arXiv:0709.2444v1 [math.RT] 15 Sep 2007 Unitary and Euclidean Representations of a Quiver Vladimir V. Sergeichuk Institute of Mathematics, Tereshchenkivska 3, Kiev, Ukraine sergeich@imath.kiev.ua Abstract A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations of canonical form, and reduce the problem of classifying Euclidean representations to the problem of classifying unitary representations. We also describe the set of dimensions of all indecomposable unitary (Euclidean) representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension. 1 Introduction Many problems of linear algebra can be formulated and studied in terms of quivers and their representations, which were proposed by Gabriel [1] (see also [2]). A quiver is a directed graph. Its representation A is given by assigning to each vertex i a vector space Ai and to each arrow α : i → j a linear mapping Aα : Ai → Aj . For example, the canonical form problems ✄ q ♣ correspond to the canonical for representations of the quivers ✯ ✂ ♣ and ♣ ✶ This is the author’s version of a work that was published in Linear Algebra Appl. 278 (1998) 37–62. Partially supported by Grant No. U6E000 from the International Science Foundation. 1 form problems for linear operators (whose solution is the Jordan normal form) and for pairs of linear mappings from one space to another (the matrix pencil problem, solved by Kronecker). In this chapter we study unitary and Euclidean representations of a quiver up to isometry. A unitary (Euclidean) representation A is given by assigning to each vertex i a finite dimensional unitary (Euclidean) space Ai and to each arrow α : i → j a linear mapping Aα : Ai → Aj . We say that two unitary (Euclidean) representations A and B are isometric and write A ≃ B if there exists a system of isometries Φi : Ai → Bi such that Φj Aα = Bα Φi for each α : i → j. Our main tool is Littlewood’s algorithm [3] for reducing matrices to triangular canonical form via unitary similarity. In [4] I rediscovered Littlewood’s algorithm and applied it to the canonical form problem for unitary representations of a quiver. Various algorithms for reducing matrices to different canonical forms under unitary similarity were also proposed by Brenner, Mitchell, McRae, Radjavi, Benedetti and Gragnolini, and others; see Shapiro’s survey [5]. In Section 2 we recall briefly Littlewood’s algorithm and study the structure of canonical matrices much as it was made in [4] for the matrices of linear operators in a unitary space. We say that a matrix problem is unitarily wild if it contains the problem of classifying linear operators in a unitary space. In Section 2.3 we show that the last problem contains the problem of classifying unitary representations of an arbitrary quiver (i.e., it is hopeless in a certain sense) and give examples of unitarily wild matrix problems. The vector dim A = (dimA1 , dimA2 , . . . , dimAp ) ∈ Np0 is called the dimension of a representation A of a quiver Q with vertices 1, 2, . . . , p (we denote N = {1, 2, . . . }, N0 = {0, 1, 2, . . . }). In Section 3 we describe the set of dimensions of direct-sum-indecomposable unitary representations of a quiver, and establish the number of parameters in an indecomposable unitary representation of a given dimension. Analogous, but much more fundamental and complicated, results for non-unitary 2 representations of a quiver were obtained by V. G. Kac [6, 7, 8] (see also [2, Sect 7.4]). In particular, if z ∈ Np and Q is a connected quiver other than • and • → •, then there exists an indecomposable unitary representation of dimension z if and only if zMQ ≥ z, where MQ = [mij ] is the p × p matrix whose entry mij is the number of arrows of the form i → j and i ← j, where (t1 , . . . , tp ) ≥ (z1 , . . . , zp ) means t1 ≥ z1 , . . . , tp ≥ zp . In Section 4 we study Euclidean representations of a quiver. Let AC denote the unitary representation obtained from a Euclidean representation A by complexification (A and AC are given by the same set of real matrices). In Section 4.1 we prove intuitively obvious facts that (i) AC ≃ BC implies A ≃ B, and (ii) if A is indecomposable and AC is decomposable, then AC ≃ U ⊕ Ū , where U is an indecomposable unitary representation. This will imply that unitary and Euclidean representations have the same sets of dimensions of indecomposable representations. In Section 4.2 we study, when a given unitary representation of a quiver can be obtained by complexification. In particular, let A be a complex matrix that is not unitarily similar to a direct sum of matrices, and let S −1 AS = Ā for a unitary matrix S (such S exists if A is unitarily similar to a real matrix). Then A is unitarily similar to a real matrix if and only if S is symmetric. 2 Unitary matrix problems We suppose that the complex numbers are lexicographically ordered: a + bi  a′ + b′ i if either a = a′ and b ≤ b′ , or a < a′ ; (1) and that the set of blocks of a block matrix A = [Aij ] are linearly ordered: Aij ≤ Ai′ j ′ if either i = i′ and j ≤ j ′ , or i > i′ . 3 (2) A block complex matrix with a given (perhaps empty) set of marked square blocks will be called a marked block matrix ; a square block is marked by a line along its principal diagonal. By a unitary matrix problem we mean the classification problem for marked block matrices A = [Aij ], 1 ≤ i ≤ l, 1 ≤ j ≤ r, up to transformations A 7→ B := R−1 AS = [Ri−1 Aij Sj ], (3) where R = R1 ⊕ · · · ⊕ Rl , S = S1 ⊕ · · · ⊕ Sr are unitary matrices, and Ri = Sj whenever the block Aij is marked. The transformation (3) is called an admissible transformation; we say that these marked block matrices A and B (with the same disposition of marked blocks) are equivalent and write A ∼ B or (R, S) : A ❀ B. (4) Notice that a matrix consisting of a single block is reduced by transformations of unitary similarity if the block is marked, and by transformations of unitary equivalence otherwise. Moreover, the matrices of every unitary representation A of a quiver can be placed into a marked block matrix A such that the admissible transformations with A correspond to reselections of the orthogonal bases in the spaces of A, for example, 2 µ ✞ ✠ λ ✝ ✿1 ✛ 2.1 ξ ν ❘ 3 S1 S2 S3 S1−1 Aλ Aµ Aν S3−1 0 (5) Aξ An algorithm The algorithm is based on the following two lemmas: 4 0 Lemma 2.1. (a) Each complex matrix A is unitarily equivalent to the matrix D = a1 I ⊕ · · · ⊕ ak−1 I ⊕ 0, ai ∈ R, a1 > · · · > ak−1 > 0. (6) (b) If R−1 DS = D ′ , where R and S are unitary matrices and D, D ′ are of the form (6), then D = D ′ , S = S1 ⊕ · · · ⊕ Sk−1 ⊕ S ′ , and R = S1 ⊕ · · · ⊕ Sk−1 ⊕ R′ , where each Si has the same size as ai I. Lemma 2.2. (a) Each square complex matrix A is unitarily similar to the block-triangular matrix   λ1  · · ·  λk λ1 I F12 · · · F1k (see (1)), the columns  λ2 I · · · F2k    F = (7) ..  , of Fi,i+1 are linearly ..  . .  independent if λi = 0 λk I λ . i+1 (b) If S −1 F S = F ′ , where S is a unitary matrix and F and F ′ have the form (7), then λi I = λ′i I and S = S1 ⊕ · · · ⊕ Sk , where each Si has the same size as λi I. Proof. These lemmas were proved in many articles, see, for example, [3, 4, 5], so we give only an outline of their proofs. Part (a) of Lemma 2.1 is the singular value decomposition; part (b) follows from D = D′, S ∗ D ∗ R∗−1 = D ∗ , D 2 R = RD 2 , S −1 DR = D, D 2 S = SD 2 . The matrix (7) is the matrix of an arbitrary linear operator A : Cn → Cn in an orthogonal basis f1 , . . . , fn such that f1 , . . . , ftr is a basis of Ker(A − λ1 I) · · · (A − λr I), 1 ≤ r ≤ k, where (x − λ1 ) · · · (x − λk ), λ1  · · ·  λk , is the minimal polynomial of A; it proves part (a) of Lemma 2.2. Successively equating the blocks of F S = SF ′ ordered with (2), we prove part (b). 5 By the canonical part of the matrix (6) or (7), we mean the matrix (6) or, respectively, the collection of blocks Fij , i ≥ j. According to Lemmas 2.1 and 2.2, the canonical part is uniquely determined by the initial matrix A and does not change if A is replaced by a unitarily equivalent or, respectively, similar matrix. The algorithm for reducing a marked block matrix A = [Aij ] to canonical form: Let Apq be the first (in the ordering (2)) block of A that changes under admissible transformations (3). Depending on the arrangement of the marked blocks, it is reduced by the transformations of unitary equivalence or similarity. Respectively, we reduce A = [Aij ] to the matrix à = [Ãij ] with Ãpq of the form (6) or (7), and then restrict ourselves to those admissible transformations with à that preserve the canonical part of Ãpq . As follows from Lemmas 2.1(b) and 2.2(b), they are exactly the admissible transformations with the marked block matrix A′ that is obtained in the following way: The block Ãpq of the form (6) or (7) consists of k horizontal and k vertical strips; we extend this partition to the whole p-th horizontal and the whole q-th vertical strips of Ã. If new k divisions pass through the marked block Ãij , we carry out k perpendicular divisions such that Ãij is partitioned into k × k subblocks with square diagonal blocks (they are crossed by the marking line) and repeat this for all new divisions. We additionally mark the subblocks a1 I, . . . , ak−1 I of Ãpq if it has the form (6). The obtained marked block matrix A′ will be called the derived matrix of A. Clearly, A ∼ B implies A′ ∼ B ′ . Let us consider the sequence of derived matrices A(0) := A, A′ , A′′ , . . . , A(s) . (8) This sequence ends with a certain matrix A(s) , s ≥ 0, for which the admissible transformations do not change any of its blocks, i.e, A(s) is equivalent only to itself. Then A ∼ B implies A(s) ∼ B (s) , i.e., A(s) = B (s) . Remove from A(s) all additional divisions into subblocks and additional marking lines that have appeared during the reduction of A to A(s) . The obtained marked block matrix will be called a canonical matrix or the canonical form of A and will be denoted by A∞ . We have the following Theorem 2.1. Each marked block matrix A is equivalent to the uniquely determined canonical matrix A∞ ; moreover, A ∼ B if and only if A∞ = B ∞ . ✷ 6 We will take under consideration the null matrices 00n , 0m0 , and 000 of size 0 × n, m × 0, and 0 × 0, putting for a p × q matrix M     M , M ⊕ 00n = M 0pn , M ⊕ 0m0 = 0mq 0m0 ⊕ 00n = 0mn . Respectively, we will consider block matrices with “empty” horizontal and/or vertical strips. Let A = [Aij ] and B = [Bij ] (1 ≤ i ≤ l, 1 ≤ j ≤ r) be marked block matrices with the same set of indices (i, j) of the marked blocks. By the block direct sum of A and B we mean the marked block matrix A ⊎ B := [Aij ⊕ Bij ] with the same disposition of marked blocks. If T1 = (R1 , S1 ) : A ❀ C and T2 = (R2 , S2 ) : B ❀ D (see (4)), then R1 , R2 and, respectively, S1 , S2 are block diagonal matrices with l and, respectively, r diagonal square blocks, and T1 ⊎ T2 := (R1 ⊎ R2 , S1 ⊎ S2 ) : A ⊎ B ❀ C ⊎ D. A marked block matrix A is said to be indecomposable if (i) its size other than 0 × 0, and (ii) A ∼ B ⊎ C implies that B or C has size 0 × 0. For every matrices M1 , . . . , Mn , N, we define (M1 , . . . , Mn ) ⊗ N := (M1 ⊗ N, . . . , Mn ⊗ N), (9) where Mi ⊗ N is obtained from Mi by replacing its entries a with aN. Theorem 2.2. (a) Each marked block matrix A is equivalent to a matrix of the form B = (P1 ⊗ Im1 ) ⊎ · · · ⊎ (Pt ⊗ Imt ) ∼ P1 ⊎ · · · ⊎ P1 ⊎ · · · ⊎ Pt ⊎ · · · ⊎ Pt , | | {z } {z } m1 copies mt copies 7 where P1 , . . . , Pt are nonequivalent indecomposable marked block matrices, uniquely determined up to equivalence (we may take P1 = P1∞ , . . . , Pt = Pt∞ ), and m1 , . . . , mt are uniquely determined natural numbers. Every admissible transformation T : B ❀ B that preserves B has the form T = (1P1 ⊗ U1 ) ⊎ · · · ⊎ (1Pt ⊗ Ut ), where 1Pi = (I, I) : Pi ❀ Pi is the identity transformation of Pi , and Ui is a unitary mi × mi matrix (1 ≤ i ≤ t). (b) A marked block matrix A of size 6= 0 × 0 is indecomposable if and only if every preserving it admissible transformation T : A ❀ A has the form T = a1A , a ∈ C, |a| = 1. (c) A canonical matrix can be reduced to an equivalent block direct sum of indecomposable canonical matrices using only admissible permutations of rows and columns. Proof. (a) We may take A = A∞ . Since admissible transformations with A(i) , 1 ≤ i ≤ s, (see (8)) are exactly the admissible transformations with A that preserve the already reduced part of A(i) (preserve A(s) if i = s), the set of admissible transformations with A(s) consists of all (R, S) : A ❀ A. By (3), R = R1 ⊕ · · · ⊕ Rl , S = S1 ⊕ · · · ⊕ Sr , where l × r is the number of blocks of A. Since (R, S) : A(s) ❀ A(s) , we have Ri = Uf (i,1) ⊕ · · · ⊕ Uf (i,li ) , Sj = Ug(j,1) ⊕ · · · ⊕ Ug(j,rj ) , (10) where f (i, α), g(j, β) ∈ {1, . . . , t} and U1 , . . . , Ut are arbitrary unitary matrices of fixed sizes. A(s) differs from A only by additional divisions of its strips into substrips (and by additional marking lines). We transpose substrips within each strip of A to obtain a matrix B ∼ A such that, for all (R, S) : B ❀ B, we have (10) with f (i, 1) ≤ · · · ≤ f (i, li), g(j, 1) ≤ · · · ≤ g(j, rj ). Clearly, B satisfies (a). (b)&(c) These statemants are obvious. 