On indices and eigenvectors of quivers
We study formulas for eigenvectors of strongly connected simply laced quivers in terms of eigenvalues. The relation of these formulas to the isomorphism of quivers is investigated.
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | On indices and eigenvectors of quivers / I. Dudchenko, M. Plakhotnyk // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 12–19. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1884182023-03-01T01:26:56Z On indices and eigenvectors of quivers Dudchenko, I. Plakhotnyk, M. We study formulas for eigenvectors of strongly connected simply laced quivers in terms of eigenvalues. The relation of these formulas to the isomorphism of quivers is investigated. 2019 Article On indices and eigenvectors of quivers / I. Dudchenko, M. Plakhotnyk // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 12–19. — Бібліогр.: 5 назв. — англ. 1726-3255 2010 MSC: 15A18 http://dspace.nbuv.gov.ua/handle/123456789/188418 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We study formulas for eigenvectors of strongly connected simply laced quivers in terms of eigenvalues. The relation of these formulas to the isomorphism of quivers is investigated. |
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Article |
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Dudchenko, I. Plakhotnyk, M. |
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Dudchenko, I. Plakhotnyk, M. On indices and eigenvectors of quivers Algebra and Discrete Mathematics |
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Dudchenko, I. Plakhotnyk, M. |
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Dudchenko, I. |
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On indices and eigenvectors of quivers |
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On indices and eigenvectors of quivers |
| title_full |
On indices and eigenvectors of quivers |
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On indices and eigenvectors of quivers |
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On indices and eigenvectors of quivers |
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on indices and eigenvectors of quivers |
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Інститут прикладної математики і механіки НАН України |
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2019 |
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On indices and eigenvectors of quivers / I. Dudchenko, M. Plakhotnyk // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 12–19. — Бібліогр.: 5 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT dudchenkoi onindicesandeigenvectorsofquivers AT plakhotnykm onindicesandeigenvectorsofquivers |
| first_indexed |
2025-07-16T10:27:08Z |
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2025-07-16T10:27:08Z |
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1837798920665694208 |
| fulltext |
“adm-n1” — 2019/3/22 — 12:03 — page 12 — #20
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 27 (2019). Number 1, pp. 12–19
c© Journal “Algebra and Discrete Mathematics”
On indices and eigenvectors of quivers∗
Iryna Dudchenko and Makar Plakhotnyk
Communicated by A. V. Zhuchok
Abstract. We study formulas for eigenvectors of strongly
connected simply laced quivers in terms of eigenvalues. The relation
of these formulas to the isomorphism of quivers is investigated.
1. Introduction
In this work we study a possibility to use indices and eigenvectors
of strongly connected simply laced quivers as characteristics, which can
provide the conclusion whether quivers are isomorphic or not.
Following P. Gabriel and [1] we use the term quiver for an oriented
graph. The term “quiver” was introduced in [3], which is devoted to finite
dimensional algebras over an algebraically closed field, with zero square
radical (see details in [4, §11.10]).
Recall that a quiver is called strongly connected, if for every two
vertices of it there exists an oriented path from one to other. A quiver is
called simply laced, if it has no loops and multiple arrows.
The maximum root of the characteristic polynomial of the adjacency
matrix of a quiver is called its index. In this work we use the terms
eigenvector, eigenvalue, index and characteristic polynomial of a quiver,
meaning the notions, which correspond to its adjacency matrix. An attempt
to reduce the question about the isomorphism of quivers to the properties
of their characteristic polynomials and eigenvalues was already made in [2].
We call two vectors with n coordinates permutationally equivalent (or,
simply, equivalent), if they are equal up to multiplication by a constant
and permutation of coordinates. Also we call a non-zero vector normalized,
∗This work was partially supported by FAPESP (Brazil), project 13/11350-2.
2010 MSC: 15A18.
Key words and phrases: strongly connected quiver, eigenvector.
“adm-n1” — 2019/3/22 — 12:03 — page 13 — #21
I . Dudchenko, M. Plakhotnyk 13
if its Euclidean norm is 1 and the first positive coordinate is non-negative.
The unique normalized eigenvector of a strongly connected quiver, which
corresponds to the index, will be called an index-vector. We write SCSL-
quiver for a strongly connected simply laced quiver.
The importance of the assumption that a quiver is strongly connected,
is provided by the following classical fact reformulated using the notion a
quiver.
Theorem 1 (Frobenius theorem, Theorem 6.5.2 in [5]). The adjacency
matrix of a strongly connected quiver has a positive eigenvalue r which is
a simple root of the characteristic polynomial. This vector is the unique
positive eigenvector up to multiplication by a constant. The absolute values
of all the other eigenvalues do not exceed r. To the maximal eigenvalue r
there corresponds an eigenvector with all positive coordinates.
