Some Classification of Prehomogeneous Vector Spaces Associated with Dynkin Quivers of Exceptional Type

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1 International Journal of Algebra, Vol. 7, 2013, no. 7, HIKARI Ltd, Some Classification of Prehomogeneous Vector Spaces Associated with Dynkin Quivers of Exceptional Type Yukimi Ishii Institute of Mathematics University of Tsukuba Tsukuba, Ibaraki, , Japan Copyright c 2013 Yukimi Ishii. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we give a complete list of tilting Λ-modules where Λ is the tensor algebra of an oriented K-modulation (M, Ω) of a valued graph of exceptional type. Our result gives some kind of classification of prehomogeneous vector spaces associated with (M, Ω). Mathematics Subject Classification: 11S90, 16G20 Keywords: valued quiver, exceptional Dynkin quiver, tilting module, prehomogeneous vector spaces 1 Introduction Let (Γ, v) be a valued graph, i.e., Γ = {1, 2,...,n} is a finite set of vertices, and v =(v ij ) i,j Γ the set of nonnegative integers which satisfy the following condition: v ii = 0 for i Γ, and there exist strictly positive integers f i (i Γ) satisfying v ij f j = v ji f i for each i, j Γ with i j. In the case of v ij 0 which (v is equivalent to v ji 0, we use the symbol ij, v ji ) i j and we call it an edge of (Γ, v). If v ij = v ji = 1, we simply write i the exceptional types, i.e., G 2 : 1 E l (l =6, 7, 8). (1, 3) 2, F 4 : 1. In this paper, we deal with j 2 (1, 2) 3, and 4

2 306 Y. Ishii An orientation Ω of (Γ, v) is given by prescribing an ordering for every edge, (v indicated by an arrow ij, v ji ) i j or (v ij, v ji i )j. Sometimes we write i j (v instead of ij, v ji ) i j for simplicity. A valued graph (Γ, v) with an orientation Ω is called a valued quiver Q = (Γ, v; Ω). For an oriented K-modulation (M, Ω) of a valued graph (Γ, v) with M =((F i ) i Γ, ( i M j ) i,j Γ ) (see Definition 2.1) and a dimension vector d =(d i ) Z n 0, we can define a pair (G d,r d ) with G d = i Γ GL(d i; F i ) and R d = i j Hom F j (F d i i Fi im j,f d j j ) where G d acts on R d (see Definition 2.3). A point x R d is called a generic point of (G d,r d ) if dim K (G d ) x = dim K G d dim K R d where (G d ) x = {g G d ; gx = x} is the isotropy subgroup of G d at x. In our case, (G d,r d ) is a prehomogeneous vector space (abbrev. PV) in the sense that it has generic points (cf. [K]). If characteristic of K is 0, by Proposition 2.7, (G d,r d ) has a dense orbit. Moreover we show that for these type of PVs (G d,r d ), some partial tilting Λ(M, Ω)-modules correspond. For the definition of partial tilting modules, see [ASS; p. 192]. Note that although (G d,r d ) depends on the choice of the orientation and the modulation, the number of the isomorphism classes of basic tilting modules does not depend on the choice of the orientation (see from the below of Proposition 2.12) and the choice of modulation (see Remark 5.5). In this paper, we give a complete list of tilting Λ-modules of type G 2 and F 4 (see Sections 3 and 4); and these results give a classification (see Theorems 3.2 and 4.2) of PVs (G d,r d ) according to corresponding partial tilting modules. For E n -type (n =6, 7, 8), by using a computer, we count the number of isomorphism classes of (partial) tilting modules (see Sections 5,6,7). This paper consists of the following 7 sections. Section 1. Introduction Section 2. Preliminaries Section 3. The Case for exceptional type G 2 Section 4. The Case for exceptional type F 4 Section 5. The Case for exceptional type E 6 Section 6. The Case for exceptional type E 7 Section 7. The Case for exceptional type E 8 2 Preliminaries In the following, we shall always assume that a valued graph (Γ, v) is connected, i.e., for any i, j Γ, there is a sequence of vertices k 1 = i, k 2,...,k t = j such that v ksk s+1 0 for all s =1, 2,...,t 1. An orientation Ω is called ad-

3 Some classification of PVs ass. with Dynkin quivers of exceptional type 307 missible if Q =(Γ, v; Ω) has no oriented cycle. In this paper, we assume that an orientation is admissible. Definition 2.1. (An oriented K-modulation (M, Ω)) We say that M =((F i ) i Γ, ( i M j ) i,j Γ ) is a K-modulation of a valued graph (Γ, v) with v =(v ij ) when it satisfies the following conditions (see [D; p. 32]). 1. Each F i (i Γ) is a finite-dimensional division algebra over a commutative field K. 2. Each i M j is a F i -F j -bimodule on which K acts centrally, and satisfies the following conditions. (a) The dimension over F j of i M j as a right F j -module is v ij, while the dimension over F i of i M j as a left F i -module is v ji. (b) j M i = HomFi ( i M j,f i ) = Hom Fj ( i M j,f j ) as a F j -F i -bimodule. Note that if we put f i = dim K F i, then we have v ij f j = v ji f i from K ( i M j )= ( i M j ) K. For a valued quiver (Γ, v;ω), if we give a F i -F j -module i M j satisfying (v the condition (a) for each ij, v ji ) i j, then we can define j M i by the condition (b), so that we obtain a K-modulation (see [D; p. 32]). Now let (M, Ω) be a pair of a K-modulation of a valued graph (Γ, v) and an orientation Ω of (Γ, v) which we call an oriented K-modulation of a valued graph (Γ, v). Definition 2.2. (The abelian category rep(m, Ω)) A representation W =( j ϕ i,w i ) of an oriented K-modulation (M, Ω) is a pair of finite-dimensional right F i -vector spaces W i for i Γ and F j -linear mappings j ϕ i : W i Fi im j W j for each arrow i j. A morphism from W =( j ϕ i,w i ) to W =( j ϕ i,w i ) is a set α =(α i) i Γ of F i -linear mappings α i : W i W i (i Γ) such that the following diagram is commutative for every arrow i j. jϕ i W i Fi im j Wj α i 1 W i Fi im j α j jϕ i W j We denote by Hom(W, W ) the totality of all morphisms from W to W. Thus we have an abelian category rep(m, Ω) (see [D]). The category rep(m, Ω) is said to be of finite (representation) type if it possesses only a finite number of non-isomorphic indecomposable representations. It is known that rep(m, Ω) is of finite representation type if and only if (Γ, v) is a Dynkin graph (see [D; Theorem 3.1]).

4 308 Y. Ishii Definition 2.3. (A representation (G d,r d ) associated with (M, Ω)) Let (M, Ω) be an oriented K-modulation of a valued graph (Γ, v) with Card(Γ) = n and M =(F i, i M j ). For a dimension vector d =(d i ) Z n 0, the direct product G d = i Γ GL(d i; F i ) acts on the space R d = i j Hom F j (F d i i im j,f d j j ) of representations ( j x i,f d i i ) of (M, Ω) by R d x =( j x i ) i j x = ( j x i ) i j R d with j x i = g j jx i (g i 1) 1 for each i j. F d i jx i d i Fi im j F j j g i 1 g j F d i jx i Fi im i j F d j j By fixing a basis of F d i i Fi im j over F j, we can identify Hom Fj (F d i i Fi im j,f d j j ) with M(d j,v ij d i ; F j ) and hence we have R d = i j M(d j,v ij d i ; F j ). We call (G d,r d ) a representation associated with an oriented K-modulation (M, Ω). In the case where (Γ, v) is a Dynkin graph, such a representation (G d,r d ) is a finite PV, i.e., R d has only a finitely many G d -orbits. Note that dim K G d = i Γ f id 2 i and dim K R d = i j v ijf j d i d j. Definition 2.4. (The tensor algebra Λ of an oriented K-modulation (M, Ω)) We define the tensor algebra of an oriented K-modulation (M, Ω) by Λ= Λ(M, Ω) = t 0 M(t) where M (0) = F 1 F n, M (1) = i j im j, and inductively M (t+1) = M (t) M (0) M(1) where M (1) has the structure of the M (0) M (0) bimodule induced by the projections M (0) F i. Note that if j k, then we have i M j k M l =0since i M j k M l = i M j e j e k km l = i M j (e j e k ) km l =0where e j =(0,...,0, 1 Fj, 0,...,0) M (0). Hence we have M (t) = k 1 M k2 k2 M k3 kt M kt+1 where the summation runs over the paths k 1 k 2 k t+1. Therefore if the maximal length of the paths in the valued quiver Q =(Γ, v, Ω) is s, we have Λ=M (0) M (1) M (s). Multiplications in Λ is defined by the tensor product by identifying M (r) M (s) = M (r+s) and distributively. We have a category mod Λ of right Λ-module of finite length. Remark 2.5. (Equivalence of categories between rep(m, Ω) and mod Λ(M,Ω)) It is well-known (see [D; Proposition 1.2]) that there is an equivalence between the category rep(m, Ω) of representations of an oriented K-modulation (M, Ω) and the category mod Λ of right Λ-modules of finite length with Λ= Λ(M, Ω). For an object W =( j ϕ i,w i ) of rep(m, Ω), we construct a right Λ-module W as follows. Put W = i Γ W i which is naturally a Z-module. We define the right action of Λ = t 0 M(t) on W inductively. For (f i ) M (0) = F 1 F n and (w i ) i Γ W, define the right action W M (0) W by ((w i ) i Γ, (f i ) i Γ ) (w i f i ) i Γ. Fi

