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Combinatorial aspects of extensions of Kronecker modules
Journal of Pure and Applied Algebra 219 (2015) 4378–4391
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Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa
Combinatorial aspects of extensions of Kronecker modules Csaba Szántó “Babeş-Bolyai” University Cluj-Napoca, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, R0-400084 Cluj-Napoca, Romania
a r t i c l e
i n f o
Article history: Received 27 June 2014 Received in revised form 11 January 2015 Available online 6 March 2015 Communicated by S. Iyengar MSC: 16G20
a b s t r a c t Let kK be the path algebra of the Kronecker quiver and consider the category mod-kK of finite dimensional right modules over kK (called Kronecker modules). We prove that extensions of Kronecker modules are field independent up to Segre classes, so they can be described purely combinatorially. We use in the proof explicit descriptions of particular extensions and a correspondence between exact “crosses” and “frames”, which (in case k is finite) is the main tool in the proof of the well known Green formula for Ringel–Hall numbers. We end the paper with some results on extensions of preinjective (or dually preprojective) Kronecker modules, involving the dominance ordering from partition combinatorics and its various generalizations. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Let K be the Kronecker quiver and k a field. We will consider the path algebra kK of K over k (called Kronecker algebra) and the category mod-kK of finite dimensional right modules over kK (called Kronecker modules). The isomorphism class of the module M will be denoted by [M ]. For d ∈ N2 let Md be the set of isomorphism classes of Kronecker modules of dimension d. Following Reineke in [9] for subsets A ⊂ Md , B ⊂ Me we define A ∗ B = {[X] ∈ Md+e |there exists an exact sequence 0 → N → X → M → 0 for some [M ] ∈ A, [N ] ∈ B}. So the product A ∗ B is the set of isoclasses of all extensions of modules M with [M ] ∈ A by modules N with [N ] ∈ B. This is in fact Reineke’s extension monoid product using isomorphism classes of modules instead of modules. It is important to know (see [9]) that the product above is associative, i.e. for A ⊂ Md , B ⊂ Me , C ⊂ Mf , we have (A ∗ B) ∗ C = A ∗ (B ∗ C). We also have {[0]} ∗ A = A ∗ {[0]} = A and the product ∗ is distributive over the union of sets. Recall that in case when k is finite, the rational Ringel–Hall algebra H(Λ, Q) associated to the algebra kK, is the free Q-module having as basis the isomorphism classes of Kronecker modules together E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jpaa.2015.02.022 0022-4049/© 2015 Elsevier B.V. All rights reserved.
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M M with a multiplication defined by [N1 ][N2 ] = [M ] FN1 N2 [M ], where the structure constants FN1 N2 = ∼ ∼ |{M ⊇ U | U = N2 , M/U = N1 }| are called Ringel–Hall numbers. Note that in this case the extension monoid product is {[N1 ]} ∗ {[N2 ]} = {[M ]|FNM1 N2 = 0}. In [15] we have proved that extensions of preinjective (preprojective) Kronecker modules are field independent, so the extension monoid product of two preinjectives (preprojectives) can be described combinatorially. In this paper we generalize this result by showing that extensions of arbitrary Kronecker modules are field independent up to Segre classes, so all extension monoid products can be described combinatorially. In order to prove this result we will use a correspondence between exact “crosses” and “frames” and we describe explicit formulas for some particular extension monoid products. Working over finite fields, this correspondence between exact “crosses” and “frames” is the main tool in the proof of the well known Green formula for Ringel–Hall numbers. The last section surveys some combinatorial properties related with the extensions and embeddings of preinjective modules. We show that particular orderings known from partition combinatorics and their generalizations (such as dominance, generalized majorization, refined generalized majorization) play an important role in this context. 2. Correspondence between exact “crosses” and “frames” Consider an acyclic quiver Q and the path algebra kQ, where k is an arbitrary field. Denote by mod-kQ the category of finite dimensional right modules over kQ. The aim of this section is to formulate a correspondence between exact “crosses” and “frames” valid over an arbitrary field k. As we will see this correspondence can be derived from the proof of the well known Green’s formula for Ringel–Hall algebras, which is valid over finite fields. We will follow the proof presented in [11] using its notations. Proposition 2.1. For N1 , N2 , N1 , N2 ∈ mod-kQ ∃M ∈ mod-kQ such that the cross N2
N2
M
N1
N1 is exact (i.e. both the row and the column are short exact sequences) iff ∃R, S, S , T ∈ mod-kQ such that the frame T
N2
N2
S
S
N1
N1
R
is exact (i.e. the edges of the frame are short exact sequences).
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Proof. We need first some preliminary notions taken from [11]. We may suppose that dimk N1 + dimk N2 = dimk N1 + dimk N2 = m. Let M ∈ mod-Λ with dimk M = m and Q(M ) be the set of quartets (a, b, a , b ) such that the cross below is exact. N2 b
b
N2
a
M
N1
a
N1 Let R be the set of all sextets (S, T, e1 , e2 , e3 , e4 ), where S is a submodule of N1 , T is a submodule of N2 and the frame below is exact (with uT , uS inclusion maps and qT , qS projections). uT
T
N2
qT
e1
e2
N2
N1
e3
e4 uS
S
N2 /T
N1
qS
N1 /S
Fix a sextet Ξ = (S, T, e1 , e2 , e3 , e4 ) from R. We denote by Y (Ξ) the pushout of the maps uT , e1 and by uN2 : N2 → Y (Ξ) and uN2 : N2 → Y (Ξ) the canonical inclusion maps. Also let X(Ξ) be the pullback of the maps qS , e4 and denote by pN1 : X(Ξ) → N1 and pN1 : X(Ξ) → N1 the canonical projections. Then it is proven in [11] that there exists a unique map f = f (Ξ) : Y (Ξ) → X(Ξ) satisfying pN1 f uN2 = 0, pN1 f uN2 = uS e3 , pN1 f uN2 = e2 qT , pN1 f uN2 = 0. Moreover, the kernel of f is isomorphic to T and the cokernel of f is isomorphic to N1 /S. uT
T
qT
N2
B/T
uN2
Y (Ξ)
e1
e2
uN
2
f =f (Ξ)
N2
N1 pN
1
X(Ξ)
e3
e4
pN1
S
uS
N1
qS
N1 /S
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Note that dimk M = m = dimk X(Ξ) + dimk Y (Ξ) − dimk Im f . Let F(f ; M ) be the set of all pairs (c, d) where c : M → X(Ξ) is an epimorphism, d : Y (Ξ) → M a monomorphism and cd = f . Finally, let O(M ) be the set of octets (S, T, e1 , e2 , e3 , e4 , c, d) where Ξ = (S, T, e1 , e2 , e3 , e4 ) ∈ R and (c, d) ∈ F(f ; M ). An important result proven in [11] is the existence of a bijection η : Q(M ) → O(M ). We are ready now to prove the proposition. Suppose ∃M ∈ mod-kQ such that Q(M ) is nonempty. Fix (a, b, a , b ) ∈ Q(M ). Let η(a, b, a, b ) = (S, T, e1 , e2 , e3 , e4 , c, d) ∈ O(M ) and we are done. Conversely suppose we have an exact frame of the given form. This implies the existence of an element Ξ = (S, T, e1 , e2 , e3 , e4 ) ∈ R. Consider f = f (Ξ) : Y (Ξ) → X(Ξ) and note that dimk X(Ξ) + dimk Y (Ξ) − dimk Im f = dimk N1 + dimk N2 = dimk N1 + dimk N2 = m. Then using the fact that kQ is hereditary (so Ext2 vanishes), it follows by [5] that there exists a module M ∈ mod-Λ with dimk M = m and a pair (c, d) such that (c, d) ∈ F(f ; M ) (see the Remark in [11] Section 4). So (S, T, e1 , e2 , e3 , e4 , c, d) ∈ O(M ) and then η −1 (S, T, e1 , e2 , e3 , e4 , c, d) = (a, b, a , b ) ∈ Q(M ). 2 There is an important corollary of Proposition 2.1: Corollary 2.2. For M, N, X, Y ∈ mod-kQ with Ext1 (M, N ) = 0 there is an exact sequence 0 → Y → M ⊕ N → X → 0 iff ∃A, B, C, D ∈ mod-kQ such that the frame below is exact. D
Y
N
C
B
M
X
A
3. Facts on Kronecker modules Throughout the article by a partition we mean a finite, decreasing sequence of strictly positive integers. The indecomposables in mod-kK are divided into three families: the preprojectives, the regulars and the preinjectives (see [1,2,10]). The preprojective (respectively preinjective) indecomposable modules (up to isomorphism) will be denoted by Pn (respectively In ), where n ∈ N∗ . The dimension vector of Pn is (n, n − 1) and that of In is (n − 1, n). A module is preprojective (preinjective) if it is the direct sum of preprojective (preinjective) indecomposables. For a partition a = (a1 , . . . , an ) we will use the notation Pa := Pan ⊕· · ·⊕Pa1 and Ia := Ia1 ⊕· · ·⊕Ian . The indecomposables which are neither preinjective nor preprojective are called regular. A module is regular if it is the direct sum of regular indecomposables. The category of regular modules is an abelian, exact subcategory which decomposes into a direct sum of uniserial categories, each of them having Auslander– Reiten quiver of the form ZA∞ /1, called homogeneous tube. These tubes are indexed by the closed points x in the scheme P1k = Proj k[X, Y ]. We denote by Hk the set of these points. A regular indecomposable of regular length t lying on the tube Tx will be denoted by Rx (t). Note that its unique regular composition series is Rx (1) ⊂ · · · ⊂ Rx (t −1) ⊂ Rx (t). Also note that for the regular simple Rx (1) its endomorphism ring End(Rx (1)) is the residue field at the point x. The degree of this field over k is called the degree of the point x and denoted by deg x. It follows that dimRx (t) = (t deg x, t deg x). In the case when k is algebraically closed, the closed points of the scheme above all have degree 1 and can be identified with the points of the classical projective line over k. For a partition λ = (λ1 , . . . , λn ) we define Rx (λ) = Rx (λ1 ) ⊕ · · · ⊕ Rx (λn ) and denote by P (respectively I, R) a preprojective (respectively preinjective, regular) module. We also define the set
