in S$, the condition (9) holds if and only if the matrix of Krein parameters $Q_{s,t}^{u}$ is positive semidefinite.

Proof.

To prove this equivalence, we simplify (9). Let $B=(b_{i,j})\in\mathcal{P}_{F_{s}},C=(c_{i,j})\in\mathcal{P}_{F_{t}}$. By Lemma 2.8, if $(m,n)\not\in F_{s}^{2}$ or $(m,n)\not\in F_{t}^{2}$, then the $(m,n)$-entry (8) of $\tilde{Q}_{s,t}^{u}(B,C)$ is [math]. If $(m,n)\in F_{s,t,u}^{2}$, then (8) is

[TABLE]

where $B^{\prime},C^{\prime}\in\mathrm{M}_{F_{s,t,u}}(\mathbb{C})$ are the principal submatrices of $B,C$ indexed by $F_{s,t,u}$. Thus $\tilde{Q}_{s,t}^{u}(B,C)$ has $B^{\prime}\circ C^{\prime}\circ Q_{s,t}^{u}$ as a principal submatrix and all other entries are [math]. This implies that $\tilde{Q}_{s,t}^{u}(B,C)\in\mathcal{P}_{F_{u}}$ if and only if $B^{\prime}\circ C^{\prime}\circ Q\in\mathcal{P}_{F_{s,t,u}}$. In particular, taking $B$ and $C$ to be the all-ones matrices, (9) implies $Q_{s,t}^{u}\in\mathcal{P}_{F_{s,t,u}}$.

Conversely, if $Q_{s,t}^{u}\in\mathcal{P}_{F_{s,t,u}}$, then $B^{\prime}\circ C^{\prime}\circ Q_{s,t}^{u}\in\mathcal{P}_{F_{s,t,u}}$ for any $B\in\mathcal{P}_{F_{s}},C\in\mathcal{P}_{F_{t}}$ by [1, Lemma 3.9], and (9) holds. ∎

Hobart [5] applied the Krein condition of the coherent configuration defined by a quasi-symmetric design by setting $B$ and $C$ to be all-ones matrices. She commented that there are no choices of $B,C$ which lead to other consequences. Indeed, since the coherent configuration defined by a quasi-symmetric design is fiber-commutative, considering the case $B=C=J$ is sufficient by Theorem 3.7.

4. Generalized quadrangles

Definition 4.1**.**

Let $P,L$ be finite sets and $I\subset P\times L$ be an incidence relation. An incidence structure $(P,L,I)$ is called a generalized quadrangle with parameters $(s,t)$ if

1. (i)

for any $l\in L$, $\#\{p\in P\mid(p,l)\in I\}=s+1$, 2. (ii)

for any $p\in P$, $\#\{\l\in L\mid(p,l)\in I\}=t+1$, 3. (iii)

for any $p\in P$ and $l\in L$ with $(p,l)\not\in I$, there exist unique $q\in P$ and unique $m\in L$ such that $(p,m),(q,m),(q,l)\in I$.

Elements of $P$ and $L$ are called *points * and lines, respectively.

Let $(P,L,I)$ be a generalized quadrangle with parameters $(s,t)$. For $p,q\in P$, if there exists $l\in L$ such that $(p,l),(q,l)\in I$, then we write $p\sim q$ and say that $p$ and $q$ are collinear. Similarly, for $l,m\in L$, if there exists $p\in P$ such that $(p,l),(p,m)\in I$, then we write $l\sim m$ and say that $l$ and $m$ are concurrent.

In this section, we apply Theorem 3.7 to generalized quadrangles and obtain the following inequalities: If $s,t>1$, then $s\leq t^{2}$ and $t\leq s^{2}$ hold. These inequalities are established in [2, 3], as a consequence of the Krein condition for the strongly regular graph defined by a generalized quadrangle. We also show that no other consequences can be obtained from Theorem 3.7 by computing all matrices of Krein parameters.