8 2.2 The structure of canonical matrices In this section we divide the set of canonical m × n matrices into disjoint subsets of canonical matrices with the same “scheme” (the number of such schemes is finite for each size m × n), and show how to construct all the canonical matrices with a given scheme (for matrices under unitary similarity this was made briefly in [4]). We partition a canonical matrix into zones, which illustrate the reduction process. Let A = A∞ be a canonical matrix. Then all its derived matrices (8) differ from A only by additional divisions and marking lines. Denote by Pl (0 ≤ l < s) the first block of A(l) that changes under admissible transformations (it is reduced when we construct A(l+1) ). (l) (l) Let Aij be a block of A(l) such that either Aij ≤ Pl or l = s. The admissible transformations with A(l) induce the unitary equivalence or similarity (l) (l) transformations with Aij . Respectively, Aij has the form (6) or (7); we de(l) note by Z(Aij ) its canonical part (see page 6). Defining by induction in l, (l) (l) we call Z(Aij ) by a zone and l by its depth if either l = 0 or Z(Aij ) is not contained in a zone of depth < l. (l) (l) For each zone Z = Z(Aij ), we put Bl(Z) := Aij and call Z by an equivalence (similarity) zone if Bl(Z) is transformed by unitary equivalence (similarity) transformations. Clearly, every canonical matrix A is partitioned into equivalence and similarity zones; for example (for a marked block matrix of the form ❅❅ ), 9 A= i 0 0 0 2 0 0 0 3 ❅ 0 ❅i 0 0 0 2 0 0 0 0 0❅i 0 0 0 2 0 0 0 0 0❅i❅0 0 0 0 0 0 0 0 0 0❅ 0 0 0 5 0 0 0 0 0❅ 0 0 0 0 0 0 0 0 0 0❅ 0 0 0 0 0 0 0 0 0 0❅ 0 0 ❅ 0 3 0 0 0 5 0 0 2 0 i 0 0 0 5 0 i 0 4 4 0 0 0 3 7 8 8 5 6 4 4 3 = 1 (11) 2 is partitioned into 10 zones, their depths are indicated on the right of (11). Let A be a canonical matrix partitioned into zones. For each similarity zone, we replace all its diagonal elements by stars. For each equivalence zone, we replace all its nonzero elements by circles, and join with a line its circles corresponding to equal elements (this line does not coincide with a marking line because the marking lines connect stars). The other elements of A are zeros, we replace theirs by points. The obtained picture will be called the scheme S(A) of A. For example, the canonical matrix (11) has the scheme S(A) = ⋆ · · · · · · · · ⋆ · · · · · · · · ⋆ · · · · · · · · ⋆ · · · · • · · · ⋆ · · · · • · · · ⋆ · · · · • · · · ⋆ · · · · · · · · ⋆ ⋆ · · · • · · · · ⋆ · · · • · · • · ⋆ · · · • · ⋆ · • • · · · • Theorem 2.3. Each canonical matrix A = [aij ] with a given scheme S = [sij ] can be constructed by successive filling of its zones by numbers starting with 10 the zones of greatest depth as follows: Let Z be a zone of depth d(Z) and let all entries in zones of depth > d(Z) be replaced by numbers. Then we replace all points, circles, and stars of Z, respectively, by zeros, positive real numbers, and complex numbers such that the following conditions hold: 1) Let sij and si+1,j+1 be circles in Z. Then aij = ai+1,j+1 if sij and si+1,j+1 are linked by a line, and aij > ai+1,j+1 otherwise. 2) Let sα,β , . . . , sα+k,β+k be all stars of Z that lie under a certain stair of Z. Then aα,β = · · · = aα+k,β+k . If sα+k+1,β+k+1, . . . , sα+t,β+t are all stars of Z that lie under the next stair of Z, then aα,β  aα+t,β+t ; moreover, aα,β ≻ aα+t,β+t whenever the columns of the block [ aij | α ≤ i ≤ α + k, β + k + 1 ≤ j ≤ β + t] are linearly dependent (this block has been filled by numbers because all its entries are located in zones of depth > d(Z)). ✷ This theorem gives a convenient way to present solutions of unitary matrix problems in small sizes by their sets of schemes. Thus, the list of schemes of canonical 5 × 5 matrices under unitary similarity was obtained by Klimenko [11]. 2.3 Unitarily wild matrix problems The canonical form problem for pairs of n × n matrices under simultaneous ✄ similarity (i.e., for representations of the quiver ✯ ✂ ♣ ❨✁ ) plays a special role in the theory of (non-unitary) matrix problems. It may be proved that its solution implies the clasification of representations of every quiver (and even representations of every finite dimensional algebra). For this reason, the classification problem for pairs of matrices under simultaneous similarity is used as a yardstick of the complexity; Donovan and Freislich [12] (see also [2]) suggested to name a classification problem wild if it contains the problem of simultaneous similarity, and otherwise to name it tame (in accordance with the partition of animals into wild and tame ones). The canonical form problem for an n × n matrix under unitary similarity ✄ ♣ ) plays the same role in the (i.e., for unitary representations of the quiver ✯ ✂ 11 theory of unitary matrix problems: it contains the problem of classifying unitary representations of every quiver. For example, the problem of classifying unitary representations of the quiver (5) can be regarded (by Lemma 2.2) as the problem of classifying, up to unitary similarity, matrices of the form:   5I I Aλ Aν Aµ  0 4I I 0 0     0 . 0 3I 0 0    0 0 0 2I Aξ  0 0 0 0 I A matrix problem is called unitarily wild (or *-wild, see [13]) if it contains the problem of classifying matrices via unitary similarity, and unitarily tame otherwise. For each unitary problem, one has an alternative: to solve it or to prove that it is unitarily wild (and hence is hopeless in a certain sense). In this section we give some examples of such alternatives. (i) Let us consider the problems of classifying nilpotent linear operators ϕ, ϕn = 0, in a unitary space. For n = 2 this problem is unitarily tame; the canonical matrix of ϕ (see page 6) is   0 D , 0 0 where D is of the form (6) without zero columns. Indeed, a matrix F = [Fij ] of the form (7) satisfies F 2 = 0 only if k = 2, F11 = 0, and F22 = 0; we can reduce F12 to the form (6). For n > 2 this problem is unitarily wild since the matrices     0 I X 0 I Y  0 0 I  and  0 0 I  0 0 0 0 0 0 are unitarily similar if and only if X and Y are unitarily similar (see also [14]). (ii) Let us consider the problem of classifying m-tuples (p1 , . . . , pm ) of projectors p2i = pi in a unitary space. 