On finite posets of injinj-finite type and their Tits forms
We prove one theorem on connection between inj-finiteness of a finite poset and positive definiteness of its Tits form.
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irk-123456789-1573822019-06-21T01:26:26Z On finite posets of injinj-finite type and their Tits forms Bondarenko, V.M. Styopochkina, M.V. We prove one theorem on connection between inj-finiteness of a finite poset and positive definiteness of its Tits form. 2006 Article On finite posets of injinj-finite type and their Tits forms / V.M. Bondarenko, M.V. Styopochkina // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 2. — С. 17–21. — Бібліогр.: 7 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 15A63, 16G20, 16G60. http://dspace.nbuv.gov.ua/handle/123456789/157382 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We prove one theorem on connection between
inj-finiteness of a finite poset and positive definiteness of its Tits
form. |
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Article |
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Bondarenko, V.M. Styopochkina, M.V. |
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Bondarenko, V.M. Styopochkina, M.V. On finite posets of injinj-finite type and their Tits forms Algebra and Discrete Mathematics |
| author_facet |
Bondarenko, V.M. Styopochkina, M.V. |
| author_sort |
Bondarenko, V.M. |
| title |
On finite posets of injinj-finite type and their Tits forms |
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On finite posets of injinj-finite type and their Tits forms |
| title_full |
On finite posets of injinj-finite type and their Tits forms |
| title_fullStr |
On finite posets of injinj-finite type and their Tits forms |
| title_full_unstemmed |
On finite posets of injinj-finite type and their Tits forms |
| title_sort |
on finite posets of injinj-finite type and their tits forms |
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Інститут прикладної математики і механіки НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/157382 |
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On finite posets of injinj-finite type and their Tits forms / V.M. Bondarenko, M.V. Styopochkina // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 2. — С. 17–21. — Бібліогр.: 7 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT bondarenkovm onfiniteposetsofinjinjfinitetypeandtheirtitsforms AT styopochkinamv onfiniteposetsofinjinjfinitetypeandtheirtitsforms |
| first_indexed |
2025-07-14T09:49:14Z |
| last_indexed |
2025-07-14T09:49:14Z |
| _version_ |
1837615341308477440 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2006). pp. 17 – 21
c© Journal “Algebra and Discrete Mathematics”
On finite posets of injinjinj-finite type and their Tits
forms
Vitalij M. Bondarenko, Marina V. Styopochkina
Communicated by V. V. Kirichenko
Dedicated to the memory of V. M. Usenko
Abstract. We prove one theorem on connection between
inj-finiteness of a finite poset and positive definiteness of its Tits
form.
The quadratic Tits form, introduced by P. Gabriel [1] for quivers, S.
Brenner [2] for quivers with relations, Yu. A. Drozd [3] for posets, etc.
plays an important role in representation theory (see e.g. the introduction
to [4]). In particular, in [2] it is proved that a finite poset A (without an
element 0) has only finitely many isomorphism classes of indecomposable
representations (over a field) if and only if its Tits form qA : Z
A∪0 → Z,
defined by the equality
qA(z) = z2
0 +
∑
i∈A
z2
i +
∑
i<j,
i,j∈A
zizj − z0
∑
i∈A
zi,
is weakly positive (a quadratic form f(z) : Z
n → Z is called weakly
positive if f(z) > 0 for all nonzero z = (z1, . . . , zn) with z1, . . . , zn ≥ 0;
if f(z) > 0 for all z 6= 0, the form is called positive definite or simply
positive).
In [5] the authors proved that the category of representations of the
category InjA (of injective representations of A) has only finitely many
isomorphism classes of indecomposable objects if and only if the Tits
form of InjA is weakly positive. In this paper we continue to study the
categories InjA of such type.
2000 Mathematics Subject Classification: 15A63, 16G20, 16G60.
Key words and phrases: category of representations, inj-finite poset, positive
definite form, the quadratic Tits form.
18 On finite posets of injinjinj-finite type and their Tits forms
1. Injective representations of posets
Throughout the paper, all posets (partially ordered sets) are finite and
all vector spaces are finite-dimensional; k denotes a fixed field.
Recall some well-known definitions in terms of vector spaces graded
by posets (see [5], [6]).