5 Some classification of PVs ass. with Dynkin quivers of exceptional type 309 For m ij i M j M (1) = i j im j and (w i ) i Γ W, define the right action W i M j W by ((w k ) k Γ,m ij ) (w t) t Γ where w j = j ϕ i (w i m ij ) if t = j and w t =0otherwise. Then inductively by w i(m ij m pq m qs )= s ϕ q (w i (m ij m pq ) m qs ), we obtain the right action of M (t) = M (t 1) M (0) M(1), and hence we have the right action of Λ= t 0 M(t) on W, i.e., W is a right Λ-module of finite length. For α =(αi ) i Γ : W W, the map α : W W defined by α((w i ) i )=(α i (w i )) i is clearly Λ-linear. Conversely for a right Λ-module X of finite length, i.e., an object of mod Λ, put X i = XF i which is a finite dimensional right F i -vector space for i Γ. Note that F i Λ. Then the action of i M j ( Λ) on each X i induces the F j -linear maps j ϕ i : X i Fi im j X j for each arrow i j. Thus we have an object ( j ϕ i,x i ) of the category rep(m, Ω). We define the dimension vector of a right Λ-module X by dim X = (dim Fi X i ) i Γ Z n 0 with n = Card(Γ). By this correspondence, we can regard each point x =( j x i ) i j R d = i j M(d j,v ij d i ; F j ) as a Λ-module X = F d 1 1 Fn dn on which (m ij) i j M (1) = i j im j ( Λ) acts as X x =(x 1,...,x n ) j 1 i j ( {}}{ 0,...,0, j x i (x i m ij ), 0,...,0) X. The orbit G d -orbit G d x corresponds to the isomorphism class of Λ-module X. Definition 2.6. Let, be the Ringel form of an oriented K-modulation (M, Ω) of a valued diagram (Γ, v), i.e., for vectors x =(x i ) i Γ, y =(y i ) i Γ Z n 0, x, y = xr t y, where R =(r ij ) i,j Γ with r ii = f i (i Γ),r ij = dim Ki M j (= (v v ij f j ) if ij, v ji ) i j ; and rij =0otherwise. Then we have dim X, dim Y = dim K Hom Λ (X, Y ) dim K Ext 1 Λ(X, Y ) for each Λ-modules X, Y (see [D; Proposition 2.2]). Proposition 2.7. Let K be a field of characteristic 0. Let G be a connected algebraic group defined over K acting on the vector space V with a K-structure V K. Then the K-rational points G K of G acts on V K. For a point x V, let G x = {g G; g x = x} be the isotropy subgroup of G at x. If a point x V K is a generic point, i.e., it satisfies dim K (G K ) x = dim K G K dim K V K, then we have G K x = V K with the Zariski topology in V K. Proof. Let Ω be a universal field. Since G = G K K Ω and V = V K K Ω, we have dim G x = dim G dim V, and hence we have G x = V (see [K]). By [B; p. 220], we have G K = G. Since the map f : G V defined by f(g) =g x is continuous with respect to the Zariski topology, we have G x = f(g) = f(g K ) f(g K )=G K x. Since G x = V, we have G K x = V with the Zariski topology in V. Since G K x V K and the Zariski topology in V K is weaker than the induced topology on V K from the Zariski topology in V,we obtain our result.

6 310 Y. Ishii Definition 2.8. A non-zero rational function f(x) on V K is called a relative invariant of (G K,V K ) if there exists a rational character χ of G K satisfying f(g x) = χ(g)f(x) as a rational function for all g G. By Proposition 2.7, if (G K,V K ) is a prehomogeneous vector space, i.e., there exists a generic point, then a relative invariant f(x) is uniquely determined by its corresponding character χ up to a constant multiple (cf. [K]). Let X(G K ) be the group of rational characters of G K. Proposition 2.9. Let (G K,V K ) be a prehomogeneous vector space with a generic point x 0. Let f 1,...,f l be relative invariants corresponding to characters χ 1,...,χ l respectively. Assume that these characters χ 1,...,χ l generate the subgroup X 1 (G K )={χ X(G K ); χ (GK ) x0 =1}. Then these f 1,...,f l generate all relative invariants of (G K,V K ). Proof. Let f(x) be any relative invariant of (G K,V K ) corresponding to a character χ. For any g (G K ) x0, we have f(g x 0 )=f(x 0 )=χ(g)f(x 0 ). By Proposition 2.7, we have f(x 0 ) 0,, and hence χ(g) = 1, i.e., χ X 1 (G K ). Hence we can express χ = χ m 1 1 χ m l l f(x) =cf m 1 1 f m l l with some (m 1,...,m l ) Z l, we have with some constant multiple. Lemma Let (G d,r d ) be a PV associated with an oriented K-modulation (M, Ω) of a valued graph (Γ, v), and X a Λ-module corresponding to a point x R d where Λ=Λ(M, Ω). Then x is a generic point of the PV if and only if Ext 1 Λ (X, X) =0, i.e., X is a partial tilting Λ-module. Proof. Since Λ is hereditary, we have dim K G d dim K R d = dim K Hom Λ (X, X) dim K Ext 1 Λ (X, X) = dim K(G d ) x dim K Ext 1 Λ (X, X). Hence, x is a generic point if and only if Ext 1 Λ (X, X) =0. By Proposition 2.7 and Lemma 2.10, we see that (the dense orbit of) a PV (G d,r d ) corresponds to the isomorphism class of some partial tilting Λ- module. Definition (Positive roots and indecomposable representations) Let (Γ, v) be a valued graph with Γ = {1, 2,...,n}. For each vertex k Γ, define the reflection r k : Q n Q n by x = (x i ) (y i ) where y k = x k + i Γ v ikx i and y i = x i (i k). The group W generated by the reflections is called the Weyl group of (Γ, v). An element x Q n is called arootof(γ, v) if there exists w W and i Γ satisfying x = we i where i e i =(0,...,0, ˇ1, 0,...,0). An element x =(x i ) Q n is called positive if all x i 0, and we write x 0. Note that positive roots belong to Z n 0. It is known that, in our case, the map rep(m, Ω) Z n 0 defined by W dim W induces the bijection between the isomorphism classes of indecomposable representation

7 Some classification of PVs ass. with Dynkin quivers of exceptional type 311 and the positive roots of (Γ, v) (see [DR; p. 2]). On the other hand, by Lemma 2.10, for a positive root d, the corresponding indecomposable representations x R d are generic points of (G d,r d ) since the corresponding Λ-module X of an indecomposable representation x has the property Ext 1 Λ (X, X) =0. By Theorem of Krull-Schmidt, any representation is uniquely decomposed to the direct sum of indecomposable representations. Let us decompose a representation x R d into the direct sum x = m 1 x 1 m s x s of indecomposable representations x 1,...,x s which are not isomorphic of each other and m k x k denotes the m k -copies of x k. Then x is a generic point if and only if the corresponding Λ-module m 1 X 1 m s X s is a partial tilting module. When m 1 = = m s = 1, such a module is called basic. It is known that Λ-module m 1 X 1 m s X s is a partial tilting module if and only if a basic Λ-module X 1 X s is a partial tilting modules. Hence x = x 1 x s R d is a generic point of a PV (G d,r d ) if and only if x = m 1 x 1 m s x s R d is a generic point of a PV (G d,r d ) for any m 1 1,...,m s 1. In general, for a partial tilting Λ-module X, there exists a Λ-module Y such that X Y is a tilting module which is, by definition, the number of its non-isomorphic direct summand is equal to n = Card(Γ) (see [ASS; p. 196]) where Λ = Λ(M, Ω) and (M, Ω) is a K-modulation of a valued graph (Γ, v). Therefore, to find a generic point of a PV (G d,r d ) associated with an oriented K-modulation (M, Ω) of a valued graph (Γ, v), it is sufficient to give a complete list of basic tilting Λ(M, Ω)-modules. Proposition For x =( j x i ) R d = i j Hom F j (F d i i x =( j x i) R d = i j Hom F j (F d i i and X = F d 1 1 F d n n Fi im j,f d j j ) and Fi im j,f d j j ), let X = F d 1 1 Fn dn be corresponding Λ-modules. Then a linear map α : X X with α =(α 1,...,α n ) M(d 1,d 1; F 1 ) M(d n,d n; F n ) is a Λ-homomorphism if and only if α j jx i = j x i (α i 1) for each arrow i j. Proof. Since Λ is generated by i M j, it is enough to show that α(m ij x) = m ij α( x) for m ij i M j and x =(x 1,...,x n ) X. Since α(m ij x) = (0,...,0,α j ( j x i (x i m ij )), 0,...,0) and m ij α( x) =m ij (α 1 x 1,...,α n x n )= (0,...,0, j x i ((α ix i ) m ij ), 0,...,0), we obtain our result. Now we shall consider the relation with the case of another orientation Ω. Definition Let Q =(Γ, v;ω) be a valued quiver with Γ={1, 2,...,n}. We call k Γ a sink (resp. source) if i k (resp. j k) for any arrow i j. For an orientation Ω and k Γ, we define the new orientation s k Ω by changing the direction of all arrows which contain k, and keep the arrows which do not contain k. An ordered set (k 1,...,k n ) of all vertices is called an

