Weighted partially orderd sets of finite type

We define representations of weighted posets and construct for them reflection functors. Using this technique we prove that a weighted poset is of finite representation type if and only if its Tits form is weakly positive; then indecomposable representations are in one-to-one correspondence with...

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Datum:	2006
1. Verfasser:	Drozd-Koroleva, O.
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Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2006
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spelling	irk-123456789-1573582019-06-21T01:26:44Z Weighted partially orderd sets of finite type Drozd-Koroleva, O. We define representations of weighted posets and construct for them reflection functors. Using this technique we prove that a weighted poset is of finite representation type if and only if its Tits form is weakly positive; then indecomposable representations are in one-to-one correspondence with the positive roots of the Tits form. 2006 Article Weighted partially orderd sets of finite type / O. Drozd-Koroleva // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 2. — С. 36–49. — Бібліогр.: 12 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 16G20, 16G60. http://dspace.nbuv.gov.ua/handle/123456789/157358 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We define representations of weighted posets and construct for them reflection functors. Using this technique we prove that a weighted poset is of finite representation type if and only if its Tits form is weakly positive; then indecomposable representations are in one-to-one correspondence with the positive roots of the Tits form.
format	Article
author	Drozd-Koroleva, O.
spellingShingle	Drozd-Koroleva, O. Weighted partially orderd sets of finite type Algebra and Discrete Mathematics
author_facet	Drozd-Koroleva, O.
author_sort	Drozd-Koroleva, O.
title	Weighted partially orderd sets of finite type
title_short	Weighted partially orderd sets of finite type
title_full	Weighted partially orderd sets of finite type
title_fullStr	Weighted partially orderd sets of finite type
title_full_unstemmed	Weighted partially orderd sets of finite type
title_sort	weighted partially orderd sets of finite type
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/157358
citation_txt	Weighted partially orderd sets of finite type / O. Drozd-Koroleva // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 2. — С. 36–49. — Бібліогр.: 12 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT drozdkorolevao weightedpartiallyorderdsetsoffinitetype
first_indexed	2025-07-14T09:48:05Z
last_indexed	2025-07-14T09:48:05Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Number 2. (2006). pp. 36 – 49 c© Journal “Algebra and Discrete Mathematics” Weighted partially orderd sets of finite type Olena Drozd-Koroleva Communicated by M. Ya. Komarnytskyj Abstract. We define representations of weighted posets and construct for them reflection functors. Using this technique we prove that a weighted poset is of finite representation type if and only if its Tits form is weakly positive; then indecomposable repre- sentations are in one-to-one correspondence with the positive roots of the Tits form. Representations of posets (partially ordered sets) were introduced in [9]. In [7, 8] a criterion was given for a poset to be representation finite, i.e. having only finitely many indecomposable representations (up