12 For m = 1 this problem is unitarily time; the canonical matrix of a projector p = p2 was obtained in [15] and [16]. Of course, it is   I D , 0 0 where D is of the form (6), since a matrix F = [Fij ] of the form (7) satisfies F 2 = F only if k = 2, F11 = I, and F22 = 0. As was proved in [16], for m ≥ 2 this problem is unitarily wild even if p1 is an orthoprojector, i.e. p1 = p21 = p∗1 , (since the pairs of idempotent matrices     I 0 X I −X , 0 0 X I −X and  I 0 0 0   Y , Y I −Y I −Y  are unitarily similar if and only if X and Y are unitarily similar), or if p1 p2 = p2 p1 = 0. (iii) The problems of classifying the following operators and systems of operators in unitary spaces are unitarily wild: • Pairs of linear operators (ϕ, ψ) such that ϕ2 = ψ 2 = ϕψ = ψϕ = 0 since  0 I 0 0    0 X , 0 0 and  0 I 0 0    0 Y , 0 0 are unitarily similar if and only if X and Y are unitarily similar. • Pairs of selfadjoint operators (ϕ, ψ) because ϕ + iψ is an arbitrary operator. The tame-wild dichotomy for satisfying quadratic relation pairs of selfadjoint operators in a Hilbert space was studied in [17]. • Pairs of unitary operators (ϕ, ψ) since (i(ϕ + 1)(ϕ − 1)−1 , i(ψ + 1)(ψ − 1)−1 ) is a pair of selfadjoint operators (the Cayley transformation). 13 • Partial isometries (i.e., linear operators ϕ such that (ϕ∗ ϕ)2 = ϕ∗ ϕ), it was proved in [18]. (iv) The problem of classifying unitary representations of a connected quiver Q is unitarily tame if Q ∈ {•, • → •} and unitarily wild otherwise. Indeed, the classification of unitary representations of the quiver • → • is given by the singular value decomposition (Lemma 2.1). The problem of classifying unitary representations of the quiver • → • ← • is unitarily wild because it reduces to the unitary matrix problem for marked block matrices of the form , and two block matrices ♣ 2I 0 0 ♣I♣ ♣ ♣ ♣♣♣♣♣ X ♣♣♣♣ 0 I 0 ♣I♣ ♣ ♣ ♣♣♣♣ ♣I♣ ♣ ♣ 0 0 0 I 0 ♣ and 2I 0 0 ♣I♣ ♣ ♣ ♣♣♣♣♣ Y ♣♣♣♣ 0 I 0 ♣I♣ ♣ ♣ ♣♣♣♣ ♣I♣ ♣ ♣ 0 0 0 I 0 are equivalent if and only if X and Y are unitarily similar. We can change the direction of an arrow in a quiver by replacing in each representation the corresponding linear mapping by the adjoint one. (v) Let us consider the problem of classifying n-tuples (V1 , . . . , Vn ) of subspaces of a unitary space U up to the following equivalence: (V1 , . . . , Vn ) ∼ (V1′ , . . . , Vn′ ) if there exists an isometry ϕ : U → U such that ϕV1 = V1′ , . . . , ϕVn = Vn′ . Fixing an orthogonal basis in U and (non-orthogonal) bases in V1 , . . . , Vn , we reduce it to the canonical form problem for block matrices A = [A1 | . . . |An ] (the columns of Ai are the basis vectors of Vi and hence are linearly independent) up to unitary transformations of rows of A and elementary (nonunitary) transformations of columns of Ai (i = 1, . . . , n). If n = 1, then A = [A1 ] reduces to I ⊕ 0p0; this follows from Lemma 2.1. If n = 2, then A = [A1 |A2 ] reduces to ♣ I 0 I ♣♣♣♣♣ ♣0♣ ♣ ♣ ♣♣ ♣ 0♣ ♣ ♣ ♣♣ D ♣♣♣♣ , 0 I 0 0 14 where D is of the form (6). This block matrix reduces to a block direct sum of matrices  1 αi (α > 0), [1|1], [1|010 ], [010 |1], [010 |010 ]. 0 1 (The problem of classifying pairs of subspaces in a complex or real vector space with scalar product given by a symmetric, or skew-symmetric, or Hermitian form was solved in [19].) For n = 3 this problem is unitarily wild even if we restrict our consideration to the triples (V1 , V2 , V3 ) with V1 ⊥ V2 since     I 0 X I 0 X′  0 I Y  reduces to  0 I Y ′  0 0 I 0 0 I if and only if (X, Y ) and (X ′ , Y ′ ) determine isometric unitary representations of the quiver • ← • → • (see page 14). An analogous statement was proved in [20] and [13]: the problem of classifying triples (p1 , p2 , p3 ) of orthoprojectors pi = p2i = p∗i in a unitary space is unitarily wild even if p1 p2 = p2 p1 = 0; such a triple determines, in one-to-one manner, a triple (V1 , V2 , V3 ) with V1 ⊥ V2 by means of Vi = Im pi . 3 Unitary representations of a quiver From now on, Q denotes a quiver with vertices 1, . . . , p and arrows α1 , . . . , αq . A unitary representation of dimension d = (d1 , . . . , dp ) ∈ Np0 (in short, a unitary d representation) will be given by assigning a matrix Aα ∈ Cdj ×di to each arrow α : i → j, i.e., by the sequence A = (Aα1 , . . . , Aαq ) (assigning to each vertex i the unitary vector space Cdi with scalar product (x, y) = x̄1 y1 + · · · + x̄di ydi , 15 we obtain a unitary representation; see page 2). ∼ An isometry A → B of d representations A and B (an autometry if A = B) is given by a sequence S = (S1 , . . . , Sp ) of unitary di × di matrices Si such that Sj Aα = Bα Si for each arrow α : i → j; we say also that B is obtained from A by admissible transformations and write A ≃ B. An autometry S : A→A ˜ is scalar if S = a1A , where a ∈ C, 1A = (Id1 , . . . , Idp ). For two sequences of matrices M = (M1 , . . . , Mt ), N = (N1 , . . . , Nt ), we denote M ⊕ N = (M1 ⊕ N1 , . . . , Mt ⊕ Nt ). A unitary d representation A of Q is indecomposable if (i) d 6= (0, . . . , 0) and (ii) A ≃ B ⊕ C implies B or C has dimension (0, . . . , 0). 3.1 Canonical representations Let A be a unitary representation of Q. Using the algorithm from page 6, we reduce Aα1 to its canonical form A∞ α1 , then restrict the set of admissible transformations with A to those that preserve A∞ α1 (it gives certain unitary matrix problems for Aα2 , . . . , Aαq with partitions them into blocks) and reduce Aα2 to its canonical form A∞ α2 , and so on. The obtained representation ∞ A∞ = (A∞ α1 , . . . , Aαq ) (we omit the marking lines) will be called a canonical representation of the quiver Q; the sequence of the schemes ∞ S(A∞ ) = (S(A∞ α1 ), . . . , S(Aαq )) will be called the scheme of A∞ . Clearly, A ≃ A∞ and A ≃ B if and only if A∞ = B ∞ . 16 Theorem 3.1. (a) Every unitary representation is isometric to a representation of the form B = (P1 ⊗ Im1 ) ⊕ · · · ⊕ (Pt ⊗ Imt ) ≃ P1 ⊕ · · · ⊕ P1 ⊕ · · · ⊕ Pt ⊕ · · · ⊕ Pt | | {z } {z } m1 copies mt copies (see (9)), where P1 , . . . , Pt are nonisometric indecomposable representations, uniquely determined up to isometry, and m1 , . . . , mt are uniquely determined ∼ natural numbers. Every autometry S : B → B has the form S = (1P1 ⊗ U1 ) ⊕ · · · ⊕ (1Pt ⊗ Ut ), where Ui is a unitary mi × mi matrix (1 ≤ i ≤ t). (b) A unitary representation of dimension 6= (0, . . . , 0) is indecomposable if and only if all its autometries are scalar. Proof. Analogously (5), the matrices of every unitary representation A of Q can be accommodated in a block diagonal matrix A = diag(Aαq , Aαq−1 , . . . , Aα1 , 0, . . . , 0) with a certain set of marked blocks such that the admissible transformations with A correspond to the admissible transformations with M(A). Then M(A∞ ) = M(A)∞ and we can apply Theorem 2.2. 