8 312 Y. Ishii admissible sequence of sinks if k 1 is a sink with respect to Ω and k t is a sink with respect to s kt 1 s k1 Ω for all t with 2 t n. Then the following facts hold (see [D; p. 30]). 1. s kn s kn 1 s k1 Ω=Ω 2. s k1 s k2 s kn Ω=Ω 3. Each k t is a source with respect to s kt+1 s kn Ω for 1 t n. Proposition For any sink k Γ, the reflection functor Δ + k : rep(m, Ω) rep(m,s k Ω) induces the categorical equivalence rep (k) (M, Ω) = rep (k) (M,s k Ω) where rep (k) (M, Ω) (resp. rep (k) (M,s k Ω)) is a full subcategory of rep(m, Ω) (resp. rep(m,s k Ω)) consisting of representations which do not contain, as their direct summands, the representation k e k =( j ϕ i,w i ) with the dimension vector e k =(0,...,0, ˇ1, 0,...,0), i.e., W k = F k,w i =0for all i k and all j ϕ i =0. 2. Similarly for any source k Γ, the reflection functor Δ k : rep(m, Ω) rep(m,s k Ω) is defined. For an admissible sequence of sinks (k 1,...,k n ), the covariant functor Δ : rep(m, Ω) rep(m, Ω) called the Coxeter functor defined by Δ =Δ k 1 Δ k 2 Δ k n induces the categorical equivalence rep (M, Ω) = rep (M, Ω) where rep (M, Ω) (resp. rep (M, Ω)) is a full subcategory of rep(m, Ω) consisting of representations without injective (resp. projective) direct summand. For X, X Ob(rep (M, Ω)), we have a K-linear isomorphism Hom(X, X ) = Hom(Δ X, Δ X ). Proof. For 1, see [D; p. 62]. For 2, see [D; Proposition 2.10]. Proposition ([D; Lemma 2.5, Corollary 2.7 (i)]) Let k Γ be a sink with respect to Ω. 1. If X Ob(rep (k) (M, Ω)) is indecomposable, then Δ + k X Ob(rep(k) (M,s k Ω)) is also indecomposable, and we have dim Δ + k X = r k(dim X). 2. For X, X Ob(rep (k) (M, Ω)), we have Ext 1 (X, X ) = Ext 1 (Δ + k X, Δ+ k X ). Proposition Let k Γ be a sink with respect to Ω. Then we have Ext 1 Λ(M,s k Ω) (X, e k)=0for any Λ(M,s k Ω)-module X. Proof. Since k Γ is a source of s k Ω, and it is known that there exists an admissible sequence of sinks (i 1,...,i n ) with i n = k with respect to s k Ω (see [ASS; p. 279]), we have e k Ob(rep(M,s k Ω)) is injective by [D; Lemma 2.1]. Hence we obtain our result.

9 Some classification of PVs ass. with Dynkin quivers of exceptional type 313 Proposition ([D; Corollary 2.4 (ii), p. 62]) Let k Γ be a sink with respect to Ω. LetY Ob(rep(M,s k Ω)) be a representation without e k as its direct summand. Then we have dim Fk Ext 1 Λ(M,s k Ω)(e k,y)= [r k (dim Y )] k where [(d 1,...,d n )] k = d k. Note that dim K Ext 1 Λ(M,s k Ω)(e k,y)= f k dim Fk Ext 1 Λ(M,s k Ω) (e k,y) where f k = dim K F k. Theorem If we obtain the table of (dim K Ext 1 Λ(X i,x j )) with respect to some orientation Ω, then we can obtain the similar table (dim K Ext 1 Λ (Y i,y j )) with respect to another orientation Ω from (dim K Ext 1 Λ (X i,x j )). Proof. By Propositions , we obtain our result. Hence it is essentially enough to calculate the table of (dim K Ext 1 Λ (X i,x j )) with respect to any one of orientations Ω. Our results show that the number of (partial) tilting Λ-modules does not depend on the choice of an orientation of arrows. The following proposition is well-known. Proposition In general, if End R (M) is a local ring for an R-module M, then M is indecomposable. The converse also holds if M has the composition series. 3 The Case for exceptional type G 2 In ( this) section, we consider the valued graph (Γ, v) with Γ = {1, 2} and v = 0 1, with an orientation Ω: (1, 3) 2, i.e., of type G 2. We take a K- modulation M = {F 1 = K, F 2 = L, 1M 2 = K L L } where L/K is a cubic extension obtained by adding α to K satisfying α, α 2 / K and α 3 K. We denote L as a K-L-bimodule by K L L. Then, for a dimension vector d = (d 1,d 2 ) Z 2 0, we have the representation (G d,r d ) with G d = GL(d 1 ; K) GL(d 2 ; L) and R d = M(d 2,d 1 ; L). The action of G d on R d is given by R d x BxA 1 R d for (A, B) G d. For a positive root α =(d 1,d 2 ), let x = 2 ϕ 1 : K d 1 K L L d 2 be the corresponding indecomposable representation. If we identify K d 1 K L with L d 1, we may regard x = 2 ϕ 1 as an element of M(d 2,d 1 ; L). Then the Λ-action of the corresponding Λ-module X = K d 1 L d 2 is given by x 1 y 1 x 1 l X (.,. ) ( 0., 2 ϕ 1 (. )) 0 x d1 l x d1 y d2

10 314 Y. Ishii for l L = 1 ( M 2 = ) M (1) Λ. There are 6 positive roots α k (1 k 6) of (Γ = 0 1 {1, 2}, v = ), which is listed below. We denote by x 3 0 k (resp. X k )an indecomposable representation (resp. Λ-module) corresponding to a positive root α k. Note that x k is a generic point of (G αk,r αk ) (see Definition 2.11). An indecomposable representation x k and the Λ-module X k corresponding to a positive root α k is given as follows where Λ = Λ(M, Ω) (see Definition 2.4). α 1 =(0, 1) x 1 =0:{0} L X 1 = L α 2 =(1, 0) x 2 =0:L {0} X 2 = K α 3 =(1, 1) x 3 =1:L L X 3 = K L α 4 =(2, 1) x 4 =(1α) :L 2 L X 4 = K 2 L α 5 =(3, 1) x 5 =(1αα ( 2 ):L) 3 L X 5 = K 3 L α 0 1 α 6 =(3, 2) x 6 = 0 α 2 : L 1 3 L 2 X 6 = K 3 L 2 Theorem 3.1. (The basic tilting Λ(M, Ω)-modules of type G 2 ) For the tensor algebra Λ of the oriented K-modulation (M, Ω), there exist exactly 5 isomorphism classes of basic tilting Λ-modules whose complete list is given by the following X 1 X 3 = L (K L) 2. X 2 X 5 = K (K 3 L) 3. X 3 X 6 =(K L) (K 3 L 2 ) 4. X 4 X 5 =(K 2 L) (K 3 L) 5. X 4 X 6 =(K 2 L) (K 3 L 2 ) Proof. In our case, we have r 11 = dim K K =1,r 22 = dim K L =3,r 12 = dim K ( 1 M 2 = 3, )( and ) hence the Ringel form is given by (x 1,x 2 ), (y 1,y 2 ) = 1 3 y1 (x 1,x 2 ) (see Definition 2.6). Since dim X 0 3 y k = α k (k =1,...,6), 2 we have the following matrix (3.1) of dim X i, dim X j (3.1)

11 Some classification of PVs ass. with Dynkin quivers of exceptional type 315 We can calculate dim K Hom Λ (X i,x j ) by Proposition 2.12, and its matrix is given by (3.2) Since dim K Ext 1 Λ (X i,x j ) = dim K Hom Λ (X i,x j ) dim X i, dim X j, we have the following matrix (3.3) of dim K Ext 1 Λ(X i,x j ) from (3.2) (3.3) Since X = X i X j is a tilting module if and only if dim K Ext 1 Λ (X, X) = dim K Ext 1 Λ (X i,x i )+dim K Ext 1 Λ (X i,x j )+dim K Ext 1 Λ (X j,x i )+dim K Ext 1 Λ (X j,x j )= 0, we obtain our result by (3.3). Theorem 3.2. (PVs corresponding to (partial) tilting modules of type G 2 ) For a dimension vector d =(d 1,d 2 ) Z 2 >0, a (partial) tilting module corresponding to a prehomogeneous vector space (GL(d 1 ; K) GL(d 2 ; L),M(d 2,d 1 ; L)) is given as follows. Note that if it corresponds to a 1 X i1 a 2 X i2 (resp. b j X j ), then the basic tilting module X i1 X i2 (resp. the basic partial tilting module X j ) is its invariant. 1. If d 2 >d 1, it corresponds to (d 2 d 1 )X 1 d 1 X If d 1 > 3d 2, it corresponds to (d 1 3d 2 )X 2 d 2 X If d 1 >d 2 > 2 3 d 1, it corresponds to (3d 2 2d 1 )X 3 (d 1 d 2 )X If 3d 2 >d 1 > 2d 2, it corresponds to (3d 2 d 1 )X 4 (d 1 2d 2 )X If 2d 2 >d 1 > 2 3 d 1 >d 2, it corresponds to (2d 1 3d 2 )X 4 (2d 2 d 1 )X If d 1 = d 2, it corresponds to d 1 X If d 1 =3d 2, it corresponds to d 2 X If d 1 = 3 2 d 2 (and hence d 2 is even), it corresponds to d 2 2 X 6.