Let A be a poset. An A-graded k-vector space is by definition the
direct sum U =
⊕
a∈A Ua of k-vector spaces Ua. A linear map ϕ : U → U ′
between A-graded vector spaces U and U ′
is called an A-map if ϕa∗a∗ = ϕaa for each a ∈ A and ϕbc = 0 for each
b, c ∈ A not satisfying b 6 c, where ϕxy denotes the linear map of Ux into
U ′
y induced by the map ϕ.
A representation of a poset A over k is by definition a triple W =
(V, U, γ) formed by a k-vector space V , an A-graded k-vector space U
and a linear map γ : V → U ; a morphism of representations
W → W ′ is given by a pair (µ, ν), formed by a linear map µ : V →
V ′ and an A-map ν : U → U ′ such that γν = µγ′. The category of
representations of A will be denoted by RepA. For a morphism α =
(µ, ν) : X → Y in RepA, we write 0 ⇒ X
α
→ Y if µ and all νxx are
injective. A representation X of a poset A is said to be injective if any
diagram
0 ⇒ R′ → R
↓
X
can be embedded in a commutative diagram
0 ⇒ R′ → R
↓ ւ
X
.
The full subcategory of RepA consisting of all injective objects will
be denoted by InjA. We say that A is of inj-finite type if the category
Funct(InjA,mod k) of representations of InjA has only finitely many
isomorphism classes of indecomposable objects.
We associate to a poset A the quiver
−→
A = (
−→
A0,
−→
A1) with the set of
vertices
−→
A0 = A and the set of arrows
−→
A1 = {i → j | i < j, i and j are adjacent}
(elements i and j > i is called adjacent if there is not an element s, such
that j > s > i). We will consider
−→
A as a commutative quiver, that is,
any two non-trivial path in
−→
A with the same starting and terminating
V. M. Bondarenko, M. V. Styopochkina 19
vertices are equal (and
−→
A has no other relations). An arrow x → y is
denoted by (x, y), and we write [x, y] if there is an arrow x → y or y → x.
In [5] the authors proved the following theorem.
Theorem 1. Let A be a poset and B = A ∪ ∞, where x < ∞
for any x ∈ A. The poset A is of inj-finite type if and only if the
commutative quiver
−→
B contains no subquiver (with relations) isomorphic
or antiisomorphic to one of the following connected commutative quiver
Q = (Q0, Q1):
I. Q1 = {(1, 3), (1, 4), (2, 3), (2, 4)};
II. Q1 = {[1, 2], [1, 3], [1, 4], [1, 5]};
III. Q1 = {[1, 2], [2, 3], [1, 4], [4, 5], [1, 6], [6, 7]};
IV. Q1 = {[1, 2], [2, 3], [3, 4], [1, 5], [5, 6], [6, 7], [1, 8]};
V. Q1 = {[1, 2], [2, 3], [1, 4], [4, 5], [5, 6], [6, 7], [7, 8], [1, 9]};
VI. Q1 = {[1, 2], [2, 3], [3, 4], [4, 5], (6, 5), (5, 8), (6, 7), (7, 8), [7, 9]};
VII. Q1 = {[1, 2], [2, 3], [3, 4], [4, 5], (6, 5), (5, 8), (6, 7), (7, 8), [8, 9]};
VIII. Q1 = {[1, 2], [2, 3], [3, 4], [4, 5], (6, 5), (7, 6), (8, 5), (7, 8), [8, 9]};
IX. Q1 = {[1, 2], [2, 3], (4, 3), (3, 8), (4, 5), (5, 8), [5, 6], [6, 7]};
X. Q1 = {[1, 2], [2, 3], [3, 4], (5, 4), (4, 8), (5, 6), (6, 7), (7, 8), [7, 9]};
XI. Q1 = {(1, 2), (2, 5), (1, 3), (3, 4), (4, 5), [4, 6], [6, 7], [7, 8], [8, 9]};
XII. Q1 = {[1, 2], [2, 3], (4, 3), (3, 8), (4, 5), (5, 6), (6, 7), (7, 8), [6, 9]}.
Recall that a quiver Q is called a subquiver of a commutative quiver P
if it can be obtained from P by combination of the following operations:
a) rejection of a (+)- or (-)-admissible vertex (i.e., such that is not,
respectively, starting or terminating for any arrow), together with all
arrows that contain it;
b) identification of the ends of an arrow α, together with rejection
of α and any unnecessary arrow β (i.e., such that is equal to a path
γ = γ1 . . . γs, where γi 6= β).
