aa r X i v : . [ m a t h . R T ] J a n Wildness for tensors
Vyacheslav Futorny a , Joshua A. Grochow b , Vladimir V. Sergeichuk c, ∗ a Department of Mathematics, University of S˜ao Paulo, Brazil b Departments of Computer Science and Mathematics, University of Colorado Boulder,Boulder, CO, USA c Institute of Mathematics, Kiev, Ukraine
Abstract
In representation theory, a classification problem is called wild if it containsthe problem of classifying matrix pairs up to simultaneous similarity. Thelatter problem is considered hopeless; it contains the problem of classifying anarbitrary finite system of vector spaces and linear mappings between them.We prove that an analogous “universal” problem in the theory of tensors oforder at most 3 over an arbitrary field is the problem of classifying three-dimensional arrays up to equivalence transformations[ a ijk ] mi =1 nj =1 tk =1 hX i,j,k a ijk u ii ′ v jj ′ w kk ′ i mi ′ =1 nj ′ =1 tk ′ =1 in which [ u ii ′ ], [ v jj ′ ], [ w kk ′ ] are nonsingular matrices: this problem containsthe problem of classifying an arbitrary system of tensors of order at mostthree. Keywords:
Wild matrix problems, Systems of tensors, Three-dimensionalarrays ∗ Corresponding author.
Email addresses: [email protected] (Vyacheslav Futorny), [email protected] (Joshua A. Grochow), [email protected] (Vladimir V.Sergeichuk)
Preprint submitted to Elsevier January 10, 2019 . Introduction and main result
We prove thatthe problem of classifying three-dimensional arrays up toequivalence transformations[ a ijk ] mi =1 nj =1 tk =1 hX i,j,k a ijk u ii ′ v jj ′ w kk ′ i mi ′ =1 nj ′ =1 tk ′ =1 in which [ u ii ′ ], [ v jj ′ ], [ w kk ′ ] are nonsingular matrices (1)“contains” the problem of classifying an arbitrary system of tensors of orderat most three; which means that the solution of the second problem canbe derived from the solution of the first (see Definition 1.2 of the notion“contains”).In some precise, the problem of classifying matrix pairs up to simultaneoussimilarity contains all classification problems for systems of linear mappings(see Section 1.1). We show that (1) is an analogous universal problem forsystems of tensors of order at most three.We are essentially concerned with systems of arrays. However, the maintheorem is formulated in Section 1.3 in terms of systems of tensors thatare considered as representations of directed bipartite graphs (i.e., directedgraphs in which the set of vertices is partitioned into two subsets and all thearrows are between these subsets). The vertices t , . . . , t p on the left representtensors and the vertices 1 , . . . , q on the right represent vector spaces.For example, a representation of the graph t ) ) G : t r r o o u u t / / is a system T ) ) A : T V r r o o u u T V / / of vector spaces V and V over a field F and tensors T ∈ V ∗ , T ∈ V ⊗ V ⊗ V ∗ , T ∈ V ⊗ V ∗ ⊗ V ∗ ,
2n which V ∗ denotes the dual space of V .The dimension of a representation is the vector (dim V , . . . , dim V q ). Tworepresentations A and A ′ of the same dimension are isomorphic if A is trans-formed to A ′ by a system of linear bijections V → V ′ , . . . , V q → V ′ q . Allrepresentations of dimension n := ( n , . . . , n q ) are isomorphic to representa-tions with V = F n , . . . , V q = F n q , whose sequences ( T , . . . , T p ) of tensorsare given by sequences A = ( A , . . . , A p ) of arrays over F . These sequencesof arrays form the vector space, which we denote by W n ( G ).Let G be a directed bipartite graph, in which each vertex has at most 3arrows (and so each representation of G consists of tensors of order at most3). The aim of this paper is to construct an affine injection F : W n ( G ) → F m × m × m with the following property: two representations A, A ′ ∈ W n ( G ) are isomorphic if and only if the array F ( A ) is transformed to F ( A ′ ) byequivalence transformations (1) (see Theorem 1.1).Note that the problem of classifying tensors of order 3 is motivated fromseemingly independent questions in mathematics, physics, and computationalcomplexity. Each finite dimensional algebra is given by a (1 , F . Our paper was motivated by the theory of wild matrix problems; in thissection we recall some known facts.A classification problem over a field F is called wild if it containsthe problem of classifying pairs ( A, B ) of square matrices ofthe same size over F up to transformations of simultaneoussimilarity ( S − AS, S − BS ), in which S is a nonsingular ma-trix; (2)see formal definitions in [6], [8], and [9, Section 14.10].Gelfand and Ponomarev [10] proved that the problem (2) (and even theproblem of classifying pairs ( A, B ) of commuting nilpotent matrices up to3imultaneous similarity) contains the problem of classifying t -tuples of squarematrices of the same size up to transformations of simultaneous similarity( M , . . . , M t ) ( C − M C, . . . , C − M t C ) , C is nonsingular . Example 1.1.
Gelfand and Ponomarev’s statement, but without the con-dition of commutativity of A and B , is easily proved. For each t -tuple M = ( M , . . . , M t ) of m × m matrices, we define two ( t + 1) m × ( t + 1) m nilpotent matrices A := I m
00 . . .. . . I m , B ( M ) := M
00 . . .. . . M t . Let N = ( N , . . . , N t ) be another t -tuple of m × m matrices. Then M and N are similar if and only if ( A, B ( M )) and ( A, B ( N )) are similar. Indeed,let ( A, B ( M )) and ( A, B ( N )) be similar; that is, AS = SA, B ( M ) S = SB ( N ) (3)for a nonsingular S . The first equality in (3) implies that S has an upperblock-triangular form S = C ∗ C . . .0 C , C is m × m. Then the second equality in (3) implies that M C = CN , . . . , M t C = CN t .Therefore, M is similar to N . Conversely, if M is similar to N via C , then( A, B ( M )) is similar to ( A, B ( N )) via diag( C, . . . , C ). Example 1.2.
The problem of classifying pairs (
M, N ) of m × n and m × m matrices up to transformations( M, N ) ( C − M R, C − N C ) , C and R are nonsingular, (4)looks simpler than the problem of classifying matrix pairs up to similaritysince (4) has additional admissible transformations. However, these problems4ave the same complexity since for each two pairs ( A, B ) and ( A ′ , B ′ ) of n × n matrices the pair (cid:18)(cid:20) I n n (cid:21) , (cid:20) n AI n B (cid:21)(cid:19) is reduced to (cid:18)(cid:20) I n n (cid:21) , (cid:20) n A ′ I n B ′ (cid:21)(cid:19) by transformations (4) if and only if ( A, B ) is similar to ( A ′ , B ′ ).Moreover, by [2] the problem (2) contains the problem of classifying rep-resentations of an arbitrary quiver over a field F (i.e., of an arbitrary finite setof vector spaces over F and linear mappings between them) and the problemof classifying representations of an arbitrary partially ordered set. Analo-gously, by [5] the problem of classifying pairs ( A, B ) of commuting complexmatrices of the same size up to transformations of simultaneous consimilarity( ¯ S − AS, ¯ S − BS ), in which S is nonsingular, contains the problem of classi-fying an arbitrary finite set of complex vector spaces and linear or semilinearmappings between them.Thus, all wild classification problems for systems of linear mappings havethe same complexity and a solution of any one of them would imply a solutionof every wild or non-wild problem.The universal role of the problem (2) is not extended to systems of tensors:Belitskii and Sergeichuk [2] proved that the problem (2) is contained in theproblem of classifying three-dimensional arrays up to equivalence but doesnot contain it. The main theorem is formulated in Section 1.3. Its proof is given in Sec-tions 2–4, in which we successively prove special cases of the main theorem.We describe them in this section.