12 316 Y. Ishii 9. If d 1 =2d 2, it corresponds to d 2 X 4. Proof. Since m i X i m j X j (i j) is a partial tilting module if and only if X i X j is a partial tilting module, we obtain all partial tilting modules by Theorem 3.1. For example, a tilting module m 1 X 1 m 3 X 3 corresponds to d =(d 1,d 2 )=m 1 α 1 + m 3 α 3 =(m 3,m 1 + m 3 ). Hence we have d 1 = m 3 <d 2 = m 1 + m 3 and X =(d 2 d 1 )X 1 d 1 X 3, i.e., we obtain 1. The remaining parts are obtained similarly. Remark 3.3. (The other orientation) Since the vertex 2 is the sink for the orientation Ω, we can apply Proposition 2.15 for k =2. Now we shall consider the other orientation Ω : 1 (1, 3) 2, and put Λ =Λ(M,s 2 Ω). Let Y 1,...,Y 6 be indecomposable Λ -modules corresponding to positive roots α 1,...,α 6. Since s 2 ((d 1,d 2 )) = (d 1, d 2 + d 1 ), we have the correspondence: s 2 ((1, 0)) = (1, 1), Δ + 2 X 2 = Y3 ; s 2 ((1, 1)) = (1, 0), Δ + 2 X 3 = Y 2 ; s 2 ((2, 1)) = (2, 1), Δ + 2 X 4 = Y 4 ; s 2 ((3, 1)) = (3, 2), Δ + 2 X 5 = Y 6 ; s 2 ((3, 2)) = (3, 1), Δ + 2 X 6 = Y5. Hence by Proposition 2.15, we have Ext 1 Λ (Y i,y j )(i, j 2). By Proposition 2.16, we have Ext 1 Λ (Y i,y 1 )=0(i 1). By Proposition 2.17, we have dim F2 Ext 1 Λ (Y 1,Y 2 )=[s 2 (dim Y 2 )] 2 =[s 2 (1, 0)] 2 = [(1, 1)] 2 =1. Since F 2 = L and [L : K] =3, we have dim K Ext 1 Λ (Y 1,Y 2 )=3. We can calculate similarly dim K Ext 1 Λ (Y 1,Y j )(j 3), and hence we obtain the table of dim K Ext 1 Λ(M,Ω )(Y i,y j ), which is actually given by the transposed matrix of (2.3). In particular, the number of (partial) tilting Λ-modules does not depend on the choice of an orientation of arrows. Remark 3.4. (The relative invariants) We define the injection ϕ : L M(3; K) by a (1,α,α 2 )=(1,α,α 2 )ϕ(a), p rα 3 qα 3 i.e., ϕ(a) = q p rα 3 for a = p + qα + rα 2 L = K + Kα + Kα 2. r q p Define the injective homomorphism Φ:GL(d 2 ; L) GL(3d 2 ; K) by Φ((z ij )= (ϕ(z ij )). We also define the K-isomorphism Ψ:M(d 2,d 1 ; L) = M(3d 2,d 1 ; K) by s Ψ((z ij )) = (ψ(z ij )) with ψ : L K 3 given by ψ(s + tα + uα 2 )= t. u Now let d =(d 1,d 2 ) be the one of the positive roots α 3,α 4,α 5,α 6. Then we define the pair (G K d,rk d ) by G K d = GL(d 1; K) Φ(GL(d 2 ; L)) ( GL(d 1 ; K) GL(3d 2 ; K)) and Rd K =Ψ(M(d 2,d 1 ; L)) = M(3d 2,d 1 ; K). The action of G K d on RK d is given by (g 1, Φ(g 2 )) Ψ(x) =Ψ(g 2 xg1 1 ) for (g 1,g 2 )

13 Some classification of PVs ass. with Dynkin quivers of exceptional type 317 G d and x R d. Since (G d,r d ) is a PV, by Proposition 2.7, there exists a dense G K d -orbit. If d = α i, then Ψ(x i ) is a generic point (i =3, 4, 5, 6). To obtain the all relative invariants (see Definition 2.8), it is enough to obtain the relative invariants whose corresponding characters generate the group X 1 (G K ) by Proposition 2.9. Actually we can construct them similarly to [S]. As an example, we show the case d =(3, 2). In this case, the relative invariant f(ψ(x)) with X = (x ij + y ij α + z ij α 2 ) M(2, 3; L) is given by det X 11 X 12 X 13 O O O A 11 O B 11 O C 11 O X 21 X 22 X 23 O O O O A 11 O B 11 O C 11 O O O X 11 X 12 X 13 A 12 O B 12 O C 12 O O O O X 21 X 22 X 23 O A 12 O B 12 O C 12 where X ij = ψ(x ij + y ij α + z ij α 2 ) M(3, 1), A ij = ψ(a ij + b ij α + c ij α 2 ) M(3, 1; K), B ij = ψ(α(a ij + b ij α + c i,j α 2 )) M(3, 1; K), C ij = ψ(α 2 (a ij + b ij α + c ij α 2 )) M(3, 1; K). We should choose constants so that f(ψ(x)) is not identically zero. As far as it satisfies this condition, the choice of constant gives just a constant multiple of the relative invariant ([S]). For example, we may choose as a 11 + b 11 α + c 11 α 2 =1and a 12 + b 12 α + c 12 α 2 = α. 4 The Case for exceptional type F 4 In this section, we consider the valued graph (Γ, v) with Γ = {1, 2, 3, 4} and v = with an orientation Ω : 1 2 (1, 2) 3 4, i.e., of type F 4. We take a K-modulation M = {F 1 = F 2 = K, F 3 = F 4 = L, 1 M 2 = K K K, 2 M 3 = K L L, 3 M 4 = L L L } where L/K is a quadratic extension obtained by adding a square root α of an element of K. Then, for a dimension vector d =(d 1,d 2,d 3,d 4 ) Z 4 0, we have the representation (G d,r d ) with G d = GL(d 1 ; K) GL(d 2 ; K) GL(d 3 ; L) GL(d 4 ; L) and R d = M(d 2,d 1 ; K) M(d 3,d 2 ; L) M(d 4,d 3 ; L). The action of G d on R d is given by R d x = (x, y, z) g x =(BxA 1,CyB 1,DzC 1 ) R d for g =(A, B, C, D) G d. For a positive root α =(d 1,d 2,d 3,d 4 ), the corresponding indecomposable representation x =( 2 ϕ 1, 3 ϕ 2, 4 ϕ 3 ) is given by maps 2 ϕ 1 : K d 1 K K( = K d 1 ) K d 2, 3ϕ 2 : K d 2 K L( = L d 2 ) L d 3 and 4 ϕ 3 : L d 3 L L( = L d 3 ) L d 4. We