to isomor- phism), and all indecomposable representations of posets of finite type were described. Further, in [4] Coxeter transformations were constructed for representations of posets, following the framework of [1]. It implied another criterion for a poset to be representation finite, not involving ex- plicit calculations, but using the Tits quadratic form, also analogous to that of [1]. Note that this paper did not give all reflections, corresponding to the Tits form. They were constructed in [6], using a generalization of representations of posets, namely, representations of bisected posets. Note that all these matrix problems are “split,” i.e. do not involve extensions of the basic field. Some cases, when such extensions arise, were considered by Dlab and Ringel [2, 3]. The problems considered in [3] generalize representations of posets, though this generalization seems insufficient, especially when compared with [2]. Our aim is to present a more adequate generalization of representa- tions of posets, which involves field extensions (even non-commutative), to construct the corresponding reflection functors and thus to obtain a 2000 Mathematics Subject Classification: 16G20, 16G60. Key words and phrases: weighted partially ordered sets, finite representation type, reflection functors, Tits form. O. Drozd-Koroleva 37 criterion of representation finite ness, as well as a description of indecom- posable representations in representation finite case. We call the arising problems representations of weighed bisected posets. They seem to be the most natural generalization of representations of posets allowing these constructions. By the way, even in “split” case they include the so called Schurian vector space categories (though nothing new arises in represen- tation finite split case). Since most proofs are quite similar to those of [6], we mainly only sketch them, though we give the details of all constructions, since they are not so evident. 1. Definitions and the Main Theorem Recall [6] that a bisected poset is a poset S with a fixed partition S = S− ∪ S+ (S− ∩ S+ = ∅) such that if i ∈ S− and j < i, also j ∈ S−. We introduce a new symbol 0 /∈ S and set Ŝ = S ∪ { 0 } , Ŝ+ = S+ ∪ { 0 } , Ŝ− = S−∪{ 0 }. It is convenient, and we always do so, to set 0 < i for i ∈ S− and i < 0 for i ∈ S+. Note that < is an order on Ŝ− and on Ŝ+, but not an order on Ŝ. We write • i⋖ j if i < j and either both i, j ∈ Ŝ− or both i, j ∈ Ŝ+; • i≪ j if i < j, i ∈ S−, j ∈ S+; • i ≶ j if i < j or j < i for i, j ∈ S. Let k be a fixed field (basic field). We consider finite dimensional skewfields (division algebras) over k and finite dimensional bimodules over such skewfields. If V is a K-L-bimodule and W is a L-F-bimodule, we write VW for the K-F-bimodule V LW . We also set V ∗ = homk(V, k) and naturally identify it with homK(V,K) and with homL(V,L) asL-K- bimodule s. We also use the natural isomorphisms HomK-F(UV,W ) ≃ HomK-L(U,WV ∗) ≃ HomL-F(V,U∗W ) ≃ (1.1) ≃ HomF-L(W ∗U, V ∗) ≃ HomL-K(VW ∗, U∗), where U, V,W are, respectively, K-L-bimodule,L-F-bimodule and K-F- bimodule, as well as the duality isomorphism V ≃ V ∗∗. If a map f belongs to one of these spaces, we usually denote by f̃ its image in another one under the corresponding isomorphism. Definition 1. A weighted bisected poset, or WBS, consists of: • A finite poset S = S− ∪ S+. We 38 Weighted partially orderd sets of finite type • a map i 7→ K(i), where i ∈ Ŝ and K(i) is a finite dimensional skewfield