Definition 1.1. An array of size d × · · · × d r over a field F is an indexedcollection A = [ a i ...i r ] d i =1 . . . d r i r =1 of elements of F . (We denote arrays byunderlined capital letters.) Let A = [ a i ...i r ] and B = [ b i ...i r ] be two arraysof size d × · · · × d r over a field F . If there exist nonsingular matrices S =[ s ij ] ∈ F d × d , . . . , S r = [ s rij ] ∈ F d r × d r such that b j ...j r = X i ,...,i r a i ...i r s i j . . . s ri r j r (5)for all j , . . . , j r , then we say that A and B are equivalent and write( S , . . . , S r ) : A ∼ −→ B. (6)5e define partitioned three-dimensional arrays by analogy with blockmatrices as follows. Let A = [ a ijk ] mi =1 nj =1 tk =1 be an array of size m × n × t .Each partition of its index sets { , . . . , m } = { , . . . , i | i + 1 , . . . , i | . . . | i ¯ m − + 1 , . . . , m }{ , . . . , n } = { , . . . , j | j + 1 , . . . , j | . . . | j ¯ n − + 1 , . . . , n }{ , . . . , t } = { , . . . , k | k + 1 , . . . , k | . . . | k ¯ t − + 1 , . . . , t } (7)(we set i = j = k := 0, i ¯ m := m , j ¯ n := n , and k ¯ t := t ) defines the partitioned array A = [ A αβγ ] ¯ mα =1 ¯ nβ =1 ¯ tγ =1 with ¯ m · ¯ n · ¯ t spatial blocks A αβγ := [ a ijk ] i α i = i α − +1 j β j = j β − +1 k γ k = k γ − +1 . Thus, A is partitioned into spatial blocks A αβγ by frontal, lateral, and hori-zontal planes.Two partitioned arrays A = [ A αβγ ] ¯ mα =1 ¯ nβ =1 ¯ tγ =1 and B = [ B αβγ ] ¯ mα =1 ¯ nβ =1 ¯ tγ =1 (8)of the same size are conformally partitioned if the sizes of the space blocks A αβγ and B αβγ are equal for each α, β, γ .Two conformally partitioned three-dimensional arrays (8) whose partitionis given by (7), are block-equivalent if there exists an equivalence ( S , S , S ) : A ∼ −→ B (see (6)) in which S = S ⊕ · · · ⊕ S m , S = S ⊕ · · · ⊕ S n , S = S ⊕ · · · ⊕ S t (9)and the sizes of diagonal blocks in (9) are given by (7).In Section 2 we prove Theorem 2.1, which implies thatthe problem (1) contains the problem of classifying parti-tioned three-dimensional arrays up to block-equivalence. (10)Theorem 2.1 is our main tool in the proof of Theorem 1.1.In Section 3, we prove Corollary 3.2, which implies thatfor an arbitrary t , the problem (1) contains the problem ofclassifying t -tuples of three-dimensional arrays up to simulta-neous equivalence, (11)6hich is a three-dimensional analogue of Gelfand and Ponomarev’s statementfrom [10] about the problem (2).In Section 4, we consider linked block-equivalence transformations of three-dimensional arrays; that is, block-equivalence transformations( S , S , S ) : A ∼ −→ B of the form (9), in which some of the diagonal blocksare claimed to be equal ( S ij = S i ′ j ′ ) and some of the diagonal blocks areclaimed to be mutually contragredient ( S ij = ( S − i ′ j ′ ) T ). We prove Theorem4.1, which implies thatthe problem (1) contains the problem of classifying par-titioned three-dimensional arrays up to linked block-equivalence. (12)The main result of the article is Theorem 1.1, which generalizes (10)–(12)and means that the problem (1) contains the problem of classifying an arbi-trary system of tensors of order at most 3. (13)Note that the second problem in (13) contains both the problem of clas-sifying systems of linear mappings and bilinear forms (i.e., representationsof mixed graphs) and the problem of classifying finite dimensional algebras;see [14, 21] and Example 6.2. Remark . Because of the potential applications in computational com-plexity, we remark that all of the containments we construct are easily seento be uniform p -projections in the sense of Valiant [26]. In this way, ourcontainments not only show that mathematically one classification problemcontains another, but also that this holds in an effective, computational sense.In particular, a polynomial-time algorithm for testing equivalence of three-dimensional arrays would yield a polynomial-time algorithm for all the otherproblems considered in this paper. (Perhaps the only caveat to be aware ofis that for partitioned arrays with t parts, the reduction is polynomial in thesize of the array and 2 t .) All systems of tensors of fixed orders two systems of tensors are isomorphic if andonly if their images are equivalent arrays ; see Remark 1.2.7n array A = [ a i ...i r ] d i =1 . . . d r i r =1 is a subarray of an array B =[ b j ...j ρ ] δ j =1 . . . δ ρ j ρ =1 if r ρ and there are nonempty (possible, one-element)subsets J ⊂ { , . . . , δ } , . . . , J ρ ⊂ { , . . . , δ ρ } such that A coincides with [ b j ...j ρ ] j ∈ J ,...,j ρ ∈ J ρ up to deleting the indices j k with one-element J k . The size of a p -tuple A = ( A , . . . , A p ) of arrays is thesequence d := ( d , . . . , d p ) of their sizes. Each array classification problem C d that we consider is given by a set of C d -admissible transformations on the setof array p -tuples of size d .We use the following definition of embedding of one classification problemabout systems of aggregates to another, which generalizes the constructionsfrom Examples 1.1 and 1.2. This definition is general; its concrete realizationis given in Theorem 1.1. Definition 1.2.
Let X = ( X , . . . , X p ) be a variable array p -tuple of size d , in which the entries of X , . . . , X p are independent variables (withoutrepetition). We say that an array classification problem C d is contained inan array classification problem D δ if there is a variable array π -tuple F ( X ) =( F , . . . , F π ) of size δ , in which every X i is a subarray of some F j and eachentry of F , . . . , F π outside of X , . . . , X p is 0 or 1, such that A is reduced to B by C d -admissible transformations if and only if F ( A ) is reduced to F ( B ) by D δ -admissible transformationsfor all array p -tuples A and B of size d .Note that F ( X ) defines an affine map A 7→ F ( A ) of the vector space ofall array p -tuples of size d to the vector space of all array π -tuples of size δ .Gabriel [7] (see also [9, 14]) suggested to consider systems of vector spacesand linear mappings as representations of quivers: a quiver is a directedgraph; its representation is given by assigning a vector space to each vertexand a linear mapping of the corresponding vector spaces to each arrow. Gen-eralizing this notion, Sergeichuk [23] suggested to study systems of tensorsas representations of directed bipartite graphs. These representations areanother form of Penrose’s tensor diagrams [20], which are studied in [4, 25].A directed bipartite graph G is a directed graph in which the set of verticesis partitioned into two subsets and all the arrows are between these subsets.8e denote these subsets by T and V , and write the vertices from T on theleft and the vertices from V on the right. For example, t ) ) t r r o o u u t o o (14)is a directed bipartite graph, in which T = { t , t , t } and V = { , } .The following definition of representations of directed bipartite graphs isgiven in terms of arrays. We show in Section 6 that it is equivalent to thedefinition from [23] given in terms of tensors that are considered as elementsof tensor products. Definition 1.3.
Let G be a directed bipartite graph with T = { t , . . . , t p } and V = { , . . . , q } . • An array-representation A of G is given by assigning – a nonnegative integer number d v to each v ∈ V , and – an array A α of size d v × · · · × d v k to each t α ∈ T with arrows λ , . . . , λ k of the form v v t α λ λ ❡❡❡❡❡❡❡❡❡❡❡❡ λ k ... v k (15)(each line is −→ or ←− and some of v , . . . , v k may coincide).The vector d = ( d , . . . , d q ) is the dimension of A . (For example, anarray-representation A of (14) of dimension d = ( d , d ) is given bythree arrays A , A , and A of sizes d , d × d × d , and d × d × d .) • We say that two array-representations A = ( A , . . . , A p ) and B =( B , . . . , B p ) of G of the same dimension d = ( d , . . . , d q ) are isomorphic and write S : A ∼ −→ B (or A ≃ B for short) if there exists a sequence For each sequence of nonnegative integers d , . . . , d q with min { d , . . . , d q } = 0, thereis exactly one array of size d × · · · × d q . In particular, the “empty” matrices of sizes 0 × n and m × F n → → F m . := ( S , . . . , S q ) of nonsingular matrices of sizes d × d , . . . , d q × d q such that ( S τ v , . . . , S τ k v k ) : A α ∼ −→ B α (16)(see (6)) for each t α ∈ T with arrows (15), where τ i := (cid:26) λ i : t α ←− v i − T if λ i : t α −→ v i for all i = 1 , . . . , k, and S − T := ( S − ) T (which is called the contragredient matrix of S ). Example 1.3. • Each array-representation of dimension d = ( d ) of t w w g g t ( ( g g d × d matrix A = [ a ij ], which is isomorphic to an array-representation B = [ b ij ] if and only if there exists a d × d nonsingularmatrix S = [ s ij ] such that b i ′ j ′ = X i,j a ij s ii ′ s jj ′ or b i ′ j ′ = X i,j a ij r ii ′ s jj ′ , respectively, where [ r ii ′ ] := S − T . Thus, B = S T AS or B = S − AS, and so we can consider each array-representation of (17) as the matrixof a bilinear form or linear operator , respectively. • Each array-representation of dimension d = ( d ) of t t t o o (18)is a d × d × d array A = [ a ijk ]. It is reduced by transformations( S, S, S − T ) : A h X i,j,k a ijk s ii ′ s jj ′ r kk ′ i i ′ j ′ k ′ , [ r kk ′ ] := S − T , in which S = [ s ij ] is a d × d nonsingular matrix.10 By (16), each array-representation of (15) defines an ( m, n )-tensor (i.e.,an m times contravariant and n times covariant tensor), where m is thenumber of arrows −→ and n is the number of arrows ←− ; see Section 6.In particular, each array-representation of (18) defines a (1 , T ∈ V ∗ ⊗ V ∗ ⊗ V , with defines a multiplication in V converting V intoa finite dimensional algebra ; see Example 6.2.In Section 3 we show that the problem (1) contains the problems of classifying(1 , , Theorem 1.1.
