Specht's criterion for systems of linear mappings

TL;DR

This paper generalizes Specht's criterion for unitary similarity of matrices to systems of linear mappings between inner product spaces, using graph-based invariants involving traces of closed walks.

Contribution

It extends Specht's criterion to arbitrary systems of linear mappings represented by directed graphs, incorporating adjoint mappings and trace invariants for classification.

Findings

Systems are unitarily equivalent if traces of all closed walks match.

The criterion applies to complex and real inner product spaces.

Graph-based invariants fully characterize system equivalence under isometries.

Abstract

W.Specht (1940) proved that two $n\times n$ complex matrices $A$ and $B$ are unitarily similar if and only if $\operatorname{trace} w(A,A^{\ast}) = \operatorname{trace} w(B,B^{\ast})$ for every word $w(x,y)$ in two noncommuting variables. We extend his criterion and its generalizations by N.A.Wiegmann (1961) and N.Jing (2015) to an arbitrary system $\mathcal A$ consisting of complex or real inner product spaces and linear mappings among them. We represent such a system by the directed graph $Q(\mathcal A)$, whose vertices are inner product spaces and arrows are linear mappings. Denote by $\widetilde Q(\mathcal A)$ the directed graph obtained by enlarging to $Q(\mathcal A)$ the adjoint linear mappings. We prove that a system $\mathcal A$ is transformed by isometries of its spaces to a system $\mathcal B$ if and only if the traces of all closed directed walks in $\widetilde Q(\mathcal A)$ and $\widetilde Q(\mathcal B)$ coincide.

Equations

$$\mathrm{trace}w(A,A^{\ast })=\mathrm{trace}w(B,B^{\ast })$$

$$\mathrm{trace}w(A_1,A_1^{\ast },\dots ,A_k,A_k^{\ast })=\mathrm{trace}w(B_1,B_1^{\ast },\dots ,B_k,B_k^{\ast })$$

$$\begin{array}{rl}& \mathrm{trace}w(A_1^{\ast }{A}_1,\dots ,A_i^{\ast }{A}_j,\dots ,A_k^{\ast }{A}_k)\\ =& \mathrm{trace}w(B_1^{\ast }{B}_1,\dots ,B_i^{\ast }{B}_j,\dots ,B_k^{\ast }{B}_k)\end{array}$$

$$-2+\frac{n}{2}+n\cdot \sqrt{\frac{2n^2}{n-1}+\frac{1}{4}}$$

$$\Vert X\otimes I_n+Y\otimes A\Vert =\Vert X\otimes I_n+Y\otimes B\Vert$$

$$\dim \mathcal{A}:=(\dim {\mathcal{U}}_1,\dots ,\dim {\mathcal{U}}_t)$$

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$$1$$

$$\pi:\quad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.003pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\crcr}}}\ignorespaces{\hbox{\kern-7.003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{v_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 55.56706pt\raise-18.79861pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8125pt\hbox{$\scriptstyle{\alpha_{\ell}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 116.56705pt\raise-4.94202pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 7.003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 11.01405pt\raise 5.15138pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\ \alpha_{1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 31.003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{v_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 45.00902pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 51.39355pt\raise 5.15138pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\alpha_{2}\ }$}}}\kern 3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 69.00902pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\ \cdots\ }$}}}}}}}{\hbox{\kern 116.50902pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{v_{\ell}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 89.0299pt\raise 5.20139pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8125pt\hbox{$\scriptstyle{\quad\alpha_{\ell-1}\ }$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 92.50902pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces$$

$$\mathcal{A}(\pi):\quad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 9.00769pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\crcr}}}\ignorespaces{\hbox{\kern-9.00769pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathcal{U}_{v_{1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 58.12767pt\raise-17.62222pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.40556pt\hbox{$\scriptstyle{\mathcal{A}_{\alpha_{\ell}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 123.66632pt\raise-5.3161pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 9.00769pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 13.3033pt\raise 6.33777pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.44556pt\hbox{$\scriptstyle{\mathcal{A}_{\alpha_{1}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 33.00769pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{U}_{v_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 51.02307pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 56.68983pt\raise 6.33777pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.44556pt\hbox{$\scriptstyle{\mathcal{A}_{\alpha_{2}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 75.02307pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\ \cdots\ }$}}}}}}}{\hbox{\kern 122.52307pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathcal{U}_{v_{\ell}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 98.43085pt\raise 6.37778pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.40556pt\hbox{$\scriptstyle{\ \mathcal{A}_{\alpha_{\ell-1}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 98.52307pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces$$

$$\mathrm{trace}\mathcal{A}(\pi ):=\mathrm{trace}\left({\mathcal{A}}_{{\alpha }_1}{\mathcal{A}}_{{\alpha }_2}\cdots {\mathcal{A}}_{{\alpha }_{\ell }}\right);$$

$$\mathrm{trace}({\mathcal{A}}_{{\alpha }_1}{\mathcal{A}}_{{\alpha }_2}\cdots {\mathcal{A}}_{{\alpha }_{\ell }})=\mathrm{trace}({\mathcal{A}}_{{\alpha }_{\ell }}{\mathcal{A}}_{{\alpha }_1}\cdots {\mathcal{A}}_{{\alpha }_{\ell -1}})=\mathrm{trace}({\mathcal{A}}_{{\alpha }_{\ell -1}}{\mathcal{A}}_{{\alpha }_{\ell }}\cdots {\mathcal{A}}_{{\alpha }_{\ell -2}})=\cdots$$