over k; • a set of finite dimensional K(i)-K(j)-bimodules V (ij), where i, j ∈ Ŝ and either j ⋖ i or i≪ j; • a set of K(i)-K(j)-linear maps µ(ikj) : V (ik)V (kj) → V (ij) given for any triple i, j, k ∈ Ŝ such that all these bimodules are defined. We write uv for µ(ikj)(uv). These maps must satisfy the following conditions: 1. “associativity”: µ(ilj)(µ(ikl)1) = µ(ikj)(1µ(klj)) as soon as these maps are defined (it means that (uv)w = u(vw)); 2. “non-degeneracy”: • if j < i, i, j ∈ S− and v ∈ V (ij), v 6= 0, there is an element u ∈ V (j0) such that vu 6= 0; • if j < i, i, j ∈ S+ and v ∈ V (ij), v 6= 0, there is an element u ∈ V (0i) such that uv 6= 0; • if j ≪ i, the map µ(j0i) is surjective. We often write “a WBS S” not mentioning the ingredients S±, K(i), V (ij) and µ(ikj). Definition 2. 1. A representation (M,f) of a WBS S consists of: • finite dimensional K(i)-vector spaces M(i) given for each i ∈ Ŝ; • K(i)-linear maps f(i) : M(i) → V (i0)M(0) given for each i ∈ S−; • K(0)-linear maps f(i) : M(0) → V (0i)M(i) given for each i ∈ S+, such that the product M(i) f(i) −−→ V (i0)M(0) 1f(j) −−−→ V (i0)V (0j)M(j) µ(i0j)1 −−−−→ V (ij)M(j) is zero for every pair i≪ j. Again, we often write “a representation M ” not mentioning f . 2. A morphism φ : (M,f) → (M, g) is a set of K(i)-linear maps φ(i) : M(i) → N(i) for all i ∈ Ŝ, φ(ji) : M(i) → V (ij)N(j) for j ⋖ i, O. Drozd-Koroleva 39 that satisfy the following conditions: g(i)φ(0) = (1φ(i))f(i) + ∑ i<j (µ(0ji)1)(1φ(ij))f(j) for i ∈ S+ and g(i)φ(i) = (1φ(0))f(i) + ∑ j<i (µ(ij0)1)(1g(j))φ(ji) for i ∈ S−. We denote by homS(M,N) the set of such morphisms. Remark. If all skewfields K(i) as well as all bimodules V (ij) coincide with the basic field k and all maps µ(ikj) are identities, these definitions coincide with the definitions of representations of bisected posets from [6]. If all K(i) = k but not necessarily V (ij) = k, we get a slight generalization of subspace categories of Schurian vector space categories [11]. Note that in the latter case the problem is never representation finite. Representations of a WBS S and their morphisms form a k-linear, fully additive category repS. The unit morphism IdM in this category is such that IdM (i) = IdM(i) for each i and all IdM (ij) = 0. Since all spaces homS(M,N) are finite dimensional, it is a Krull–Schmidt cate- gory, i.e. every representation uniquely decomposes into a direct sum of indecomposable ones. Definition 3. We call a WBS S representation finite if it only has finitely many non-isomorphic indecomposable representations. Otherwise we call it representation infinite. We are going to find a criterion for a WBS to be representation finite and to describe indecomposable representations in representation finite case. To do it, just as in [1, 2, 4, 6], we use the Tits form and reflection functors. Definition 4. For a WBS S we set di = dimk K(i), dij = dji = dimk V (ij), consider the real vector space RŜ of functions x : Ŝ → R and define the Tits form QS as the quadratic form on the space RŜ such that QS(x) = ∑ i∈Ŝ dix(i)2 + ∑ i<j i,j∈S dijx(i)x(j) − ∑ i∈S di0x(i)x(0). 