A Finite-Tame-Wild Trichotomy Theorem for Tensor Diagrams
AA FINITE-TAME-WILD TRICHOTOMY THEOREM FORTENSOR DIAGRAMS
JACOB TURNER : Abstract.
In this paper, we consider the problem of determining when twotensor networks are equivalent under a heterogeneous change of basis. In par-ticular, to a string diagram in a certain monoidal category (which we calltensor diagrams), we formulate an associated abelian category of representa-tions. Each representation corresponds to a tensor network on that diagram.We then classify which tensor diagrams give rise to categories that are finite,tame, or wild in the traditional sense of representation theory. For those tensordiagrams of finite and tame type, we classify the indecomposable representa-tions. Our main result is that a tensor diagram is wild if and only if it containsa vertex of degree at least three. Otherwise, it is of tame or finite type.
Keywords:
Tensor Networks, Representation Theory, Wild Representations, TameRepresentations Introduction
Tensor networks were first introduced by Roger Penrose in the 1970’s as a wayof studying systems in physics [34]. They have exploded in the past few decades asan active area of research with various different applications. In quantum physics,they generalize the quantum circuit model in quantum computing, are used in thestudy of approximating ground states of Hamiltonians [43, 42, 33], appear as modelsin statistical physicals, including the Potts and Ising models [10], and appear asinvariants of quantum entanglement [38, 15, 16, 17, 22, 24, 25, 28].In algebraic complexity theory, tensor network contraction is a P -hard prob-lem which can be used to design algorithms for combinatorial counting problems.Leslie Valiant found a certain class of tensor networks that always had a polyno-mial time evaluation, which he called holographic algorithms [40, 39, 23, 30]. Withthis framework, Valiant found new polynomial time algorithms not known to haveany [41]. Later attempts at generalization also found new polynomial time algo-rithms for computing partition functions and Tutte polynomials [32, 31]. Thesealgorithms have lead to interests in dichotomy theorems regarding the hardness oftensor network contraction [4, 13, 5, 9, 8, 7, 6]. Other notable examples of tensornetworks are graphical models in algebraic statistics and phylogenetic tree modelsin evolutionary biology.Given a tensor network, there is natural set of basis changes that typically leavewhat is considered most important about the tensor network invariant. This groupaction is sometimes called a heterogeneous change of basis. When tensor networkstates are used to approximate the ground states of Hamiltonians, these states can : Korteweg-de Vries Institute for Mathematics, University of Amsterdam, 1098 XG Amster-dam, Netherlands. a r X i v : . [ m a t h . R T ] M a y FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 2 be viewed as covariant maps, with respect to this group action, from the repre-sentation space of some tensor diagram to a variety of states. Furthermore, thevariety of such states carries a similar group action which is poorly understood.Studying the orbits of these tensor networks under the natural group action is acentral problem in this area.Tensor networks are themselves invariant polynomials that are invariant underthe induced group action (cf. [2]). Invariant theory and the study of entanglementhave a close relationship [12, 26, 21]. Recently, tensor networks we shown to givecomplete sets of invariants of density operators under local unitary equivalence [38],a specific case of determining the orbits of certain tensor networks under heteroge-neous changes of basis. This makes them an important measure of entanglement.In the study of tensor networks as circuits, the question of which tensor networksare equivalent to those with a known polynomial time evaluation procedure is ofgreat interest. This includes the question of both classifying and recognizing Holantproblems, holographic circuits, and matchgates (cf. [29, 37]). Recognizing a holo-graphic circuit in an arbitrary basis is still an open problem as they are formulatedwith respect to a specific basis.In SLOCC paradigm of studying entanglement as well as the study of tensorrank in applied algebraic geometry both consider the the natural group action onvery simple types of tensor networks. In this setting, determining the orbits isalready known to be intractable in most cases [1]. More specifically, it was shownthat determining the orbits of trivalent tensors under the natural group action isas difficult as determining simultaneous similarity of pairs of matrices.As such, the orbit classification problem of trivalent tensors is a wild problemand is as difficult to solve as classifying the orbits of any finite dimensional algebra.In this paper, we expand the classification not only to tensors of every arity, butto all tensor diagrams. A tensor diagram is a string diagram in the graphicallanguage of free monoidal category. We will consider all tensor networks that canbe associated to a given tensor diagram and give a complete classification of whichtensor diagrams have finitely many orbits in every dimension, a finite number ofone parameter families in every dimension, and which are wild.We first reinterpret tensor networks on a tensor diagram as an abelian categoryof representations of a combinatorial object, similar to the approach used in quivertheory. We then construct explicit inclusions of wild subcategories in the categoryof representations of tensor diagrams of wild type. Otherwise, as we are in anabelian category, we have a notion of indecomposable objects, and for those tensordiagrams of finite and tame type, we classify the indecomposable representations.Our main theorem states that a tensor diagram is wild if and only if it have vertexof degree at least three and otherwise it is of tame or finite type. Furthermore, westate explicitly which tensor diagrams are of tame type and which are of finite type.2.