14 318 Y. Ishii may regard x as an element of M(d 2,d 1 ; K) M(d 3,d 2 ; L) M(d 4,d 3 ; L). The Λ-action on the corresponding Λ-module X = K d 1 K d 2 L d 3 L d 4 is given by x 1 y 1 z 1 w 1 x 1 k y 1 l z 1 l (.,.,.,. ) ( 0., 2 ϕ 1., 3 ϕ 2., 4 ϕ 3. ) 0 x d1 k y d2 l z d3 l x d1 y d2 z d3 w d4 for (k, l, l ) K L L = 1 M 2 2 M 3 3 M 4 = M (1) Λ. There are 24 positive roots α k (1 k 24) of the valued graph (Γ, v) of type F 4, which is listed below. We denote by x k (resp. X k ) an indecomposable representation (resp. Λ-module) corresponding to a positive root α k. Note that x k is a generic point of a PV (G αk,r αk ). An indecomposable representation x k corresponding to a positive root α k is given as follows. α 1 =(0, 0, 0, 1) x 1 =(0, 0, 0) X 1 = {0} {0} {0} L α 2 =(0, 0, 1, 0) x 2 =(0, 0, 0) X 2 = {0} {0} L {0} α 3 =(0, 0, 1, 1) x 3 =(0, 0, 1) X 3 = {0} {0} L L α 4 =(0, 1, 0, 0) x 4 =(0, 0, 0) X 4 = {0} K {0} {0} α 5 =(0, 1, 1, 0) x 5 =(0, 1, 0) X 5 = {0} K L {0} α 6 =(0, 1, 1, 1) x 6 =(0, 1, 1) X 6 = {0} K L L α 7 =(0, 2, 1, 0) x 7 =(0, ( 1 α ), 0) X 7 = {0} K 2 L {0} α 8 =(0, 2, 1, 1) x 8 =(0, ( 1 α ), 1) X 8 = {0} K 2 L L α 9 =(0, 2, 2, 1) x 9 =(0,I 2, ( 1 α ) ) X 9 = {0} K 2 L 2 L α 10 =(1, 0, 0, 0) x 10 =(0, 0, 0) X 10 = K {0} {0} {0} α 11 =(1, 1, 0, 0) x 11 =(1, 0, 0) X 11 = K K {0} {0} α 12 =(1, 1, 1, 0) x 12 =(1, 1, 0) X 12 = K K L {0} α 13 =(1, 1, 1, 1) x 13 =(1, ( 1, ) 1) X 13 = K K L L 1 α 14 =(1, 2, 1, 0) x 14 =(, ( 1 α ), 0) X 0 14 = K K 2 L {0} ( ) 1 α 15 =(1, 2, 1, 1) x 15 =(, ( 1 α ), 1) X 0 15 = K K 2 L L ( ) 1 α 16 =(1, 2, 2, 1) x 16 =(,I 0 2, ( 1 α ) ) X 16 = K K 2 L 2 L 1 ( ) α 17 =(1, 3, 2, 1) x 17 =( 0 α 0 0,, ( 0 1 ) ) 0 1 α 1 X 17 = K K 3 L 2 L α 18 =(2, 2, 1, 0) x 18 =(I 2, ( 1 α ), 0) X 18 = K 2 K 2 L {0} α 19 =(2, 2, 1, 1) x 19 =(I 2, ( 1 α ), 1) X 19 = K 2 K 2 L L α 20 =(2, 2, 2, 1) x 20 =(I 2,I 2, ( 1 α ) ) X 20 = K 2 K 2 L 2 L 1 0 ( ) α 21 =(2, 3, 2, 1) x 21 =( 0 0 α 0 0,, ( 1 α ) ) 0 1 α 0 1

15 Some classification of PVs ass. with Dynkin quivers of exceptional type 319 X 21 = K 2 K 3 L 2 L 1 0 ( ) α 22 =(2, 4, 2, 1) x 22 =( α ,, ( 1 1 ) ) α 0 1 X 22 = K 2 K 4 L 2 L 1 0 α 23 =(2, 4, 3, 1) x 23 =( 0 0 α , 0 1 α 0, ( ) ) X 23 = K 2 K 4 L 3 L 1 0 α 24 =(2, 4, 3, 2) x 24 =( 0 0 α ( ) 0 0, 0 1 α , ) X 24 = K 2 K 4 L 3 L 2 Theorem 4.1. (The basic tilting Λ(M, Ω)-modules of type F 4 ) For the tensor algebra Λ of the oriented K-modulation (M, Ω) of type F 4, there exist exactly 66 isomorphism classes of basic tilting Λ-modules. They are given by the following list. For example, (1, 3, 6, 13) indicates the tilting module X 1 X 3 X 6 X 13 = L (L L) (K L L) (K K L L). (1) (1, 3, 6, 13) (2) (1, 3, 10, 13) (3) (1, 4, 8, 15) (4) (1, 4, 11, 19) (5) (1, 4, 15, 19) (6) (1, 6, 8, 13) (7) (1, 8, 13, 15) (8) (1, 10, 11, 19) (9) (1, 10, 13, 19) (10) (1, 13, 15, 19) (11) (2, 3, 6, 13) (12) (2, 3, 10, 13) (13) (2, 5, 9, 16) (14) (2, 5, 12, 20) (15) (2, 5, 16, 20) (16) (2, 6, 9, 13) (17) (2, 9, 13, 16) (18) (2, 10, 12, 20) (19) (2, 10, 13, 20) (20) (2, 13, 16, 20) (21) (4, 7, 8, 15) (22) (4, 7, 14, 22) (23) (4, 7, 15, 22) (24) (4, 11, 18, 19) (25) (4, 14, 18, 19) (26) (4, 14, 19, 22) (27) (4, 15, 19, 22) (28) (5, 7, 8, 17) (29) (5, 7, 12, 23) (30) (5, 7, 17, 23) (31) (5, 8, 9, 16) (32) (5, 8, 16, 24) (33) (5, 8, 17, 24) (34) (5, 12, 20, 23) (35) (5, 16, 20, 24) (36) (5, 17, 20, 23) (37) (5, 17, 20, 24) (38) (6, 8, 9, 13) (39) (7, 8, 15, 17) (40) (7, 12, 14, 22) (41) (7, 12, 21, 22) (42) (7, 12, 21, 23) (43) (7, 15, 17, 23) (44) (7, 15, 21, 22) (45) (7, 15, 21, 23) (46) (8, 9, 13, 16) (47) (8, 13, 15, 24) (48) (8, 13, 16, 24) (49) (8, 15, 17, 24) (50) (10, 11, 18, 19) (51) (10, 12, 18, 19) (52) (10, 12, 19, 20) (53) (10, 13, 19, 20) (54) (12, 14, 18, 19) (55) (12, 14, 19, 22) (56) (12, 19, 20, 21) (57) (12, 19, 21, 22) (58) (12, 20, 21, 23) (59) (13, 15, 19, 20) (60) (13, 15, 20, 24) (61) (13, 16, 20, 24) (62) (15, 17, 20, 23) (63) (15, 17, 20, 24) (64) (15, 19, 20, 21) (65) (15, 19, 21, 22) (66) (15, 20, 21, 23)

16 320 Y. Ishii Proof. To express the matrix (dim K Hom Λ (X i,x j )) simply, we introduce Y i (1 i 24) as follows (cf. 5). Y 1 = X 13,Y 2 = X 6,Y 3 = X 3,Y 4 = X 1,Y 5 = X 5,Y 6 = X 16,Y 7 = X 9,Y 8 = X 2,Y 9 = X 15,Y 10 = X 17,Y 11 = X 24,Y 12 = X 8,Y 13 = X 12,Y 14 = X 21,Y 15 = X 23,Y 16 = X 20,Y 17 = X 4,Y 18 = X 14,Y 19 = X 22,Y 20 = X 7,Y 21 = X 10,Y 22 = X 11,Y 23 = X 18,Y 24 = X 19. Then we have the following table (4.1) of dim K Hom Λ (Y i,y j ). A 1 A 2 A 3 A 4 A 5 A 6 0 A 1 A 2 A 3 A 4 A A 1 A 2 A 3 A A 1 A 2 A A 1 A A 1 (4.1) and each 4 4 matrices A 1...A 6 is given by ) ) A 1 =, A 2 =, A 3 = A 5 = ( ( ), A 6 = ( ( ). ( ), A 4 = ( ) , 0222 The table of the Ringel form Y i,y j is given by (4.2). A 1 A 2 A 3 A 4 A 5 A 6 t A 1 A 1 A 2 A 3 A 4 A 5 t A 2 t A 1 A 1 A 2 A 3 A 4 t A 3 t A 2 t A 1 A 1 A 2 A 3 t A 4 t A 3 t A 2 t A 1 A 1 A 2 t A 5 t A 4 t A 3 t A 2 t A 1 A 1 (4.2) Since dim K Ext 1 Λ (Y i,y j ) = dim K Hom Λ (Y i,y j ) dim Y i, dim Y j, we have the following table (4.3) of dim K Ext 1 Λ(Y i,y j ) from (4.1), (4.2) t A t A t 2 A t A t 3 A t 2 A (4.3) t A t 4 A t 3 A t 2 A t A t 5 A t 4 A t 3 A t 2 A 1 0 Thus we obtain our result.