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Rn = {[Rx1 (a1 ) ⊕ . . . ⊕ Rxr (ar )] | r, a1 , . . . , ar ∈ N∗ , x1 , . . . , xr ∈ Hk are pairwise different and a1 deg x1 + . . . + ar deg xr = n}. We will denote by R the full subcategory of mod-kK having as objects regular modules with indecomposable components taken from pairwise different tubes. This means that the set of isomorphism classes of the objects in R is exactly ∪n∈N∗ Rn . Note that R is not extension closed. We will describe now (in our special context) the so-called decomposition symbol used by Hubery in [6]. A decomposition symbol α = (μ, σ) consists of a pair of partitions denoted by μ (specifying a module without homogeneous regular summands) and a multiset σ = {(λ1 , d1 ), . . . , (λr , dr )}, where λi are partitions and di ∈ N∗ . The multiset σ will be called a Segre symbol. Given a decomposition symbol α = (μ, σ) (where μ = (c = (c1 , . . . , ct ), d = (d1 , . . . , ds ))) and a field k, we define the decomposition class S(α, k) to be the set of isomorphism classes of modules of the form M (μ, k) ⊕ R, where M (μ, k) = Pct ⊕ · · · ⊕ Pc1 ⊕ Id1 ⊕ · · · ⊕ Ids is the kK-module (up to isomorphism uniquely) determined by μ and R = Rx1 (λ1 ) ⊕ · · · ⊕ Rxr (λr ) for some distinct points x1 , . . . xr ∈ Hk such that deg xi = di . For a Segre symbol σ let S(σ, k) := S((∅, σ), k). Trivially S(α, k) ∩ S(β, k) = ∅ for decomposition symbols α = β. Also note that Rn = ∪S(σ, k), where the union is taken over all Segre symbols σ = {((a1 ), d1 ), . . . , ((ar ), dr )} with r ∈ N∗ and a1 d1 + . . . + ar dr = n. Remark 3.1. For k finite with q elements the number of points x ∈ Hk of degree 1 is q + 1. The number of points x ∈ Hk of degree l ≥ 2 is N (q, l) = 1l d|l μ( dl )q d , where μ is the Möbius function and N (q, l) is the number of monic, irreducible polynomials of degree l over a field with q elements. We can conclude that for a decomposition symbol α the polynomial nα (q) = |S(α, k)| is strictly increasing in q > 1 (see [6]). The following well-known lemma summarizes some facts on Kronecker modules: Lemma 3.2. a) Let P be preprojective, I preinjective and R regular module. Then Hom(R, P ) = Hom(I, P ) = Hom(I, R) = Ext1 (P, R) = Ext1 (P, I) = Ext1 (R, I) = 0. b) If x = x , then Hom(Rx (t), Rx (t )) = Ext1 (Rx (t), Rx (t )) = 0 (i.e. the tubes are pairwise orthogonal). c) For n ≤ m, we have dimk Hom(Pn , Pm ) = m−n+1 and Ext1 (Pn , Pm ) = 0; otherwise Hom(Pn , Pm ) = 0 and dimk Ext1 (Pn , Pm ) = n − m − 1. In particular End(Pn ) ∼ = k and Ext1 (Pn , Pn ) = 0. d) For n ≥ m, we have dimk Hom(In , Im ) = n − m + 1 and Ext1 (In , Im ) = 0; otherwise Hom(In , Im ) = 0 and dimk Ext1 (In , Im ) = m − n − 1. In particular End(In ) ∼ = k and Ext1 (In , In ) = 0. 1 e) dimk Hom(Pn , Im ) = n + m − 2 and dimk Ext (Im , Pn ) = m + n. f) dimk Hom(Pn , Rx (t)) = dimk Hom(Rx (t), In ) = t deg x and dimk Ext1 (Rx (t), Pn ) = dimk Ext1 (In , Rx (t)) = t deg x. g) dimk Hom(Rx (t1 ), Rx (t2 )) = dimk Ext1 (Rx (t1 ), Rx (t2 )) = min (t1 , t2 ) deg x. h) For P a preprojective module every nonzero morphism f : Pn → P is a monomorphism. If R is regular then for every nonzero morphism f : Pn → R, f is either a monomorphism or Im f is regular. In particular if R is regular simple and Im f is regular then f is an epimorphism. i) For I a preinjective module every nonzero morphism f : I → In is an epimorphism. If R is regular then for every nonzero morphism f : R → In , f is either an epimorphism or Im f is regular. In particular if R is regular simple and Im f is regular then f is a monomorphism. The defect of M ∈ mod-kK with dimension vector (a, b) is defined in the Kronecker case as ∂M := b − a. Observe that if M is a preprojective (preinjective, respectively regular) indecomposable, then ∂M = −1 (∂M = 1, respectively ∂M = 0). Moreover, for a short exact sequence 0 → M1 → M2 → M3 → 0 in mod-kK we have ∂M2 = ∂M1 + ∂M3 .
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An immediate consequence of the facts above is the following: Corollary 3.3. a) For n ≥ m we have that {[In ]} ∗ {[Im ]} = {[In ⊕ Im ]} and {[Pm ]} ∗ {[Pn ]} = {[Pm ⊕ Pn ]}. b) If P is preprojective, I is preinjective and R is regular module, then {[P ]} ∗{[R]} ∗{[I]} = {[P ⊕R⊕I]}. So for the partitions c = (c1 , . . . , ct ) and d = (d1 , . . . , ds ) we have that: {[Pc ⊕ Rx1 (λ1 ) ⊕ · · · ⊕ Rxr (λr ) ⊕ Id ]} = {[Pct ]} ∗ · · · ∗ {[Pc1 ]} ∗ {[Rx1 (λ1 )]} ∗ · · · ∗ {[Rxr (λr )]} ∗ {[Id1 ]} ∗ · · · ∗ {[Ids ]}. As stated in the beginning, we focus our attention on extensions of Kronecker modules, or equivalently on the products of the form {[M ]} ∗ {[N ]}. Using the corollary above we can see that this iteratively reduces to the knowledge of the following particular products: {[In ]} ∗ {[Im ]}, {[Pm ]} ∗ {[Pn ]}, where m ≥ n, {[In ]} ∗ {[Rx (λ)]}, {[Rx (λ)]} ∗ {[Pn ]}, {[In ]} ∗ {[Pm ]}, {[Rx (λ)]} ∗ {[Rx (μ)]}. 4. Particular extension monoid products and field independence in the general case In this section we will work in the category mod-kK with k an arbitrary field. We will analyze the field independence of extensions of arbitrary Kronecker modules. For this purpose we will describe the particular extension monoid products listed at the end of the previous section. We start with the description of {[In ]} ∗ {[Im ]} and {[Pn ]} ∗ {[Pm ]}. Proposition 4.1. We have: {[In ]} ∗ {[Im ]} =
{[In ⊕ Im ]} {[Im ⊕ In ], [Im−1 ⊕ In+1 ], . . . , [Im− m−n ⊕ In+ m−n ]} 2
2
for n − m ≥ −1 . for n − m < −1
Dually we have: {[Pn ]} ∗ {[Pm ]} =
{[Pn ⊕ Pm ]} {[Pm ⊕ Pn ], [Pm+1 ⊕ Pn−1 ], . . . , [Pm+ n−m ⊕ Pn− n−m ]} 2
2
for n − m ≤ 1 . for n − m > 1
Proof. For k a finite field the formulas follow directly from the corresponding formulas for the Ringel–Hall product (see [13] for details). In [15] it is proven that the possible middle terms in preinjective or preprojective short exact sequences do not depend on the base field, so we are done. 2 We describe now the product {[Rx (λ)]} ∗ {[Rx (μ)]}, where λ and μ are partitions. This is a classical result and it was studied in the equivalent context of p-modules by T. Klein in [7]. So we have: Proposition 4.2. {[Rx (λ)]} ∗ {[Rx (μ)]} = {[Rx (μ)]} ∗ {[Rx (λ)]} = {[Rx (ν)]|cνλμ = 0}, where cνλμ is the Littlewood–Richardson coefficient (which is field independent). Using our knowledge on the Littlewood–Richardson coefficients we obtain in particular the following: Corollary 4.3. {[Rx (λ)]} ∗ {[Rx (n)]} = {[Rx (n)]} ∗ {[Rx (λ)]} = {[Rx (ν)]|ν − λ is a horizontal n-strip}.
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Using the field independence of the Littlewood–Richardson coefficients we also obtain: Corollary 4.4. For two Segre symbols σ, τ we have that S(σ, k) ∗ S(τ, k) = S(ρ, k), where the union is taken over a finite number of specific Segre symbols ρ, combinatorially (field independently) determined by the symbols σ, τ . Proof. Using Proposition 4.2 the field independent combinatorial nature of the product is clear. What remains to prove is that the product is the union of full Segre classes. Suppose that the class [Rx1 (λ1 ) ⊕ · · · ⊕ Rxr (λr )] occurs in the product for some distinct points x1 , . . . xr ∈ Hk such that deg xi = di . We prove that in this case the whole Segre class S({(λ1 , d1 ), . . . , (λr , dr )}, k) occurs in the product. For a component Rxi (λi ) we have the following three possibilities: i a) it comes from the product [Rxi (μi )][Rxi (ν i )], where (μi , di ) ∈ σ, (ν i , di ) ∈ τ , so cλμi ν i = 0. b) (λi , di ) ∈ σ, c) (λi , di ) ∈ τ . Note that in any of the cases above, the component Ryi (λi ) can be obtained in a similar way, where yi ∈ Hk is arbitrary such that deg yi = di . 2 Remark 4.5. One can observe that the previous corollary is valid also in the case when one of the Segre classes are empty (due to the smallness of the field). See also Remark 3.1. Next we consider the products {[In ]} ∗ {[Rx (λ)]} and {[Rx (λ)]} ∗ {[Pn ]}. We need the following lemma: Lemma 4.6. Let Pn , Pm be preprojective indecomposables with n < m. Then there is a short exact sequence 0 → Pn → Pm → X → 0 iff X satisfies the following conditions: i) it is a regular module with dimX = dimPm − dimPn , ii) if Rx (t) and Rx (t ) are two indecomposable components of X then x = x . Proof. Suppose we have a short exact sequence 0 → Pn → Pm → X → 0. We will check the conditions i) and ii). Condition i). Trivially, dimX = dimPm − dimPn and ∂X = 0. Note that X cannot have preprojective components, since if P would be such an indecomposable component, then Pm P Pm which is impossible due to Lemma 3.2 h). So X is regular. Condition ii). Suppose X = X ⊕Rx (t1 ) ⊕. . .