40 Weighted partially orderd sets of finite type We fix the natural base { ei \| i ∈ Ŝ } in the space RŜ, where ei(j) = δij and identify a function x : Ŝ → R with the vector (xi \| i ∈ Ŝ), where xi = x(i). For a representation M ∈ repS we define its (vector) dimen- sion dimM ∈ RŜ as the function i 7→ dimkM(i). Actually, dimM ∈ NŜ; the latter semigroup we call the semigroup of dimensions for S. The Tits form is integer in the sense of [12], since di \| dij for all possible i, j. Therefore, (real) roots of this form are defined: they are vectors that can be obtained from ei by a series of reflections. Recall that the reflection σi is defined as the unique non-identical linear map RŜ → RŜ such that σix(j) = x(j) for all j 6= i and QS(σix) = QS(x) for all x. One easily sees that d0σ0x(0) = ∑ i∈S di0x(i) − d0x(0), diσix(i) = di0x(0) − dix(i) − ∑ j≶i dijx(j) if i ∈ S. We write x > 0 and call x positive if x 6= 0 and x(i) ≥ 0 for all i ∈ Ŝ. Especially, positive roots are defined. Now we are able to formulate the main theorem of our paper. Theorem 1. A WBS S is representation finite if and only if its Tits form is weakly positive, i.e. QS(x) > 0 for each x > 0. Moreover, in this case • the dimensions of indecomposable representations of S coincide with the positive roots of the form QS; • any two indecomposable representations having equal dimensions are isomorphic. The fact that representation finite ness implies weakly positivity of the Tits form is general for matrix problems. It follows, for instance, from [5]. The proof of other assertions of Theorem 1 relies upon reflection functors, which we shall construct in the next section. Though this construction was inspired by [6], its details are more complicated, so we present them thoroughly. 2. Reflection functors First we define reflections of WBS themselves. Definition 5. 1. Given a WBS S, we set: O. Drozd-Koroleva 41 • V (ii) = K(i) and take for µ(iij) and µ(ijj) the natural iso- morphisms K(i)V (ij) ≃ V (ij) and V (ij)K(j) ≃ V (ij) as soon as V (ij) is defined; • V (ji) = V (ij)∗ as soon as V (ij) is defined; • µ(kji) and µ(jik) to be the maps corresponding to µ(ikj) via the isomorphisms (1.1) as soon as µ(ikj) is defined. One easily checks that the associativity conditions hold for these maps too, while the non-degeneracy conditions turn into surjectivity of the maps µ(j0i) for all i, j ∈ S, j < i. 2. We call an element p ∈ Ŝ a source (a sink) if it is a maximal element of Ŝ− (respectively, a minimal element of Ŝ+). Especially, 0 is a source (a sink) if and only if S− = ∅ (respectively, S+ = ∅). 3. For any source or a sink p we define the reflected WBS Sp with the same underlying poset and the same values of K(i) as follows: (a) If p ∈ S− (p ∈ S+) is a source (respectively, a sink), then S− p = S− \{ p }, S+ p = S+∪{ p } (respectively, S+ p = S+ \{ p }, S− p = S− ∪ { p }); (b) If 0 is a source (a sink), then S− = S, S+ = ∅ (respectively, S+ = S, S− = ∅). The new values of V (ij) and µ(ikj) are defined as in item (1). Note that if p is a source (a sink) in Ŝ, it becomes a sink (respectively, a source) in Ŝp. We also consider the dual WBS. Definition 6. Let S be a WBS, M = (M,f) be a representation of S. The dual WBS S◦ and the dual representation M◦(M◦, f◦) are defined as follows: 1. As an ordered set, S◦ is opposite to S, i.e. consists of the same elements, but i < j in S◦ if and only if j < i in S. The bi- section is given by the rule S◦± = S∓. The skewfields K◦(i) are opposite to K(i), V ◦(ij) = V (ji) as an K◦(i)-K◦(j)-bimodule, and µ◦(ikj) = µ(jki) under the natural identification of V ◦(ik)V ◦(kj) with V (jk)V (ki). 2. M◦(i) = M(i)∗ and f◦(i) = f̃(i)∗, namely, (a) if i ∈ S◦+ = S−, then f(i) : M(i) → V (i0)M(0), thus f(i)∗ : M(0)∗V (i0)∗ → M(i)∗ and tif(i)∗ : M(0)∗ = M◦(0) → M(i)∗V (i0) = V ◦(0i)M◦(i); 42 Weighted partially orderd sets of finite type (b) if i ∈ S◦− = S+, then f(i) : M(0) → V (0i)M(i), thus f(i)∗ : M(i)∗V (0i)∗ → M(0)∗ and f̃(i)∗ : M(i)∗ = M◦(i) → M(0)∗V (0i) = V ◦(i0)M◦(0). 