17 Some classification of PVs ass. with Dynkin quivers of exceptional type 321 From Theorem 4.1, we have immediately a complete list of PVs associated (1, 2) with the F 4 -type quiver. Although (G d,r d )isa finite PV for any dimension d and also a trivial PV in the sense of [K], our list can be regarded as a finer classification of such PVs. Theorem 4.2. (PVs corresponding to tilting modules of type F 4 ) For a dimension vector d =(d 1,d 2,d 3,d 4 ) Z 4 >0, a tilting module corresponding to a prehomogeneous vector space (GL(d 1 ; K) GL(d 2 ; K) GL(d 3 ; L) GL(d 4 ; L),M(d 2,d 1 ; K) M(d 3,d 2 ; L) M(d 4,d 3 ; L)) is given as follows. Note that if it corresponds to a 1 X i1 a 2 X i2 a 3 X i3 a 4 X i4, then the basic tilting module X i1 X i2 X i3 X i4 is its invariant. 1. If d 1 <d 2 <d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 3 d 2 )X 3 (d 2 d 1 )X 6 d 1 X If d 2 <d 1,d 2 <d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 3 d 2 )X 3 (d 1 d 2 )X 10 d 2 X If 2d 3 <d 2,d 1 <d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 2 2d 3 )X 4 (d 3 d 1 )X 8 d 1 X If 2d 3 <d 1 <d 2,d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 2 d 1 )X 4 (d 1 2d 3 )X 11 d 3 X If d 3 <d 1 < 2d 3 <d 2,d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 2 2d 3 )X 4 (2d 3 d 1 )X 15 (d 1 d 3 )X If d 3 <d 2,d 1 + d 2 < 2d 3,d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (2d 3 d 1 d 2 )X 6 (d 2 d 3 )X 8 d 1 X If d 2 < 2d 3 <d 1 + d 2,d 1 <d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 3 d 1 )X 8 (2d 3 d 2 )X 13 (d 1 + d 2 2d 3 )X If 2d 3 <d 2 <d 1,d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 1 d 2 )X 10 (d 2 2d 3 )X 11 d 3 X If d 2 <d 1,d 3 <d 2 < 2d 3,d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 1 d 2 )X 10 (2d 3 d 2 )X 13 (d 2 d 3 )X If d 3 <d 1 <d 2 < 2d 3,d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (2d 3 d 2 )X 13 (d 2 d 1 )X 15 (d 1 d 3 )X If d 1 <d 2 <d 4 <d 3, it corresponds to (d 3 d 4 )X 2 (d 4 d 2 )X 3 (d 2 d 1 )X 6 d 1 X If d 2 <d 1,d 2 <d 4 <d 3, it corresponds to (d 3 d 4 )X 2 (d 4 d 2 )X 3 (d 1 d 2 )X 10 d 2 X 13.

18 322 Y. Ishii 13. If d 1 <d 4, 2d 4 <d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (d 2 2d 4 )X 5 (d 4 d 1 )X 9 d 1 X If 2d 4 <d 1 <d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (d 2 d 1 )X 5 (d 1 2d 4 )X 12 d 4 X If d 4 <d 1 < 2d 4 <d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (d 2 2d 4 )X 5 (2d 4 d 1 )X 16 (d 1 d 4 )X If d 1 + d 2 < 2d 4,d 4 <d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (2d 4 d 1 d 2 )X 6 (d 2 d 4 )X 9 d 1 X If d 1 <d 4,d 2 < 2d 4 <d 1 + d 2,d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (d 4 d 1 )X 9 (2d 4 d 2 )X 13 (d 1 + d 2 2d 4 )X If 2d 4 <d 2 <d 1,d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (d 1 d 2 )X 10 (d 2 2d 4 )X 12 d 4 X If d 4 <d 2 < 2d 4,d 2 <d 1,d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (d 1 d 2 )X 10 (2d 4 d 2 )X 13 (d 2 d 4 )X If d 4 <d 1 <d 2 < 2d 4,d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (2d 4 d 2 )X 13 (d 2 d 1 )X 16 (d 1 d 4 )X If d 1 <d 4 <d 3, 2d 3 <d 2, it corresponds to (d 2 2d 3 )X 4 (d 3 d 4 )X 7 (d 4 d 1 )X 8 d 1 X If 2d 4 <d 1 <d 3, 2d 3 <d 2, it corresponds to (d 2 2d 3 )X 4 (d 3 d 1 )X 7 (d 1 2d 4 )X 14 d 4 X If d 4 <d 1 < 2d 4,d 1 <d 3, 2d 3 <d 2, it corresponds to (d 2 2d 3 )X 4 (d 3 d 1 )X 7 (2d 4 d 1 )X 14 (d 1 d 4 )X If d 4 <d 3, 2d 3 <d 1 <d 2, it corresponds to (d 2 d 1 )X 4 (d 1 2d 3 )X 11 (d 3 d 4 )X 18 d 4 X If d 3 + d 4 <d 1 < 2d 3 <d 2, it corresponds to (d 2 2d 3 )X 4 (2d 3 d 1 )X 14 (d 1 d 3 d 4 )X 18 d 4 X If d 3 <d 1, 2d 4 <d 1 <d 3 + d 4, 2d 3 <d 2, it corresponds to (d 2 2d 3 )X 4 (d 1 2d 4 )X 14 (d 1 d 3 )X 19 (d 3 + d 4 d 1 )X If d 4 <d 3 <d 1 < 2d 4, 2d 3 <d 2, it corresponds to (d 2 2d 3 )X 4 (2d 4 d 1 )X 15 (d 1 d 3 )X 19 (d 3 d 4 )X If d 1 <d 4,d 3 + d 4 <d 2,d 1 + d 2 < 2d 3, it corresponds to (2d 3 d 1 d 2 )X 5 (d 2 d 3 d 4 )X 7 (d 4 d 1 )X 8 d 1 X 17.

19 Some classification of PVs ass. with Dynkin quivers of exceptional type If 2d 4 <d 1,d 3 + d 4 <d 2,d 1 + d 2 < 2d 3, it corresponds to (2d 3 d 1 d 2 )X 5 (d 2 d 3 d 4 )X 7 (d 1 2d 4 )X 12 d 4 X If d 4 <d 1 < 2d 4,d 3 + d 4 <d 2,d 1 + d 2 < 2d 3, it corresponds to (2d 3 d 1 d 2 )X 5 (d 2 d 3 d 4 )X 7 (2d 4 d 1 )X 17 (d 1 d 4 )X If 2d 4 <d 2,d 3 <d 2,d 1 + d 2 <d 3 + d 4, it corresponds to (d 2 2d 4 )X 5 (d 2 d 3 )X 8 (d 3 + d 4 d 1 d 2 )X 9 d 1 X If d 1 <d 4, 2d 4 <d 2,d 1 +2d 2 < 2d 3 +2d 4,d 3 +d 4 <d 1 +d 2, it corresponds to (d 2 2d 4 )X 5 (d 4 d 1 )X 8 (2d 3 +2d 4 d 1 2d 2 )X 16 (d 1 + d 2 d 3 d 4 )X If d 1 <d 4, 2d 3 +2d 4 <d 1 +2d 2,d 2 <d 3 +d 4,d 1 +d 2 < 2d 3, it corresponds to (2d 3 d 1 d 2 )X 5 (d 4 d 1 )X 8 (d 1 +2d 2 2d 3 2d 4 )X 17 (d 3 + d 4 d 2 )X If 2d 4 <d 1,d 3 <d 2 <d 3 + d 4,d 1 + d 2 < 2d 3, it corresponds to (2d 3 d 1 d 2 )X 5 (d 1 2d 4 )X 12 (d 3 + d 4 d 2 )X 20 (d 2 d 3 )X If d 1 <d 4, 2d 4 <d 2,d 3 <d 2,d 1 +2d 2 < 2d 3 +2d 4, it corresponds to (d 2 2d 4 )X 5 (2d 3 +2d 4 d 1 2d 2 )X 16 (d 1 d 4 )X 20 (d 2 d 3 )X If d 1 < 2d 4,d 1 +d 2 <d 3 +2d 4,d 2 <d 3 +d 4,d 1 +d 2 < 2d 3, it corresponds to (2d 3 d 1 d 2 )X 5 (2d 4 d 1 )X 17 (d 3 +d 4 d 2 )X 20 (d 1 +d 2 d 3 2d4)X If d 4 <d 1,d 1 + d 2 <d 3 +2d 4,d 1 +2d 2 < 2d 3 +2d 4,d 1 + d 2 < 2d 3,it corresponds to (2d 3 d 1 d 2 )X 5 (d 1 +2d 2 2d 3 2d 4 )X 17 (d 1 d 4 )X 20 (d 3 +2d4 d 1 d 2 )X If d 4 <d 3 <d 2,d 1 + d 2 < 2d 4, it corresponds to (2d 4 d 1 d 2 )X 6 (d 2 d 3 )X 8 (d 3 d 4 )X 9 d 1 X If d 1 <d 4,d 2 < 2d 3 <d 1 + d 2,d 3 + d 4 <d 2, it corresponds to (d 2 d 3 d 4 )X 7 (d 4 d 1 )X 8 (d 1 + d 2 2d 3 )X 15 (2d 3 d 2 )X If d 2 < 2d 3, 2d 3 +2d 4 <d 1 + d 2,d 1 <d 3, it corresponds to (d 3 d 1 )X 7 (2d 3 d 2 )X 12 (d 1 + d 2 2d 3 2d 4 )X 14 d 4 X If 2d 3 +d 4 <d 1 +d 2, 2d 4 <d 1,d 1 +d 2 < 2d 3 +2d 4,d 1 <d 3, it corresponds to (d 3 d 1 )X 7 (d 1 2d 4 )X 12 (2d 3 +2d 4 d 1 d 2 )X 21 (d 1 + d 2 2d 3 d 4 )X If d 3 +d 4 <d 2, 2d 3 <d 1 +d 2, 2d 4 <d 1,d 1 +d 2 < 2d 3 +d 4, it corresponds to (d 2 d 3 d 4 )X 7 (d 1 2d 4 )X 12 (d 1 +d 2 2d 3 )X 21 (2d 3 +d 4 d 1 d 2 )X 23.