3.2 The set of dimensions of indecomposable unitary representations We will use the following notation: • MQ = [mij ] is the p × p matrix, in which mij is the number of arrows i → j and i ← j of the quiver Q; • supp(z) is the full subquiver of Q with the vertex set {i | zi 6= 0} for each z ∈ Np0 ; • ei = (0, . . . , 1, . . . , 0) ∈ Np0 with 1 in the ith position. 17 Denote by D(Q) the subset of Np0 consists of e1 , . . . , ep , all ei + ej with mij = 1, and all nonzero z with connected supp(z) ∈ / {•, • → •} such that zMQ ≥ z. In this section we prove: Theorem 3.2. D(Q) is the set of dimensions of indecomposable unitary representations of a quiver Q. Put ∆i (z) := X z ∈ Np0 , mij zj , j 1 ≤ i ≤ p. Then zMQ = (∆1 (z), . . . , ∆p (z)). Lemma 3.1. D(Q) satisfies the following conditions: (i) If z ∈ D(Q) and supp(z) ∈ / {•, • → •, • ❨✁}, then zMQ > z. (ii) If z, u ∈ D(Q) and z < u, then there exists i such that z + ei ≤ u and z + ei ∈ D(Q). Proof. (i) Let z ∈ D(Q), supp(z) ∈ / {•, • → •, • ❨✁}, zMQ = z. Fix i such that zi = max{z1 , . . . , zp }. Then mij 6= 0 for a certain j 6= i. Since zj = ∆j (z) ≥ mij zi ≥ zi , we have zi = zj , mij = 1, mkj zk = 0 for all k 6= i. Taking zj and zi instead of zi and zj , we obtain mki zk = 0 for all k 6= j. Hence supp(z) = • → •, a contradiction. (ii) Let z, u ∈ D(Q) and z < u. If supp(z) 6= supp(u), then there exists a nonzero mij with i ∈ supp(u) \ supp(z), j ∈ supp(u) ∩ supp(z). The z + ei satisfies the requirements. 18 We may assume that supp(z) = supp(u) = Q. Then Q 6∈ {•, • → •}. Fix a vertex l such that zl < ul . We will suppose that ∆l (z) = zl and mll = 0 (otherwise z + el satisfies the requirements). Assume first that zl ≤ zj for some mlj 6= 0. The condition ∆l (z) = zl implies zl = zj , mlj = 1, and mlk = 0 for all k 6= j. Hence zj = zl < ul ≤ ∆l (u) = uj . Since Q 6= • → •, mjk 6= 0 for some k 6= l, and we can take z + ej . Next, let zl > zj (and hence z + ej ∈ D(Q)) for all nonzero mlj . If zj = uj for all mlj 6= 0, then ul ≤ ∆l (u) = ∆l (z) = zl , a contradiction. Hence zj < uj for a certain mlj 6= 0, and we can take z + ej . Lemma 3.2. If A is a unitary d representation of Q and d ∈ / D(Q), then A is decomposable. Proof. Assume to the contrary, that A is indecomposable. Then supp(d) is connected; Lemma 2.1 and d ∈ / D(Q) imply supp (d) ∈ / {•, • → •}, dMQ  d, that is, there exists l such that ∆l (d) < dl . Then mll = 0 and we can assume that there are no arrows starting from l (otherwise we replace each arrow α : l → i by α∗ : i → l, simultaneously replacing Aα by the adjoint matrix). Let α, β, . . . , γ be all arrows stopping at l; combine the corresponding them matrices of A into a single dl × ∆l (d) matrix [Aα |Aβ | · · · |Aγ ]. The number of its rows is greater than the number of its columns; making a zero row by unitary transformations of rows, we obtain A ≃ B ⊕ P , where P is the zero representation of dimension el , a contradiction. 19 Lemma 3.3. If there exists an indecomposable unitary z representation and zi < ∆i (z) for a certain vertex i, then there exists an indecomposable unitary z + ei representation. Proof. Let A be an indecomposable unitary z representation and z1 < ∆1 (z). We can assume that each starting from the vertex 1 arrow is a loop (replacing each λ : 1 → j, j 6= 1, by λ∗ : j → 1 and, respectively, Aλ by A∗λ∗ ). 1) Assume first that there is a loop α : 1 → 1 and define a unitary z + e1 representation H in which • Hα is the nilpotent Jordan block of size (z1 + 1) × (z1 + 1), • Hβ := Aβ ⊕ 011 for each β : 1 → 1, β 6= α; • Hγ := Aγ ⊕ 010 for each γ : j → 1, j 6= 1; and • Hδ := Aδ for each δ : j → k, k 6= 1. The representation H is indecomposable. Indeed, let A− and H − denote the restrictions of A and H on the subquiver Q− := Q \ α. By Theorem 3.1(a), we may assume that A− = (P1 ⊗ Im1 ) ⊕ · · · ⊕ (Pt ⊗ Imt ), where P1 , . . . , Pt are nonisometric indecomposable representations of Q− , P1 is the zero representation of dimension e1 , and m1 ≥ 0, m2 > 0, . . . , mt > 0. Clearly, H − = (P1 ⊗ Im1 +1 ) ⊕ (P2 ⊗ Im2 ) ⊕ · · · ⊕ (Pt ⊗ Imt ). Let ∼ ∼ S = (S1 , S2 , . . . ) : H → H. Since S : H − → H − , by Theorem 3.1(a) S = (1P1 ⊗ U1 ) ⊕ · · · ⊕ (1Pt ⊗ Ut ), (d11 ) S1 = U1 (d1t ) ⊕ · · · ⊕ Ut 20 , (12) where (d1j , . . . , dpj ) = dim(Pj ) and Uj is a unitary matrix (1 ≤ j ≤ t). Since S1 H α = H α S1 , Hα is a Jordan block and S1 is a unitary matrix, we have S1 = aI, a ∈ C. The representation A is indecomposable, so that d1j 6= 0 and by (12) Uj = aI for all 1 ≤ j ≤ t. Hence S = a1H and H is indecomposable by Theorem 3.1(b). 2) There remains the case m11 = 0. Let α1 : j1 → 1, . . . , αl : jl → 1 be all the arrows stopping at 1. We denote by A− the restriction of A on the subquiver Q− := Q \ {1; α1 , . . . , αl }. By Theorem 3.1(a), we may assume that A− = (P1 ⊗ Im1 ) ⊕ · · · ⊕ (Pt ⊗ Imt ), where P1 , . . . , Pt are nonisometric indecomposable unitary representations of Q− . ∼ Let (S2 , . . . , Sp ) : A− → A− . By Theorem 3.1(a), Si = (Idi1 ⊗ U1 ) ⊕ · · · ⊕ (Idit ⊗ Ut ), where (d2j , . . . , dpj ) = dimPj . For an arbitrary unitary z1 × z1 matrix S1 , we define à by means of ∼ S = (S1 , S2 , . . . , Sp ) : A → à (then Ã− = A− ). Taking into account that Ãατ = S1−1 Aατ Sjτ and partitioning the sets of columns of every Aατ and Ãατ in the same manner as Sjτ , we obtain B := [Aα1 | · · · |Aαl ] = [B1 | · · · |Bm ] 21 and B̃ := [Ãα1 | · · · |Ãαl ] = [B̃1 | · · · |B̃m ], where m= l X τ =1 (djτ 1 + · · · + djτ t ) and B̃i = S1−1 Bi Uf (i) for a certain f (i) ∈ {1, . . . , t}. Let z1 × ui be the size of Bi and put ri = rank[B1 | · · · |Bi−1 |Bi+1 | · · · |Bm ]. B is a z1 × ∆1 (z) matrix and z1 < ∆1 (z), so z1 − ri < ui for a certain i. Since S1 and Uf (i) are arbitrary unitary matrices, by Lemma 2.1(a) there exists S such that   C1 · · · Ci−1 Ci Ci+1 · · · Cm B̃ = , 0 ··· 0 D 0 ··· 0 where the rows of [C1 | · · · |Ci−1 |Ci+1 | · · · |Cm ] are linearly independent and D is a (z1 − ri ) × ui