⊕Rx (tl ). Then we have a monomorphism Hom(X, Rx (1)) → l Hom(Pm , Rx (1)), so dimk Hom(X, Rx (1)) = dimk Hom(X , Rx (1)) + i=1 dimk Hom(Rx (ti ), Rx (1)) ≤ dimk Hom(Pm , Rx (1)) = deg x and dimk Hom(Rx (ti ), Rx (1)) = deg x. It follows that l = 1. Conversely, suppose now that X is a regular module satisfying conditions i) and ii). It is enough to show that Pm projects on X, since for an epimorphism f : Pm → X we have that ∂ Ker f = −1, so Ker f ∼ = Pn . Notice first that there are no monomorphisms Pm → X because dimX = dimPm − dimPn < dimPm . For a nonzero f : Pm → X we have the short exact sequence 0 → Ker f → Pm → Im f → 0. Since Ker f ⊆ Pm we have that Ker f is preprojective (so with negative defect) and is not 0 (because f is not mono) and Im f ⊆ X implies that Im f may contain preprojectives and regulars as direct summands (and it is nonzero since f is nonzero). The equality ∂ Ker f + ∂ Im f = ∂Pm = −1 gives us ∂ Im f = 0, so Im f is regular. For X = Rx (t) we have that Hom(Pm , X) = 0 (see Lemma 3.2 f)). If there are no epimorphisms in Hom(Pm , Rx (t)) then using the remarks above and the uniseriality of regulars we would have Hom(Pm , Rx (t)) ∼ = Hom(Pm , Rx (t − 1)) a contradiction. So we have an epimorphism Pm → X. Suppose now that X = Rx1 (t1 ) ⊕ . . . ⊕ Rxl (tl ). From the discussion above we have the epimorphisms fi : Pm → Rxi (ti ). Let f : Pm → X, f (x) = fi (x) the diagonal map. We have that Im f is regular so due
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to uniseriality Im f = Rx1 (t1 ) ⊕ . . . ⊕ Rxl (tl ) with Rxi (ti ) ⊆ Rxi (ti ). Since fi = pi f are epimorphisms we have that Rxi (ti ) = Rxi (ti ) so f is an epimorphism. 2 Proposition 4.7. We have: {[Rx (λ)]} ∗ {[Pn ]} = {[Pn+t deg x ⊕ Rx (μ)] | where λ − μ is a horizontal strip of length t, for some t ∈ N}. Dually we have: {[In ]} ∗ {[Rx (λ)]} = {[Rx (μ) ⊕ In+t deg x ] | where λ − μ is a horizontal strip of length t, for some t ∈ N}. Proof. We prove the first formula. Suppose we have a short exact sequence g
0 → Pn → X −→ Rx (λ) → 0. Note that we can’t have preinjective components in X (since due to Lemma 3.2 a) they would embed into Ker g ∼ = Pn ). Since ∂X = −1, it follows using Lemma 3.2 c) that X is of the form X = Pn+t deg x ⊕ Rx (μ) where μ is a partition with |μ| ≤ |λ| and t = |λ| − |μ|. If μ = (0) then by Lemma 4.6 we have an exact sequence 0 → Pn → Pn+t deg x → Rx (λ) → 0 iff λ = (t) i.e. iff λ − (0) is a horizontal t-strip. If μ = (0) then we apply Corollary 2.2 with choices X = Rx (λ), Y = Pn , M = Pn+t deg x and N = Rx (μ). It follows that we have an exact sequence 0 → Pn → Pn+t deg x ⊕ Rx (μ) → Rx (λ) → 0 iff ∃A, B, C, D ∈ mod-kK such that the frame below is exact. D
Pn
Rx (μ)
C
B
Pn+t deg x
Rx (λ)
A
By Lemma 3.2 a) B, D are preprojectives or 0. Note that B, D can’t be both preprojectives (due to the defect) and also if B = 0 then A = Pn+t deg x , a contradiction since Rx (λ) would project on a preprojective. This means that we must have D = 0, so B = Pn , C = Rx (μ) and using Lemma 4.6 it follows that A = Rx (t) (where t = |λ − μ|). So we have an exact sequence 0 → Pn → Pn+t deg x ⊕ Rx (μ) → Rx (λ) → 0 iff the frame below is exact. 0
Pn
Rx (μ)
Rx (μ)
Pn
Pn+t deg x
Rx (λ)
Rx (t)
Applying Corollary 4.3 it follows that λ − μ must be a horizontal t-strip. 2
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For λ = (m) we have in particular: Corollary 4.8. {[Rx (m)]} ∗ {[Pn ]} = {[Pn+i deg x ⊕ Rx (m − i)]| where i = 0, . . . , m}. Dually, {[In ]} ∗ {[Rx (m)]} = {[Rx (m − i) ⊕ In+i deg x ]| where i = 0, . . . , m}. Applying the previous corollary inductively, we obtain the following: Corollary 4.9. a) We have Rn ∗ {[Pm ]} = ({[Pm ]} ∗ Rn ) ∪ ({[Pm+1 ]} ∗ Rn−1 ) ∪ · · · ∪ {[Pm+n ]}. b) For a Segre symbol σ = {(λ1 , d1 ), . . . , (λr , dr )} we have that S(σ, k) ∗ {[Pm ]} = {[Pm+t ]} ∗ S(τ, k), where the union is taken over all Segre symbols of the form τ = {(μ1 , d1 ), . . . , (μr , dr )} with λi − μi a horizontal strip of length ti and t = t1 d1 + · · · + tr dr . The preinjective version of the formulas above follows dually. Finally we consider the product {[In ]} ∗ {[Pm ]}. Proposition 4.10. We have {[In ]} ∗ {[Pm ]} = Rn+m+1 ∪ {[Pm ⊕ In ]}. Proof. Suppose first that X Pm ⊕ In . Then we prove that there is an exact sequence of the form 0 → Pm → X → In → 0 iff X is a regular module having indecomposable components from pairwise different tubes and dimX = dimPm + dimIn . f g Suppose we have a short exact sequence 0 → Pm −→ X −→ In → 0. Then dimX = dimPm + dimIn and ∂X = ∂Pm + ∂In = 0. Suppose X = P ⊕ R ⊕ I (where P , R and I are preprojective, preinjective and regular modules). Note that pP f : Pm → P must be nonzero so it is a monomorphism (see Lemma 3.2 h)) which means that dimPm ≤ dimP . In the same way f qI : I → In must be nonzero so it is an epimorphism (see Lemma 3.2 i)) which means that dimIn ≤ dimI . But dimPm + dimIn = dimP + dimR + dimI which implies R = 0 and pP f , f qI are isomorphisms, so X ∼ = Pm ⊕ In a contradiction. This means that X is regular. Suppose X = X ⊕ Rx (t1 ) ⊕ . . . ⊕ Rx (tl ), then we have the monomorphism 0 → Hom(X, Rx (1)) → Hom(Pm , Rx (1)), since Hom(In , Rx (1)) = 0. It follows that dimk Hom(X, Rx (1)) = dimk Hom(X , Rx (1)) +
l
dimk Hom(Rx (ti ), Rx (1))
i=1
≤ dimk Hom(Pm , Rx (1)) = deg x and dimk Hom(Rx (ti ), Rx (1)) = deg x, so l = 1. Conversely, suppose that X is a regular module having indecomposable components from pairwise different tubes and dimX = dimPm + dimIn . Repeating the proof of Lemma 3.2 in [14] the existence of an exact sequence 0 → Pm → X → In → 0 follows. 2 Using the previous results on particular extension monoid products (more precisely Corollaries 3.3, 4.4, 4.9 and Proposition 4.10) it follows inductively that the extension monoid product of Kronecker modules is field independent in general up to Segre classes. More precisely we obtain the following theorem: Theorem 4.11. For two decomposition symbols α, β we have that S(α, k) ∗ S(β, k) = S(γ, k), where the union (which is disjoint) is taken over a finite number of specific decomposition symbols γ combinatorially (field independently) determined by the symbols α, β.
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Remark 4.12. One can observe that the theorem above is valid also in the case when one of the decomposition classes is empty (due to the small size of the field). See also Remark 3.1. Another result which follows from the description of particular extension monoid products gives some nice characterizations of the category R. Proposition 4.13. a) The category R consists exactly of those regular modules which are images of epimorphisms starting from an indecomposable preprojective module. b) The category R consists exactly of those regular modules which can occur as middle terms in extensions of preinjective indecomposables by preprojective indecomposables. c) The category R consists exactly of those regular modules for which the endomorphism ring is a product of uniserial rings. 5. Combinatorial aspects of extensions of preinjective Kronecker modules There are some very interesting combinatorial properties related with the extensions of decomposable preinjective modules. The dominance partial ordering known from partition combinatorics and its generalization (used in matrix pencil theory) play an important role in this context. We recall first the definition of these orderings. Consider the partitions a = (a1 , . . . , an ) and b = (b1 , . . . , bn ). The dominance partial ordering is defined as follows (see [8]): a b iff a1 ≤ b1 , a1 + a2 ≤ b1 + b2 , . . . , a1 + . . . + an−1 ≤ b1 + . . . + bn−1 and a1 + . . . . + an ≤ b1 + . . . + bn . In case a1 + . . . . + an = b1 + . . . + bn (when a, b are partitions of the same number) we will use the notation a b. Remark 5.1. The definition above works also for a, b ∈ Zn . Following Baragaña, Zaballa, Mondié, Dodig, Sto˘sić one can define the so-called generalized majorization (see [3,4]). This generalization of the dominance ordering of partitions was used by matrix theorists to describe the difficult combinatorial background of matrix pencil completion problems. Consider the partitions a = (a1 , . . . , an ), b = (b1 , . . . , bm ), c = (c1 , . . . , cm+n ). Then we say that the pair (b, a) is a generalized majorization of c (and denote it by c ≺ (b, a)) iff bi ≥ ci+n , i = 1, . . . , m, m
bi +
i=1 hq i=1
n i=1
hq −q
ci −
i=1
ai =
bi ≤
q
m+n
ci ,
i=1
ai , q = 1, . . . , n,
i=1
where hq := min{i|bi−q+1 < ci }, q = 1, . . . , n. Adopting the convention that ci , bi = +∞ for i ≤ 0, ci = −∞, for i > m + n bi = −∞, for i > m and the empty sums are zero, one can see that the indices hq and the sums above are all well defined. Moreover, we have that q ≤ hq ≤ q + m and h1 < h2 < · · · < hn . The term generalized majorization is motivated by the fact, that for m = 0 the generalized majorization reduces to the dominance ordering c a.