Let G be a directed bipartite graph with the set T = { t , . . . , t p } of left vertices and the set V = { , . . . , q } of right vertices, inwhich each left vertex has at most three arrows. Let d = ( d , . . . , d q ) be anarbitrary sequence of nonnegative integers, and let X = ( X , . . . , X p ) be avariable array-representation of G of dimension d , in which the entries ofarrays X , . . . , X p are independent variables.Then there exists a partitioned three-dimensional variable array F ( X ) inwhich • p spatial blocks are X , . . . , X p , and • each entry of the other spatial blocks is or ,such that two array-representations A and B of dimension d of the graph G over a field are isomorphic if and only if F ( A ) and F ( B ) are equivalent as unpartitioned arrays. (19) Remark . The variable array F ( X ) defines an embedding of the vectorspace of array-representations of dimension d of the graph G into the vectorspace of three-dimensional arrays of some fixed size. This embedding satisfiesDefinition 1.2 and its image (which consists of all F ( A ) with A of dimension d ) is an affine subspace. Two representations are isomorphic if and only iftheir images are equivalent.
2. Proof of the statement (10)The statement (10) is proved in the following theorem.11 heorem 2.1.
For each partition (7) , there exists a partitioned three-dimensional variable array F ( X ) in which (i) one spatial block is an m × n × t variable array X whose entries areindependent variables, and (ii) each entry of the other spatial blocks is or ,such that two m × n × t arrays A and B partitioned into ¯ m · ¯ n · ¯ t spatial blocksare block-equivalent if and only if F ( A ) and F ( B ) are equivalent.2.1. Slices and strata of three-dimensional arrays We give a three-dimensional array A = [ a ijk ] mi =1 nj =1 tk =1 by the sequence ofmatrices A −→ := ( A , A , . . . , A t ) , A k := [ a ijk ] ij , (20)which are the frontal slices of A . For example, a 3 × × A =[ a ijk ] i =13 j =13 k =1 can be given by its frontal slices A a a a a a a A a a a a a a a a A a a a a a a a a a a a (21)An array A = [ a ijk ] mi =1 nj =1 tk =1 can be also given by the sequence of lat-eral slices [ a i k ] ik , . . . , [ a ink ] ik , and by the sequence of horizontal slices [ a jk ] jk , . . . , [ a mjk ] jk .A linear reconstruction of a sequence ( A , . . . , A t ) of matrices of the samesize given by a nonsingular matrix U = [ u ij ] is the transformation( A , . . . , A t ) ❜ U := ( A u + · · · + A t u t , . . . , A u t + · · · + A t u tt ) . (22)Clearly, every linear reconstruction of ( A , . . . , A t ) is a sequence of the fol-lowing elementary linear reconstructions:(a) interchange of two matrices, 12b) multiply any matrix by a nonzero element of F ,(c) add a matrix multiplied by an element of F to another matrix.The following lemma is obvious. Lemma 2.1.
Given two three-dimensional arrays A and B of the same size. (a) A and B are equivalent if and only if B can be obtained from A bylinear reconstructions of frontal slices, then of lateral slices, and finallyof horizontal slices. (b) ( R, S, U ) : A ∼ −→ B if and only if B −→ = ( R T A S, R T A S, . . . , R T A t S ) ❜ U. (23)Let A = [ a ijk ] mi =1 nj =1 tk =1 be a three-dimensional array, whose partition A = [ A αβγ ] ¯ mα =1 ¯ nβ =1 ¯ tγ =1 into ¯ m · ¯ n · ¯ t spatial blocks is defined by partitions (7)of its index sets. The partition of { , . . . , t } in (7) into ¯ t disjoint subsetsdefines also the division of A by frontal planes into ¯ t frontal strata [ a ijk ] mi =1 nj =1 k k =1 , [ a ijk ] mi =1 nj =1 k k = k +1 , . . . , [ a ijk ] mi =1 nj =1 k ¯ t k = k ¯ t − +1 ;each frontal stratum is the union of frontal slices corresponding to the samesubset. In the same way, A is divided into ¯ n lateral strata and into ¯ m horizontal strata .Two partitioned three-dimensional arrays A and B are block-equivalentif and only if B can be obtained from A by linear reconstructions of frontalstrata, of lateral strata, and of horizontal strata. Lemma 2.2.
For each partition (7) , there exists a three-dimensional variablearray G ( X ) partitioned into ¯ m ′ · ¯ n ′ · ¯ t ′ spatial blocks and satisfying (i) and (ii) from Theorem 2.1 such that two m × n × t arrays A and B partitioned into ¯ m · ¯ n · ¯ t spatial blocks are block-equivalent if and only if G ( A ) and G ( B ) areblock-equivalent with respect to partition into ¯ m ′ · ¯ n ′ · t ′ spatial blocks (i.e., wedelete the horizontal partition). This lemma implies Theorem 2.1 since we can delete the horizontal par-tition, then the lateral partition, and finally the frontal partition in the sameway. 13 .3. Proof of Lemma 2.2 for a partitioned array of size m × n × m × n × m × n × A can be given by the sequence A −→ = ( A , A , A | A , A , A )of its frontal slices, which are m × n matrices (see (20)). Its two spatial blocksare given by the sequences ( A , A , A ) and ( A , A , A ). Construct by A theunpartitioned array M A given by the sequence of frontal slices M −→ A = ( M A , . . . , M A ):= " I r A , " I r A , " I r
00 0 00 A , " I r A , " I r A , " I r A , (24)in which r := min { m, n } + 1 . (25)Let B −→ := ( B , B , B | B , B , B ) give another array B of the same size as A , which is partitioned by the frontal plane conformally to A . Let us provethat A and B are block-equivalent ⇐⇒ M A and M B are equivalent. (26)= ⇒ . In view of Lemma 2.1(b), B can be obtained from A by a sequenceof the following transformations:(i) simultaneous equivalence transformations of A , . . . , A ,(ii) linear reconstructions (a)–(c) (from Section 2.1) of ( A , A , A ),(iii) linear reconstructions (a)–(c) of ( A , A , A ).14t suffices to consider the case when B is obtained from A by one of thetransformations (i)–(iii).If this transformation is (i), then M B can be obtained from M A by asimultaneous equivalence transformation of its frontal slices M A , . . . , M A .Hence, M A and M B are equivalent.If this transformation is (ii), then we make a linear reconstruction of( M A , M A , M A ). It spoils the blocks [ I r , [0 I r , [0 0 I r ] in (24), which canbe restored by simultaneous elementary transformations of M A , . . . , M A thatdo not change the new A , . . . , A . Hence M A and M B are equivalent.The case of transformation (iii) is considered analogously. ⇐ =. Let M A and M B be equivalent. By Lemma 4(b), there existsa matrix sequence N −→ = ( N , . . . , N ) such that the matrices in M −→ A and N −→ are simultaneously equivalent and N −→ is reduced to M −→ B by some linearreconstruction N −→ ❜ S − = M −→ B given by a nonsingular 6 × S = [ s jk ](see (22)). Then ( N , . . . , N ) = ( M B , . . . , M B ) ❜ S and so N k = I r s k I r s k I r s k I r s k I r s k I r s k C k , k = 1 , . . . , , (27)where ( C , . . . , C ) := ( B , . . . , B ) ❜ S. (28)Since the matrices in M −→ A are simultaneously equivalent to the matrices in N −→ , rank M Ak = rank N k for all k . Since A k , B k , C k are m × n , (25) shows thatrank A k < r and rank C k < r . If k = 1 , ,
3, then rank N k = rank M Ak < r ,and so by (27) s k = s k = s k = 0. If k = 4 , ,