3. If φ ∈ homS(M,N), we define φ◦ : N◦ →M◦ setting φ◦(i) = φ̃(i)∗ and φ◦(ij) = φ̃(ji)∗. The following result is then evident. Proposition 1. Definition 6 establishes a duality functor ◦ : repS → repS◦, i.e. an equivalence repS → (repS◦)op such that there is a natural isomorphism M ≃ (M◦)◦. Thus there is a one-to-one correspondence between indecomposable representations of S and S◦. In particular, S is representation finite if and only if so is S◦. We introduce some useful notations. Definition 7. Let M = (M,f) be a representation of a WBS S, p ∈ S. We set: M+(p) = ⊕ p≤i, i∈S+ V (pi)M(i), M−(p) = ⊕ i≤p, i∈S− V (pi)M(i), f+(p) : V (p0)M(0) →M+(p) is the map with the components f+(pi) : V (p0)M(0) 1f(i) −−−→ V (p0)V (0i)M(i) µ(p0i)1 −−−−→ V (pi)M(i), f−(p) : M−(p) → V (p0)M(0) is the map with the components f−(pi) : V (pi)M(i) 1f(i) −−−→ V (pi)V (i0)M(0) µ(pi0)1 −−−−→ V (p0)M(0). We define M±(0) and f±(0) by analogous formulae, just omitting condi- tions “p ≤ i” and “i ≤ p” under the summation sign. Now we construct the reflection functors Σp : repS → repSp. Definition 8. Let M = (M,f) be a representations of a WBS S, p ∈ Ŝ is a source or a sink. We define a representation ΣpM = (M ′, f ′) of the WBS Sp as follows (in all cases M ′(i) = M(i) for all i 6= p): 1. If p ∈ S− is a source, we set f ′(i) = f(i) for i 6= p, M ′(p) = ker f+(p)/ Im f−(p), choose a retraction ρM : V (p0)M(0) → ker f+(p) and set f ′(p) = π̃MρM , where πM is the natural surjec- tion ker f+(p) →M ′(p). O. Drozd-Koroleva 43 2. If p = 0 is a source, we set M ′(0) = Cok f+ and f ′(i) = π̃M (i), where πM (i) is the i-th component of the natural surjection πM : M+(p) →M ′(0). 3. If p ∈ S+ is a sink, we set f ′(i) = f(i) if i 6= p, M ′(p) = ker f+(p)/ Im f−(p), choose a section σM : Cok f−p → V (p0)M(0) and set f ′(p) = σMεM , where εM is the natural injection M ′(p) → Cok f−(p). 4. If 0 is a sink, we set M ′(0) = ker f−(0) and f ′(i) = ε̃M (i), where εM (i) is the i-th component of the embedding εM : M ′(0) →M−(0). Evidently, M ′ is indeed a representation of Sp. In cases 1 and 3 these definitions depend on the choice of ρM and σM . Nevertheless, Corollary 10 below will show that another choice of ηM and σM gives isomorphic representations of Sp. We also define reflected morphisms morphisms. Definition 9. Keep the notations of Definition 8, and let φ : M → N be a morphism of representations, where N = (N, g). We define a morphism Σpφ = φ′ : ΣpM → Σp(N) as follows (again we set φ′(i) = φ(i) and φ′(ij) = φ(ij) if i 6= p, j 6= p): 1. Let p ∈ S− be a source. Then f+(p) induces an injection Im(1 − θρM ) →M+(p), where θ is the embedding ker f+(p) → V (p0)M(0), so we can choose a homomorphism ξ : M+(p) → V (p0)M(0) such that ξf+(p) = θρM − 1. We set • φ′(p)(x + Im f−(p)) = (1φ(0))(x) + Im g−(p) for every x ∈ ker f+(p). Note that the definition of morphisms implies that 1φ(0) maps ker f+(p) to ker g+(p) and Im f−(p) to Im g−(p). • φ′(pi) = ψ̃(i), where i > p, ψ(i) = πNρN (1φ(0))ξ(i) and ξ(i) is the i-th component of ξ. 2. Let p = 0 be a source. Then we choose a section η : M ′(0) →M+(0) and set • φ′(0) = πNφ +η, where φ+ : M+(0) → N+(0) has the (ij)-th component 1φ(i) if i = j, (µ(pji)1)(1φ(ij)) if i < j, and 0 if j < i. 3. Let p ∈ S+ be a sink. Then g−(p) induces an surjection N−(p) → Im(1 − σNτ), where τ is the natural surjection V (p0)N(0) → Cok g−(p), so we can choose a homomorphism η : V (p0)N(0) → N−(p) such that g−(p)η = σNτ − 1. We set 44 Weighted partially orderd sets of finite type • φ′(p)(x + Im f−(p)) = (1φ(0))(x) + Im g−(p) for every x ∈ ker f+(p). • φ′(ip) = η(i)(1φ(0))f ′(p), where i < p and η(i) is the i-th component of η. (Recall that f ′(p) = σMεM .) 