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Consider the partitions a = (a1 , . . . , an ), b = (b1 , . . . , bm ) and c = (c1 , . . . , cm+n ). Using the definition of h1 h1 −1 hq hq −q the generalized majorization we define inductively x1 := i=1 ci − i=1 bi , xq := i=1 ci − i=1 bi − x1 − · · · − xq−1 for q = 1, . . . , n. If x := (x1 , . . . , xn ) ∈ Zn then observe that x depends only on b, c and not on a. The following proposition gives another connection between the notion of generalized majorization and the dominance ordering. Proposition 5.2. Suppose that for the partitions a, b, c above we have that bi ≥ ci+n for i = 1, . . . , m and n m+n m i=1 bi + i=1 ai = i=1 ci . Then c ≺ (b, a) iff x a. n m+n m Proof. If c ≺ (b, a) then it follows from the definition that x a. Since i=1 bi + i=1 ai = i=1 ci , we m+n−hn n n obtain that i=1 xi + i=1 (chn +i − bhn −n+i ) = i=1 ai . But we have that chn +i − bhn −n+i ≤ 0 and n n n n x ≤ a , so c = b hn +i hn −n+i for i = 1, . . . , m + n − hn and i=1 i i=1 i i=1 xi = i=1 ai , which means that x a. The converse statement is trivial. 2 It follows from the proof above that for n = 1 the condition c ≺ (b, a) is equivalent to m
bi + a1 =
i=1
m+1
ci ,
i=1
bi ≥ ci+1 , i = 1, . . . , m, bi = ci+1 , i = h1 , . . . , m, where h1 := min{i|bi < ci }, or to m i=1
bi + a1 =
m+1
ci ,
i=1
∃h1 ∈ {1, . . . m + 1} such that bi ≥ ci , i = 1, . . . h1 − 1, bh1 < ch1 , bi = ci+1 , i = h1 , . . . , m. In this case we speak about an elementary generalized majorization and denote it by c ≺1 (b, a1 ). A result by Dodig and Sto˘sić in [3] shows that one can decompose the generalized majorization into a “composition” of elementary generalized majorizations. More precisely we have: Proposition 5.3. (See [3].) We have c ≺ (b, a) iff there is a sequence of partitions dj = (dj1 , . . . , djm+j ), j = 1, . . . , n with d0 = b and dn = c such that dj ≺1 (dj−1 , aj ) for j = 1, . . . , n. Using the combinatorial notions above, we are now ready to formulate some interesting results on extensions of decomposable preinjective modules. The first result was already proven by Szántó in [12]. Proposition 5.4. (See [12].) Suppose a = (a1 , . . . , an ) is a partition. Then {[Ian ]} ∗ · · · ∗ {[Ia1 ]} = {[Iα ]|α a}.
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Remember that {[Ia1 ]} ∗ · · · ∗ {[Ian ]} = {[Ia ]} and denote the reversed product {[Ian ]} ∗ · · · ∗ {[Ia1 ]} = {[Iα ]|α a} by Ia . The following result connects the notion of elementary generalized majorization to extensions of preinjectives, generalizing Proposition 4.1. Proposition 5.5. Consider the partitions b = (b1 , . . . , bm ), c = (c1 , . . . , cm+1 ) and let a1 ∈ N∗ . Then c ≺1 (b, a1 ) iff [Ic ] ∈ {[Ia1 ]} ∗ {[Ib ]} Proof. Use Proposition 4.1 iteratively. 2 Example 5.6. Using the proposition above and the definition of the elementary generalized majorization we obtain that the elements in {[I3 ]} ∗ {[I11 ⊕ I7 ⊕ I7 ⊕ I7 ⊕ I4 ⊕ I4 ⊕ I1 ]} are given by the following partitions: (11, 7, 7, 5, 5, 4, 4, 1), (11, 7, 6, 6, 5, 4, 4, 1), (10, 7, 7, 6, 5, 4, 4, 1), (9, 7, 7, 7, 5, 4, 4, 1), (11, 6, 6, 6, 6, 4, 4, 1), (10, 7, 6, 6, 6, 4, 4, 1), (9, 7, 7, 6, 6, 4, 4, 1), (8, 7, 7, 7, 6, 4, 4, 1), (7, 7, 7, 7, 7, 4, 4, 1), with h1 = 5, (11, 7, 7, 7, 4, 4, 3, 1), (11, 7, 7, 6, 4, 4, 4, 1), (10, 7, 7, 7, 4, 4, 4, 1), with h1 = 7. We are ready now to connect the notion of generalized majorization to extensions and embeddings of preinjectives. Proposition 5.7. Consider the partitions a = (a1 , . . . , an ), b = (b1 , . . . , bm ), c = (c1 , . . . , cm+n ). Let x = h1 h1 −1 hq hq −q (x1 , . . . , xn ) ∈ Zn with x1 = i=1 ci − i=1 bi , xq = i=1 ci − i=1 bi − x1 − · · · − xq−1 for q = 1, . . . , n. Then we have: a) [Ic ] ∈ Ia ∗ {[Ib ]} iff c ≺ (b, a). b) If the partition a is minimal (using the ordering ), then [Ic ] ∈ {[Ia ]} ∗ {[Ib ]} iff c ≺ (b, a). c) There is a monomorphism Ib → Ic iff bi ≥ ci+n , for i = 1, . . . , hn −n, bi = ci+n , for i = hn −n+1, . . . , m and there is a partition a such that x a . Moreover if a is minimal (using the ordering ), then [Ic ] ∈ {[Ia ]} ∗ {[Ib ]}. Proof. a) The case n = 1 is Proposition 5.5. The general case is then a consequence of Proposition 5.3. b) Use a) and Proposition 5.4. c) If there is a monomorphism Ib → Ic , then we have an exact sequence 0 → Ib → Ic → Ia → 0 with a a partition. But then [Ic ] ∈ {[Ia ]} ∗ {[Ib ]} ⊆ Ia ∗ {[Ib ]}, so c ≺ (b, a ) and the assertion follows using Proposition 5.2. Conversely, the given conditions guarantee by Proposition 5.2 that c ≺ (b, a ), so [Ic ] ∈ Ia ∗ {[Ib ]}, which means that we have an exact sequence 0 → Ib → Ic → Ia → 0, for a partition a with a a . In case a is minimal, then of course a = a , so [Ic ] ∈ {[Ia ]} ∗ {[Ib ]}. 2 Remark 5.8. The result a) above is also proved in [16] by Szöllősi without the use of Proposition 5.3. This means that it is possible (by using the result a)) to obtain an independent proof for Proposition 5.3. It is natural to try to describe combinatorially the product {[Ia ]} ∗ {[Ib ]}, where a, b are arbitrary partitions. The results above show us that the introduced combinatorial notions (including the notion of generalized majorization) generally are not enough for this task. This is the reason why we introduce the following refinement of the generalized majorization, inspired from some technical results by Szöllősi in [16] and [17].
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First of all we need an another characterization of the generalized majorization. Proposition 5.9. (See [16].) Consider the partitions a = (a1 , . . . , an ), b = (b1 , . . . , bm ), c = (c1 , . . . , cm+n ) j l+j l and let Bj = {l ∈ {0, . . . , m}| k=1 bk + k=1 ak ≥ k=1 ck } for 1 ≤ j ≤ n. Then c ≺ (b, a) iff m+n
ci =
i=1
n
ai +
i=1
m
bi ,
i=1
Bj = ∅, for 1 ≤ j ≤ n, bi ≥ cβi , for 1 ≤ i ≤ m, where αj =
min B1 + 1 max{αj−1 + 1, min Bj + j}
j=1 1
and βi =
min{l ∈ {1, . . . , r}|l = αj , 1 ≤ j ≤ n} min{l ∈ {βi−1 + 1, . . . , r}|l = αj , 1 ≤ j ≤ n}
i=1 . 1
We are ready now to define the refinement of the generalized majorization. Consider the partitions j l a = (a1 , . . . , an ), b = (b1 , . . . , bm ), c = (c1 , . . . , cm+n ) and let Bj = {l ∈ {0, . . . , m}| k=1 bk + k=1 ak ≥ l+j k=1 ck }. We say that the pair (b, a) is a refined generalized majorization of c (and denote it by c ≺ (b, a)) iff m+n i=1
ci =
n i=1
ai +
m
bi ,
i=1
Bj = ∅, for 1 ≤ j ≤ n, (∗) aj ≤ cαj for 1 ≤ j ≤ n bi ≥ cβi , for 1 ≤ i ≤ m, where αj =
j=1 min B1 + 1 max{αj−1 + 1, min Bj + j} 1 < j ≤ n
and βi =
min{l ∈ {1, . . . , r}|l = αj , 1 ≤ j ≤ n} min{l ∈ {βi−1 + 1, . . . , r}|l = αj , 1 ≤ j ≤ n}
i=1 . 1
Remark 5.10. Using the previous proposition, one can see that the difference between the generalized majorization and its refinement consist in the extra condition (∗). The definition above permits us to describe in general the extensions of preinjective modules. The following theorem is in fact a reformulation of the main theorem by Szöllősi in [17]. Theorem 5.11. (See [17].) Consider the partitions a = (a1 , . . . , an ), b = (b1 , . . . , bm ), c = (c1 , . . . , cm+n ). Then [Ic ] ∈ {[Ia ]} ∗ {[Ib ]} iff c ≺ (b, a).
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Remark 5.12. The theorem above leads to a simple linear-time algorithm (in the length of partitions) which decides whether [Ic ] ∈ {[Ia ]} ∗ {[Ib ]}. See [17] for details and examples. Remark 5.13. One can easily see that the results above dualize for preprojective modules. Acknowledgements The author is very grateful to the referee for suggestions and comments to improve the manuscript. This work was supported by the Bolyai Scholarship of the Hungarian Academy of Sciences and Grant PN-II-ID-PCE-2012-4-0100. References [1] I. Assem, D. Simson, A. Skowronski, Elements of Representation Theory of Associative Algebras, Vol. 1: Techniques of Representation Theory, LMS Student Texts, vol. 65, Cambridge Univ. Press, 2006. [2] M. Auslander, I. Reiten, S. Smalo, Representation Theory of Artin Algebras, Cambridge Stud. in Adv. Math., vol. 36, Cambridge Univ. Press, 1995. [3] M. Dodig, M. Sto˘sić, On convexity of polynomial paths and generalized majorizations, Electron. J. Comb. 17 (1) (2010) R61. [4] M. Dodig, M. Sto˘sić, On properties of the generalized majorization, Electron. J. Linear Algebra 26 (2013) 471–509. [5] D. Happel, C.M. Ringel, Tilted algebras, Trans. Am. Math. Soc. 274 (1982) 399–443. [6] A. Hubery, Hall polynomials for affine quivers, Represent. Theory 14 (2010). [7] T. Klein, The multiplication of Schur-functions and extensions of p-modules, J. Lond. Math. Soc. 43 (1968) 280–284. [8] I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1995. [9] M. Reineke, The monoid of families of quiver representations, Proc. Lond. Math. Soc. 84 (2002) 663–685. [10] C.M. Ringel, Tame Algebras and Integral Quadratic Forms, Lect. Notes Math., vol. 1099, Springer, 1984. [11] C.M. Ringel, Green’s theorem on Hall algebras, in: Representation Theory of Algebras and Related Topics, in: CMS Conference Proceedings, vol. 19, AMS, Providence, 1996, pp. 185–245. [12] Cs. Szántó, On the Hall product of preinjective Kronecker modules, Mathematica 48(71) (2) (2006) 203–206. [13] Cs. Szántó, Hall numbers and the composition algebra of the Kronecker algebra, Algebr. Represent. Theory 9 (2006) 465–495. [14] Cs. Szántó, On some Ringel–Hall products in tame cases, J. Pure Appl. Algebra 216 (2012) 2069–2078. [15] Cs. Szántó, I. Szöllősi, On preprojective short exact sequences in the Kronecker case, J. Pure Appl. Algebra 216 (2012) 1171–1177. [16] I. Szöllősi, On the combinatorics of extensions of preinjective Kronecker modules, Acta Univ. Sapientiae Math. 6 (2014) 92–106. [17] I. Szöllősi, Computing the extensions of preinjective and preprojective Kronecker modules, J. Algebra 408 (2014) 205–221.