$$(\mathcal{A}x,y)=(x,{\mathcal{A}}^{\text{medstar}}y)$$

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$$1$$

$$\mathrm{trace}\tilde{\mathcal{A}}(\pi )=\mathrm{trace}\tilde{\mathcal{B}}(\pi )$$

$$\phi ((r+2)(d_1+\cdots +d_t)),$$

$$\frac{r(r+1)}{2}⩾\max \{m_{ij}|\text{i and j are vertices of Q}\},$$

$$(U_1^{-1}{A}_{\alpha }{U}_2,U_1^{-1}{A}_{\beta }{U}_3,U_2^{-1}{A}_{\gamma }{U}_2,U_3^{-1}{A}_{\delta }{U}_2,U_3^{-1}{A}_{\epsilon }{U}_2,U_3^{-1}{A}_{\zeta }{U}_3)$$

$$B_{\alpha }=U_v^{-1}{A}_{\alpha }{U}_{u}for\; every\; arrow\alpha :u⟶v.$$

$$d_1$$

$$\mathrm{trace}\tilde{A}(\pi )=\mathrm{trace}\tilde{B}(\pi )$$

$$\cdot$$

$$1$$

$$(U{A}_1{V}_1,\dots ,U{A}_k{V}_k)=(B_1,\dots ,B_k)$$

$$\mathrm{trace}w(A_1^{\ast }{A}_1,\dots ,A_k^{\ast }{A}_k)=\mathrm{trace}w(B_1^{\ast }{B}_1,\dots ,B_k^{\ast }{B}_k)$$

$$(U{A}_1{V}_1,\dots ,U{A}_k{V}_k)=(B_1,\dots ,B_k)$$

$$\mathrm{trace}w(A_1^{⊺}{A}_1,\dots ,A_k^{⊺}{A}_k)=\mathrm{trace}w(B_1^{⊺}{B}_1,\dots ,B_k^{⊺}{B}_k)$$

$$Q_1:$$

$$A=\left[\begin{array}{ccc}{A}_{11}& \cdots & A_{1t}\\ ⋮& \ddots & ⋮\\ A_{t1}& \cdots & A_{tt}\end{array}\right],B=\left[\begin{array}{ccc}{B}_{11}& \cdots & B_{1t}\\ ⋮& \ddots & ⋮\\ B_{t1}& \cdots & B_{tt}\end{array}\right]$$

Full text

AMS classification: 15A21; 15A63; 16G20; 47A67

Keywords: Specht’s criterion; Unitary similarity; Unitary and Euclidean representations of quivers

1 Introduction

Each system of complex inner product spaces and linear mappings among them can be represented by a directed graph, in which the vertices are inner product spaces and the arrows are linear mappings. We reduce the problem of classifying such systems to the problem of classifying complex matrices up to unitary similarity, apply Specht’s criterion for unitary similarity of complex matrices, and obtain a generalization of the following criteria:

Specht’s criterion for unitary similarity

([19]; see also [8, Theorem 2.2.6], [9], [10, Theorem 6.3], and [14]). Two $n\times n$ complex matrices $A$ and $B$ are unitarily similar if and only if

[TABLE]

for every word $w(x,y)$ in two noncommuting variables.

Wiegmann’s criterion for simultaneous unitary similarity

([20]; see also [18, Theorem 6.2]). Let $(A_{1},\dots,A_{k})$ and $(B_{1},\dots,B_{k})$ be two $k$-tuples of $n\times n$ complex matrices. There exists a unitary matrix $U$ such that $(U^{-1}A_{1}U,\dots,U^{-1}A_{k}U)=(B_{1},\dots,B_{k})$ if and only if

[TABLE]

for every word $w(x_{1},y_{1},\dots,x_{k},y_{k})$ in $2k$ noncommuting variables.

Jing’s criterion for simultaneous unitary equivalence

([9]). Let $(A_{1},\dots,A_{k})$ and $(B_{1},\dots,B_{k})$ be two $k$-tuples of $m\times n$ complex matrices. There exist unitary matrices $U$ and $V$ such that $(UA_{1}V,\dots,UA_{k}V)=(B_{1},\dots,B_{k})$ if and only if

[TABLE]

for every word $w(x_{11},\dots,x_{ij},\dots,x_{kk})$ in $k^{2}$ noncommuting variables.

Complex $n\times n$ matrices $A$ and $B$ are unitarily similar if $B=U^{-1}AU$ for some unitary matrix $U$; they are complex orthogonally similar if $B=S^{-1}AS$ for some complex matrix $S$ such that $S^{T}S=I_{n}$. Real $n\times n$ matrices $A$ and $B$ are real orthogonally similar if $B=S^{-1}AS$ for some real matrix $S$ such that $S^{T}S=I_{n}$.

Pearcy [14] (see also [10, Section 2-6] and [8, Section 2.2]) noticed that Specht’s criterion also holds for real matrices with respect to real orthogonal similarity. Jing [9] proved that his, Specht’s, and Wiegmann’s criteria hold for real matrices with respect to real orthogonal similarity (real orthogonal equivalence in Jing’s criterion) and for complex matrices with respect to complex orthogonal similarity (equivalence) if transposed matrices are used instead of conjugate transposed matrices in (1)–(3).