Note that Q is considered as a quiver with relations induced by the
relations of commutativity (Q is not necessarily commutative).
2. The main result
We study connection between inj-finiteness of a finite poset and positive
definiteness of its Tits form.
A poset A is said to be quasi-primitive if
−→
A is a disjoint union of chains
(in the case when all arrows of every chain have the same direction, the
poset is called primitive).
20 On finite posets of injinjinj-finite type and their Tits forms
The main result of this paper is the following theorem.
Theorem 2. Let A be a quasi-primitive poset which is not self-dual.
Then both A and Aop are of inj-finite type if and only if the Tits form of
A is positive.
Proof. Sufficiency. Let the Tits form of A is positive. Then by
Theorem 4 [7] A is, up to isomorphism and antiisomorphism, a subposet
(proper or not) of one of the following posets: 1) 1 ≺ 2 ≺ 7, 3 ≺ 4 ≺
5 ≺ 6 ≺ 7; 2) 2 ≺ 3, 4 ≺ 5 ≺ 6 ≺ 7; 3) 2 ≺ 7, 3 ≺ 4 ≺ 5 ≺ 6 ≺ 7;
4) 2 ≺ 3, 2 ≺ 7, 4 ≺ 5 ≺ 6 ≺ 7; 5) 2 ≺ 3 ≺ 4, 2 ≺ 7, 5 ≺ 6 ≺ 7; 6)
1 ≺ 2 ≺ 3, 4 ≺ 7, 5 ≺ 6 ≺ 7; 7) 1 ≺ 2 ≺ 3, 4 ≺ 5, 4 ≺ 7, 6 ≺ 7; 8) 1 ≺
3, 2 ≺ 3, 2 ≺ 7, 4 ≺ 5 ≺ 6 ≺ 7; 9) 1 ≺ 4, 2 ≺ 3 ≺ 4, 2 ≺ 5 ≺ 6 ≺ 7; 10)
1 ≺ 4, 2 ≺ 3 ≺ 4, 2 ≺ 7, 5 ≺ 6 ≺ 7; 11) 1 ≺ 5, 2 ≺ 3 ≺ 4 ≺ 5, 2 ≺ 7, 6 ≺ 7;
12) 1 ≺ 6, 2 ≺ 3 ≺ 4 ≺ 5 ≺ 6, 2 ≺ 7; 13) 1 ≺ 2, 1 ≺ 4, 3 ≺ 4, 3 ≺ 7, 5 ≺
6 ≺ 7; 14) 1 ≺ 2, 1 ≺ 5, 3 ≺ 4 ≺ 5, 3 ≺ 7, 6 ≺ 7; 15) 1 ≺ 2 ≺ 5, 3 ≺ 4 ≺
5, 3 ≺ 6 ≺ 7; 16) 1 ≺ 2 ≺ . . . ≺ p, p + 1 ≺ p + 2 ≺ . . . ≺ p + q, 1 ≺ p + q
(the posets 1)–15) and 16) contain of elements 1,2,. . . , and are of order 7
and p + q, respectively). By Theorem 1 the poset A is of inj-finite type
(even when it is self-dual).