Theorem 2 ([2]). SCSL-quivers with four vertices are isomorphic if
and only if their characteristic polynomials are equal and right and left
index-vectors are permutationally equivalent.
The number of vertices of a quiver cannot be increased in Theorem 2.
The matrices
0 0 0 0 1
0 0 0 0 1
0 1 0 1 0
0 1 1 0 0
1 0 0 1 0
and
0 0 0 0 1
0 0 0 0 1
0 1 0 1 0
1 0 1 0 0
0 1 0 1 0
provide a counterexample. They are non-equivalent. Nevertheless, their
characteristic polynomials are equal, their right eigenvectors coincide, and
their left eigenvectors coincide, too.
Also the equivalence of left index-vectors cannot be removed from
Theorem 2. This shows the following example.
Example 1. There are two non-isomorphic SCSL-quivers, whose normal-
ized right index-vectors and characteristic polynomials coincide [2].
Let
A1 =
0 0 0 1
0 0 1 1
1 0 0 1
0 1 1 0
and A2 =
0 0 0 1
0 0 1 1
1 1 0 0
0 1 1 0
“adm-n1” — 2019/3/22 — 12:03 — page 14 — #22
14 Indices and eigenvectors of quivers
be adjacency matrices of quivers Q1 and Q2, respectively. These quivers
are not isomorphic, because Q1 has a vertex, which is a head of three
arrows, and Q2 does not have such a vertex.
The characteristic polynomials of A1 and A2 are equal to λ4 − 2λ2 −
2λ − 1. Right index-vectors of Q1 and Q2 quivers coincide and are ap-
proximately equal to (0.314, 0.577, 0.484, 0.577)t. The approximate
values of left index-vectors of A1 and A2 are (0.307, 0.366, 0.565, 0.673)
and (0.314, 0.577, 0.577, 0.484) respectively. This fact shows that the
condition of the equivalence of left index-vectors is necessary.
Remark 1. Example 1, which shows that the equivalence of left index-
vectors cannot be removed from Theorem 2, is unique (up to isomorphism
of quivers).
We improve Theorem 2, using the method of computing of eigenvectors
and characteristic polynomials, which has been presented in [1]. Let A be an
adjacency matrix of a quiver with n vertices. Any characteristic vector v of
A can be treated as a solution of the equation A−λE = 0, considered as a
system of linear equations with respect to unknown coordinates of v. Thus,
the formulas for coordinates of v1, . . . , vn in terms of the eigenvalues λ can
be found, and this computation is algorithmically equivalent to solving a
system of homogeneous linear equations. The formulas for v1, . . . , vn in
terms of an arbitrary characteristic number λ of A can be considered as a
new characteristic of a quiver (of the adjacency matrix).
Example 2 ([1, p. 94]). For the Dynkin graph Ẽ7,
3
7 5 2 1 4 6 8
the general formulas for the eigenvector v = (v1, . . . , v8)
t in terms of
eigenvalues are v1 = λ3 − 2λ, v2 = λ4 − 4λ2 + 3, v3 = λ2 − 2, v4 = λ2 − 1,
v5 = λ5 − 5λ3 + 5λ, v6 = λ, v7 = λ4 − 5λ2 + 5, v8 = 1.
Example 3 ([1, p. 97]). For the Dynkin graph D̃5,
4 6
1 2
3 5
“adm-n1” — 2019/3/22 — 12:03 — page 15 — #23
I . Dudchenko, M. Plakhotnyk 15
the general formulas for the eigenvector in terms of eigenvalues are v1 = λ,
v2 = λ2 − 2, v3 = v4 = 1, v5 = v6 =
λ3
−3λ
2 .
2. Gauss-DGFKK expressions
We show that for any strongly connected simply laced quiver there
exist expressions for the coordinates of eigenvectors in terms of eigenvalues.
Recall the following fact.
Remark 2 (Remark 6.5.2 in [5]). A permutationally irreducible matrix
A > 0 cannot have two linearly independent non-negative eigenvectors
corresponding to the same eigenvalue.
Lemma 1. Let Q be a strongly connected simply laced quiver on n ver-
tices with a characteristic polynomial ξ. For any eigenvalue λ0 there
exist elements f1, . . . , fn of the field of fractions of R[x], such that vt =
(f1(λ0), . . . , fn(λ0))
t for the eigenvector v of Q, which corresponds to λ0.
Proof. If λ0 ∈ R, then v is also real, so its coordinates can be considered
as the required polynomials f1, . . . , fn, whence the lemma is trivial.
Suppose that λ0 ∈ C \ R. Denote by λ0 the complex conjugate to λ0
and let f0(x) = (x−λ0)(x−λ0) ∈ R[x]. Denote by F0 the field of fractions
of the factor-ring R[x]/(f0). Apply the classical Gaussian elimination
algorithm of diagonalization to the matrix A− λE over the field F0. This
process leads to the general form of g ∈ Fn
0 such that (A−λE)g = 0 ∈ F0.