20 324 Y. Ishii 43. If d 3 +d 4 <d 2, 2d 3 <d 1 +d 2,d 4 <d 1, 2d 1 +d 2 < 2d 3 +2d 4, it corresponds to (d 2 d 3 d 4 )X 7 (d 1 + d 2 2d 3 )X 15 (2d 3 +2d 4 2d 1 d 2 )X 17 (d 1 d 4 )X If d 2 < 2d 3,d 1 < 2d 4, 2d 3 + d 4 <d 1 + d 2,d 1 <d 3, it corresponds to (d 3 d 1 )X 7 (2d 4 d 1 )X 15 (2d 3 d 2 )X 21 (d 1 + d 2 2d 3 d 4 )X If d 1 + d 2 < 2d 3 + d 4,d 3 + d 4 <d 2,d 1 < 2d 4, 2d 3 +2d 4 < 2d 1 + d 2,it corresponds to (d 2 d 3 d 4 )X 7 (2d 4 d 1 )X 15 (2d 1 + d 2 2d 3 2d 4 )X 21 (2d 3 + d 4 d 1 d 2 )X If d 3 <d 2,d 1 + d 2 <d 3 + d 4,d 2 < 2d 4 <d 1 + d 2, it corresponds to (d 2 d 3 )X 8 (d 3 + d 4 d 1 d 2 )X 9 (2d 4 d 2 )X 13 (d 1 + d 2 2d 4 )X If d 1 <d 4 <d 3,d 2 < 2d 4, 2d 3 <d 1 + d 2, it corresponds to (d 4 d 1 )X 8 (2d 4 d 2 )X 13 (d 1 + d 2 2d 3 )X 15 (d 3 d 4 )X If d 1 <d 4,d 2 < 2d 4,d 3 + d 4 <d 1 + d 2 < 2d 3, it corresponds to (d 4 d 1 )X 8 (2d 4 d 2 )X 13 (2d 3 d 1 d 2 )X 16 (d 1 + d 2 d 3 d 4 )X If d 1 <d 4, 2d 4 <d 2 <d 3 + d 4, 2d 3 <d 1 + d 2, it corresponds to (d 4 d 1 )X 8 (d 1 + d 2 2d 3 )X 15 (d 2 2d 4 )X 17 (d 3 + d 4 d 2 )X If d 4 <d 3, 2d 3 <d 2 <d 1, it corresponds to (d 1 d 2 )X 10 (d 2 2d 3 )X 11 (d 3 d 4 )X 18 d 4 X If d 2 < 2d 3,d 3 + d 4 <d 2 <d 1, it corresponds to (d 1 d 2 )X 10 (2d 3 d 2 )X 12 (d 2 d 3 d 4 )X 18 d 4 X If d 3 <d 2 <d 3 + d 4, 2d 4 <d 2 <d 1, it corresponds to (d 1 d 2 )X 10 (d 2 2d 4 )X 12 (d 2 d 3 )X 19 (d 3 + d 4 d 2 )X If d 4 <d 3 <d 2 <d 1,d 2 < 2d 4, it corresponds to (d 1 d 2 )X 10 (2d 4 d 2 )X 13 (d 2 d 3 )X 19 (d 3 d 4 )X If d 3 + d 4 <d 1 <d 2 < 2d 3, it corresponds to (2d 3 d 2 )X 12 (d 2 d 1 )X 14 (d 1 d 3 d 4 )X 18 d 4 X If 2d 3 +2d 4 <d 1 + d 2,d 3 <d 1 <d 3 + d 4,d 2 < 2d 3, it corresponds to (2d 3 d 2 )X 12 (d 1 +d 2 2d 3 2d 4 )X 14 (d 1 d 3 )X 19 (d 3 +d 4 d 1 )X If 2d 4 <d 1 <d 2 <d 3 + d 4,d 3 <d 1, it corresponds to (d 1 2d 4 )X 12 (d 1 d 3 )X 19 (d 3 + d 4 d 2 )X 20 (d 2 d 1 )X If 2d 4 <d 1,d 3 <d 1,d 3 + d 4 <d 2,d 1 + d 2 < 2d 3 +2d 4, it corresponds to (d 1 2d 4 )X 12 (d 1 d 3 )X 19 (2d 3 +2d 4 d 1 d 2 )X 21 (d 2 d 3 d 4 )X 22.

21 Some classification of PVs ass. with Dynkin quivers of exceptional type If 2d 4 < d 1,d 1 < d 3,d 2 < d 3 + d 4, 2d 3 < d 1 + d 2, it corresponds to (d 1 2d 4 )X 12 (d 3 + d 4 d 2 )X 20 (d 1 + d 2 2d 3 )X 21 (d 3 d 1 )X If d 4 <d 3 <d 1 <d 2 < 2d 4, it corresponds to (2d 4 d 2 )X 13 (d 2 d 1 )X 15 (d 1 d 3 )X 19 (d 3 d 4 )X If 2d 3 <d 1 +d 2,d 4 <d 1 <d 3,d 2 < 2d 4, it corresponds to (2d 4 d 2 )X 13 (d 1 + d 2 2d 3 )X 15 (d 1 d 4 )X 20 (d 3 d 1 )X If d 1 +d 2 < 2d 3,d 4 <d 1,d 3 <d 2 < 2d 4, it corresponds to (2d 4 d 2 )X 13 (2d 3 d 1 d 2 )X 16 (d 1 d 4 )X 20 (d 2 d 3 )X If 2d 1 + d 2 < 2d 3 +2d 4,d 2 <d 3 + d 4,d 3 +2d 4 <d 1 + d 2, 2d 3 <d 1 + d 2, it corresponds to (d 1 + d 2 2d 3 )X 15 (2d 3 +2d 4 2d 1 d 2 )X 17 (d 3 + d 4 d 2 )X 20 (d 1 + d 2 d 3 2d 4 )X If 2d 3 <d 1 + d 2 <d 3 +2d 4,d 4 <d 1, 2d 4 <d 2, it corresponds to (d 1 + d 2 2d 3 )X 15 (d 2 2d 4 )X 17 (d 1 d 4 )X 20 (d 3 +2d 4 d 1 d 2 )X If d 3 <d 1 < 2d 4 <d 2 <d 3 + d 4, it corresponds to (2d 4 d 1 )X 15 (d 1 d 3 )X 19 (d 3 + d 4 d 2 )X 20 (d 2 2d 4 )X If d 3 <d 1 < 2d 4,d 3 + d 4 <d 2 < 2d 3, it corresponds to (2d 4 d 1 )X 15 (d 1 d 3 )X 19 (2d 3 d 2 )X 21 (d 2 d 3 d 4 )X If d 1 <d 3,d 1 < 2d 4,d 2 <d 3 + d 4, 2d 3 +2d 4 < 2d 1 + d 2, it corresponds to (2d 4 d 1 )X 15 (d 3 +d 4 d 2 )X 20 (2d 1 +d 2 2d 3 2d 4 )X 21 (d 3 d 1 )X 23. Proof. We can obtain the results similarly to Theorem 3.2. Remark 4.3. PVs corresponding to a partial tilting module whose number of direct summands is less than 4 appears as the boundary of the above list. For example, the case of the boundary d 1 <d 2 <d 3 = d 4 (resp. d 1 <d 2 = d 3 = d 4 ) of 1. corresponds to a partial tilting module (d 3 d 2 )X 3 (d 2 d 1 )X 6 d 1 X 13 (resp. (d 2 d 1 )X 6 d 1 X 13 ). Since we assume that d Z 4 >0, the partial tilting module X k alone does not appear for k =1,...,12, 14, 18. Remark 4.4. Define ( the injective ) homomorphism Φ d : GL(d; L) GL(2d; K) pij q by (p ij + q ij α) ij ij α 2 (i, j =1,...,d) and the K-isomorphism q ij p ij ij ( ) Ψ d,d : M(d, d ; L) M(2d, d pst ; K) by (p st +q st α) st (s =1,...,d; t = 1,...,d ). Then define the pair (G K d,rk d ) by q st st