4. Let p = 0 be a sink. Then we choose a retraction ξ : N−(0) → N ′(0) and set • φ′(0) = ξφ−εM , where φ− : M−(0) → N−(0) has the (ij)-th component 1φ(i) if i = j, (µ(pji)1)(1φ(ij)) if i < j, and 0 if j < i. Again, this construction depends on the choice of ξ or η. Never- theless, we shall show that, after some non-essential factorization, this dependence disappears. Definition 10. We denote by T p the trivial representation at the point p, i.e. such that T p(p) = k, T p(i) = 0 for i 6= p, by Ip the ideal of repS generated by the identity morphism of T p and by rep(p) S the factor- category repS/Ip. We call a representation M T p-free if it has no direct summands isomorphic to T p. The construction of ΣpM implies that this representation is always T p-free. The following result is also evident. Proposition 2. 1. If p ∈ S−, M is T p-free if and only if f(p)−1 ( ∑ i<p Im f−(p)(i) ) = 0. 2. If p ∈ S+, M is T p-free if and only if f̃(p) ( ⋂ i>p ker f+(p)(i) ) = M(p). 3. M is Tω-free if and only if ker f+(ω) ⊆ Im f−(ω). Proposition 3. We keep the notations of Definitions 8 and 9. 1. Σpφ is indeed a morphism ΣpM → ΣpN . 2. If we choose another homomorphism ξ′ or η′ instead of ξ or η, satis- fying the same conditions. Denote the obtained morphism ΣpM → ΣpN by φ′′. Then φ′ − φ′′ ∈ Ip. O. Drozd-Koroleva 45 Proof. We check the case (3); the case (1) is quite similar and the cases (2) and (4) are even easier. To prove that φ′ is a morphism, we only have to verify that g′(p)φ′(p) = (1φ′(0))f ′(p) + ∑ i<p (µ(pi0)1)(1g(i))φ′(ip). First note that φ′(p) coincides with ρ′τ(1φ(0))σMεM , where ρ′ : Cok g−(p) → N ′(p) is any retraction. Thus g′(p)φ′(p) = σNεNρ ′τ(1φ(0))σMεM = σNτ(1φ(0))f ′(p). On the other hand, (µ(pi0)1)(1g(i)) is the i-th component g−(p)(i) of g−(p). Therefore (µ(pi0)1)(1g(i))φ′(ip) = g−(p)(i)η(i)(1φ(0))f ′(p) = = (σN (i)τ(i) − 1)(1φ(0))f ′(p). Thus also (1φ′(0))f ′(p) + ∑ i<p (µ(pi0)1)(1g(i))φ′(ip) = σNτ(1φ(0))f ′(p). If we choose another η′ such that g−(p)η′ = σNτ − 1 then δ = φ′−φ′′ has all components zero except maybe δ(ip) = γ(i)(1φ(0))f ′(p), where γ = η − η′ and g−(p)γ = 0. Hence, δ = δ′δ′′, where δ′′ : M ′ → rT p (r = dimK(p)M ′(p)) has all components zero except δ′′(p) = 1, while δ′ : rT p → N ′ has all components zero except δ′(ip) = δ(ip). All relations that we have to verify to show that δ′ and δ′′ are indeed morphisms are trivial, except the only one for δ′ at the point p. But the latter coincide with the corresponding relation for δ. Corollary 1. The constructions of subsections 7 and 8 actually defines a functor Fp : rep(p) S → rep(p) Sp. In particular, the isomorphism class of FpM does not depend on the choice of ρM in case 1 or σM in case 3. Proposition 4. If p is a source or a sink, Fpp ≃ Id, the identity func- tor of the category rep(p) S. Therefore Fp : rep(p) S → rep(p) Sp is an equivalence. Proof. Again we only consider the case 1, when p ∈ S− is a source. Let M = (M,f) be a T p-free representation of rep(S), M ′ = (M ′, f ′) = FpM and M ′′ = (M ′′, f ′′) = FpM ′. All com- ponents of M ′ and M ′′ coincide with those of M except M ′(p) = ker f+(p)/ Im f−(p), f ′(p) = π̃MρM and