By Remark 2, each of vectors g(λ0) and g(λ0) is an eigenvector of Q, and
they correspond to λ0 and λ0, respectively.
Lemma 2. For any strongly connected simply laced quiver there exist
real fractional-polynomial formulas, which express the coordinates of an
eigenvector in terms of eigenvalues. In other words, let v(1), . . . , v(k) be
all linearly independent vectors of Q, which, by Remark 2, correspond to
eigenvalues λ1, . . . , λk. Then there exist functions f1, . . . , fn from the field
of fractions of R[x] such that
(v(s))t = (f1(λs), . . . , fn(λs))
for every s, 1 6 s 6 k.
Proof. Denote by ξ = ξ1 · . . . · ξm the decomposition of the characteristic
polynomial of Q into the product of indecomposable multipliers over R[x].
By Lemma 1, for every i, 1 6 i 6 m there exist elements f
(i)
1 , . . . , f
(i)
n of the
“adm-n1” — 2019/3/22 — 12:03 — page 16 — #24
16 Indices and eigenvectors of quivers
field of fractions of R[x] such that for every root λ of ξi the corresponding
eigenvector v(λ) can be expressed as v(λ) = (f
(i)
1 (λ), . . . , f
(i)
n (λ))t. Thus,
v =
m∑
i=1
∑
λ: ξi(λ)=0
v(λ) ·
ξ
ξi
,
which is the required vector by the construction.
The proof of Lemma 2 motivates the following construction.
1. For a strongly connected simply laced quiver Q with the adjacency
matrix A consider the matrix B = A−λE as one over the field of fractions
of the ring of polynomials R[λ].
2. Apply the Gaussian elimination algorithm to reduce the matrix B to
the upper diagonal form B̃ = (̃bij).
3. Take the matrix C = (cij) such that cnn = 0 and cij = b̃ij otherwise.
4. Use the matrix C, as the matrix of a system of linear homogeneous
equations to express variables x1, . . . , xn−1 in terms of xn. Thus, we obtain
a vector v(λ) = (x1(λ), . . . , xn−1(λ), xn)
t, which provides a characteristic
of Q.
Since Lemma 2 was motivated by [1], the vector v(λ), constructed
above, will be called the Gauss-DGFKK expression for eigenvectors of Q.
Example 4. Find Gauss-DGFKK expressions for the eigenvectors of the
quiver 1 // 2oo . The necessary transformations of the matrix are
(
−λ 1
1 −λ
)
⇒
(
−λ 1
1 + −λ
λ
−λ+ 1
λ
)
=
(
−λ 1
0 1−λ2
λ
)
⇒
(
1 −λ
0 λ2 − 1
)
.
Claim now that λ2 − 1 is the characteristic polynomial of our quiver.
Then the equation x1 − λx2 = 0 with the assumption x2 = 1 gives the
Gauss-DGFKK expression v = (λ, 1).
The following example provides an improvement of Theorem 2 by
replacing the condition of equality of left eigenvectors to the condition of
equality of the Gauss-DGFKK expressions.
Example 5. Gauss-DGFKK expressions for left eigenvectors of quivers
in Example 1 are different.
Our computations show, that the Gauss-DGFKK expression for the
right eigenvector of A1 from Example 1 is
(
1
λ
, λ2+λ+1
λ3 , λ+1
λ2 , 1
)t
. Analo-
gously, the Gauss-DGFKK expression for the right eigenvector of A2 is(
1
λ
, −1−λ2
λ(1−λ2)
, −2
1−λ2 , 1
)t
.
“adm-n1” — 2019/3/22 — 12:03 — page 17 — #25
I . Dudchenko, M. Plakhotnyk 17
Nevertheless, note that numerical values of given eigenvectors are equal
if λ is an eigenvalue of the corresponding matrix. Moreover, the numerical
values of these expressions are equal if and only if λ is the eigenvalue of
the quiver.
The next theorem follows from Theorem 2 and Remark 1.
Theorem 3. If SCSL-quivers with four vertices have either different
characteristic polynomials, or distinct Gauss-DGFKK expressions for the
right eigenvector, then they are non-isomorphic.
The following example shows that Theorem 3 is not a criterion.
Example 6. There are equivalent SCSL-quivers with four vertices, whose
Gauss-DGFKK expressions for the right eigenvectors are different.
Let
A =
0 0 0 1
0 0 0 1
0 1 0 0
1 0 1 0
be the adjacency matrix of a SCSL-quiver and B the adjacency matrix of
the quiver, obtained from the SCSL-quiver by renumbering of vertices 3
and 4.