6, then rank N k = rank M Ak > r and so ( s k , s k , s k ) = (0 , , N k = rank M Ak < r and so s k = s k = s k = 0. Thus, S = s s s s s s s s s ⊕ s s s s s s s s s . (29)By (28) and (29), the array B is block-equivalent to the conformallypartitioned array C given by C −→ = ( C , C , C | C , C , C ). It remains to15rove that the arrays A and C are block-equivalent. By (27), N k = M Ck Q, k = 1 , . . . , , where Q := s I r s I r s I r s I r s I r s I r s I r s I r s I r ⊕ s I r s I r s I r s I r s I r s I r s I r s I r s I r ⊕ I n . Therefore, the matrices in N −→ and M −→ C are simultaneously equivalent. Sincethe matrices in M −→ A and N −→ are simultaneously equivalent, we have that thematrices in M −→ A = ( M A , . . . , M A ) and M −→ C = ( M C , . . . , M C ) (30)are simultaneously equivalent. Hence, the representations F r + n M A M A ... , , M A ; ; F r + m and F r + n M C M C ... , , M C ; ; F r + m of the quiver r $ $ * * ... : : r (6 arrows) (31)are isomorphic.By (24), the sequences (30) have the form( E , . . . , E ) ⊕ ( A , . . . , A ) and ( E , . . . , E ) ⊕ ( C , . . . , C ) . By the
Krull–Schmidt theorem (see [12, Corollary 2.4.2] or [14, Section 43.1]),each representation of a quiver is isomorphic to a direct sum of indecompos-able representations; this sum is uniquely determined, up to permutation andisomorphisms of direct summands. Hence, the sequences ( A , . . . , A ) and( C , . . . , C ) gave isomorphic representations of the quiver (31), and so theirmatrices are simultaneously equivalent. Thus, A and C are block-equivalent,which proves that A and B are block-equivalent.16 .4. Proof of Lemma 2.2 for an arbitrary partitioned three-dimensional array Let us prove that Lemma 2.2 holds for a partitioned array A =[ A αβγ ] ¯ mα =1 ¯ nβ =1 ¯ tγ =1 of size m × n × t whose partition into ¯ m · ¯ n · ¯ t spatial blocksis given by (7). There is nothing to prove if ¯ t = 1. Assume that ¯ t > . We give A by the sequence A −→ = ( A , . . . , A k | A k +1 , . . . , A k | . . . | A k ¯ t − +1 , . . . , A k ¯ t ) , k ¯ t = t of frontal slices A = [ A αβ ] ¯ mα =1 ¯ nβ =1 , A = [ A αβ ] ¯ mα =1 ¯ nβ =1 , . . . They are block matrices of size m × n with the same partition into ¯ m · ¯ n blocks.By analogy with (24), we consider the array M A given by the sequence M −→ A = ( M A , . . . , M At ) of block matrices M Ak := ∆ k . . . ∆ kk k,k +1 . . . ∆ kk . . . ∆ k,k ¯ t − +1 . . . ∆ kk ¯ t A k in which the blocks∆ k , . . . , ∆ kk | {z } each of size r × r, ∆ k,k +1 , . . . , ∆ kk | {z } each of size 2 r × r, . . . , ∆ k,k ¯ t − +1 , . . . , ∆ kk ¯ t | {z } each of size 2 ¯ t − r × ¯ t − r are defined as follows: r := min { m, n } + 1 and(∆ , . . . , ∆ t ) := ( I, , . . . , , . . . , ∆ t ) := (0 , I, . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (∆ t , . . . , ∆ tt ) := (0 , . . . , , I ) . We partition M A by lateral and horizontal planes that extend the partitionof its spatial block A , but we do not partition M A by frontal planes. Thus,17ach matrix M Ak is partitioned as follows: M Ak = ∆ k ∆ k . . . . . . . . . . . . . . . . . . ... ... ... ...0 0 . . . ∆ k,t − ∆ kt . . . A k A k . . . A nk . . . A k A k . . . A nk ... ... ... ... ... ... ...0 0 . . . A ¯ m k A ¯ m k . . . A ¯ m ¯ nk . (32)Let B = [ B αβγ ] ¯ mα =1 ¯ nβ =1 ¯ tγ =1 be an array of the same size and with the samepartition into spatial blocks as A . Let us prove that A and B are block-equivalent ⇐⇒ M A and M B are block-equivalent. (33)= ⇒ . It is proved as for (26). ⇐ =. Let M A and M B be block-equivalent. Then there exists a sequence N −→ = ( N , . . . , N t ) of matrices of the same size and with the same partition as(32) such that the matrices in M −→ A and N −→ are simultaneously block-equivalentand N −→ ❜ S − = M −→ B for some nonsingular matrix S = [ s jk ] ∈ F t × t . Thus,( N , . . . , N t ) = ( M B , . . . , M Bt ) ❜ S and so N k = (cid:2) I r s k . . . I r s k k (cid:3) ⊕ (cid:2) I r s k +1 ,k . . . I r s k k (cid:3) ⊕ · · · ⊕ (cid:2) I ¯ t − r s k ¯ t − +1 ,k . . . I ¯ t − r s k ¯ t k (cid:3) ⊕ C k , where C −→ = ( C , . . . , C t ) := ( B , . . . , B t ) ❜ S. (34)Denote by C the array defined by (34) and partitioned conformally with thepartitions of A and B .Reasoning as in Section 2.3, we prove that S = S ⊕ S ⊕ · · · ⊕ S ¯ t , S ∈ F k × k , S ∈ F ( k − k ) × ( k − k ) , . . . . C and B are block-equivalent. It remains to prove thatthe arrays A and C are block-equivalent. (35)We can reduce N −→ to M −→ C = ( M C , . . . , M Ct ) by simultaneous elementarytransformations of columns 1 , . . . , k of N , . . . , N t , simultaneous elementarytransformations of columns k + 1 , . . . , k , . . . , and simultaneous elementarytransformations of columns k ¯ t − + 1 , . . . , k ¯ t (these transformations do notchange C ). Hence, the matrices in N −→ and M −→ C are simultaneously block-equivalent. Since the matrices in M −→ A and N −→ are simultaneously block-equivalent, we have thatthe matrices in M −→ A and M −→ C are simultaneously block-equivalent. (36)By the block-direct sum of two block matrices M = [ M αβ ] pα =1 qβ =1 and M = [ N αβ ] pα =1 qβ =1 , we mean the block matrix M ⊞ N := [ M αβ ⊕ N αβ ] pα =1 qβ =1 . This operation was studied in [22]; it is naturally extended to t -tuples ofblock matrices:( M , . . . , M t ) ⊞ ( N , . . . , N t ) := ( M ⊞ N , . . . , M t ⊞ N t ) . By (32), M Ak = ∆ k ⊞ A k for k = 1 , . . . , t , where∆ k := ∆ k ∆ k . . . . . . . . . . . . . . . . . . . . . . . . . . . ∆ k,t − ∆ kt (the empty strips do not have entries). Thus, M −→ A = ∆ −→ ⊞ A −→ .Each matrix M Ck has the form (32) with C ijk instead of A ijk , and so M −→ C = ∆ −→ ⊞ C −→ .Let us prove that the matrices in A −→ and C −→ are simultaneously block-equivalent. Define the quiver Q with ¯ m + ¯ n vertices 1 , . . . , ¯ m, ′ , . . . , ¯ n ′ andwith ¯ m ¯ nt arrows: with t arrows β ′ λ αβ λ αβ ... , , λ αβt α (37)19rom each vertex β ′ ∈ { ′ , . . . , ¯ n ′ } to each vertex α ∈ { , . . . , ¯ m } .Let K −→ = ( K , . . . , K t ) be an arbitrary sequence of block matrices K k =[ K αβk ] ¯ mα =1¯ nβ =1 , in which K αβk is of size m α × n β . This sequence defines thearray K partitioned into ¯ m · ¯ n · R ( K −→ ) of Q by assigning mappings to the arrows (37) as follows: F n β K αβ % % K αβ ... - - K αβt F m α Let L −→ = ( L , . . . , L t ) be another sequence of block matrices L k =[ L αβk ] ¯ mα =1 ¯ nβ =1 , in which all matrices have the same size and the same par-tition into blocks as the matrices of K −→ . Clearly, the matrices in K −→ and L −→ are simultaneously block-equivalent if and only if the representations R ( K −→ )and R ( L −→ ) are isomorphic.By (36), the matrices in M −→ A = ∆ −→ ⊞ A −→ and M −→ C = ∆ −→ ⊞ C −→ are simultane-ously block-equivalent. Hence, the representations R ( M −→ A ) = R ( ∆ −→ ) ⊕ R ( A −→ )and R ( M −→ C ) = R ( ∆ −→ ) ⊕ R ( C −→ ) of Q are isomorphic. By the Krull–Schmidttheorem for quiver representations, R ( A −→ ) and R ( C −→ ) are isomorphic too.Thus, the matrices in A −→ and C −→ are simultaneously block-equivalent, whichproves (35).We have proved (33), which finishes the proof of Lemma 2.2 and hencethe proof of Theorem 2.1.