Specht’s criterion requires infinitely many tests. Pearcy [14, Theorem 1] proved that it suffices to verify the condition (1) for all words of length at most $2n^{2}$. Laffey [11] showed that it is sufficient to verify (1) for all words of length at most the smallest integer that is greater than or equal to $(2n^{2}+4)/3$, and hence for all words of length at most $n^{2}$. A better bound

[TABLE]

on the sufficient word length was given by Pappacena [13]; see also [8, Theorem 2.2.8].

Two alternative approaches to the problem of unitary similarity involve different ideas:

• •

Arveson [1, Theorems 2 and 3] (see also [3, 4, 5]) proved that if $A,B\in\mathbb{C}^{n\times n}$ and $A$ is not unitarily similar to a direct sum of square matrices of smaller sizes, then $A$ and $B$ are unitarily similar if and only if

[TABLE]

for all $X,Y\in\mathbb{C}^{n\times n}$. Here, $\|M\|$ is the spectral norm (largest singular value) of $M$.

• •

Littlewood [12] (see also [18]) constructed an algorithm that reduces each square complex matrix $A$ by unitary similarity transformations to a “canonical” matrix $A_{\text{can}}$ in such a way that $A$ and $B$ are unitarily similar if and only if $A_{\text{can}}=B_{\text{can}}$. Littlewood’s algorithm was extended in [16, 17] to unitary representations of a quiver.

2 Representations and ${}^{\!\medstar\!}$unitary representations of a quiver

2.1 Representations of a quiver

Classification problems for systems of linear mappings can be formulated in terms of quivers and their representations introduced by P. Gabriel in [6]. A quiver is a directed graph (loops and multiple arrows are allowed); we suppose that its vertices are $1,\dots,t$. Its representation $\mathcal{A}=(\mathcal{A}_{\alpha},\mathcal{U}_{v})$ over a field $\mathbb{F}$ is given by assigning to each vertex $v$ a vector space $\mathcal{U}_{v}$ over $\mathbb{F}$ and to each arrow $\alpha:u\longrightarrow v$ a linear mapping $\mathcal{A}_{\alpha}:\mathcal{U}_{u}\to\mathcal{U}_{v}$. The vector

[TABLE]

is the dimension of the representation $\mathcal{A}$.

For example, each representation

[TABLE]

of the quiver

[TABLE]

consists of vector spaces $\mathcal{U}_{1},\ \mathcal{U}_{2},\ \mathcal{U}_{3}$ and linear mappings $\mathcal{A}_{\alpha}$, $\mathcal{A}_{\beta}$, $\mathcal{A}_{\gamma}$, $\mathcal{A}_{\delta}$, $\mathcal{A}_{\varepsilon}$, $\mathcal{A}_{\zeta}$.

An oriented cycle $\pi$ of length $\ell\geqslant 1$ in a quiver $Q$ is a sequence of arrows of the form

[TABLE]

in which some of the vertices $v_{1},v_{2},\dots,v_{\ell}$ and some of the arrows $\alpha_{1},\dots,\alpha_{\ell}$ may coincide; see [15, Section 2.1]. Thus, an oriented cycle is a closed directed walk, in which vertices and arrows may repeat.

For each representation $\cal A$ of $Q$ and any cycle (8), define the cycle of linear mappings

[TABLE]

on (8). Write

[TABLE]

this number does not depend on the choice of the initial vertex $v_{1}$ in the cycle since the trace is invariant under cyclic permutations:

[TABLE]

2.2 ${}^{\!\medstar\!\!}$Unitary representations of a quiver

We extend Specht’s criterion to systems of linear mappings on complex inner product spaces. A complex inner product space (also called a Hermitian space, a finite-dimensional Hilbert space, or a unitary space) is a complex vector space with scalar product given by a positive definite Hermitian form. We also extend Specht’s criterion to systems of linear mappings on complex Euclidean spaces, which are complex vector spaces with scalar product given by a nonsingular symmetric bilinear form.

For convenience in studying the respective spaces simultaneously, a complex inner product space is called a ∗​unitary space, and a complex Euclidean space is called a ${}^{\intercal\!}$unitary space.

Let \!\medstar\!\in\{\intercal,\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{\ast}}}\}. For each linear mapping $\mathcal{A}:\mathcal{U}\to\mathcal{V}$ between ${}^{\!\medstar\!}$unitary spaces $\mathcal{U}$ and $\mathcal{V}$, we define the adjoint mapping $\mathcal{A}^{\!\medstar\!}:\mathcal{V}\to\mathcal{U}$ via

[TABLE]

The following definition generalizes the definition of unitary representations of quivers given in [16, 17].

Definition 1**.**

Let $Q$ be a quiver with vertices $1,\dots,t$, and let \!\medstar\!\in\{\intercal,\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{\ast}}}\} be fixed.

• •

A ${}^{\!\medstar\!\!}$unitary representation $\mathcal{A}=(\mathcal{A}_{\alpha},\mathcal{U}_{v})$ of $Q$ is given by assigning to each vertex $v$ a ${}^{\!\medstar\!}$unitary space $\mathcal{U}_{v}$ and to each arrow $\alpha:u\longrightarrow v$ a linear mapping $\mathcal{A}_{\alpha}:\mathcal{U}_{u}\to\mathcal{U}_{v}$.