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Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa
Combinatorial aspects of extensions of Kronecker modules Csaba Szántó “Babeş-Bolyai” University Cluj-Napoca, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu nr. 1, R0-400084 Cluj-Napoca, Romania
a r t i c l e
i n f o
Article history: Received 27 June 2014 Received in revised form 11 January 2015 Available online 6 March 2015 Communicated by S. Iyengar MSC: 16G20
a b s t r a c t Let kK be the path algebra of the Kronecker quiver and consider the category mod-kK of finite dimensional right modules over kK (called Kronecker modules). We prove that extensions of Kronecker modules are field independent up to Segre classes, so they can be described purely combinatorially. We use in the proof explicit descriptions of particular extensions and a correspondence between exact “crosses” and “frames”, which (in case k is finite) is the main tool in the proof of the well known Green formula for Ringel–Hall numbers. We end the paper with some results on extensions of preinjective (or dually preprojective) Kronecker modules, involving the dominance ordering from partition combinatorics and its various generalizations. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Let K be the Kronecker quiver and k a field. We will consider the path algebra kK of K over k (called Kronecker algebra) and the category mod-kK of finite dimensional right modules over kK (called Kronecker modules). The isomorphism class of the module M will be denoted by [M ]. For d ∈ N2 let Md be the set of isomorphism classes of Kronecker modules of dimension d. Following Reineke in [9] for subsets A ⊂ Md , B ⊂ Me we define A ∗ B = {[X] ∈ Md+e |there exists an exact sequence 0 → N → X → M → 0 for some [M ] ∈ A, [N ] ∈ B}. So the product A ∗ B is the set of isoclasses of all extensions of modules M with [M ] ∈ A by modules N with [N ] ∈ B. This is in fact Reineke’s extension monoid product using isomorphism classes of modules instead of modules. It is important to know (see [9]) that the product above is associative, i.e. for A ⊂ Md , B ⊂ Me , C ⊂ Mf , we have (A ∗ B) ∗ C = A ∗ (B ∗ C). We also have {[0]} ∗ A = A ∗ {[0]} = A and the product ∗ is distributive over the union of sets. Recall that in case when k is finite, the rational Ringel–Hall algebra H(Λ, Q) associated to the algebra kK, is the free Q-module having as basis the isomorphism classes of Kronecker modules together E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jpaa.2015.02.022 0022-4049/© 2015 Elsevier B.V. All rights reserved.
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M M with a multiplication defined by [N1 ][N2 ] = [M ] FN1 N2 [M ], where the structure constants FN1 N2 = ∼ ∼ |{M ⊇ U | U = N2 , M/U = N1 }| are called Ringel–Hall numbers. Note that in this case the extension monoid product is {[N1 ]} ∗ {[N2 ]} = {[M ]|FNM1 N2 = 0}. In [15] we have proved that extensions of preinjective (preprojective) Kronecker modules are field independent, so the extension monoid product of two preinjectives (preprojectives) can be described combinatorially. In this paper we generalize this result by showing that extensions of arbitrary Kronecker modules are field independent up to Segre classes, so all extension monoid products can be described combinatorially. In order to prove this result we will use a correspondence between exact “crosses” and “frames” and we describe explicit formulas for some particular extension monoid products. Working over finite fields, this correspondence between exact “crosses” and “frames” is the main tool in the proof of the well known Green formula for Ringel–Hall numbers. The last section surveys some combinatorial properties related with the extensions and embeddings of preinjective modules. We show that particular orderings known from partition combinatorics and their generalizations (such as dominance, generalized majorization, refined generalized majorization) play an important role in this context. 2. Correspondence between exact “crosses” and “frames” Consider an acyclic quiver Q and the path algebra kQ, where k is an arbitrary field. Denote by mod-kQ the category of finite dimensional right modules over kQ. The aim of this section is to formulate a correspondence between exact “crosses” and “frames” valid over an arbitrary field k. As we will see this correspondence can be derived from the proof of the well known Green’s formula for Ringel–Hall algebras, which is valid over finite fields. We will follow the proof presented in [11] using its notations. Proposition 2.1. For N1 , N2 , N1 , N2 ∈ mod-kQ ∃M ∈ mod-kQ such that the cross N2
N2
M
N1
N1 is exact (i.e. both the row and the column are short exact sequences) iff ∃R, S, S , T ∈ mod-kQ such that the frame T
N2
N2
S
S
N1
N1
R
is exact (i.e. the edges of the frame are short exact sequences).
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Proof. We need first some preliminary notions taken from [11]. We may suppose that dimk N1 + dimk N2 = dimk N1 + dimk N2 = m. Let M ∈ mod-Λ with dimk M = m and Q(M ) be the set of quartets (a, b, a , b ) such that the cross below is exact. N2 b
b
N2
a
M
N1
a
N1 Let R be the set of all sextets (S, T, e1 , e2 , e3 , e4 ), where S is a submodule of N1 , T is a submodule of N2 and the frame below is exact (with uT , uS inclusion maps and qT , qS projections). uT
T
N2
qT
e1
e2
N2
N1
e3
e4 uS
S
N2 /T
N1
qS
N1 /S
Fix a sextet Ξ = (S, T, e1 , e2 , e3 , e4 ) from R. We denote by Y (Ξ) the pushout of the maps uT , e1 and by uN2 : N2 → Y (Ξ) and uN2 : N2 → Y (Ξ) the canonical inclusion maps. Also let X(Ξ) be the pullback of the maps qS , e4 and denote by pN1 : X(Ξ) → N1 and pN1 : X(Ξ) → N1 the canonical projections. Then it is proven in [11] that there exists a unique map f = f (Ξ) : Y (Ξ) → X(Ξ) satisfying pN1 f uN2 = 0, pN1 f uN2 = uS e3 , pN1 f uN2 = e2 qT , pN1 f uN2 = 0. Moreover, the kernel of f is isomorphic to T and the cokernel of f is isomorphic to N1 /S. uT
T
qT
N2
B/T
uN2
Y (Ξ)
e1
e2
uN
2
f =f (Ξ)
N2
N1 pN
1
X(Ξ)
e3
e4
pN1
S
uS
N1
qS
N1 /S
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Note that dimk M = m = dimk X(Ξ) + dimk Y (Ξ) − dimk Im f . Let F(f ; M ) be the set of all pairs (c, d) where c : M → X(Ξ) is an epimorphism, d : Y (Ξ) → M a monomorphism and cd = f . Finally, let O(M ) be the set of octets (S, T, e1 , e2 , e3 , e4 , c, d) where Ξ = (S, T, e1 , e2 , e3 , e4 ) ∈ R and (c, d) ∈ F(f ; M ). An important result proven in [11] is the existence of a bijection η : Q(M ) → O(M ). We are ready now to prove the proposition. Suppose ∃M ∈ mod-kQ such that Q(M ) is nonempty. Fix (a, b, a , b ) ∈ Q(M ). Let η(a, b, a, b ) = (S, T, e1 , e2 , e3 , e4 , c, d) ∈ O(M ) and we are done. Conversely suppose we have an exact frame of the given form. This implies the existence of an element Ξ = (S, T, e1 , e2 , e3 , e4 ) ∈ R. Consider f = f (Ξ) : Y (Ξ) → X(Ξ) and note that dimk X(Ξ) + dimk Y (Ξ) − dimk Im f = dimk N1 + dimk N2 = dimk N1 + dimk N2 = m. Then using the fact that kQ is hereditary (so Ext2 vanishes), it follows by [5] that there exists a module M ∈ mod-Λ with dimk M = m and a pair (c, d) such that (c, d) ∈ F(f ; M ) (see the Remark in [11] Section 4). So (S, T, e1 , e2 , e3 , e4 , c, d) ∈ O(M ) and then η −1 (S, T, e1 , e2 , e3 , e4 , c, d) = (a, b, a , b ) ∈ Q(M ). 2 There is an important corollary of Proposition 2.1: Corollary 2.2. For M, N, X, Y ∈ mod-kQ with Ext1 (M, N ) = 0 there is an exact sequence 0 → Y → M ⊕ N → X → 0 iff ∃A, B, C, D ∈ mod-kQ such that the frame below is exact. D
Y
N
C
B
M
X
A
3. Facts on Kronecker modules Throughout the article by a partition we mean a finite, decreasing sequence of strictly positive integers. The indecomposables in mod-kK are divided into three families: the preprojectives, the regulars and the preinjectives (see [1,2,10]). The preprojective (respectively preinjective) indecomposable modules (up to isomorphism) will be denoted by Pn (respectively In ), where n ∈ N∗ . The dimension vector of Pn is (n, n − 1) and that of In is (n − 1, n). A module is preprojective (preinjective) if it is the direct sum of preprojective (preinjective) indecomposables. For a partition a = (a1 , . . . , an ) we will use the notation Pa := Pan ⊕· · ·⊕Pa1 and Ia := Ia1 ⊕· · ·⊕Ian . The indecomposables which are neither preinjective nor preprojective are called regular. A module is regular if it is the direct sum of regular indecomposables. The category of regular modules is an abelian, exact subcategory which decomposes into a direct sum of uniserial categories, each of them having Auslander– Reiten quiver of the form ZA∞ /1, called homogeneous tube. These tubes are indexed by the closed points x in the scheme P1k = Proj k[X, Y ]. We denote by Hk the set of these points. A regular indecomposable of regular length t lying on the tube Tx will be denoted by Rx (t). Note that its unique regular composition series is Rx (1) ⊂ · · · ⊂ Rx (t −1) ⊂ Rx (t). Also note that for the regular simple Rx (1) its endomorphism ring End(Rx (1)) is the residue field at the point x. The degree of this field over k is called the degree of the point x and denoted by deg x. It follows that dimRx (t) = (t deg x, t deg x). In the case when k is algebraically closed, the closed points of the scheme above all have degree 1 and can be identified with the points of the classical projective line over k. For a partition λ = (λ1 , . . . , λn ) we define Rx (λ) = Rx (λ1 ) ⊕ · · · ⊕ Rx (λn ) and denote by P (respectively I, R) a preprojective (respectively preinjective, regular) module. We also define the set