M ′′(p) = ker f ′+(p)/ Im f ′−(p), 46 Weighted partially orderd sets of finite type f ′′(p) = σM ′εM ′ . By definition, M ′+(p) = M+(p) ⊕ M ′(p) and f ′+(p)(p) = πMρM , hence ker f ′+(p) = ker f+(p) ∩ kerπMρM = Im f−(p). Thus M ′′(p) = Im f−(p)/ ∑ i<p Im f−(p)(i). By 1 (1), f(p) is injective and Im f−(p) = Im f(p) ⊕ ∑ i<p Im f−(p)(i). Therefore the natural map ι : M(p) → M ′′(p) is bijective. Moreover, we can choose a section σM ′ so that εM ′σM ′ coincides with this bijection. Then we obtain an isomorphism φ : M → M ′′ setting φ(p) = ι, φ(i) = 1 for i 6= p and φ(ij) = 0 for all possible i, j. Obviously, this construction is functorial modulo the ideal Ip, so we get an isomorphism of functors Id ≃ Fpp. Definition 11. 1. Let p = (p1, p2, . . . , pm) be a sequence of elements of Ŝ. We call it admissible and define Sp by the following recursive rules: • If m = 1, p is admissible if and only if p1 is a source or a sink; then Sp = Sp1 . • If m > 1, p is admissible if and only if p1 is a source or a sink in Sq, where q = (p2, p3, . . . , pm); then Sp = (Sq)p1 . 2. If pm is a source (a sink) and, for every k < m, pk is a source (respectively, a sink) in S(pk+1,pk+2,...,pm), we call the sequence p a source sequence (respectively, a sink sequence). 3. We set p∗ = (pm, pm−1, . . . , p1). 4. If p is admissible, we denote by Σp the composition Σp1 Σp2 . . . Σpm and by Ip the ideal in repS generated by the identity morphisms of the representations T (p1,p2,...,pk) = Σ(p1,p2,...,pk−1)T pk (1 ≤ k ≤ m). We set rep(p) S = repS/Ip. Corollary 2. If a sequence p is admissible, the functor Σp establishes an equivalence rep(p) S → rep(p∗) Sp, the inverse equivalence being Σp∗ . In particular, there is a one-to-one correspondence between indecomposable representations of S and Sp; thus S is representation finite if and only if so is Sp. 3. Proof of the Main Theorem Now we are able to prove the sufficiency in Theorem 1. In this section S denotes a WBS with a weakly positive Tits form. For any dimension vector d ∈ NŜ we consider the set rep(d,S) of representations of S of dimension d, namely such representations M ∈ repS that M(i) is a fixed K(i)-vector space U(i) of dimension d(i). This set can be considered O. Drozd-Koroleva 47 as the set of k-valued points of an affine algebraic variety over k. The dimension of this variety is at most Q− S (d) = ∑ i∈S di0d(i)d(0) − ∑ i≪j dijd(i)d(j). Isomorphisms between these representations can be considered as k-valued elements of a linear algebraic group G(d) of dimension Q+ S (d) = ∑ i∈Ŝ did(i)2 + ∑ i⋖j, i,j∈S dijd(i)d(j). The isomorphism classes are just the orbits of this group. Note that QS = Q+ S − Q− S . We denote by ind(d,S) the subset of indecomposable representations from rep(d,S). In what follows we suppose that the field k is infinite (the case of finite fields can be then treated as in [1], and we omit the details, which are quite standard). Then one easily sees that the k-valued points are dense in the variety of representations, as well as in the group G(d). Especially, if rep(d,S) has finitely many orbits, each component of this variety is actually a closure of some orbit. Recall that a representation M of a WBS S is called sincere if M(i) 6= 0 for each i ∈ Ŝ. We prove the sufficiency using induction on \|S\|. Especially, we can suppose that S only has finitely many non-sincere indecomposable rep- resentations. More precisely, we prove the following result. Theorem 2. Let S be a WBS with weakly positive Tits form. Then 1. S is representation finite. 