• •

Two ${}^{\!\medstar\!}$unitary representations $\mathcal{A}=(\mathcal{A}_{\alpha},\mathcal{U}_{v})$ and $\mathcal{B}=(\mathcal{B}_{\alpha},\mathcal{V}_{v})$ of $Q$ are isometric if there exists a family of isometries (linear bijections that preserve the scalar products) $\varphi_{1}:\mathcal{U}_{1}\to\mathcal{V}_{1},\dots,\varphi_{t}:\mathcal{U}_{t}\to\mathcal{V}_{t}$ such that the diagram

[TABLE]

is commutative ($\varphi_{v}\mathcal{A}_{\alpha}=\mathcal{B}_{\alpha}\varphi_{u}$) for each arrow $\alpha:u\longrightarrow v$.

For example, the problem of classifying ${}^{\!\medstar\!}$unitary representations of the quiver (7) is the problem of classifying systems (6) consisting of ${}^{\!\medstar\!}$unitary spaces $\mathcal{U}_{1},\ \mathcal{U}_{2},\ \mathcal{U}_{3}$ and linear mappings $\mathcal{A}_{\alpha}$, $\mathcal{A}_{\beta}$, …, $\mathcal{A}_{\zeta}$.

It is customary to omit the asterisk in the terms “{}^{\mathop{\scalebox{1.5}{\raisebox{-0.60275pt}{\ast}}}\!}unitary space” and “{}^{\mathop{\scalebox{1.5}{\raisebox{-0.60275pt}{\ast}}}\!}unitary representation”.

3 The main theorem and its corollaries

3.1 The main theorem

For each quiver $Q$, denote by $\widetilde{Q}$ the quiver with double the number of arrows obtained by attaching to $Q$ the arrows $\alpha^{\star}:v\longrightarrow u$ for all arrows $\alpha:u\longrightarrow v$ of $Q$. For example, if $Q$ is the quiver (7), then $\widetilde{Q}$ is

[TABLE]

For each ${}^{\!\medstar\!}$unitary representation $\mathcal{A}$ of $Q$, we define the ${}^{\!\medstar\!}$unitary representation $\widetilde{\mathcal{A}}$ of $\widetilde{Q}$ that coincides with $\mathcal{A}$ on $Q\subset\widetilde{Q}$ and that assigns to each new arrow $\alpha^{\star}:v\longrightarrow u$ the mapping $\widetilde{\mathcal{A}}_{\alpha^{\star}}:=\mathcal{A}_{\alpha}^{\!\medstar\!}:\mathcal{U}_{v}\to\mathcal{U}_{u}$, which is the adjoint of $\mathcal{A}_{\alpha}:\mathcal{U}_{u}\to\mathcal{U}_{v}$ (see (9)).

The main result of the article is the following theorem, which is proved in Section 4.

Theorem 1**.**

Let \!\medstar\!\in\{\intercal,\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{\ast}}}\}.

• (a)

*Two *${}^{\!\medstar\!}$unitary representations $\mathcal{A}$ and $\mathcal{B}$ of a quiver $Q$ are isometric if and only if

[TABLE]

for each oriented cycle $\pi$ in the quiver $\widetilde{Q}$.

• (b)

It suffices to verify (11) for all cycles $\pi$ of length at most

[TABLE]

in which $(d_{1},\dots,d_{t})$ is the dimension of the representations $\mathcal{A}$ and $\mathcal{B}$ $($see (5)$)$, $\varphi(n)$ is any bound for the sufficient word length in Specht’s criterion $($for example, $\varphi(n)$ is $n^{2}$ or Pappacena’s bound (4)$)$, and $r$ is the minimal natural number such that

[TABLE]

in which $m_{ij}$ is the number of arrows from $j$ to $i$ in $Q$.

3.2 The main theorem in matrix form

We say that a square complex matrix $A$ is ${}^{\!\medstar\!}$unitary (\!\medstar\!\in\{\intercal,\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{\ast}}}\}) if $A^{\!\medstar\!}A=I$. Thus, *⊺*unitary matrices are complex orthogonal matrices and *∗*unitary matrices are unitary matrices.

A basis of a ${}^{\!\medstar\!}$unitary space $\mathcal{U}$ is orthonormal if the scalar product in this basis is given by the identity matrix. The change of basis matrix from an orthonormal basis to an orthonormal basis is a ${}^{\!\medstar\!}$unitary matrix. If $[x]$ is the coordinate vector of $x\in\mathcal{U}$ in an orthonormal basis, then $(x,y)=[x]^{\!\medstar\!\,}[y]$ for all $x,y\in\mathcal{U}$. If $\mathcal{A}:\mathcal{U}\to\mathcal{V}$ is a linear mapping between complex inner ${}^{\!\medstar\!}$product spaces and $A$ is its matrix in some orthonormal bases of $\mathcal{U}$ and $\mathcal{V}$, then $A^{\!\medstar\!}$ is the matrix of the adjoint mapping $\mathcal{A}^{\!\medstar\!}:\mathcal{V}\to\mathcal{U}$ (see (9)).

Each ${}^{\!\medstar\!}$unitary representation $\mathcal{A}$ in (6) can be given by the sequence $A=(A_{\alpha},\,A_{\beta},\,\dots,\,A_{\zeta})$ of matrices of the linear mappings $\mathcal{A}_{\alpha},\,\mathcal{A}_{\beta},\,\dots,\,\mathcal{A}_{\zeta}$ in some orthonormal bases of the spaces $\mathcal{U}_{1},\ \mathcal{U}_{2},\ \mathcal{U}_{3}$. The representation $\cal A$ in other orthonormal bases is given by the sequence

[TABLE]

in which $U_{1},\,U_{2},\,U_{3}$ are the change of basis matrices; they are arbitrary ${}^{\!\medstar\!}$unitary matrices of suitable sizes. Thus, the problem of classifying ${}^{\!\medstar\!}$unitary representations of the quiver (7) reduces to the problem of classifying matrix sequences $(A_{\alpha},\,\dots,\,A_{\zeta})$ up to transformations of the form (14). This example leads to the following definition.