3. Proof of the statement (11)
In this section, we prove several corollaries of Theorem 2.1.
Corollary 3.1.
Theorem 1.1 holds if and only if it holds with the condition F ( A ) and F ( B ) are block-equivalent (38) instead of (19) .Proof. Suppose Theorem 1.1 with (38) instead of (19) holds for some parti-tioned three-dimensional variable array F ( X ). Reasoning as in Section 2.4,20e first construct an array F ( X ) := M F ( X ) that is not partitioned by frontalplanes and satisfies the conditions of Theorem 1.1 with (38) instead of (19).Then we apply an analogous construction to F ( X ) and obtain an array F ( X ) that is not partitioned by frontal and lateral planes and satisfies theconditions of Theorem 1.1 with (38) instead of (19). At last, we construct anunpartitioned array F ( X ) that satisfies the conditions of Theorem 1.1.We draw U rr rr SR AAA rrrrrrrrrrrr ∼ −→ rrrrrr BBB rrrrrrrrrrrr (39)if (
R, S, U ) : A ∼ −→ B . We denote arrays in illustrations by bold letters.Gelfand and Ponomarev [10] proved that the problem of classifying pairsof matrices up to simultaneous similarity contains the problem of classifying p -tuples of matrices up to simultaneous similarity for an arbitrary p . A three-dimensional analogue of their statement is the following corollary, in whichwe use the notion “contains” in the sense of Definition 1.2. Corollary 3.2.
The problem (1) contains the problem of classifying p -tuplesof three-dimensional arrays up to simultaneous equivalence for an arbitrary p .Proof. Due to Theorem 2.1, it suffices to prove that the second problem iscontained in the problem of classifying partitioned three-dimensional arraysup to block-equivalence.Let A = ( A , . . . , A p ) be a sequence of unpartitioned arrays of size m × n × t . Define the partitioned array N A = [ N A αβγ ] α =12 β =1 pγ =1 = I t ✐✐✐✐✐✐✐✐ ✐✐✐✐✐✐✐✐ ✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇ ✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇ ✇✇✇✇✇✇✇✇✇✇✇✇ ✇✇✇✇ ✇✇✇✇ A p A p A p ✇✇✇✇ I t ✐✐✐✐✐✐✐✐ ✐✐✐✐✐✐✐✐ ✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇ ✇✇✇✇✇✇✇✇ ✇✇✇✇✇✇✇✇ ✇✇✇✇ A A A ✇✇✇✇✇✇✇✇
21n which N A = · · · = N A p = I t of size 1 × t × t, ( N A , . . . , N A p ) = A , and the other spatial blocks are zero (the spatial blocks are indexed as theentries in (21); the diagonal lines in N A and N A p denote the main diagonalof I n consisting of units).Let B = ( B , . . . , B p ) be another sequence of unpartitioned arrays of thesame size m × n × t . Let us prove that the arrays in A and B are simultaneouslyequivalent if and only if N A and N B are block-equivalent.= ⇒ . It is obvious. ⇐ =. Let N A and N B be block-equivalent; that is, there exists([ r ] ⊕ R, S ⊕ S , U ⊕ · · · ⊕ U p ) : N A ∼ −→ N B . In the notation (39), U p ⑥⑥⑥ ⑥⑥⑥ I t ❦❦❦❦❦❦❦ ❦❦❦❦❦❦❦ ⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥⑥ ⑥⑥⑥⑥ A p A p A p ⑥⑥⑥⑥ I t ❦❦❦❦❦❦❦ ❦❦❦❦❦❦❦ S U ⑥⑥⑥ ⑥⑥⑥ ⑥⑥⑥⑥⑥⑥⑥⑥ S ⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥⑥ R A A A ⑥⑥⑥⑥⑥⑥⑥⑥ ∼ −→ ⑥⑥⑥⑥⑥⑥⑥⑥ I t ❦❦❦❦❦❦❦ ❦❦❦❦❦❦❦ ⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥⑥ ⑥⑥⑥⑥ B p B p B p ⑥⑥⑥⑥ I t ❦❦❦❦❦❦❦ ❦❦❦❦❦❦❦⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥⑥ B B B ⑥⑥⑥⑥⑥⑥⑥⑥ Equating the spatial blocks (1 , , , . . . , (1 , , p ), we get S T I t U r = I t , . . . , S T I t U p r = I t (which follows from (5) in analogy to (23)). Hence, U = · · · = U p and so ( R, S , U ) : A ∼ −→ B , . . . , A p ∼ −→ B p .In the next two corollaries, we consider two important special cases ofTheorem 1.1. Their proofs may help to understand the proof of Theorem1.1. If ( R, S, U ) : A ∼ −→ B , then we write that A is reduced by ( R, S, U ) -transformations . Recall that the problems of classifying (1 , , S, S, S − T )-transformations and ( S, S, S )-transformations, respectively. Re-call also that each partitioned three-dimensional array is partitioned intostrata by frontal, lateral, and horizontal planes, and each stratum consists ofslices; see Section 2.1. By a plane stratum , we mean a stratum consisting ofone slice. 22 orollary 3.3.
The problem (1) contains the problem of classifying (1 , -tensors.Proof. Due to Theorem 2.1, it suffices to prove that the second problem iscontained in the problem of classifying partitioned three-dimensional arraysup to block-equivalence.For each unpartitioned array A of size n × n × n , define the partitionedarray K A = [ K Aαβγ ] α =12 β =11 γ =1 := ⑧⑧⑧⑧⑧⑧ ☎☎☎☎☎ I n ✿✿ ✿✿ AAA ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ☎☎☎☎☎⑧⑧⑧⑧⑧⑧ I n ❥❥❥❥❥ ❥❥❥❥❥ ⑧⑧⑧⑧⑧⑧ (40)of size ( n + 1) × ( n + 1) × n . It is obtained from A by attaching under it andon the right of it the plane strata that are the identity matrices: K A = A, K A = I n , K A = I n , K A = 0(the diagonal lines in K A and K A denote the main diagonal of I n consistingof units). Let B be another unpartitioned array of size n × n × n , and let K A and K B be block-equivalent. This means that there exists( R ⊕ [ r ] , S ⊕ [ s ] , U ) : K A ∼ −→ K B ;that is, ⑧⑧⑧⑧⑧⑧ U ☎☎ ☎☎ RI n ✿✿ ✿✿ AAA ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ☎☎☎☎☎ SU ⑧⑧ ⑧⑧ I n ❥❥❥❥❥ ❥❥❥❥❥ ⑧⑧⑧⑧⑧⑧ ∼ −→ ⑧⑧⑧⑧⑧⑧ ☎☎☎☎☎ I n ✿✿ ✿✿ BBB ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ☎☎☎☎☎⑧⑧⑧⑧⑧⑧ I n ❥❥❥❥❥ ❥❥❥❥❥ ⑧⑧⑧⑧⑧⑧ Equating the spatial blocks (1 , ,
1) and (2 , , R T I n U s = I n and S T I n U r = I n . Hence U − T = Rs = Sr , and so ( R, S, U ) =(