First, we construct a coherent configuration from a generalized quadrangle. Let $X_{1}=P$ and $X_{2}=L$ be fibers. Adjacency relations on $X=X_{1}\sqcup X_{2}$ are defined as

[TABLE]

Then $\mathcal{C}=(X,\{R_{I}\}_{I\in\mathcal{I}})$ is a coherent configuration, where $\mathcal{I}=\{(i,j,k)\mid 1\leq i,j\leq 2,\;1\leq k\leq r_{i,j}\}$ and $r_{1,1}=r_{2,2}=3$, $r_{1,2}=r_{2,1}=2$. Let $A_{i,j,k}$ be the adjacency matrix of the relation $R_{i,j,k}$, and let $\mathfrak{A}$ be the adjacency algebra of $\mathcal{C}$. Then $\mathfrak{A}$ is decomposed as

[TABLE]

where $\mathfrak{C}_{1},\mathfrak{C_{2}}\simeq\mathrm{M}_{2}(\mathbb{C})$ and $\mathfrak{C}_{3},\mathfrak{C}_{4}\simeq\mathbb{C}$. Moreover, $F_{1}=F_{2}=\{1,2\}$, $F_{3}=\{1\}$, $F_{4}=\{2\}$.

For each $\mathfrak{C}_{i}$, a basis of matrix units can be expressed as follows: For $\mathfrak{C}_{1}$,

[TABLE]

For $\mathfrak{C}_{2}$,

[TABLE]

For $\mathfrak{C}_{3}$,

[TABLE]

For $\mathfrak{C}_{4}$,

[TABLE]

For these bases of matrix units, the matrices of Krein parameters $Q_{3,3}^{3}$ and $Q_{4,4}^{4}$ are the $1\times 1$ matrices given by

[TABLE]

By Theorem 3.7, both $Q_{3,3}^{3}$ and $Q_{4,4}^{4}$ are positive semidefinite, so $s^{2}\geq t$ and $t^{2}\geq s$ hold, provided $s,t>1$. The consequences of Theorem 3.7 for all other matrices of Krein parameters are trivial. Indeed, the other matrices of Krein parameters are given as follows (we omit those matrices determined by Proposition 3.4, and those determined to be zero by Proposition 3.5):

[TABLE]

where

[TABLE]

and

[TABLE]

5. Absolute bounds for fiber-commutative coherent configurations

Let $\mathfrak{A}$ be the adjacency algebra of a coherent configuration $\mathcal{C}=(X,\{R_{I}\}_{I\in\mathcal{I}})$, and let $\{\Delta_{s}\mid s\in S\}$ be a set of representatives of all irreducible matrix representations of $\mathfrak{A}$ over $\mathbb{C}$ satisfying $\Delta_{s}(A)^{*}=\Delta_{s}(A^{*})$ for any $A\in\mathfrak{A}$. Denote by $h_{s}$ the multiplicity of $\Delta_{s}$ in the standard module $\mathbb{C}^{X}$. In this section, we assume that $\Delta_{s}(\varepsilon_{i,j}^{s})=E_{i,j}$ for a basis of matrix units $\{\varepsilon_{i,j}^{s}\}$ for $\mathfrak{C}_{s}$, where $E_{i,j}$ is $e_{s}\times e_{s}$ matrix with $(i,j)$-entry $1$ and all other entries [math]. The following bound is known as the absolute bound.

Lemma 5.1** ([6, Theorem 5]).**

For any $s,t\in S$, we have

[TABLE]

For fiber-commutative coherent configurations, we can simplify this inequality.

Theorem 5.2**.**

Let $Q_{s,t}^{u}$ $(s,t,u\in S)$ be the matrices of Krein parameters for $\mathcal{C}$. For any $s,t\in S$, we have

[TABLE]

Proof.

By (11), for any $u\in S$, we have

[TABLE]

and the rank of this matrix is $\mathrm{rank}(Q_{s,t}^{u})$. By Lemma 5.1, the result follows. ∎