Definition 2**.**

Let \!\medstar\!\in\{\intercal,\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{\ast}}}\} and let $Q$ be a quiver with vertices $1,\dots,t$.

• •

A complex matrix representation $A$ of dimension $(d_{1},\dots,d_{t})$ of $Q$ is given by assigning to each arrow $\alpha:u\longrightarrow v$ a complex matrix ${A}_{\alpha}$ of size $d_{v}\times d_{u}$ (we take $d_{i}:=0$ if the vertex $i$ does not have arrows).

• •

Two complex matrix representations $A$ and $B$ of $Q$ are ${}^{\!\medstar\!\!}$unitarily isometric if there exist ${}^{\!\medstar\!}$unitary matrices $U_{1},\dots,U_{t}$ such that

[TABLE]

For example, two complex matrix representations

[TABLE]

of the quiver (7) are ${}^{\!\medstar\!}$unitarily isometric if and only if $B=(B_{\alpha},\dots,B_{\zeta})$ is of the form (14).

The principle formalized in the following obvious lemma reduces the problem of classifying ${}^{\!\medstar\!}$unitary representations up to isometry to the problem of classifying complex matrix representations up to ${}^{\!\medstar\!}$unitary isometry.

Lemma 1**.**

Let $\cal A$ and $\cal B$ be two ${}^{\!\medstar\!}$unitary representations of a quiver. Choosing orthonormal bases in their spaces, we get two complex matrix representations $A$ and $B$. Then $\cal A$ and $\cal B$ are isometric if and only if $A$ and $B$ are ${}^{\!\medstar\!}$unitarily isometric.

For each oriented cycle (8) in a quiver $Q$ and each complex matrix representation $A$ of $Q$, we write $A(\pi):=A_{\alpha_{1}}A_{\alpha_{2}}\cdots A_{\alpha_{\ell}}.$ The following theorem is equivalent to Theorem 1 due to Lemma 1.

Theorem 2**.**

Let \!\medstar\!\in\{\intercal,\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{\ast}}}\}.

• (a)

*Two complex matrix representations $A$ and $B$ of a quiver $Q$ are *${}^{\!\medstar\!}$unitarily isometric if and only if

[TABLE]

for each oriented cycle $\pi$ in the quiver $\widetilde{Q}$.

• (b)

It suffices to verify (16) for all cycles $\pi$ of length at most (12).

3.3 Corollaries

A Euclidean representation $\mathcal{A}=(\mathcal{A}_{\alpha},\mathcal{U}_{v})$ of a quiver $Q$ is defined in [17] as a list of Euclidean spaces $\mathcal{U}_{v}$ assigned to all vertices $v$ and linear mappings $\mathcal{A}_{\alpha}:\mathcal{U}_{u}\to\mathcal{U}_{v}$ assigned to all arrows $\alpha:u\longrightarrow v$. Two Euclidean representations $\mathcal{A}=(\mathcal{A}_{\alpha},\mathcal{U}_{v})$ and $\mathcal{B}=(\mathcal{B}_{\alpha},\mathcal{V}_{v})$ of $Q$ are isometric if there exists a family of isometries $\varphi_{1}:\mathcal{U}_{1}\to\mathcal{V}_{1},\dots,\varphi_{t}:\mathcal{U}_{t}\to\mathcal{V}_{t}$ such that the diagram (10) is commutative for each arrow $\alpha:u\longrightarrow v$.

We say that a matrix representation $A$ of $Q$ is real if all its matrices are real. Two real matrix representations $A$ and $B$ are real orthogonally isometric if there exist real orthogonal matrices $U_{1},\dots,U_{t}$ such that (15) holds.

Corollary 1**.**

• (a)

Two Euclidian representations $\mathcal{A}$ and $\mathcal{B}$ of a quiver $Q$ are isometric if and only if $\operatorname{trace}{\widetilde{\mathcal{A}}}(\pi)=\operatorname{trace}{\widetilde{\mathcal{B}}}(\pi)$ for each oriented cycle $\pi$ in the quiver $\widetilde{Q}$.

• (b)

Two real matrix representations $A$ and $B$ of a quiver $Q$ are isometric if and only if $\operatorname{trace}{\widetilde{A}}(\pi)=\operatorname{trace}{\widetilde{B}}(\pi)$ for each oriented cycle $\pi$ in the quiver $\widetilde{Q}$.

• (c)

It suffices to verify the equalities in (a) and (b) for all cycles $\pi$ of length at most (12).

Proof.

Let $\cal A$ and $\cal B$ be two Euclidean representations of a quiver. Choosing orthonormal bases in their spaces, we get two real matrix representations $A$ and $B$. Then $\cal A$ and $\cal B$ are isometric if and only if $A$ and $B$ are orthogonally isometric, and so (a) follows from (b). The statement (b) follows from Theorem 2 due to the following statement proved in [17, Theorem 4.1(a)]:

two real matrix representations of a quiver are real orthogonally isometric if and only if they are unitarily isometric.