22 326 Y. Ishii G K d = GL(d 1; K) GL(d 2 ; K) Φ d3 (GL(d 3 ; L)) Φ d4 (GL(d 4 ; L)) R K d = M(d 2,d 1 ; K) Ψ d3 d 2 (M(d 3,d 2 ; L)) Ψ d4 d 3 (M(d 4,d 3 ; L)). Similarly to the case of G 2, we can construct the relative invariants of type F 4 of the above PV (G K d,rk d ). 5 The Case for exceptional type E 6 In this section, we considerthe valued graph (Γ, v) with Γ = {1, 2, 3, 4, 5, 6} and v = with an orientation Ω : , i.e., of type E 6. There exist 36 positive roots α 1,...,α 36 given by α 1 =(0, 0, 0, 0, 0, 1), α 2 =(0, 0, 0, 0, 1, 0), α 3 =(0, 0, 0, 1, 0, 0), α 4 =(0, 0, 0, 1, 1, 0), α 5 =(0, 0, 1, 0, 0, 0), α 6 =(0, 0, 1, 0, 0, 1), α 7 =(0, 0, 1, 1, 0, 0), α 8 =(0, 0, 1, 1, 0, 1), α 9 =(0, 0, 1, 1, 1, 0), α 10 =(0, 0, 1, 1, 1, 1), α 11 =(0, 1, 0, 0, 0, 0), α 12 =(0, 1, 1, 0, 0, 0), α 13 =(0, 1, 1, 0, 0, 1), α 14 =(0, 1, 1, 1, 0, 0), α 15 =(0, 1, 1, 1, 0, 1), α 16 =(0, 1, 1, 1, 1, 0), α 17 =(0, 1, 1, 1, 1, 1), α 18 =(0, 1, 2, 1, 0, 1), α 19 =(0, 1, 2, 1, 1, 1), α 20 =(0, 1, 2, 2, 1, 1), α 21 =(1, 0, 0, 0, 0, 0), α 22 =(1, 1, 0, 0, 0, 0), α 23 =(1, 1, 1, 0, 0, 0), α 24 =(1, 1, 1, 0, 0, 1), α 25 =(1, 1, 1, 1, 0, 0), α 26 =(1, 1, 1, 1, 0, 1), α 27 =(1, 1, 1, 1, 1, 0), α 28 =(1, 1, 1, 1, 1, 1), α 29 =(1, 1, 2, 1, 0, 1), α 30 =(1, 1, 2, 1, 1, 1), α 31 =(1, 1, 2, 2, 1, 1), α 32 =(1, 2, 2, 1, 0, 1), α 33 =(1, 2, 2, 1, 1, 1), α 34 =(1, 2, 2, 2, 1, 1), α 35 =(1, 2, 3, 2, 1, 1), α 36 =(1, 2, 3, 2, 1, 2). We take a K-modulation M = {F 1 = = F 6 = K, 1 M 2 = K K K, 3 M 2 = KK K, 3M 4 = K K K, 5M 4 = K K K, 3M 6 = K K K } where K K K is a commutative field K as a K-K-bimodule. Take an admissible sequence of sinks (k 1,...,k 6 )=(6, 4, 5, 2, 1, 3) with respect to Ω (see Definition 2.13). For any t satisfying 1 t 6, let e kt Ob(rep(M,s kt s k6 Ω)) be the representation k t with the dimension vector e kt =(0,...,0, ˇ1, 0,..., 0). Then define the representation P kt Ob(rep(M, Ω)) by Δ k 1 Δ k t 1 e kt. Let Δ =Δ k 1 Δ k 6 = Δ 6 Δ 4 Δ 5 Δ 2 Δ 1 Δ 3 : rep(m, Ω) rep(m, Ω) be the Coxeter functor. Take the Coxeter element δ = r k6 r k1 of the Weyl group (see Definition 2.11 and see [D, p. 44]).

23 Some classification of PVs ass. with Dynkin quivers of exceptional type 327 Proposition 5.1. The 36 representations Δ s P t (1 t 6 and 0 s 5) are the complete representatives of isomorphism classes of indecomposable representations in rep(m, Ω), and the 6 representations Δ 5 P t (1 t 6) are the complete representatives of isomorphism classes of indecomposable injective representations. Moreover we have dim Δ s P t = δ s (dim P t )(1 t 6 and 0 s 5) and dim P kt = r k1 r kt 1 (e kt ). Proof. For any indecomposable representation X Ob(rep(M, Ω)), there exist t (1 t 6) and a non-negative integer s satisfying X = Δ s P t (see [D; p. 71 and p. 79 Theorem 2.19 (v)]). For any indecomposable representation Y Ob(rep(M, Ω)), the following (a), (b) hold. (a) If Δ Y 0, then Δ Y is indecomposable and dim Δ Y = δ 1 (dim Y ) (see [D; p. 69, Lemma 2.9]). (b) Δ Y =0 Y is injective δ 1 (dim Y ) /Z 6 0 (see [D; p. 73, Proposition 2.12]) By [D; Lemma 2.5], we have dim P kt = r k1 r kt 1 (e kt ), and hence dim P 1 = α 22, dim P 2 = α 11, dim P 3 = α 15, dim P 4 = α 3, dim P 5 = α 4, dim P 6 = α 1.For example, dim Δ 1 P 1 = δ 1 α 22 = α 8, dim Δ 2 P 1 = δ 1 α 8 = α 16, dim Δ 3 P 1 = δ 1 α 16 = α 24, dim Δ 4 P 1 = δ 1 α 24 = α 7, dim Δ 5 P 1 = δ 1 α 7 = α 2, and δ 1 α 2 =(0, 0, 0, 1, 1, 0) / Z 6 0. By similar direct calculation, we can show that the set of dim Δ s P t (1 t 6 and 0 s 5) coincides the set of all positive roots and δ 1 (dim Δ 5 P t ) / Z 6 0 (1 t 6). Hence we obtain our result. Now we shall calculate dim K Hom(Δ s 1 P t1, Δ s 2 P t2 )(1 t 1,t 2 6 and 0 s 1,s 2 5). Proposition 5.2. The following assertions 1, 2, 3 hold. 1. For 1 t 1,t 2 6 and 0 s 5, we have dim K Hom(P t1,p t2 ) = dim K Hom(Δ s P t1, Δ s P t2 ). 2. For 1 s 2 4, 1 s 5 s 2 and 1 t 1,t 2 6, we have dim K Hom(P t1, Δ s 2 P t2 ) = dim K Hom(Δ s P t1, Δ s 2 s P t2 ). 3. For 1 s 1 4, 1 s 5 s 1 and 1 t 1,t 2 6, we have dim K Hom(Δ s 1 P t1,p t2 ) = dim K Hom(Δ s 1 s P t1, Δ s P t2 ).

24 328 Y. Ishii Proof. By 2 of Proposition 2.14, we have our result. Note that {Δ 5 P t 1 t 6} is the set of complete representatives of isomorphism classes of indecomposable injective representations (see [D; Proposition 2.8]). Proposition 5.3. For 1 s 1 4 and 1 t 1,t 2 6, we have dim K Hom(Δ s 1 P t1,p t2 )=0. Proof. Since Δ s 1 P t1 is a non-projective indecomposable representation, and P t2 is an indecomposable projective representation, by [D; p. 75, Lemma 2.13], we obtain our result. Proposition 5.4. For 1 t 6 and 0 s 5, we have (dim K Hom(P 1, Δ s P t ), dim K Hom(P 2, Δ s P t ),...,dim K Hom(P 6, Δ s P t )) = δ s (dim P t ). Proof. Since the maximal length of paths is 1 in our orientation Ω, we have Λ=Λ(M, Ω) = M (0) M (1) =(F 1 F 6 ) ( 1 M 2 3 M 2 3 M 4 3 M 6 5 M 4 ) (see Definition 2.4). Put e t =(0,...,0, 1 Ft, 0,...,0) F 1 F 6 Λ. Then we have 1 Λ = e e 6, e 2 t = e t (1 t 6) and e t1 e t2 = e t2 e t1 = 0(1 t 1 t 2 6), and hence we obtain Λ = e 1 Λ+ + e 6 Λ. By [ASS; p. 19, Lemma 4.2], we have End Λ (e t Λ) = e t Λe t as K-algebras. Since e t Λe t = (0,...,0,F t, 0,...,0) = F t, the endomorphism ring End Λ (e t Λ) is a field, and hence by Proposition 2.19, each e t Λ is indecomposable. By Remark 2.5, the dimension vector of the indecomposable representation corresponding to e t Λ Ob( mod Λ )(1 t 6) is (dim K e t Λe 1, dim K e t Λe 2,...,dim K e t Λe 6 ). This coincides with dim P t. For example, we have e 3 Λ=F 3 ( 3 M 2 3 M 4 3 M 6 ) and hence e 3 Λe k =0(k =1, 5), e 3 Λe k = 3 M k (k =2, 4, 6), e 3 Λe 3 = F 3. This implies that (dim K e 3 Λe 1,...,dim K e 3 Λe 6 )=(0, 1, 1, 1, 0, 1) = α 15 = dim P 3. Hence the indecomposable representation corresponding to e t Λ(1 t 6) coincides with P t up to isomorphisms. The right Λ-module corresponding the representation Δ s P t =( j ϕ (s,t) i,w (s,t) i )(1 t 6and 0 s 5) is given by M (s,t) = i Γ W (s,t) i (see Remark 1.5). By [ASS; p. 19, Lemma 4.2], we have dim K Hom Λ (e t Λ,M (s,t) ) = dim K M (s,t) e t (1 t, t 6 and 0 s 5). Since dim K M (s,t) e t = dim K W (s,t) t = [dim Δ s P t ] t where [dim Δ s P t ] t stands for the t -th component of dim Δ s P t, we have (dim K Hom Λ (e 1 Λ,M (s,t) ),...,dim K Hom Λ (e 6 Λ,M (s,t) )) = dim Δ s P t = δ s (dim P t ). Since e t Λ (resp. M (s,t) ) corresponds to P t (resp. Δ s P t ), we obtain (dim K Hom(P 1, Δ s P t ), dim K Hom(P 2, Δ s P t ),...,dim K Hom(P 6, Δ s P t )) = δ s (dim P t ) for 1 t 6 and 0 s 5.