matrix of the form diag(a1 , . . . , an ) ⊕ 0kh with real a1 ≥ · · · ≥ an > 0. Since à is indecomposable and z1 − ri < ui , we have k = 0 and h > 0. Let an+1 be a real number such that an > an+1 > 0. The replacement D by D ′ = diag(a1 , . . . , an , an+1 ) ⊕ 00,h−1 changes B̃ to a new matrix B̃ ′ and à to a new representation H of dimension z + e1 . ∼ Let R : H → H. Since H and A coincide on Q− and ∼ (R2 , . . . , Rp ) : A− → A− , by Theorem 3.1(a) the matrices R2 , . . . , Rp have the form Rj = (Idj1 ⊗ V1 ) ⊕ · · · ⊕ (Idjτ ⊗ Vτ ) 22 with unitary V1 , . . . , Vτ . By R1−1 B̃ ′ (Rj1 ⊕ · · · ⊕ Rjl ) = B̃ ′ , R1 has the form R11 ⊕ R12 , where −1 ′ R12 D Vf (i) = D ′ . Lemma 2.1 implies R12 = R13 ⊕ [c]. Putting R̃1 = R11 ⊕ R13 , ∼ R̃j = Rj (j > 1), we have R̃ : à → Ã. By Theorem 3.1(b), for some a ∈ C, so R̃j = aI, 1 ≤ j ≤ p, Vj = aI, 1 ≤ j ≤ τ. In particular, Vf (i) = aI and, since −1 ′ R12 D Vf (i) = D ′ , c = a and R1 = aI. Therefore R = a1H and H is indecomposable by Theorem 3.1(b). Proof of Theorem 3.2. Let U(Q) denote the set of dimensions of indecomposable unitary representations of Q. Lemma 3.2 implies U(Q) ⊂ D(Q). Let u ∈ D(Q). Then ui 6= 0 for a certain i. Using Lemma 3.1(ii), we select a sequence u1 := ei , u2 , . . . , ut := u in D(Q) such that u2 − u1 , . . . , ut − ut−1 ∈ {e1 , . . . , ep }. By Lemma 3.3, {u1, . . . , ut } ⊂ U(Q), 23 D(Q) ⊂ U(Q). 3.3 The number of parameters in an indecomposable unitary representation By the number of real (complex) parameters of a unitary representation A we mean the number of circles (stars) in the scheme S(A∞ ). Recall that to circles correspond positive real numbers in A∞ , and to stars correspond complex numbers; the other entries in A∞ are zeros. Kac [7, Theorem C] proved that the maximal number of paremeters in an indecomposable (non-unitary) representation of dimension d over an algebraically closed field is 1 − ϕQ (d), where ϕQ (x) = x21 + · · · + x2p − p X mij xi xj i,j=1 is a Z-bilinear form called the Tits form of the quiver Q, and mij is the number of arrows i → j and i ← j. We say that a zone (see page 9) is in general position if all its diagonal entries are distinct and, if it is an equivalence zone, nonzero. A unitary representation A is said to be in general position if all zones in A∞ are in general position. Theorem 3.3. (a) For every d ∈ D(Q) (see page 18) there exists an indecomposable canonical unitary d representation of general position, its scheme is uniquely determined by d. P (b) An indecomposable unitary d representation A has di − 1 real parameters and at most 1X 1 − ϕQ (d) + di (di − 1) 2 complex parameters; this number is reached if and only if A is in general position. Proof. We consider the set of zones of a canonical unitary representation ∞ A∞ = (A∞ α1 , . . . , Aαq ) as linearly ordered: Z1 < Z2 if i1 < i2 ; or i1 = i2 and l1 < l2 ; or i1 = i2 , l1 = l2 and Bl(Z1 ) < Bl(Z2 ) (13) (see (2) and Section 2.2); where Zk (k = 1, 2) is a zone of depth lk in A∞ αi . k 24 (a) Let d ∈ D(Q). By Theorem 3.2, there exists an indecomposable unitary d representation A. Let A be not in general position, and let Z be the first (in the sense of (13)) zone of A∞ that is not in general position. Changing diagonal entries of Z, we transform it into a zone Z̃ of general position and A∞ into a new representation Ã. This exchange narrows down the set of admissible transformations that preserve all zones ≤ Z, and, by Theorem 3.1(b), A∞ has only scalar autometries (as an indecomposable representation), therefore, à has only scalar autometries and is indecomposable too. If à is not in general position, we repeat this process for it, and so on, until we obtain an indecomposable d representation B of general position. Its scheme is uniquely determined since, for each zone Z of B, the set of admissible transformations that preserve all zones ≤ Z (and hence the matrix problem for the remaining part of B) does not depend on diagonal entries of Z such that it is in general position. (b) Let A be an indecomposable canonical d representation, and Z be its zone or the symbol ∞. Denote by J(Z) the set of all isometries of the form ∼ S : A → à that preserve all zones < Z (all zones if Z = ∞). As follows from the algorithms from pages 6 and 16, J(Z) consists of all sequences of the form S = (S1 , . . . , Sp ), where Si = Uσ(i1) ⊕ Uσ(i2) ⊕ · · · ⊕ Uσ(iti ) , σ : {(ij) | 1 ≤ i ≤ p, 1 ≤ j ≤ ti } → {1, . . . , t} is a fixed surjection, and U1 , . . . , Ut are arbitrary unitary matrices of fixed sizes m1 × m1 , . . . , mt × mt (we will write S = S(U1 , . . . , Ut )). Put ∆1 (Z) = m1 + · · · + mt , ∆2 (Z) = m21 + · · · + m2t . Let Z 6= ∞ and Z ′ be the first zone after Z (Z ′ = ∞ if Z is the last zone of A). We will prove that ∆1 (Z) − ∆1 (Z ′ ) = n• (Z), ∆2 (Z) − ∆2 (Z ′ ) ≤ 2n(Z) − n• (Z) − 2n⋆ (Z), (14) (15) and that the equality in (15) holds if and only if Z is a zone of general position; where n(Z) is the number of entries in Z, and n• (Z) (resp., n⋆ (Z)) is the number of circles (resp., stars) that correspond to the diagonal entries of Z. 25 As follows from the algorithms from pages 6 and 16, the block Bl(Z) is reduced by transformations Bl(Z) 7→ Ui−1 Bl(Z)Uj , (16) where S = S(U1 , . . . , Ut ) ∈ J(Z) and i and j are determined by Z; moreover, this S is contained in J(Z ′ ) if and only if (16) preserves Z. (i) Let i 6= j, say, i = 1 and j = 2. Then, by Lemma 2.1, Z = Bl(Z) = a1 Ir1 ⊕ · · · ⊕ ak−1 Irk−1 ⊕ 0xy , rα ≥ 1, x ≥ 0, and y ≥ 0. The transformation (16) preserves Z if and only if U1 = V1 ⊕ · · · ⊕ Vk and U2 = V1 ⊕ · · · ⊕ Vk−1 ⊕ Vk+1 , (17) where V1 , . . . , Vk+1 are unitary matrices of sizes r1 × r1 , . . . , rk−1 × rk−1 , x × x, y × y. Hence, J(Z ′ ) consists of all S ∈ J(Z) with U1 and U2 of the form (17), that is, S = S(V1 , . . . Vk+1, U3 . . . Ut ). Therefore, ∆1 (Z ′ ) = r1 + · · · + rk−1 + x + y + m3 + · · · + mt , 2 ∆2 (Z ′ ) = r12 + · · · + rk−1 + x2 + y 2 + m23 + · · · + m2t . By (16), Bl(Z) has size m1 × m2 , m1 = r1 + · · · + rk−1 + x, m2 = r1 + · · · + rk−1 + y, so n(Z) = m1 m2 , n• (Z) = r1 + · · · + rk−1 , n⋆ (Z) = 0. We have ∆1 (Z) − ∆1 (Z ′ ) = r1 + · · · + rk−1 = n• (Z) 26 and ∆2 (Z) − ∆2 (Z ′ ) = (r1 + · · · + rk−1 + x)2 2 + (r1 + · · · + rk−1 + y)2 − r12 − · · · − rk−1 − x2 − y 2 = [(r1 + · · · + rk−1 + x) − (r1 + · · · + rk−1 + y)]2 + 2(r1 + · · · + rk−1 + x)(r1 + · · · + rk−1 + y) 2 − r12 − · · · − rk−1 − x2 − y 2 2 = (x − y)2 + 2n(Z) − r12 − · · · − rk−1 − x2 − y 2 2 = −2xy + 2n(Z) − r12 − · · · − rk−1 ≤ 2n(Z) − r1 − · · · − rk−1 = 2n(Z) − n• (Z). Moreover, we have the equality if and only if r1 = · · · = rk−1 = 1, xy = 0, i.e, Z is in general position. (ii) Let i = j, say, i = j = 1. Then, by Lemma 2.2, Bl(Z) = [Fαβ ], where Fαβ = 0 if α > β, and Fαα = λα Irα , rα ≥ 1, r1 + · · · + rk = m1 . The transformation (16) preserves Z = {Fαβ | α ≤ β} if and only if U1 = V1 ⊕ · · · ⊕ Vk , where V1 , . . . , Vk are unitary matrices of sizes r 1 × r 1 , . . . , rk × r k . Hence, J(Z ′ ) consists of all S ∈ J(Z) with U1 = V1 ⊕ · · · ⊕ Vk , that is, S = S(V1 , . . . , Vk , U2 , . . . , Ut ). So ∆1 (Z) − ∆1 (Z ′ ) = m1 − r1 − · · · − rk = 0 = n• (Z) 27 and ∆2 (Z) − ∆2 (Z ′ ) = (r1 + · · · + rk )2 − r12 − · · · − rk2 X rα rβ − 2(r12 + · · · + rk2 ) =2 α≤β = 2n(Z) − 2(r12 + · · · + rk2 ) ≤ 2n(Z) − 2(r1 + · · · + rk ) = 2n(Z) − 2n⋆ (Z). Moreover, we have the equality if and only if r1 = · · · = rk = 1, that is, Z is in general position. Hence, the relations (14) and (15) hold. Let Z1 < · · · < Zr be all zones of A ordered by (13), and let dim A = (d1 , . . . , dp ). Then Zi′ = Zi+1 for i < r, and Zr′ = ∞. Since J(Z1 ) consists of all sequences S = (S1 , . . . , Sp ) of unitary d1 × d1 , . . . , dp × dp matrices, ∆2 (Z1 ) = d21 + · · · + d2p . ∆1 (Z1 ) = d1 + · · · + dp , Since A is indecomposable, by Theorem 3.1(b) J(∞) consists of all sequences λ ∈ C, S = λ(Id1 , . . . , Idp ), |λ| = 1, so S = S([λ]), ∆1 (∞) = ∆2 (∞) = 1. By (14), d1 + · · · + dp − 1 = ∆1 (Z1 ) − ∆1 (∞) r r X X ′ = (∆1 (Zi ) − ∆1 (Zi )) = n• (Zi ) i=1 i=1 is the number of circles in S(A∞ ), that is, the number of real parameters in A. 28 By (15), d21 + . . . + d2p − 1 = ∆2 (Z1 ) − ∆2 (∞) r X (∆2 (Zi ) − ∆2 (Zi′ )) = i=1 r X ≤2 But Pr i=1 i=1 n(Zi ) − r X i=1 n• (Zi ) − 2 r X n⋆ (Zi ). i=1 n(Zi ) is the number of entries in Aα1 , . . . , Aαq , hence, it is equal to p X mij di dj , i,j=1 where mij is the number of arrows i → j and i ← j; ⋆ n (A) := r X n⋆ (Zi ) i=1 is the number of complex parameters in A. Therefore, X 1 X 2 1 X n⋆ (A) ≤ mij di dj − ( di − 1) − ( di − 1) 2 2 X X 1X 2 (di − di) = 1−[ d2i − mij di dj ] + 2 1X = 1 − ϕQ (d) + di (di − 1). 2 We have the equality if and only if all Zi are in general position, i.e., A is in general position. The proof implies Corollary 3.1. (a) Let d ∈ D(Q) and m = max{d1 , . . . , dp }. Then there exists an indecomposable canonical d representation of general position with entries in {0, 1, . . . , m}. P (b) A decomposable unitary d representation has less than di − 1 real parameters and less than 1X di (di − 1) 1 − ϕQ (d) + 2 complex parameters. 29 Proof. (a) This statement follows from Theorems 3.3(a) and 2.3. (b) This statement is proved as Theorem 3.3(b), but, in the last two paragraphs of its proof, we must use ∆1 (∞) > 1 and ∆2 (∞) > 1 instead of ∆1 (∞) = ∆2 (∞) = 1 since J(∞) contains a non-scalar authometry in the case of a decomposable representation A. 4 Euclidean representations of a quiver Let Q be a quiver with vertices 1, . . . , p and arrows α1 , . . . , αq . A Euclidean representation A of dimension d = (d1 , . . . , dp ) ∈ Np0 will be given by assigning a matrix Aα ∈ Rdj ×di to each arrow α : i → j; i.e., by the sequence A = (Aα1 , . . . , Aαq ). ∼ An R-isometry A →R B of Euclidean representations A and B will be given by a sequence S = (S1 , . . . , Sp ) of real orthogonal matrices such that Sj Aα = ∼ Bα Si for each arrow α : i → j (analogously, R : A →C B denotes an isometry in the sense of Section 3). A Euclidean d representation A is said to be Rindecomposable if (i) d 6= (0, . . . , 0) and (ii) A ≃R B ⊕ C implies that B or C has dimension (0, . . . , 0). For a sequence of complex matrices M = (M1 , . . . , Mn ), we define the conjugate sequence M̄ = (M̄1 , . . . , M̄n ), the transposed sequence M T = (M1T , . . . , MnT ), and the adjoint sequence M ∗ = M̄ T . Clearly, the Euclidean representations are the selfconjugate unitary representations. 30 4.1 A reduction to unitary representations We give a standard reduction of the problem of classifying Euclidean representations to the problem of classifying unitary representations. Let ind(Q) and indR (Q) denote complete systems of nonisometric indecomposable unitary representations and non-R-isometric R-indecomposable Euclidean representations respectively. Let us replace each representation in ind(Q) that is isometric to a Euclidean representation by a Euclidean one, ∼ and denote the set of such by ind0 (Q) (if A ∈ ind(Q) and S : A→C Ā, then A is isometric to a Euclidean representation if and only if S T = S; see Theorem 4.2). Denote by ind1 (Q) the set consisting of all representations from ind(Q) that are isometric to their conjugates, but not to a selfconjugate, together with one representation from each pair {A, B} ⊂ ind(Q) such that A6≃C Ā ≃C B. For a unitary d representation A = (Aα1 , . . . , Aαq ), we define the Euclidean 2d representation AR = (ARα1 , . . . , ARαq ), where ARα , is obtained from Aα by replacing each entry a + bi (a, b ∈ R) by the block a b −b a Since U with the unitary −1     a + bi 0 a b U= 0 a − bi −b a   1 1 −1 U=√ , 2 i i we have AR ≃C A ⊕ Ā. 31 (18) Theorem 4.1. (a) Let A and B be Euclidean representations of a quiver Q. Then A ≃R B if and only if A ≃C B. (b) Every Euclidean representation is R-isometric to a direct sum of indecomposable Euclidean representations, uniquely determined up to R-isometry of summands. Moreover, indR (Q) = ind0 (Q) ∪ {AR | A ∈ ind1 (Q)}. (19) (c) The set of dimensions of R-indecomposable Euclidean representations of Q coincides with the set of dimensions of indecomposable unitary representations and is equal to D(Q) (see page 18). A homomorphism (R-homomorphism) S : A → B of representations A and B of Q is a sequence of complex (real) matrices S = (S1 , . . . , Sp ) such that Sj Aα = Bα Si for each arrow α : i → j. Clearly, an isomorphism S is an isometry if and only if S ∗ = S −1 . Lemma 4.1. The following properties are equivalent for a unitary (Euclidean) representation A: (i) A is decomposable (R-decomposable). (ii) There exists an endomorphism (R-endomorphism) F : A → A such that F = F ∗ = F 2 ∈ / {0A , 1A }. (iii) There exists a nonscalar selfadjoint endomorphism (Rendomorphism) S = S ∗ : A → A. ∼ Proof. (i)⇒(ii) Let S : A → B ⊕ C be an isometry of unitary representations (R-isometry of Euclidean representations) and B 6= 0 6= C. Then F := S −1 (1B ⊕ 0C )S : A → A satisfies (ii). (ii)⇒(iii) Put S := F . (iii)⇒(i) Since every Si in S is a Hermitian (resp., real symmetric) matrix, there exists a unitary (real orthogonal) matrix Ui such that Ri := Ui Si Ui−1 = diag(ai1 , . . . , aiti ), 32 where aij ∈ R and ai1 ≥ · · · ≥ aiti . Define the unitary (Euclidean) representation B by means of the isometry ∼ U := (U1 , . . . , Up ) : A → B. Then ∼ R := USU −1 : B → B is an isometry and R = a1 I1 ⊕ · · · ⊕ at It , where a1 > · · · > at , Ii = (Ini1 , . . . , Inip ), nij ≥ 0. Clearly, B = B1 ⊕ · · · ⊕ Bt , where dim(Bi ) = (ni1 , . . . , nip ). Proof of Theorem 4.1. 