(In particular, two lists of real matrices are simultaneously real orthogonally similar if and only if they are simultaneously unitarily similar; see [10, Theorem 65] or [8, Theorem 2.5.21].) ∎

Corollary 2**.**

Applying Theorem 2 and Corollary 1 to complex and real matrix representations of the quivers

[TABLE]

we get Specht’s, Wiegmann’s, and Jing’s criteria (see the beginning of Section 1).

Applying Theorem 2 and Corollary 1 to complex and real matrix representations of the quiver

[TABLE]

with $k\geqslant 1$, we obtain the following criterion.

Corollary 3**.**

• (a)

Let $A_{1},\dots,A_{k}$ and $B_{1},\dots,B_{k}$ $(k\geqslant 1)$ be complex matrices with $m$ rows. Suppose that $A_{i}$ and $B_{i}$ have $n_{i}$ columns, $i=1,2,\dots,k$. Then there exist unitary matrices $U,V_{1},\dots,V_{k}$ such that

[TABLE]

if and only if

[TABLE]

for every word $w(x_{1},\dots,x_{k})$ in noncommuting variables.

• (b)

Let $A_{1},\dots,A_{k}$ and $B_{1},\dots,B_{k}$ $(k\geqslant 1)$ be real (respectively, complex) matrices with $m$ rows. Suppose that $A_{i}$ and $B_{i}$ have $n_{i}$ columns, $i=1,2,\dots,k$. Then there exist real orthogonal (respectively, complex orthogonal) matrices $U,V_{1},\dots,V_{k}$ such that

[TABLE]

if and only if

[TABLE]

for every word $w(x_{1},\dots,x_{k})$ in noncommuting variables.

• (c)

*It suffices to verify (17) and (18) for all words of length at most $\varphi(3(m+n_{1}+\dots+n_{k}))$, in which $\varphi(n)$ is any bound for the sufficient word length in Specht’s criterion *$($see Theorem 1(b)).

Denote by $Q_{t}$ the complete quiver with vertices $1,2,\dots,t$; that is, the quiver in which each vertex has exactly one loop and every pair of distinct vertices is connected by a pair of arrows (one in each direction). For example,

[TABLE]

Applying Theorem 1 to representations of $Q_{t}$, we get the following criterion.

Corollary 4**.**

Let

[TABLE]

be conformally partitioned $m\times m$ complex matrices, in which all diagonal blocks are square. Let \!\medstar\!\in\{\intercal,\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{\ast}}}\} be fixed. The following statements are equivalent:

• (i)

$U^{-1}AU=B$*, in which $U=U_{1}\oplus\dots\oplus U_{t}$ and each $U_{i}$ is *${}^{\!\medstar\!}$unitary and the same size as $A_{ii}$.

• (ii)

The equality

[TABLE]

with

[TABLE]

holds for all $\varepsilon_{1},\dots,\varepsilon_{\ell}\in\{1,\!\medstar\!\}$, all $i_{1},\dots,i_{\ell}\in\{1,\dots,t\}$, and every natural number $\ell$.

• (iii)

*The equality (19) holds for all $\varepsilon_{1},\dots,\varepsilon_{\ell}\in\{1,\!\medstar\!\}$, all $i_{1},\dots,i_{\ell}\in\{1,\dots,t\}$, and all $\ell\leqslant\varphi(3m)$, in which $m\times m$ is the size of $A$ and $B$, and $\varphi(n)$ is any bound for the sufficient word length in Specht’s criterion *$($see Theorem 1(b)).

4 Proof of Theorems 1 and 2

It suffices to prove Theorem 2 since it is equivalent to Theorem 1.

To prove Theorem 2, we reduce the problem of classifying complex matrix representations of a quiver up to ${}^{\!\medstar\!}$unitary isometry to the problem of classifying complex matrices up to ${}^{\!\medstar\!}$unitary similarity. We then apply Specht’s criterion for matrices under unitary similarity and its generalization by Jing [9] to matrices under complex orthogonal similarity.

4.1 From matrix representations of a quiver up to ${}^{\!\medstar\!\!}$unitary isometry to matrices up to ${}^{\!\medstar\!\!}$unitary similarity

Let \!\medstar\!\in\{\intercal,\mathop{\scalebox{1.5}{\raisebox{-0.86108pt}{\ast}}}\}. For each quiver $Q$ and its complex matrix representation $A$, we construct a square complex matrix $M_{Q}(A)$ such that

[TABLE]

Examples of $M_{Q}(A)$ are given in [16] and [17, Section 2.3], in which Littlewood’s algorithm for reducing complex matrices to canonical form under unitary similarity is extended to unitary representations of quivers. An analogous construction was used in [7, Lemma 2] to reduce the problem of classifying $(p+q)$-tuples of complex $n\times n$ matrices $(A_{1},\dots,A_{p};B_{1},\dots,B_{q})$ up to transformations

[TABLE]

to the problem of classifying square matrices up to unitary similarity.

4.1.1 An example

Let $A=(A_{\alpha},\dots,A_{\zeta})$ be a complex matrix representation of the quiver (6). Define the matrix

[TABLE]

For each complex matrix representation $B=(B_{\alpha},\dots,B_{\zeta})$ of $Q$ of the same dimension as $A$, we replace the blocks $A_{\alpha},\dots,A_{\zeta}$ of $M(A)$ with $B_{\alpha},\dots,B_{\zeta}$ and denote the matrix obtained by $M(B)$. Let us prove that (21) holds with $M(A)$ and $M(B)$ instead of $M_{Q}(A)$ and $M_{Q}(B)$.