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Rn = {[Rx1 (a1 ) ⊕ . . . ⊕ Rxr (ar )] | r, a1 , . . . , ar ∈ N∗ , x1 , . . . , xr ∈ Hk are pairwise different and a1 deg x1 + . . . + ar deg xr = n}. We will denote by R the full subcategory of mod-kK having as objects regular modules with indecomposable components taken from pairwise different tubes. This means that the set of isomorphism classes of the objects in R is exactly ∪n∈N∗ Rn . Note that R is not extension closed. We will describe now (in our special context) the so-called decomposition symbol used by Hubery in [6]. A decomposition symbol α = (μ, σ) consists of a pair of partitions denoted by μ (specifying a module without homogeneous regular summands) and a multiset σ = {(λ1 , d1 ), . . . , (λr , dr )}, where λi are partitions and di ∈ N∗ . The multiset σ will be called a Segre symbol. Given a decomposition symbol α = (μ, σ) (where μ = (c = (c1 , . . . , ct ), d = (d1 , . . . , ds ))) and a field k, we define the decomposition class S(α, k) to be the set of isomorphism classes of modules of the form M (μ, k) ⊕ R, where M (μ, k) = Pct ⊕ · · · ⊕ Pc1 ⊕ Id1 ⊕ · · · ⊕ Ids is the kK-module (up to isomorphism uniquely) determined by μ and R = Rx1 (λ1 ) ⊕ · · · ⊕ Rxr (λr ) for some distinct points x1 , . . . xr ∈ Hk such that deg xi = di . For a Segre symbol σ let S(σ, k) := S((∅, σ), k). Trivially S(α, k) ∩ S(β, k) = ∅ for decomposition symbols α = β. Also note that Rn = ∪S(σ, k), where the union is taken over all Segre symbols σ = {((a1 ), d1 ), . . . , ((ar ), dr )} with r ∈ N∗ and a1 d1 + . . . + ar dr = n. Remark 3.1. For k finite with q elements the number of points x ∈ Hk of degree 1 is q + 1. The number of points x ∈ Hk of degree l ≥ 2 is N (q, l) = 1l d|l μ( dl )q d , where μ is the Möbius function and N (q, l) is the number of monic, irreducible polynomials of degree l over a field with q elements. We can conclude that for a decomposition symbol α the polynomial nα (q) = |S(α, k)| is strictly increasing in q > 1 (see [6]). The following well-known lemma summarizes some facts on Kronecker modules: Lemma 3.2. a) Let P be preprojective, I preinjective and R regular module. Then Hom(R, P ) = Hom(I, P ) = Hom(I, R) = Ext1 (P, R) = Ext1 (P, I) = Ext1 (R, I) = 0. b) If x = x , then Hom(Rx (t), Rx (t )) = Ext1 (Rx (t), Rx (t )) = 0 (i.e. the tubes are pairwise orthogonal). c) For n ≤ m, we have dimk Hom(Pn , Pm ) = m−n+1 and Ext1 (Pn , Pm ) = 0; otherwise Hom(Pn , Pm ) = 0 and dimk Ext1 (Pn , Pm ) = n − m − 1. In particular End(Pn ) ∼ = k and Ext1 (Pn , Pn ) = 0. d) For n ≥ m, we have dimk Hom(In , Im ) = n − m + 1 and Ext1 (In , Im ) = 0; otherwise Hom(In , Im ) = 0 and dimk Ext1 (In , Im ) = m − n − 1. In particular End(In ) ∼ = k and Ext1 (In , In ) = 0. 1 e) dimk Hom(Pn , Im ) = n + m − 2 and dimk Ext (Im , Pn ) = m + n. f) dimk Hom(Pn , Rx (t)) = dimk Hom(Rx (t), In ) = t deg x and dimk Ext1 (Rx (t), Pn ) = dimk Ext1 (In , Rx (t)) = t deg x. g) dimk Hom(Rx (t1 ), Rx (t2 )) = dimk Ext1 (Rx (t1 ), Rx (t2 )) = min (t1 , t2 ) deg x. h) For P a preprojective module every nonzero morphism f : Pn → P is a monomorphism. If R is regular then for every nonzero morphism f : Pn → R, f is either a monomorphism or Im f is regular. In particular if R is regular simple and Im f is regular then f is an epimorphism. i) For I a preinjective module every nonzero morphism f : I → In is an epimorphism. If R is regular then for every nonzero morphism f : R → In , f is either an epimorphism or Im f is regular. In particular if R is regular simple and Im f is regular then f is a monomorphism. The defect of M ∈ mod-kK with dimension vector (a, b) is defined in the Kronecker case as ∂M := b − a. Observe that if M is a preprojective (preinjective, respectively regular) indecomposable, then ∂M = −1 (∂M = 1, respectively ∂M = 0). Moreover, for a short exact sequence 0 → M1 → M2 → M3 → 0 in mod-kK we have ∂M2 = ∂M1 + ∂M3 .
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An immediate consequence of the facts above is the following: Corollary 3.3. a) For n ≥ m we have that {[In ]} ∗ {[Im ]} = {[In ⊕ Im ]} and {[Pm ]} ∗ {[Pn ]} = {[Pm ⊕ Pn ]}. b) If P is preprojective, I is preinjective and R is regular module, then {[P ]} ∗{[R]} ∗{[I]} = {[P ⊕R⊕I]}. So for the partitions c = (c1 , . . . , ct ) and d = (d1 , . . . , ds ) we have that: {[Pc ⊕ Rx1 (λ1 ) ⊕ · · · ⊕ Rxr (λr ) ⊕ Id ]} = {[Pct ]} ∗ · · · ∗ {[Pc1 ]} ∗ {[Rx1 (λ1 )]} ∗ · · · ∗ {[Rxr (λr )]} ∗ {[Id1 ]} ∗ · · · ∗ {[Ids ]}. As stated in the beginning, we focus our attention on extensions of Kronecker modules, or equivalently on the products of the form {[M ]} ∗ {[N ]}. Using the corollary above we can see that this iteratively reduces to the knowledge of the following particular products: {[In ]} ∗ {[Im ]}, {[Pm ]} ∗ {[Pn ]}, where m ≥ n, {[In ]} ∗ {[Rx (λ)]}, {[Rx (λ)]} ∗ {[Pn ]}, {[In ]} ∗ {[Pm ]}, {[Rx (λ)]} ∗ {[Rx (μ)]}. 4. Particular extension monoid products and field independence in the general case In this section we will work in the category mod-kK with k an arbitrary field. We will analyze the field independence of extensions of arbitrary Kronecker modules. For this purpose we will describe the particular extension monoid products listed at the end of the previous section. We start with the description of {[In ]} ∗ {[Im ]} and {[Pn ]} ∗ {[Pm ]}. Proposition 4.1. We have: {[In ]} ∗ {[Im ]} =
{[In ⊕ Im ]} {[Im ⊕ In ], [Im−1 ⊕ In+1 ], . . . , [Im− m−n ⊕ In+ m−n ]} 2
2
for n − m ≥ −1 . for n − m < −1
Dually we have: {[Pn ]} ∗ {[Pm ]} =
{[Pn ⊕ Pm ]} {[Pm ⊕ Pn ], [Pm+1 ⊕ Pn−1 ], . . . , [Pm+ n−m ⊕ Pn− n−m ]} 2
2
for n − m ≤ 1 . for n − m > 1
Proof. For k a finite field the formulas follow directly from the corresponding formulas for the Ringel–Hall product (see [13] for details). In [15] it is proven that the possible middle terms in preinjective or preprojective short exact sequences do not depend on the base field, so we are done. 2 We describe now the product {[Rx (λ)]} ∗ {[Rx (μ)]}, where λ and μ are partitions. This is a classical result and it was studied in the equivalent context of p-modules by T. Klein in [7]. So we have: Proposition 4.2. {[Rx (λ)]} ∗ {[Rx (μ)]} = {[Rx (μ)]} ∗ {[Rx (λ)]} = {[Rx (ν)]|cνλμ = 0}, where cνλμ is the Littlewood–Richardson coefficient (which is field independent). Using our knowledge on the Littlewood–Richardson coefficients we obtain in particular the following: Corollary 4.3. {[Rx (λ)]} ∗ {[Rx (n)]} = {[Rx (n)]} ∗ {[Rx (λ)]} = {[Rx (ν)]|ν − λ is a horizontal n-strip}.
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Using the field independence of the Littlewood–Richardson coefficients we also obtain: Corollary 4.4. For two Segre symbols σ, τ we have that S(σ, k) ∗ S(τ, k) = S(ρ, k), where the union is taken over a finite number of specific Segre symbols ρ, combinatorially (field independently) determined by the symbols σ, τ . Proof. Using Proposition 4.2 the field independent combinatorial nature of the product is clear. What remains to prove is that the product is the union of full Segre classes. Suppose that the class [Rx1 (λ1 ) ⊕ · · · ⊕ Rxr (λr )] occurs in the product for some distinct points x1 , . . . xr ∈ Hk such that deg xi = di . We prove that in this case the whole Segre class S({(λ1 , d1 ), . . . , (λr , dr )}, k) occurs in the product. For a component Rxi (λi ) we have the following three possibilities: i a) it comes from the product [Rxi (μi )][Rxi (ν i )], where (μi , di ) ∈ σ, (ν i , di ) ∈ τ , so cλμi ν i = 0. b) (λi , di ) ∈ σ, c) (λi , di ) ∈ τ . Note that in any of the cases above, the component Ryi (λi ) can be obtained in a similar way, where yi ∈ Hk is arbitrary such that deg yi = di . 2 Remark 4.5. One can observe that the previous corollary is valid also in the case when one of the Segre classes are empty (due to the smallness of the field). See also Remark 3.1. Next we consider the products {[In ]} ∗ {[Rx (λ)]} and {[Rx (λ)]} ∗ {[Pn ]}. We need the following lemma: Lemma 4.6. Let Pn , Pm be preprojective indecomposables with n < m. Then there is a short exact sequence 0 → Pn → Pm → X → 0 iff X satisfies the following conditions: i) it is a regular module with dimX = dimPm − dimPn , ii) if Rx (t) and Rx (t ) are two indecomposable components of X then x = x . Proof. Suppose we have a short exact sequence 0 → Pn → Pm → X → 0. We will check the conditions i) and ii). Condition i). Trivially, dimX = dimPm − dimPn and ∂X = 0. Note that X cannot have preprojective components, since if P would be such an indecomposable component, then Pm P Pm which is impossible due to Lemma 3.2 h). So X is regular. Condition ii). Suppose X = X ⊕Rx (t1 ) ⊕. . .⊕Rx (tl ). Then we have a monomorphism Hom(X, Rx (1)) → l Hom(Pm , Rx (1)), so dimk Hom(X, Rx (1)) = dimk Hom(X , Rx (1)) + i=1 dimk Hom(Rx (ti ), Rx (1)) ≤ dimk Hom(Pm , Rx (1)) = deg x and dimk Hom(Rx (ti ), Rx (1)) = deg x. It follows that l = 1. Conversely, suppose now that X is a regular module satisfying conditions i) and ii). It is enough to show that Pm projects on X, since for an epimorphism f : Pm → X we have that ∂ Ker f = −1, so Ker f ∼ = Pn . Notice first that there are no monomorphisms Pm → X because dimX = dimPm − dimPn < dimPm . For a nonzero f : Pm → X we have the short exact sequence 0 → Ker f → Pm → Im f → 0. Since Ker f ⊆ Pm we have that Ker f is preprojective (so with negative defect) and is not 0 (because f is not mono) and Im f ⊆ X implies that Im f may contain preprojectives and regulars as direct summands (and it is nonzero since f is nonzero). The equality ∂ Ker f + ∂ Im f = ∂Pm = −1 gives us ∂ Im f = 0, so Im f is regular. For X = Rx (t) we have that Hom(Pm , X) = 0 (see Lemma 3.2 f)). If there are no epimorphisms in Hom(Pm , Rx (t)) then using the remarks above and the uniseriality of regulars we would have Hom(Pm , Rx (t)) ∼ = Hom(Pm , Rx (t − 1)) a contradiction. So we have an epimorphism Pm → X. Suppose now that X = Rx1 (t1 ) ⊕ . . . ⊕ Rxl (tl ). From the discussion above we have the epimorphisms fi : Pm → Rxi (ti ). Let f : Pm → X, f (x) = fi (x) the diagonal map. We have that Im f is regular so due