1) We first prove the statement (a) for an Rindecomposable Euclidean representation A. Let ∼ S = Φ + iΨ : A →C B, where Φ and Ψ are real matrices and B is a Euclidean representation. Then Φ and Ψ are R-homomorphisms A → B. Since 1A = S ∗ S = (ΦT − iΨT )(Φ + iΨ) = (ΦT Φ + ΨT Ψ) + i(ΦT Ψ − ΨT Φ), we have ΦT Φ + ΨT Ψ = 1A . By Lemma 4.1, the selfadjoint R-endomorphisms ΦT Φ and ΨT Ψ are scalar, that is, ΦT Φ = λ1A , ΨT Ψ = µ1A , λ + µ = 1. 33 Obviously, λ and µ are non-negative real numbers. For definiteness, λ > 0, 1 ∼ then λ− 2 Φ : A →R B. 2) Let A be an R-indecomposable Euclidean representation that is decomposable as a unitary representation. We prove that A ≃R B R ≃C B ⊕ B̄, where B is an indecomposable unitary representation that is not isometric to a Euclidean representation. Indeed, by Lemma 4.1 there exists an endomorphism F : A → A such that F = F∗ = F2 ∈ / {0A , 1A }. Let F = Φ + iΨ, where Φ and Ψ are sequences of real matrices. Since F = F ∗ = ΦT − iΨT , it follows that Φ = ΦT and Ψ = −ΨT . By Lemma 4.1, the endomorphism Φ is scalar, i.e., Φ = λ1A , λ ∈ R. If λ = 0, then iΨ = F = F 2 = −Ψ2 and Ψ = 0A , a contradiction. Hence λ 6= 0. Since F = F 2 = (λ1A + iΨ)2 = (λ2 1A − Ψ2 ) + 2λiΨ, we have λ2 1A − Ψ2 = λ1A , 2λΨ = Ψ. 1 λ= , 2 1 Ψ2 = − 1A . 4 The condition F 6= 1A implies Ψ 6= 0A , By [21, Sect. 4.4, Exercise 25], every nonsingular skew-symmetric real matrix is real orthogonally similar to a direct sum of matrices of the form   0 a , a > 0. −a 0 Since 1 Ψ2 = − 1A , 4 ΨT = −Ψ, 34 there exists a sequence S of real orthogonal matrices such that   1 0 1 −1 SΨS = I ⊗ −1 0 2 (see (9)), where I = (I, . . . , I). Put G := SF S −1   1 i 1 . = I⊗ 2 −i 1 ∼ Define the Euclidean representation C by means of S : A →R C. Then G : C → C is an R-endomorphism. It follows from the form of G and the definition of homomorphisms, that C = B R for a certain B. If B is a decomposable unitary representation, say, B ≃C X ⊕ Y , then by (18) A ≃R B R ≃C B ⊕ B̄ ≃C X ⊕ Y ⊕ X̄ ⊕ Ȳ ≃C X R ⊕ Y R , by 1) A ≃R X R ⊕ Y R , a contradiction. If B is isometric to a Euclidean representation, say, B ≃C D = D̄, then A ≃R B R ≃C B ⊕ B̄ ≃C D ⊕ D, by 1) A ≃R D ⊕ D, a contradiction. This proves 2). (a)–(b). Let A and B be Euclidean representations, A ≃R B, A ≃R A1 ⊕ · · · ⊕ Al , B ≃R B1 ⊕ · · · ⊕ Br , where Ai and Bj are R-indecomposable. From 2) and Theorem 3.1(a), l = r and, after a permutation of summands, Ai ≃C Bi . By 1), Ai ≃R Bi . The equality (19) is obvious. (c). By Corollary 3.1(a), there exists an R-indecomposable Euclidean representation (with entries in N0 ) of dimension z for every z ∈ D(Q). Conversely, let A be an R-indecomposable Euclidean representation. If A is indecomposable as a unitary representation, then by Theorem 3.2 dim(A) ∈ D(Q). Otherwise by 2) A ≃C B ⊕ B̄, where B is an indecomposable unitary representation, i.e., d := dim(B) ∈ D(Q). Since B is not isometric to a Euclidean representation, supp (d) ∈ / {•, • → •}. 35 Applying twice the definition of D(Q) (see page 3.2), we have dMQ ≥ d, 4.2 2dMQ ≥ 2d, dim(A) = 2d ∈ D(Q). Unitary representations that are isometric to Euclidean representations Theorem 4.1(b) reduces the problem of classifying Euclidean representations of a quiver Q to the following two problems: • classify unitary representations of Q (i.e., construct the set ind(Q)); • bring to light for each A ∈ ind(Q) whether it is isometric to a Euclidean representation and to construct that representation. In this section we consider the second problem. Lemma 4.2. (a) If S is a symmetric unitary matrix, then there exists a unitary matrix U such that S = U T U. (b) If S is a skew-symmetric unitary matrix, then there exists a unitary matrix U such that     0 1 0 1 T U. ⊕···⊕ S=U −1 0 −1 0 Proof. Analogous statement for a non-unitary matrix S is given in [21, Sect. 4.4, Corollary 4.4.4 and Exercise 26]. The condition of unitarity makes its proof much more easy. We give it sketchily since an explicit form of U is needed for the applications of the next theorem. Given a symmetric (skew-symmetric) unitary matrix Sn with rows s1 , . . . , sn . If s1 6= e1 := (1, 0, . . . , 0), we take a unitary matrix Un with rows u1 , . . . , un such that u1 = α(e1 + s1 ), α ∈ C, (resp., Cu1 + Cu2 = Ce1 + Cs1 ). Then ū1 Sn = α(e1 + s̄1 )Sn = α(e1 Sn + s̄1 Sn ) = α(s1 + e1 ) = u1 = e1 Un , 36 hence (Un−1 )T Sn Un−1 = Ūn Sn Un−1 = [1] ⊕ Sn−1 (resp., then Ūn Sn Un−1  0 β ⊕ Sn−2 , = −β 0  |β| = 1; replacing u2 by βu2, we make β = 1). If s1 = e1 , we have Sn = [1] ⊕ Sn−1 , Un := In . We repeat this procedure until we obtain the required U := Un (I1 ⊕ Un−1 )(I2 ⊕ Un−2 ) · · · (In−1 ⊕ U1 ) (resp., U := Un (I2 ⊕ Un−2 ) · · · ). Theorem 4.2. (a) Let A be a unitary representation and A 6≃C Ā. Then A is not isometric to a Euclidean representation. ∼ (b) Let A be an indecomposable unitary representation and S : A →C Ā. (i) If S = S T , then A is isometric to a Euclidean representation B ∼ given by U : A →C B, where U1 , . . . , Up are arbitrary unitary matrices such that UiT Ui = Si (they exist by Lemma 4.2(a)). (ii) If S 6= S T , then S = −S T and A is not isometric to a Euclidean representation but is isometric to a unitary representation C of the form   X Y −Ȳ X̄ ∼ given by V : A →C C, where V1 , . . . , Vp are arbitrary unitary matrices such that   0 I T V i = Si Vi −I 0 (they exist by Lemma 4.2(b)). 37 ∼ Proof. (a) Let R : A →C B, where B is a Euclidean representation. Then ∼ ∼ RT = R̄−1 : B̄ →C Ā, H := RT R : A →C Ā (observe that H = H T ). ∼ (b) Let A be an indecomposable unitary representation and S : A →C Ā. ∼ Then S̄S : A →C A, by Theorem 3.1(b) S̄S = λ1A , S = λS̄ −1 = λS T = λ(λS T )T = λ2 S, and λ ∈ {1, −1}. ∼ (i) Let λ = 1, U : A →C B and U T U = S. 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