R, Rsr − , R − T s − ) : A ∼ −→ B . Thus, ( R, Rr − , R − T ) : A ∼ −→ B , and so( Rr − , Rr − , ( Rr − ) − T ) : A ∼ −→ B . 23 orollary 3.4. The problem (1) contains the problem of classifying (0 , -tensors.Proof. For each unpartitioned array A of size n × n × n , define the array L A := QQQ ⑧⑧⑧⑧ with P = Q = 0 AAA PPP ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ⑧⑧⑧⑧ of size n × n × n partitioned into 1 · · L A , under it, and behind it. All blocks of the newplane strata are zero except for the identity matrices I n under and directlybehind of P and under and on the right of Q , and also except for the matrix I at the intersection of these planes: N A := I n ①①①①① ①①①①① I QQQ ttttt I n ❏❏ ❏❏ttttt ttttt AAA PPP tttttttttttttttttttt ttttt tttttttttttttttttttt I n ❢❢❢❢❢❢ ❢❢❢❢❢❢ tttttttttt I n ❢❢❢❢❢❢ ❢❢❢❢❢❢ tttttttttttttttttttt (41)The obtained array N A is partitioned into 2 · · B be another unpartitioned array of size n × n × n . Let N A and N B be block-equivalent:( R ⊕ [ r ] , S ⊕ S ⊕ [ s ] , U ⊕ U ⊕ [ u ]) : N A ∼ −→ N B . Equating the spatial blocks with I n and I , we get S T U r = I n , S T U r = I n , R T U s = I n , R T S u = I n , rsu = 1 . Hence S = U − T r − = Rr − s, U = S − T r − = Rr − u and we have ( R, S , U ) = ( Rr − r, Rr − s, Rr − u ) : A ∼ −→ B. Since rsu = 1, ( Rr − , Rr − , Rr − ) : A ∼ −→ B .24 . Proof of the statement (12) Each block-equivalence transformation of a partitioned three-dimensionalarray A has the form( R, S, U ) = ( R ⊕ · · · ⊕ R ¯ m , S ⊕ · · · ⊕ S ¯ n , U ⊕ · · · ⊕ U ¯ t ) : A ∼ −→ B. (42)In this section, we consider a special case of these transformations: some ofthe diagonal blocks in R, S, U are claimed to be equal or mutually contra-gredient.Let us give formal definitions. Consider a finite set P with two relations ∼ and ⋊⋉ satisfying the following conditions:(i) ∼ is an equivalence relation,(ii) if a ⋊⋉ b , then a ≁ b ,(iii) if a ⋊⋉ b , then b ⋊⋉ c if and only if a ∼ c .Taking a = c in (iii), we obtain that a ⋊⋉ b implies b ⋊⋉ a .It is clear that the relation ⋊⋉ can be extended to the set P / ∼ of equiv-alence classes such that a ⋊⋉ b if and only if [ a ] ⋊⋉ [ b ], where [ a ] and [ b ] arethe equivalence classes of a and b . Moreover, if an equivalence relation ∼ on P is fixed and ∗ is any involutive mapping on P / ∼ (i.e., [ a ] ∗∗ = [ a ] for each[ a ] ∈ P / ∼ ), then the relation ⋊⋉ defined on P as follows: a ⋊⋉ b ⇐⇒ [ a ] = [ a ] ∗ = [ b ]satisfies (ii) and (iii), and each relation ⋊⋉ satisfying (ii) and (iii) can be soobtained.Let P := { , . . . , ¯ m ; 1 ′ , . . . , ¯ n ′ ; 1 ′′ , . . . , ¯ t ′′ } (43)be the disjoint union of the set of first indices, the set of second indices, andthe set of third indices of A = [ A αβγ ] ¯ mα =1 ¯ nβ =1 ¯ tγ =1 . Since these sets correspondto nonintersecting subsets of P , we can denote all transforming matrices in(42) by the same letter:( S ⊕ · · · ⊕ S ¯ m , S ′ ⊕ · · · ⊕ S ¯ n ′ , S ′′ ⊕ · · · ⊕ S ¯ t ′′ ) : A ∼ −→ B, and give the partition of A by the sequence d := ( d , . . . , d ¯ m ; d ′ , . . . , d ¯ n ′ ; d ′′ , . . . , d ¯ t ′′ ) (44)25in which the semicolons separate the sets of sizes of frontal, lateral, andhorizontal strata) such that the size of each A ijk is d i × d j ′ × d k ′′ .Let ∼ and ⋊⋉ be binary relations on (43) satisfying (i)–(iii). Let thepartition (44) of A = [ A αβγ ] ¯ mα =1 ¯ nβ =1 ¯ tγ =1 satisfies the condition: d α = d β if α ∼ β or α ⋊⋉ β for all α, β ∈ P . (45)We say that (43) is a linked block-equivalence transformation if the followingtwo conditions hold for all α, β ∈ P : S α = S β if α ∼ β , S α = S − Tβ if α ⋊⋉ β . (46)It is convenient to give the relations ∼ and ⋊⋉ on P bythe graph Q with the set of vertices P and with two typesof arrows: two vertices α and β are linked by a solid line if α ∼ β and α = β , and by a dotted line if α ⋊⋉ β . (47) Example 4.1.
The graphs1 1 ′ ′′ and 1 1 ′ ′′ give the problems of classifying partitioned arrays consisting of a singlespatial block A up to ( S, S, S − T )-transformations and up to ( S, S, S )-transformations, respectively; that is, the problems of classifying (1 , , Example 4.2.
The graph 1 1 ′ ′′ ′ gives the problem of classifying partitioned arrays A = [ A αβ ] consisting of2 · · R ⊕ R − T , S ⊕ U − T , U )-transformations; that is,up to linked block-equivalence transformations U tt tt S U − T A A A R ✇✇✇✇✇ A A A A A A R − T A A A ttttt ✇✇✇✇✇✇✇✇✇✇ A = ( A , A , A , A ) of the directed bipartite graph t q q n n k k rt o o m m st q q ❜❜❜❜❜❜❜❜❜❜❜ o o ut ? ? m m We prove the statement (12) in the following theorem. Its array H ( X )generalizes the arrays (40) and (41). Theorem 4.1.
Let P := { , . . . , ¯ m ; 1 ′ , . . . , ¯ n ′ ; 1 ′′ , . . . , ¯ t ′′ } be a set with binaryrelations ∼ and ⋊⋉ satisfying the conditions (i)–(iii) from the beginning ofSection 4. Let X = [ X αβγ ] ¯ mα =1 ¯ nβ =1 ¯ tγ =1 be a variable array whose entries areindependent parameters and whose partition into spatial blocks is given bysome sequence (44) satisfying (45) .Then there exists a partitioned array H ( X ) = [ H αβγ ] ¯¯ mα =1 ¯¯ nβ =1 ¯¯ tγ =1 in which ¯ m < ¯¯ m , ¯ n < ¯¯ n , ¯ t < ¯¯ t ; (a) X is the subarray of H ( X ) located at the first ¯ m · ¯ n · ¯ t spatial blocks(i.e. H αβγ = X αβγ if α ¯ m , β ¯ n , and γ ¯ t ); (b) H ¯¯ m ¯¯ n ¯¯ t = [1] is of size × × , and the other spatial blocks outside of X are zero except for some of H ¯¯ mβγ , H α ¯¯ nγ , H αβ ¯¯ t that are the identitymatricessuch that two three-dimensional arrays A and B over a field, partitionedconformally to X , are linked block-equivalent if and only if H ( A ) and H ( B ) are block-equivalent. Note that the disposition of the identity matrices outsideof X (see (b) ) depends only on ( P , ∼ , ⋊⋉ ) .Proof. Let ¯¯ m > ¯ m , ¯¯ n > ¯ n , and ¯¯ t > ¯ t be large enough in order to make possiblethe further arguments. Let X be the variable array from the theorem. Denoteby H ( X ) = [ H αβγ ] ¯¯ mα =1 ¯¯ nβ =1 ¯¯ tγ =1 the array that satisfies (a) and in which allthe spatial blocks outside of X are zero except for H m ¯¯ n ¯¯ t = [1] of size 1 × × Q be the graph defined in (47) that gives the relations ∼ and ⋊⋉ from27he theorem. Consider the graph 1 1 ′ ′′ ′ ′′ Q : ... ... ...