2. ind(d,S) 6= ∅ if and only if d is a root of the Tits form. In this case ind(d,S) consists of a unique orbit, which is dense in rep(d,S). 3. If M is a sincere indecomposable representation of S, there is a source (as well as a sink) sequence p such that M ≃ ΣpN for a non-sincere representation N ∈ rep(Sp∗). Our proof, like that of [6] relies on the following lemmas. (Recall that we always suppose that the Tits form is weakly positive.) Lemma 1. Suppose that the assertions of Theorem 2 hold for S. Let p be a source or a sink in Ŝ, M = (M,f) ∈ ind(d,S), where d 6= ep, d′ = σpd. Then: 1. If d(p) > 0, the map f+(p) is surjective and the map f−(p) is injective. 48 Weighted partially orderd sets of finite type 2. If d(p) = 0, ker f+(p) = Im f−(p). Proof. It obviously follows from the assertion (2), since the representa- tions satisfying the claimed conditions form an open subset in rep(d,S). Lemma 2. If S is a WBS with a weakly positive Tits form, p is a source or a sink in Ŝ, M ∈ ind(d,S) and d(p) > 0, then f+(p) is surjective and f−(p) is injective. The proof of this lemma practically coincide with that of [6, Lemma 3.3], so we omit it. Corollary 3. If S is a WBS with a weakly positive Tits form, M ∈ ind(d,S), p ∈ Ŝ is a source or a sink in Ŝ and d(p) > 0, then dimΣpM = σp dimM . Moreover, if N is another representation with the same prop- erties, homS(M,N) ≃ homSp (ΣpM,ΣpN). Since the number of positive roots is finite (it follows from [4, Ap- pendix]), Corollary 3 implies the assertion (3) of Theorem 2. Since the assertions (1) and (2) hold for non-sincere representations (by the induc- tive conjecture), we obtain them for all representations too. It accom- plishes the proof of Theorem 3. References [1] I. N.Ḃernstein, I.M. Gelfand and V. A.Ponomarev. Coxeter functors and Gabriel’s theorem. Uspechi Mat. Nauk 28, no. 2 (1973), 19–33. [2] V.Dlab, C.M.Ringel. Indecomposable representations of graphs and algebras. Mem. Amer. Math. Soc. 173 (1976). [3] V.Dlab, C.M.Ringel. On algebras of finite representation type. J. Algebra 33 (1975), 306–394. [4] Y.A. Drozd. Coxeter transformations and representations of partially ordered sets. Funk. Analiz Prilozh. 8:3 (1974), 34–42. [5] Y.A. Drozd. On tame and wild matrix problems. In: “Matrix Problems.” Inst. Math. AN UkrSSR, Kiev, 1977, p.104–114. [6] Y.A.Drozd. Representations of bisected posets and reflection functors. CMS Con- ference Proceedings 24 (1996), 153–165. [7] M. M. Kleiner. Partially ordered sets of finite type. Zapiski Nauch. Semin. LOMI 28 (1972), 32–41. [8] M. M. Kleiner. On sincere representations of partially ordered sets of finite type. Zapiski Nauch. Semin. LOMI 28 (1972), 42–60. [9] L.A. Nazarova and A. V.Roiter. Representations of partially ordered sets. Zapiski Nauch. Semin. LOMI 28 (1972), 5–31. [10] S. A. Ovsienko. Integral weakly positive forms. In: Schurian matrix problems and quadratic forms. Inst. Math. AN UkrSSR, Kiev, 1978, p.3–17. O. Drozd-Koroleva 49 [11] C.M. Ringel. Tame Algebras and Integral Quadratic Forms. Lecture Notes in Math. 1099 (1984). [12] A.V. Roiter. Roots of integral quadratic forms. Trudy Mat. Inst. Steklova AN SSSR 148 (1978), 201–210. Contact information O. Drozd-Koroleva Sosnytska 19 - 299; 02090; Kyiv, Ukraine E-Mail: olena.drozd@gmail.com URL: www.geocities.com/drozd_olena/ Received by the editors: 05.09.2006 and in final form 29.09.2006.

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