The Gauss-DGFKK expression for the right eigenvector of A is(
1
λ
, 1
λ
, 1
λ2 , 1
)t
. At the same time, the Gauss-DGFKK expression for the
right eigenvector of B is
(
1
λ2
−1
, 1
λ2
−1
, λ
λ2
−1
, 1
)t
.
Example 7. There exist quivers with 5 vertices, whose characteristic
polynomials are different, but both left and right eigenvectors can be
expressed by the same formulas.
Consider quivers, given by their adjacency matrices
A1 =
0 0 0 1
0 0 1 0
1 0 0 0
0 1 1 0
and A2 =
0 0 0 1
0 0 1 0
1 0 0 0
1 1 1 0
.
Their characteristic polynomials are
f1(λ) = λ4 − λ− 1 and f2(λ) = λ4 − λ2 − λ− 1
“adm-n1” — 2019/3/22 — 12:03 — page 18 — #26
18 Indices and eigenvectors of quivers
respectively. The Gauss-DGFKK formulas for the right eigenvectors are
v(1)r =
(
1
λ
,
1
λ3
,
1
λ2
, 1
)t
and v(2)r =
(
1
λ
,
1
λ3
,
1
λ2
, 1
)t
.
The Gauss-DGFKK formulas for the left eigenvectors are
v
(1)
l =
(
λ+ 1
λ3
,
1
λ
,
λ+ 1
λ2
, 1
)
and v
(2)
l =
(
λ2 + λ+ 1
λ3
,
1
λ
,
λ+ 1
λ2
, 1
)
.
Nevertheless, formulas for vectors v
(1)
l and v
(2)
l can be rewritten as
vl =
(
λ,
1
λ
,
λ+ 1
λ2
, 1
)
,
because λ4 = λ + 1 for each solution of the equation f1(λ) = 0, and
λ4 = λ2 + λ+ 1 for each solution of the equation f2(λ) = 0.
Example 8. There are non-isomorphic quivers such that their charac-
teristic polynomials are equal and Gauss-DGFKK expressions for right
eigenvectors are also equal.
Let SCSL-quivers Q1 and Q2 be given by their adjacency matrices
A1 =
0 0 0 1 0
0 0 0 1 0
0 0 0 1 1
0 0 1 0 1
1 1 1 1 0
and A2 =
0 0 1 0 0
0 0 0 1 0
0 0 0 1 1
0 0 1 0 1
1 1 1 1 0
.
These quivers are non-conjugated, because Q1 has a vertex, such that
for all other vertices there are arrows to it and Q2 has not such a vertex.
Characteristic polynomials of quivers coincide and are equal to
−λ5 + 3λ3 + 4λ2 + 2λ.
Gauss-DGFKK expressions for right eigenvectors v1 and v2 of A1 and
A2 are
v1 = v2 =
(
λ+ 1
λ3 − λ
,
λ+ 1
λ3 − λ
,
1
λ− 1
,
1
λ− 1
, 1
)t
.
Acknowledgement
The authors would like to thank the referee for various useful comments
and remarks, which have helped to improve the manuscript.
“adm-n1” — 2019/3/22 — 12:03 — page 19 — #27
I . Dudchenko, M. Plakhotnyk 19
References
[1] M.A. Dokuchaev, N.M. Gubareni, V.M. Futorny, M.A. Khibina and V.V.
Kirichenko, Dynkin Diagrams and Spectra of Graphs, São Paulo Journal of
Mathematical Sciences, 7, No. 1, 83-104 (2013).
[2] I. Dudchenko, V. Kirichenko and M. Plakhotnyk, Simply laced quivers and their
properties, Ukrainian Mathenatical Journal, 64, 2012, p. 291-306.
[3] P. Gabriel, Unzerlegbare Darstellungen 1, Man. Math., 1972, v.6, p.71-103;
[4] M. Hazewinkel, N. Gubareni and V.V. Kirichenko, Algebras, Rings and Modules,
V.I, Mathematics and Its Applications, Vol. 575, Kluwer Academic Publishers,
2004, 380 p.
[5] M. Hazewinkel, N. Gubareni and V.V. Kirichenko, Algebras, Rings and Modules,
V.II, Mathematics and Its Applications, Vol. 586, Springer, 2007, 400 p.
Contact information
I. Dudchenko Slovjansk, Donetsk region, Ukraine 84112.
E-Mail(s): dudchira@gmail.com
M. Plakhotnyk Instituto de Matematica e Estatistica da
Universidade de São Paulo. Rua do Matão,
1010. CEP 05508-090. São Paulo - SP. Brazil.
E-Mail(s): makar.plakhotnyk@gmail.com
Received by the editors: 11.07.2018
and in final form 19.03.2019.
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