¯¯ m ¯¯ n ¯¯ t (48)without edges, whose vertices correspond to the indices of H ( X ). Eachvertex of Q is the vertex of Q . We will consecutively join the edges of Q to Q and respectively modify H ( X ) until obtain Q r ⊃ Q and H ( X ) := H r ( X )satisfying Theorem 4.1.On each step k , we construct Q k and H k ( X ) = [ H kαβγ ] such that(1 ◦ ) the conditions (a) and (b) hold with H k ( X ) instead of H ( X ), and(2 ◦ ) for every two arrays A and B partitioned conformally to X , each block-equivalence( S ⊕ · · · ⊕ S ¯¯ m , S ′ ⊕ · · · ⊕ S ¯¯ n ′ , S ′′ ⊕ · · · ⊕ S ¯¯ t ′′ ) : H k ( A ) ∼ −→ H k ( B ) (49)with S ¯¯ m = S ¯¯ n ′ = S ¯¯ t ′′ = [1] satisfies (46) in which the relations ∼ and ⋊⋉ are given by the graph Q k .If k = 0, then (1 ◦ ) and (2 ◦ ) hold since the block-equivalence coincides withthe linked block-equivalence with respect to the relations ∼ and ⋊⋉ given by Q .Reasoning by induction, we assume that Q k and H k ( X ) satisfying (1 ◦ )and (2 ◦ ) have been constructed. We construct Q k +1 and H k +1 ( X ) = [ H k +1 αβγ ]as follows. Let λ be an edge of Q that does not belong to Q k . Denote by Q k +1 the graph obtained from Q k by joining λ and all the edges that appearautomatically due to the transitivity of ∼ and the condition (iii) from thebeginning of this section.For definiteness, we suppose that λ connects a vertex from the first columnand a vertex from the first or second column in (48). The following cases arepossible. Case 1: λ is dotted and connects a vertex from the first column and a vertexfrom the second column in (48) . Let λ : α β ′ . We replace the spatial28lock H kαβ ¯¯ t = 0 of size d α × d β ′ × H k +1 αβ ¯¯ t = I ( d α = d β ′ by (45)); the other spatial blocks of H k and H k +1 coincide.Since H k +1 αβ ¯¯ t = I and S ¯¯ t ′′ = [1], we have S Tα IS β ′ = I in (49). Hence, S − Tα = S β ′ , which ensures the conditions (1 ◦ ) and (2 ◦ ) with k +1 insteadof k . Case 2: λ is solid and connects a vertex from the first column and a vertexfrom the second column in (48) . Let λ : α β ′ and let γ ′′ ∈ { ¯ t +1 , . . . , ¯¯ t − } be a vertex from the third column in (48) that does nothave arrows (it exists since we have supposed that ¯¯ m, ¯¯ n, ¯¯ t are largeenough). Reasoning as in Case 1, we join the following two dottedarrows to Q k : α γ ′′ β ′ Then the solid arrow λ : α β ′ is joined automatically by (iii). Case 3: λ is solid and connects two vertices from the first column. Let λ : α β . Reasoning as in Case 2, we join the following two dottedarrows to Q k : α γ ′′ β Case 4: λ is dotted and connects two vertices from the first column. Let λ : α β . Let γ ′ ∈ { ¯ n + 1 , . . . , ¯¯ n − } and δ ′′ ∈ { ¯ t + 1 , . . . , ¯¯ t − } bevertices from the second and third columns in (48) that do not havearrows. We join the following three dotted arrows to Q k : α γ ′ δ ′′ β Then the dotted arrow λ : α β is attached automatically by (iii).The conditions (1 ◦ ) and (2 ◦ ) with k + 1 instead of k hold in all the cases.We repeat this construction until obtain Q r ⊃ Q .Let A and B be three-dimensional arrays partitioned conformally to X such that H r ( A ) and H r ( B ) are block-equivalent; that is, there exists( R ⊕ · · · ⊕ R ¯¯ m , R ′ ⊕ · · · ⊕ R ¯¯ n ′ , R ′′ ⊕ · · · ⊕ R ¯¯ t ′′ ) : H r ( A ) ∼ −→ H r ( B ) . H r ¯¯ m ¯¯ n ¯¯ t is of size 1 × ×
1, the summands R ¯¯ m , R ¯¯ n ′ , R ¯¯ t ′′ are 1 ×
1. Let R ¯¯ m = [ a ], R ¯¯ n ′ = [ b ], and R ¯¯ t ′′ = [ c ]. Since H r ¯¯ m ¯¯ n ¯¯ t = [1], abc = 1. Hence,( a − ( R ⊕· · ·⊕ R ¯¯ m ) , b − ( R ′ ⊕· · ·⊕ R ¯¯ n ′ ) , c − ( R ′′ ⊕· · ·⊕ R ¯¯ t ′′ )) : H r ( A ) ∼ −→ H r ( B ) . Since a − R ¯¯ m = b − R ¯¯ n ′ = c − R ¯¯ t ′′ = [1], we use (1 ◦ ) and (2 ◦ ) with k = r and get that( a − ( R ⊕ · · · ⊕ R ¯ m ) , b − ( R ′ ⊕ · · · ⊕ R ¯ n ′ ) , c − ( R ′′ ⊕ · · · ⊕ R ¯ t ′′ )) : A ∼ −→ B is a linked block-equivalence with respect to the relations ∼ and ⋊⋉ fromTheorem 4.1.
5. Proof of Theorem 1.1
In this section, we prove Theorem 1.1, which ensures the statement (13).
Lemma 5.1.
It suffices to prove Theorem 1.1 for all graphs G in which eachleft vertex has three arrows.Proof. Let G be a directed bipartite graph with T = { t , . . . , t p } and V = { , . . . , q } .(1 ◦ ) Let G have a left vertex t with exactly two arrows: ut ❡❡❡❡❡❡❡❡❡❡❡ v or t v (50)where each line is −→ or ←− . Denote by G ′ the graph obtained from G by replacing (50) with ut ❝❝❝❝❝❝❝❝❝❝❝❝❝❝ v or t vt p +1 q +1 l l o o l l ❩❩❩❩❩❩❩❩❩❩❩❩❩❩ t p +1 q +1 l l o o l l ❩❩❩❩❩❩❩❩❩❩❩❩❩❩ (51)respectively (thus, T ′ = { t , t , . . . , t p +1 } and V ′ = { , , . . . , q +1 } ). Letus extend each array-representation A of G to the array-representation A ′ of G ′ by assigning the 1 × × A ′ t p +1 := [1] to the new vertex t p +1 .Let us prove that A ≃ B if and only if A ′ ≃ B ′ . If ( S , . . . , S q ) : A ∼ −→ B ,then ( S , . . . , S q , [1]) : A ′ ∼ −→ B ′ . Conversely, let ( S , . . . , S q , [ a ]) : A ′ ∼ −→B ′ . Then ([ a ] , [ a ] , [ a ] − T ) : A t p +1 ∼ −→ B t p +1 . Since A t p +1 = B t p +1 = [1], wehave a = 1, and so ( S , . . . , S q ) : A ∼ −→ B .302 ◦ ) Let G have a left vertex t with exactly one arrow: t −→ v or t ←− v .Then by analogy with (51) we replace it with t p +1 v v q +1 , , o o r r ❞❞❞❞❞❞❞❞❞❞❞❞❞❞ t vt p +2 h h q +2 o o l l ❩❩❩❩❩❩❩❩❩❩❩❩❩❩ in which t v is t −→ v or t ←− v .We repeat (1 ◦ ) and (2 ◦ ) until extend G to a graph e G with the set ofleft vertices e T = { t , t , . . . , t p + p ′ } in which each left vertex has exactly threearrows. For array-representations A = ( A , . . . , A p ) and B = ( B , . . . , B p ) of G , define the array-representations e A := ( A , . . . , A p , [1] , . . . , [1]) , e B := ( B , . . . , B p , [1] , . . . , [1]) (52)of e G and obtain that A ≃ B if and only if e A ≃ e B .Suppose that Theorem 1.1 holds for e G and some array e F ( e X ), in which e X = ( X , . . . , X p , [ x p +1 ] , . . . , [ x p + p ′ ]) is a variable array-representation of e G .Substituting the array-representations (52), we find that e F ( e A ) is equivalentto e F ( e B ) if and only if e A ≃ e B , if and only if A ≃ B . Hence, Theorem1.1 holds for G and for the array F ( X ) := e F ( X , . . . , X p , [1] , . . . , [1]) with X := ( X , . . . , X p ).The direct sum A ⊕ B of three-dimensional arrays A and B is the par-titioned array [ C αβ ] α,β =1 , in which C := A, C := B, and the other C αβ := 0: A ⊕ B := ②②② ②②② AAA ②②②②②②
BBB ②②②②②②
Proof of Theorem 1.1.