$\Longrightarrow$. Let $A$ and $B$ be ${}^{\!\medstar\!}$unitarily isometric. Then $B$ is represented in the form (14), in which $U_{1},U_{2},U_{3}$ are ${}^{\!\medstar\!}$unitary matrices. Writing

[TABLE]

we obtain

[TABLE]

$\Longleftarrow$. Let $M(A)$ and $M(B)$ be ${}^{\!\medstar\!}$unitarily similar; that is, $M(A)U=UM(B)$ with ${}^{\!\medstar\!}$unitary $U$. Partition $U=[U_{ij}]$ conformally to $M(A)$. Equating the blocks of $M(A)U$ and $UM(B)$ along the block diagonals starting from the lower left corner, we find that $U$ is upper block triangular. Since $U$ is ${}^{\!\medstar\!}$unitary, it is block diagonal. Equating the blocks of $M(A)U$ and $UM(B)$ at the places of $I$’s in (22), we find that $U$ has the form (23). By (24), $A_{\alpha},A_{\beta},\dots,A_{\zeta}$ are transformed as in (14), which proves (21).

4.1.2 The general case

Definition 3**.**

Let $Q$ be a quiver with vertices $1,\dots,t$. For each pair of vertices $(i,j)$, let

[TABLE]

be all the arrows from $j$ to $i$ (the number $m_{ij}$ is called the multiplicity of $j\longrightarrow i$). Let $r$ be the minimal natural number such that $r(r+1)/2\geqslant\max{m_{ij}}$ (see (13)). Define the $(r+2)t\times(r+2)t$ partitioned matrix

[TABLE]

in which all blocks are $t\times t$,

[TABLE]

(thus, the main diagonal of $M_{Q}(x)$ is $(1,2,3,\dots,(r+2)t)$) and

[TABLE]

Thus, $M_{Q}(x)$ depends on parameters

[TABLE]

which correspond to the arrows (25).

Example 1**.**

If $Q$ is the quiver (7), then

[TABLE]

in which

[TABLE]

Lemma 2**.**

Let $Q$ be a quiver with vertices $1,\dots,t$ and arrows (25), and let $M_{Q}(x)$ be the parameter matrix (26). For each complex matrix representation $A$ of $Q$, denote by $M_{Q}(A)$ the block matrix obtained from $M_{Q}(x)$ by replacing the parameters (27) with

[TABLE]

replacing each other nonzero entry $a$ with the scalar block $aI$, and replacing the zero entries with the zero blocks of suitable sizes. Then $M_{Q}(A)$ is correctly constructed and (21) holds.

4.1.3 Proof of Lemma 2

Let $A$ be a complex matrix representation of dimension $(d_{1},\dots,d_{t})$ of $Q$. Substituting it into (26), we get the block matrix

[TABLE]

in which $d:=d_{1}+\dots+d_{t}$, each $t\times t$ block of (26) becomes a $d\times d$ block that is partitioned into $t$ horizontal strips of sizes $d_{1},\dots,d_{t}$ and $t$ vertical strips of the same sizes; namely,

[TABLE]

and

[TABLE]

Let us prove that (21) holds for every two complex matrix representations $A$ and $B$ of $Q$.

$\Longrightarrow$. Let $A$ and $B$ be ${}^{\!\medstar\!}$unitarily isometric; that is, (15) holds for some ${}^{\!\medstar\!}$unitary matrices $U_{1},\dots,U_{t}$. Then $M_{Q}(B)=U^{-1}M_{Q}(A)U$ with

[TABLE]

$\Longleftarrow$. Suppose that $M_{Q}(A)U=UM_{Q}(B)$ with a ${}^{\!\medstar\!}$unitary matrix $U$. Partition $U$ into $(r+2)^{2}$ blocks $U_{ij}$ conformally to (29). Equating the blocks of $M_{Q}(A)U$ and $UM_{Q}(B)$ along the block diagonals starting from the lower left corner, we find that $U$ is upper block triangular. Since $U$ is ${}^{\!\medstar\!}$unitary, $U$ is block diagonal; that is, it has the form $U=\operatorname{diag}(U^{(1)},\dots,U^{(r+2)})$. Since $\widehat{D}_{i}U^{(i)}=U^{(i)}\widehat{D}_{i}$, each $U^{(i)}$ is block diagonal too: $U^{(i)}=\operatorname{diag}(U^{(i)}_{1},\dots,U^{(i)}_{t}).$ Equating the blocks of $M_{Q}(A)U$ and $UM_{Q}(B)$ at the places of $\widehat{I}_{d}$, we find that $U^{(1)}=\dots=U^{(r+2)}$, and so

[TABLE]

in which every $U_{i}$ is a $d_{i}\times d_{i}$ ${}^{\!\medstar\!}$unitary matrix. The equalities

[TABLE]

ensure that $A_{\alpha_{ij:\xi}}U_{j}=U_{i}B_{\alpha_{ij:\xi}}$ for each arrow $\alpha_{ij:\xi}:j\longrightarrow i$ of $Q$, and so the complex matrix representations $A$ and $B$ are ${}^{\!\medstar\!}$unitarily isometric. The proof of Lemma 2 is complete.

Remark 1*.*

A matrix that is simpler than (26) can be constructed for most concrete quivers. For example, Section 4.1.1 shows that the matrix

[TABLE]

can be used in Lemma 2 instead of (28).