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to uniseriality Im f = Rx1 (t1 ) ⊕ . . . ⊕ Rxl (tl ) with Rxi (ti ) ⊆ Rxi (ti ). Since fi = pi f are epimorphisms we have that Rxi (ti ) = Rxi (ti ) so f is an epimorphism. 2 Proposition 4.7. We have: {[Rx (λ)]} ∗ {[Pn ]} = {[Pn+t deg x ⊕ Rx (μ)] | where λ − μ is a horizontal strip of length t, for some t ∈ N}. Dually we have: {[In ]} ∗ {[Rx (λ)]} = {[Rx (μ) ⊕ In+t deg x ] | where λ − μ is a horizontal strip of length t, for some t ∈ N}. Proof. We prove the first formula. Suppose we have a short exact sequence g
0 → Pn → X −→ Rx (λ) → 0. Note that we can’t have preinjective components in X (since due to Lemma 3.2 a) they would embed into Ker g ∼ = Pn ). Since ∂X = −1, it follows using Lemma 3.2 c) that X is of the form X = Pn+t deg x ⊕ Rx (μ) where μ is a partition with |μ| ≤ |λ| and t = |λ| − |μ|. If μ = (0) then by Lemma 4.6 we have an exact sequence 0 → Pn → Pn+t deg x → Rx (λ) → 0 iff λ = (t) i.e. iff λ − (0) is a horizontal t-strip. If μ = (0) then we apply Corollary 2.2 with choices X = Rx (λ), Y = Pn , M = Pn+t deg x and N = Rx (μ). It follows that we have an exact sequence 0 → Pn → Pn+t deg x ⊕ Rx (μ) → Rx (λ) → 0 iff ∃A, B, C, D ∈ mod-kK such that the frame below is exact. D
Pn
Rx (μ)
C
B
Pn+t deg x
Rx (λ)
A
By Lemma 3.2 a) B, D are preprojectives or 0. Note that B, D can’t be both preprojectives (due to the defect) and also if B = 0 then A = Pn+t deg x , a contradiction since Rx (λ) would project on a preprojective. This means that we must have D = 0, so B = Pn , C = Rx (μ) and using Lemma 4.6 it follows that A = Rx (t) (where t = |λ − μ|). So we have an exact sequence 0 → Pn → Pn+t deg x ⊕ Rx (μ) → Rx (λ) → 0 iff the frame below is exact. 0
Pn
Rx (μ)
Rx (μ)
Pn
Pn+t deg x
Rx (λ)
Rx (t)
Applying Corollary 4.3 it follows that λ − μ must be a horizontal t-strip. 2
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For λ = (m) we have in particular: Corollary 4.8. {[Rx (m)]} ∗ {[Pn ]} = {[Pn+i deg x ⊕ Rx (m − i)]| where i = 0, . . . , m}. Dually, {[In ]} ∗ {[Rx (m)]} = {[Rx (m − i) ⊕ In+i deg x ]| where i = 0, . . . , m}. Applying the previous corollary inductively, we obtain the following: Corollary 4.9. a) We have Rn ∗ {[Pm ]} = ({[Pm ]} ∗ Rn ) ∪ ({[Pm+1 ]} ∗ Rn−1 ) ∪ · · · ∪ {[Pm+n ]}. b) For a Segre symbol σ = {(λ1 , d1 ), . . . , (λr , dr )} we have that S(σ, k) ∗ {[Pm ]} = {[Pm+t ]} ∗ S(τ, k), where the union is taken over all Segre symbols of the form τ = {(μ1 , d1 ), . . . , (μr , dr )} with λi − μi a horizontal strip of length ti and t = t1 d1 + · · · + tr dr . The preinjective version of the formulas above follows dually. Finally we consider the product {[In ]} ∗ {[Pm ]}. Proposition 4.10. We have {[In ]} ∗ {[Pm ]} = Rn+m+1 ∪ {[Pm ⊕ In ]}. Proof. Suppose first that X Pm ⊕ In . Then we prove that there is an exact sequence of the form 0 → Pm → X → In → 0 iff X is a regular module having indecomposable components from pairwise different tubes and dimX = dimPm + dimIn . f g Suppose we have a short exact sequence 0 → Pm −→ X −→ In → 0. Then dimX = dimPm + dimIn and ∂X = ∂Pm + ∂In = 0. Suppose X = P ⊕ R ⊕ I (where P , R and I are preprojective, preinjective and regular modules). Note that pP f : Pm → P must be nonzero so it is a monomorphism (see Lemma 3.2 h)) which means that dimPm ≤ dimP . In the same way f qI : I → In must be nonzero so it is an epimorphism (see Lemma 3.2 i)) which means that dimIn ≤ dimI . But dimPm + dimIn = dimP + dimR + dimI which implies R = 0 and pP f , f qI are isomorphisms, so X ∼ = Pm ⊕ In a contradiction. This means that X is regular. Suppose X = X ⊕ Rx (t1 ) ⊕ . . . ⊕ Rx (tl ), then we have the monomorphism 0 → Hom(X, Rx (1)) → Hom(Pm , Rx (1)), since Hom(In , Rx (1)) = 0. It follows that dimk Hom(X, Rx (1)) = dimk Hom(X , Rx (1)) +
l
dimk Hom(Rx (ti ), Rx (1))
i=1
≤ dimk Hom(Pm , Rx (1)) = deg x and dimk Hom(Rx (ti ), Rx (1)) = deg x, so l = 1. Conversely, suppose that X is a regular module having indecomposable components from pairwise different tubes and dimX = dimPm + dimIn . Repeating the proof of Lemma 3.2 in [14] the existence of an exact sequence 0 → Pm → X → In → 0 follows. 2 Using the previous results on particular extension monoid products (more precisely Corollaries 3.3, 4.4, 4.9 and Proposition 4.10) it follows inductively that the extension monoid product of Kronecker modules is field independent in general up to Segre classes. More precisely we obtain the following theorem: Theorem 4.11. For two decomposition symbols α, β we have that S(α, k) ∗ S(β, k) = S(γ, k), where the union (which is disjoint) is taken over a finite number of specific decomposition symbols γ combinatorially (field independently) determined by the symbols α, β.
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Remark 4.12. One can observe that the theorem above is valid also in the case when one of the decomposition classes is empty (due to the small size of the field). See also Remark 3.1. Another result which follows from the description of particular extension monoid products gives some nice characterizations of the category R. Proposition 4.13. a) The category R consists exactly of those regular modules which are images of epimorphisms starting from an indecomposable preprojective module. b) The category R consists exactly of those regular modules which can occur as middle terms in extensions of preinjective indecomposables by preprojective indecomposables. c) The category R consists exactly of those regular modules for which the endomorphism ring is a product of uniserial rings. 5. Combinatorial aspects of extensions of preinjective Kronecker modules There are some very interesting combinatorial properties related with the extensions of decomposable preinjective modules. The dominance partial ordering known from partition combinatorics and its generalization (used in matrix pencil theory) play an important role in this context. We recall first the definition of these orderings. Consider the partitions a = (a1 , . . . , an ) and b = (b1 , . . . , bn ). The dominance partial ordering is defined as follows (see [8]): a b iff a1 ≤ b1 , a1 + a2 ≤ b1 + b2 , . . . , a1 + . . . + an−1 ≤ b1 + . . . + bn−1 and a1 + . . . . + an ≤ b1 + . . . + bn . In case a1 + . . . . + an = b1 + . . . + bn (when a, b are partitions of the same number) we will use the notation a b. Remark 5.1. The definition above works also for a, b ∈ Zn . Following Baragaña, Zaballa, Mondié, Dodig, Sto˘sić one can define the so-called generalized majorization (see [3,4]). This generalization of the dominance ordering of partitions was used by matrix theorists to describe the difficult combinatorial background of matrix pencil completion problems. Consider the partitions a = (a1 , . . . , an ), b = (b1 , . . . , bm ), c = (c1 , . . . , cm+n ). Then we say that the pair (b, a) is a generalized majorization of c (and denote it by c ≺ (b, a)) iff bi ≥ ci+n , i = 1, . . . , m, m
bi +
i=1 hq i=1
n i=1
hq −q
ci −
i=1
ai =
bi ≤
q
m+n
ci ,
i=1
ai , q = 1, . . . , n,
i=1
where hq := min{i|bi−q+1 < ci }, q = 1, . . . , n. Adopting the convention that ci , bi = +∞ for i ≤ 0, ci = −∞, for i > m + n bi = −∞, for i > m and the empty sums are zero, one can see that the indices hq and the sums above are all well defined. Moreover, we have that q ≤ hq ≤ q + m and h1 < h2 < · · · < hn . The term generalized majorization is motivated by the fact, that for m = 0 the generalized majorization reduces to the dominance ordering c a.
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Consider the partitions a = (a1 , . . . , an ), b = (b1 , . . . , bm ) and c = (c1 , . . . , cm+n ). Using the definition of h1 h1 −1 hq hq −q the generalized majorization we define inductively x1 := i=1 ci − i=1 bi , xq := i=1 ci − i=1 bi − x1 − · · · − xq−1 for q = 1, . . . , n. If x := (x1 , . . . , xn ) ∈ Zn then observe that x depends only on b, c and not on a. The following proposition gives another connection between the notion of generalized majorization and the dominance ordering. Proposition 5.2. Suppose that for the partitions a, b, c above we have that bi ≥ ci+n for i = 1, . . . , m and n m+n m i=1 bi + i=1 ai = i=1 ci . Then c ≺ (b, a) iff x a. n m+n m Proof. If c ≺ (b, a) then it follows from the definition that x a. Since i=1 bi + i=1 ai = i=1 ci , we m+n−hn n n obtain that i=1 xi + i=1 (chn +i − bhn −n+i ) = i=1 ai . But we have that chn +i − bhn −n+i ≤ 0 and n n n n x ≤ a , so c = b hn +i hn −n+i for i = 1, . . . , m + n − hn and i=1 i i=1 i i=1 xi = i=1 ai , which means that x a. The converse statement is trivial. 2 It follows from the proof above that for n = 1 the condition c ≺ (b, a) is equivalent to m
bi + a1 =
i=1
m+1
ci ,
i=1
bi ≥ ci+1 , i = 1, . . . , m, bi = ci+1 , i = h1 , . . . , m, where h1 := min{i|bi < ci }, or to m i=1
bi + a1 =
m+1
ci ,
i=1
∃h1 ∈ {1, . . . m + 1} such that bi ≥ ci , i = 1, . . . h1 − 1, bh1 < ch1 , bi = ci+1 , i = h1 , . . . , m. In this case we speak about an elementary generalized majorization and denote it by c ≺1 (b, a1 ). A result by Dodig and Sto˘sić in [3] shows that one can decompose the generalized majorization into a “composition” of elementary generalized majorizations. More precisely we have: Proposition 5.3. (See [3].) We have c ≺ (b, a) iff there is a sequence of partitions dj = (dj1 , . . . , djm+j ), j = 1, . . . , n with d0 = b and dn = c such that dj ≺1 (dj−1 , aj ) for j = 1, . . . , n. Using the combinatorial notions above, we are now ready to formulate some interesting results on extensions of decomposable preinjective modules. The first result was already proven by Szántó in [12]. Proposition 5.4. (See [12].) Suppose a = (a1 , . . . , an ) is a partition. Then {[Ian ]} ∗ · · · ∗ {[Ia1 ]} = {[Iα ]|α a}.