Necessity. Let the Tits form of A is not positive. We prove that
A or Aop is not of inv-finite type. By Theorem 3 [7] A contains (up
to isomorphism and antiisomorphism) one of the following posets: 1)
1 ≺ 2 ≺ 3 ≺ 7, 4 ≺ 5 ≺ 6 ≺ 7; 2) 1 ≺ 2 ≺ 8, 3 ≺ 4 ≺ 5 ≺ 6 ≺ 7 ≺ 8; 3)
1 ≺ 2, 3 ≺ 4, 5 ≺ 6; 4) 2 ≺ 3 ≺ 6, 4 ≺ 5 ≺ 6; 5) 2 ≺ 3 ≺ 4, 5 ≺ 6 ≺ 7;
6) 1 ≺ 2, 3 ≺ 7, 4 ≺ 5 ≺ 6 ≺ 7; 7) 1 ≺ 2 ≺ 4, 3 ≺ 4, 3 ≺ 7, 5 ≺ 6 ≺ 7;
8) 2 ≺ 3, 4 ≺ 5 ≺ 6 ≺ 7 ≺ 8; 9) 2 ≺ 8, 3 ≺ 4 ≺ 5 ≺ 6 ≺ 7 ≺ 8; 10)
2 ≺ 3, 2 ≺ 8, 4 ≺ 5 ≺ 6 ≺ 7 ≺ 8; 11) 1 ≺ 2 ≺ 3 ≺ 4, 5 ≺ 8, 6 ≺ 7 ≺ 8;
12) 1 ≺ 2 ≺ 3 ≺ 4, 5 ≺ 6, 5 ≺ 8, 7 ≺ 8; 13) 1 ≺ 3, 2 ≺ 3, 2 ≺ 8, 4 ≺ 5 ≺
6 ≺ 7 ≺ 8; 14) 1 ≺ 4, 2 ≺ 3 ≺ 4, 2 ≺ 5 ≺ 6 ≺ 7 ≺ 8; 15) 1 ≺ 4, 2 ≺ 3 ≺
4, 2 ≺ 8, 5 ≺ 6 ≺ 7 ≺ 8; 16) 1 ≺ 7, 2 ≺ 3 ≺ 4 ≺ 5 ≺ 6 ≺ 7, 2 ≺ 8; 17)
the four elements 1, 2, 3, 4 are pairwise incomparable (the posets 1)–16)
consist of elements 1, 2, . . . , p, where in each case p is a maximal indicated
number). By Theorem 1 each of the posets 1)–13), 14op), 15), 17) are not
of inj-finite type (even when it is self-dual). The poset P = 16) is self-
dual and of inj-finite type. But if A contains P , then it contains the poset
P ∪ 9, where either 9 is incomparable to any i ∈ P , or 9 is incomparable
to any i ∈ P \ 8 and 9 ≺ 8, or 9 is incomparable to any i ∈ P \ {2, 8} and
9 ≻ 8 (the case A = P is impossible since A is not self-dual); in the first
case P contains the poset 17), and in the second and third ones P is not
of inj-finite type by Theorem 1.
Thus A or Aop is not of inv-finite type.
Theorem 2 is proved.
V. M. Bondarenko, M. V. Styopochkina 21
REFERENCES
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V. 6. – P. 71–103.
2. Brenner S. Quivers with commutativity conditions and some phenomenol-
ogy of forms // Proc. of Intern. Conference of Representations of Alge-
bras. – Carleton Univ., Ottawa, Ontario, 1974. – Paper N5.
3. Drozd Yu. A. Coxeter transformations and representations of partially
ordered sets // Funkc. Anal. i Priložen. – 1974. – V. 8. – P. 34–42 (in
Russian).
4. Bondarenko V. M., Polishchuk A. M. Minimax sums of posets and the
quadratic Tits form // Algebra Discrete Math. – 2004. – N 1 – P. 17–36.
5. Bondarenko, V. M., Styopochkina M. V. Posets of injective-finite type //
Bull. of the University of Uzhgorod. – 2005. –. 10-11. – P. 22–33 (in
Russian).
6. Bondarenko V. M. Linear operators on S-graded vector spaces // Linear
algebra and appl., speciall issue: “Linear Algebra Methods in Represen-
tation Theory”. – 2003. – 365. – 45-90.
7. Bondarenko V. M., Styopochkina M. V. (Min, max)-equivalence of posets
and the quadratic Tits form // Theory of Functions and Algebra, Inst.
Math. NAS Ukraine. – 2,3. – 2005. – P. 3-46 (in Russian).
Contact information
V. M. Bondarenko Institute of Mathematics, National
Academy of Sciences of Ukraine,
Tereshchenkivska 3, 01601 Kyiv, Ukraine
E-Mail: vit-bond@imath.kiev.ua
M. V. Styopochkina Kyiv Taras Shevchenko University,
Volodymyrs’ka 64, Kyiv, 01033, Ukraine
E-Mail: StMar@ukr.net
Received by the editors: 05.09.2006
and in final form 29.09.2006.
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