Let G be a directed bipartite graph with left vertices t , . . . , t p and right vertices 1 , . . . , q . Due to Lemma 5.1, we can suppose thateach left vertex of G has exactly three arrows: v v p t λ λ ′ λ ′′ v ′ . . . t p λ p λ p ′ λ p ′′ v p ′ v ′′ v p ′′ A = ( A , . . . , A p ) , B = ( B , . . . , B p ) (53)be two array-representations of G of the same size. By Definition 1.3, eachisomorphism S : A ∼ −→ B is given by a sequence S := ( S , . . . , S q ) (54)of nonsingular matrices such that( S τ i v i , S τ i ′ v i ′ , S τ i ′′ v i ′′ ) : A i ∼ −→ B i , i = 1 , . . . , p, (55)where τ i ( ε ) := (cid:26) λ i ( ε ) : t i ←− v i ( ε ) − T if λ i ( ε ) : t i −→ v i ( ε ) for i = 1 , . . . , p and ε = 0 , , , in which i (0) := i , i (1) := i ′ , and i (2) := i ′′ .The array-representations (53) define the partitioned arrays A := A ⊕ · · · ⊕ A p , B := B ⊕ · · · ⊕ B p . The sequence (54) defines the isomorphism S : A ∼ −→ B if and only if (55)holds if and only if( S τ v ⊕ · · · ⊕ S τ p v p , S τ ′ v ′ ⊕ · · · ⊕ S τ p ′ v p ′ , S τ ′′ v ′′ ⊕ · · · ⊕ S τ p ′′ v p ′′ ) : A ∼ −→ B. (56)An arbitrary block-equivalence( R ⊕ · · · ⊕ R p , R ′ ⊕ · · · ⊕ R p ′ , R ′′ ⊕ · · · ⊕ R p ′′ , ) : A ∼ −→ B (57)has the form (56) if and only if the following two conditions hold: • R i ( ε ) = R j ( δ ) if v i ( ε ) = v j ( δ ) and τ i ( ε ) = τ j ( δ ) , • R i ( ε ) = R − Tj ( δ ) if v i ( ε ) = v j ( δ ) and τ i ( ε ) = τ j ( δ ) ;that is, if and only if (57) is a linked block-equivalence with respect to therelations ∼ and ⋊⋉ on P := { , . . . , p ; 1 ′ , . . . , p ′ ; 1 ′′ , . . . , p ′′ } that are defined as follows: 32 i ( ε ) ∼ j ( δ ) if v i ( ε ) = v j ( δ ) and τ i ( ε ) = τ j ( δ ) , • i ( ε ) ⋊⋉ j ( δ ) if v i ( ε ) = v j ( δ ) and τ i ( ε ) = τ j ( δ ) .Thus, A ≃ B if and only if A and B are linked block-equivalent.Let X = ( X , . . . , X p ) be a variable array-representation of G of dimen-sion d = ( d , . . . , d q ). Define the array K ( X ) := X ⊕ · · · ⊕ X p . Twoarray-representations A and B of G of dimension d are isomorphic if andonly if K ( A ) and K ( B ) are linked block-equivalent. This proves Theorem1.1 due to Corollary 3.1 and Theorem 4.1.Note that the array K ( X ) in the proof of Theorem 1.1 is somewhat large.One can construct smaller arrays in most specific cases (as in Example 4.2).
6. Tensor-representations of directed bipartite graphs
Definition 1.3 of representations of directed bipartite graphs is given interms of arrays. In this section, we give an equivalent definition in terms oftensors that are considered as elements of tensor products, which may extendthe range of validity of Theorem 1.1.Recall that a tensor on vector spaces V , . . . , V m + n over a field F (some ofthem can be equal) is an element of the tensor product T ∈ V ⊗ · · · ⊗ V m ⊗ V ∗ m +1 ⊗ · · · ⊗ V ∗ m + n , (58)in which V ∗ i denotes the dual space consisting of all linear forms V i → F .The tensor T is called m times contravariant and n times covariant , or an( m, n ) -tensor for short.Choose a basis f α , . . . , f αd α in each space V α ( α = 1 , . . . , m + n ) andtake the dual basis f ∗ α , . . . , f ∗ αd α in V ∗ α , where f ∗ αi : V i → F is defined by f ∗ αi ( f αi ) := 1 and f ∗ αi ( f αj ) := 0 if i = j . The tensor product in (58) is thevector space over F with the basis formed by f i ⊗ · · · ⊗ f mi m ⊗ f ∗ m +1 ,i m +1 ⊗ · · · ⊗ f ∗ m + n,i m + n , and so each tensor (58) is uniquely represented in the form T = X i ,...,i m + n a i ...i m + n f i ⊗ · · · ⊗ f mi m ⊗ f ∗ m +1 ,i m +1 ⊗ · · · ⊗ f ∗ m + n,i m + n (59)and can be given by its array A = [ a i ...i m + n ] over F . g α , . . . , g αd α in each V α , and let S α := [ s αij ] be thechange of basis matrix (i.e., g αj = P i,j s αij f αi ) for each α = 1 , . . . , m + n .Denote by B the array of T in the new set of bases. Then( S − T , . . . , S − Tm , S m +1 , . . . , S m + n ) : A ∼ −→ B, which allows us to reformulate Definition 1.3 as follows. Definition 6.1 (see [23, Sec. 4]) . Let G be a directed bipartite graph with T = { t , . . . , t p } and V = { , . . . , q } . • A representation ( T , V ) of G is given by assigning a finite dimensionalvector space V v over F to each v ∈ V and a tensor T α to each t α ∈ T sothat if v v t α λ λ ❡❡❡❡❡❡❡❡❡❡❡❡ λ k ... v k are all arrows in a vertex t α (each line is −→ or ←− ; some of v , . . . , v k ∈ T may coincide), then T α ∈ V ε v ⊗ · · · ⊗ V ε p v p , ε i := (cid:26) λ i : t α ←− v i , ∗ if λ i : t α −→ v i . • Two representations ( T , V ) and ( T ′ , V ′ ) of G are isomorphic if thereexists a system of linear bijections ϕ v : V v → V ′ v ( v ∈ V ) that togetherwith the contragredient bijections ϕ ∗− v : V ∗ v → V ′∗ v transform T , . . . , T p to T ′ , . . . , T ′ p . Example 6.1.
A special case of (59) is a (0 , T = P i,j a ij f ∗ i ⊗ f ∗ j ∈ V ∗ ⊗ V ∗ . It can be identified with the bilinear form T : V × V → F , ( v , v ) X i,j a ij f ∗ i ( v ) f ∗ j ( v ) . A (1 , T = P i,j a ij f i ⊗ f ∗ j ∈ V ⊗ V ∗ can be identified with thelinear mapping T : V → V , v X i,j a ij f i f ∗ j ( v ) . a ij ] of these tensors are the matrices of these bilinear forms andlinear mappings. The systems of tensors of order 2 are studied in [13, 14, 21,23, 24] as representations of graphs with undirected, directed, and doubledirected ( ←→ ) edges that are assigned, respectively, by (0,2)-, (1,1)-, and(2,0)-tensors on the vector spaces assigned to the corresponding vertices.The problem of classifying such representations (i.e., of arbitrary systemsof bilinear forms, linear mappings, and bilinear forms on dual spaces) wasreduced in [21] to the problem of classifying representations of quivers. Example 6.2 (see [23, Sec. 4]) . Each representation T ) ) T V r r o o u u T V o o (60)of the bipartite directed graph (14) consists of vector spaces V and V andtensors T ∈ V , T ∈ V ∗ ⊗ V ∗ ⊗ V , T ∈ V ∗ ⊗ V ∗ ⊗ V . Each pair (Λ , M ), consisting of a finite dimensional algebra Λ with unit 1 Λ over a field F and a Λ-module M , defines the representation T = 1 Λ * * T Λ F r r o o t t T M F o o of (14), in which T = X i a ∗ i ⊗ a ∗ i ⊗ a i , T = X i a ∗ i ⊗ m ∗ i ⊗ m i (all a ij ∈ Λ and m ij ∈ M ) are the tensors that define the multiplications inΛ and M :( a ′ , a ′′ ) X i a ∗ i ( a ′ ) a ∗ i ( a ′′ ) a i , ( a, m ) X i a ∗ i ( a ) m ∗ i ( m ) m i . Note that the identities (additivity, distributivity, . . . ) that must satisfythe operations in Λ and M can be written via tensor contractions in such35 way that each representation (60) satisfying these relations defines a pairconsisting of a finite dimensional algebra and its module. This leads tothe theory of representations of bipartite directed graphs with relations thatgeneralizes the theory of representations of quivers with relations. Acknowledgements
The authors are grateful to the referee for the useful suggestions. V. Fu-torny was supported by CNPq grant 301320/2013-6 and by FAPESP grant2014/09310-5. J.A. Grochow was supported by NSF grant DMS-1750319.The work of V.V. Sergeichuk was supported by FAPESP grant 2015/05864-9and was done during his visits to the Santa Fe Institute (the workshop “Wild-ness in computer science, physics, and mathematics”) and to the Universityof S˜ao Paulo.