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Remember that {[Ia1 ]} ∗ · · · ∗ {[Ian ]} = {[Ia ]} and denote the reversed product {[Ian ]} ∗ · · · ∗ {[Ia1 ]} = {[Iα ]|α a} by Ia . The following result connects the notion of elementary generalized majorization to extensions of preinjectives, generalizing Proposition 4.1. Proposition 5.5. Consider the partitions b = (b1 , . . . , bm ), c = (c1 , . . . , cm+1 ) and let a1 ∈ N∗ . Then c ≺1 (b, a1 ) iff [Ic ] ∈ {[Ia1 ]} ∗ {[Ib ]} Proof. Use Proposition 4.1 iteratively. 2 Example 5.6. Using the proposition above and the definition of the elementary generalized majorization we obtain that the elements in {[I3 ]} ∗ {[I11 ⊕ I7 ⊕ I7 ⊕ I7 ⊕ I4 ⊕ I4 ⊕ I1 ]} are given by the following partitions: (11, 7, 7, 5, 5, 4, 4, 1), (11, 7, 6, 6, 5, 4, 4, 1), (10, 7, 7, 6, 5, 4, 4, 1), (9, 7, 7, 7, 5, 4, 4, 1), (11, 6, 6, 6, 6, 4, 4, 1), (10, 7, 6, 6, 6, 4, 4, 1), (9, 7, 7, 6, 6, 4, 4, 1), (8, 7, 7, 7, 6, 4, 4, 1), (7, 7, 7, 7, 7, 4, 4, 1), with h1 = 5, (11, 7, 7, 7, 4, 4, 3, 1), (11, 7, 7, 6, 4, 4, 4, 1), (10, 7, 7, 7, 4, 4, 4, 1), with h1 = 7. We are ready now to connect the notion of generalized majorization to extensions and embeddings of preinjectives. Proposition 5.7. Consider the partitions a = (a1 , . . . , an ), b = (b1 , . . . , bm ), c = (c1 , . . . , cm+n ). Let x = h1 h1 −1 hq hq −q (x1 , . . . , xn ) ∈ Zn with x1 = i=1 ci − i=1 bi , xq = i=1 ci − i=1 bi − x1 − · · · − xq−1 for q = 1, . . . , n. Then we have: a) [Ic ] ∈ Ia ∗ {[Ib ]} iff c ≺ (b, a). b) If the partition a is minimal (using the ordering ), then [Ic ] ∈ {[Ia ]} ∗ {[Ib ]} iff c ≺ (b, a). c) There is a monomorphism Ib → Ic iff bi ≥ ci+n , for i = 1, . . . , hn −n, bi = ci+n , for i = hn −n+1, . . . , m and there is a partition a such that x a . Moreover if a is minimal (using the ordering ), then [Ic ] ∈ {[Ia ]} ∗ {[Ib ]}. Proof. a) The case n = 1 is Proposition 5.5. The general case is then a consequence of Proposition 5.3. b) Use a) and Proposition 5.4. c) If there is a monomorphism Ib → Ic , then we have an exact sequence 0 → Ib → Ic → Ia → 0 with a a partition. But then [Ic ] ∈ {[Ia ]} ∗ {[Ib ]} ⊆ Ia ∗ {[Ib ]}, so c ≺ (b, a ) and the assertion follows using Proposition 5.2. Conversely, the given conditions guarantee by Proposition 5.2 that c ≺ (b, a ), so [Ic ] ∈ Ia ∗ {[Ib ]}, which means that we have an exact sequence 0 → Ib → Ic → Ia → 0, for a partition a with a a . In case a is minimal, then of course a = a , so [Ic ] ∈ {[Ia ]} ∗ {[Ib ]}. 2 Remark 5.8. The result a) above is also proved in [16] by Szöllősi without the use of Proposition 5.3. This means that it is possible (by using the result a)) to obtain an independent proof for Proposition 5.3. It is natural to try to describe combinatorially the product {[Ia ]} ∗ {[Ib ]}, where a, b are arbitrary partitions. The results above show us that the introduced combinatorial notions (including the notion of generalized majorization) generally are not enough for this task. This is the reason why we introduce the following refinement of the generalized majorization, inspired from some technical results by Szöllősi in [16] and [17].
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First of all we need an another characterization of the generalized majorization. Proposition 5.9. (See [16].) Consider the partitions a = (a1 , . . . , an ), b = (b1 , . . . , bm ), c = (c1 , . . . , cm+n ) j l+j l and let Bj = {l ∈ {0, . . . , m}| k=1 bk + k=1 ak ≥ k=1 ck } for 1 ≤ j ≤ n. Then c ≺ (b, a) iff m+n
ci =
i=1
n
ai +
i=1
m
bi ,
i=1
Bj = ∅, for 1 ≤ j ≤ n, bi ≥ cβi , for 1 ≤ i ≤ m, where αj =
min B1 + 1 max{αj−1 + 1, min Bj + j}
j=1 1
and βi =
min{l ∈ {1, . . . , r}|l = αj , 1 ≤ j ≤ n} min{l ∈ {βi−1 + 1, . . . , r}|l = αj , 1 ≤ j ≤ n}
i=1 . 1
We are ready now to define the refinement of the generalized majorization. Consider the partitions j l a = (a1 , . . . , an ), b = (b1 , . . . , bm ), c = (c1 , . . . , cm+n ) and let Bj = {l ∈ {0, . . . , m}| k=1 bk + k=1 ak ≥ l+j k=1 ck }. We say that the pair (b, a) is a refined generalized majorization of c (and denote it by c ≺ (b, a)) iff m+n i=1
ci =
n i=1
ai +
m
bi ,
i=1
Bj = ∅, for 1 ≤ j ≤ n, (∗) aj ≤ cαj for 1 ≤ j ≤ n bi ≥ cβi , for 1 ≤ i ≤ m, where αj =
j=1 min B1 + 1 max{αj−1 + 1, min Bj + j} 1 < j ≤ n
and βi =
min{l ∈ {1, . . . , r}|l = αj , 1 ≤ j ≤ n} min{l ∈ {βi−1 + 1, . . . , r}|l = αj , 1 ≤ j ≤ n}
i=1 . 1
Remark 5.10. Using the previous proposition, one can see that the difference between the generalized majorization and its refinement consist in the extra condition (∗). The definition above permits us to describe in general the extensions of preinjective modules. The following theorem is in fact a reformulation of the main theorem by Szöllősi in [17]. Theorem 5.11. (See [17].) Consider the partitions a = (a1 , . . . , an ), b = (b1 , . . . , bm ), c = (c1 , . . . , cm+n ). Then [Ic ] ∈ {[Ia ]} ∗ {[Ib ]} iff c ≺ (b, a).
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Remark 5.12. The theorem above leads to a simple linear-time algorithm (in the length of partitions) which decides whether [Ic ] ∈ {[Ia ]} ∗ {[Ib ]}. See [17] for details and examples. Remark 5.13. One can easily see that the results above dualize for preprojective modules. Acknowledgements The author is very grateful to the referee for suggestions and comments to improve the manuscript. This work was supported by the Bolyai Scholarship of the Hungarian Academy of Sciences and Grant PN-II-ID-PCE-2012-4-0100. References [1] I. Assem, D. Simson, A. Skowronski, Elements of Representation Theory of Associative Algebras, Vol. 1: Techniques of Representation Theory, LMS Student Texts, vol. 65, Cambridge Univ. Press, 2006. [2] M. Auslander, I. Reiten, S. Smalo, Representation Theory of Artin Algebras, Cambridge Stud. in Adv. Math., vol. 36, Cambridge Univ. Press, 1995. [3] M. Dodig, M. Sto˘sić, On convexity of polynomial paths and generalized majorizations, Electron. J. Comb. 17 (1) (2010) R61. [4] M. Dodig, M. Sto˘sić, On properties of the generalized majorization, Electron. J. Linear Algebra 26 (2013) 471–509. [5] D. Happel, C.M. Ringel, Tilted algebras, Trans. Am. Math. Soc. 274 (1982) 399–443. [6] A. Hubery, Hall polynomials for affine quivers, Represent. Theory 14 (2010). [7] T. Klein, The multiplication of Schur-functions and extensions of p-modules, J. Lond. Math. Soc. 43 (1968) 280–284. [8] I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1995. [9] M. Reineke, The monoid of families of quiver representations, Proc. Lond. Math. Soc. 84 (2002) 663–685. [10] C.M. Ringel, Tame Algebras and Integral Quadratic Forms, Lect. Notes Math., vol. 1099, Springer, 1984. [11] C.M. Ringel, Green’s theorem on Hall algebras, in: Representation Theory of Algebras and Related Topics, in: CMS Conference Proceedings, vol. 19, AMS, Providence, 1996, pp. 185–245. [12] Cs. Szántó, On the Hall product of preinjective Kronecker modules, Mathematica 48(71) (2) (2006) 203–206. [13] Cs. Szántó, Hall numbers and the composition algebra of the Kronecker algebra, Algebr. Represent. Theory 9 (2006) 465–495. [14] Cs. Szántó, On some Ringel–Hall products in tame cases, J. Pure Appl. Algebra 216 (2012) 2069–2078. [15] Cs. Szántó, I. Szöllősi, On preprojective short exact sequences in the Kronecker case, J. Pure Appl. Algebra 216 (2012) 1171–1177. [16] I. Szöllősi, On the combinatorics of extensions of preinjective Kronecker modules, Acta Univ. Sapientiae Math. 6 (2014) 92–106. [17] I. Szöllősi, Computing the extensions of preinjective and preprojective Kronecker modules, J. Algebra 408 (2014) 205–221.