Remark 2*.*

A quiver is unitarily wild if the problem of classifying its unitary representations contains the problem of classifying unitary representations of the quiver {\begin{picture}(6.0,6.0)\put(3.0,3.6){\circle*{3.0}} \end{picture}\!\!\righttoleftarrow}; that is, it contains the problem of classifying square complex matrices up to unitary similarity. By Lemma 2, the problem of classifying unitary representations of each quiver is contained in the problem of classifying unitary representations of the quiver {\begin{picture}(6.0,6.0)\put(3.0,3.6){\circle*{3.0}} \end{picture}\!\!\righttoleftarrow}. Therefore, the problems of classifying unitary representations have the same complexity for all unitarily wild quivers. Moreover, a classification of unitary representations of any of them would imply the classification of unitary representations of each quiver. By [17, Section 2.3], all connected quivers are unitarily wild, except for the simplest quivers and {\begin{picture}(6.0,6.0)\put(3.0,3.6){\circle*{3.0}} \end{picture}\!\!\longrightarrow\!\!\begin{picture}(6.0,6.0)\put(3.0,3.6){\circle*{3.0}} \end{picture}}. The notion of unitarily wild matrix problems is analogous to the notion of wild matrix problems: a matrix problem is wild if it contains the problem of classifying matrix pairs up to similarity. By [2], the latter problem contains the problem of classifying representations of an arbitrary quiver and an arbitrary partially ordered set.

4.2 Proof of Theorem 1

(a) Let $\mathcal{A}=(\mathcal{A}_{\alpha},\mathcal{U}_{v})$ and $\mathcal{B}=(\mathcal{B}_{\alpha},\mathcal{V}_{v})$ be two ${}^{\!\medstar\!}$unitary representations of a quiver $Q$ with vertices $1,\dots,t$ (see Definition 1).

$\Longrightarrow$.  Let $\cal A$ and $\cal B$ be isometric; that is, there exist isometries $\varphi_{1}:\mathcal{U}_{1}\to\mathcal{V}_{1},\dots,\varphi_{t}:\mathcal{U}_{t}\to\mathcal{V}_{t}$ such that $\varphi_{v}\mathcal{A}_{\alpha}=\mathcal{B}_{\alpha}\varphi_{u}$ for each arrow $\alpha:u\longrightarrow v$. Let

[TABLE]

be a cycle in $\widetilde{Q}$ (thus, each $\gamma_{i}$ is either $\alpha_{i}$ or $\alpha_{i}^{\ast}$, where $\alpha_{i}$ is an arrow of $Q$). Then

[TABLE]

$\Longleftarrow$.  Let

[TABLE]

Choosing orthonormal bases in the spaces of representations $\mathcal{A}$ and $\mathcal{B}$, we obtain two matrix representations $A$ and $B$ of $Q$. By Lemma 2, it suffices to prove that the matrices $M_{Q}(A)$ and $M_{Q}(B)$ are ${}^{\!\medstar\!}$unitarily similar. Due to Specht’s criterion [19] for complex matrices under unitary similarity and its generalization by Jing [9] to complex matrices under complex orthogonal similarity (see Section 1), $M_{Q}(A)$ and $M_{Q}(B)$ are ${}^{\!\medstar\!}$unitarily similar if and only if

[TABLE]

for every word $w(x,y)$.

Let us consider the matrix

[TABLE]

that is partitioned into $(r+2)^{2}$ blocks as (29). Since

[TABLE]

it suffices to prove that

[TABLE]

Each $W_{\ell\ell}(A)$ is a linear combination of products of blocks of the form [math], $\widehat{D}_{i}$, $\widehat{A}_{j}$, and $\widehat{A}_{k}^{\,\!\medstar\!}$ in $W(A)$. Thus, it suffices to prove that

[TABLE]

for each word

[TABLE]

Let us consider the matrix

[TABLE]

that is partitioned into $t$ horizontal strips and $t$ vertical strips of sizes $d_{1},\dots,d_{t}$ (as the blocks of (30)). Since

[TABLE]

it suffices to prove that

[TABLE]

Each nonzero $V_{ll}(A)$ is a linear combination of products of the form

[TABLE]

in which $k\geqslant 1$, $i_{2},\dots,i_{k}\in\{1,\dots,t\}$, $\varepsilon_{1},\dots,\varepsilon_{k}\in\{1,\!\medstar\!\}$, and

[TABLE]

We must prove that

[TABLE]

For each natural number $n$, let $[n]\in\{1,2,\dots,t\}$ be the vertex of $Q$ such that $[n]\equiv n\pmod{t}$. The matrices of the product (32) define the complex matrix representation

[TABLE]

of the oriented cycle

[TABLE]

in $\widetilde{Q}$, in which $\alpha^{(1)}:=\alpha$ and $\alpha^{(\!\medstar\!)}:=\alpha^{\star}$ for every arrow $\alpha$ of $Q$.

The equality (33) holds by (31), which proves the statement (a) in Theorem 1.

(b) The bound (12) holds since the matrix (29) is of size $n\times n$, in which $n=(r+2)(d_{1}+\dots+d_{t})$.

Acknowledgements

V. Futorny is supported in part by CNPq grant (301320/2013-6) and by FAPESP grant (2014/09310-5). This work was done during the visit of V.V. Sergeichuk to the University of São Paulo; he is grateful to the university for hospitality and the FAPESP for financial support (grant 2015/05864-9).