Preliminaries
We first give a pedestrian definition of tensor diagram and a representation ofsuch an object. First we define a semi-graph to be a finite collection of vertices andedges p V, E q , where each edge is incident to at most two vertices. An edge may alsoonly be incident to zero or one vertex. We call such an edge dangling . If a tensordiagram has no dangling wires, it is called closed . Definition 2.1.
A tensor diagram T “ p V, E q is a directed semi-graph. FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 3
In the context of tensor diagrams, instead of saying “edges”, we will insteaduse the term “wire” instead. This is more consistent with the literature on tensornetworks, although sometimes the term string is used instead. A subdiagram of atensor diagram p V, E q is a pair V Ď V and E Ď E where p V , E q is a semi-graphwith the inherited orientation. A subdiagram p V , E q Ď p V, E q is called induced iffor every v, w P V , the set of wires between v, w in E is the same as the set ofwires between them in E . Definition 2.2.
Let T “ p V, E q be a tensor diagram. A representation of T ,denoted R p T q is a pair of maps: For every e P E , R p e q is a vector space over agiven field k . For each vertex v , let á N p v q denote set of incoming wires and à N p v q the set of outgoing wires. Then R p v q is a matrix inHom ˆ â A P á N p v q A, â B P à N p v q B ˙ . We call a representation of a tensor diagram a tensor network .As in quiver theory, representations can be completely defined in a combina-torial manner but benefit by viewing them in terms of categories. Finite quiverscorrespond to finite categories, and tensor diagrams will be morphisms in a finitelygenerated monoidal category, which we will define shortly.We assume in this paper that the reader is familiar with compact closed monoidalcategories. In particular we consider free compact closed monoidal categories; suchcategories are always equivalent to a category of diagrammatic languages whosemorphisms are in bijection with labeled tensor diagrams. Furthermore, we shallneed basic facts about cup and cap morphisms and symmetric braiding morphismsin a monoidal category. Excellent treatments of this subject can be found in [27,18, 36].Let us consider any finitely generated free compact closed monoidal category F . By finitely generated, we mean that Ob p F q is generated by taking all tensorproducts of a finite number of atomic objects that cannot be written as a tensorproduct of other objects, with the obvious exception of the natural isomorphisms b A Ñ A , A b Ñ A , for any object A and the monoidal unit in F .We note that our theory need not consider a finitely generated monoidal cate-gory; however, since a tensor diagram can only represent a finite number of objectsand morphisms, this is sufficient for our purposes. Indeed, we could consider fullsubcategories of a monoidal category induced by the objects in a tensor diagramand this would be equivalent. Using finitely generated monoidal categories simpli-fies the technical aspect of monoidal signatures that we will need.Since we assumed it was free, the category F can be visualized in terms ofdiagrams, as previously mentioned (glossing over the technicality that we may needto replace F with an equivalent category). We take the convention that primalobjects in F are given by right oriented wires, dual objects by left oriented wires,and that tensor products shall be taken vertically in order from top to bottom. Asusual, the unit is denoted by empty space.Let Ob A p F q denote the atomic primal objects of F . We will assume that theseobjects have a labeling (cid:96) : L Ñ Ob A p F q , where L are the labels and we only demandthat the unit be labeled . Any other object in F inherits a label from (cid:96) as follows:if some object can be expressed as tensor product of k atomic objects in F , then FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 4 its label is A b ¨ ¨ ¨ b A k , where A i is the label of the i th object as specified by (cid:96) .If if A is the label of a primal object, then A ˚ is the label of its dual object. Wenote that one can also form A ˚˚ , but this is naturally isomorphic to A so we donot need this label. We similarly define labels for morphisms in F . Since F is afree category, we have no non-trivial equivalence of labels. Together, these labelsform a monoidal signature; for a more precise definition of monoidal signatures, see[19, 20].Monoidal signatures formally allow us to use variables to describe objects andmorphisms in F where the variables can take any value in the monoidal signature.In a tensor diagram, one may label wires with an object variable and vertices withmorphism variables. We remark that one does not typically label the unit in atensor diagram. In this work, we shall consider all labels, both on wires and onvertices, as distinct. This coincides with the definition given above.We note that the wire denoting the object A is naturally associated with theidentity morphism on A . A tensor diagram also forms a morphism from the ten-sor product of the dangling dual wires to the tensor product of the dangling pri-mal wires. If there are no dangling wires, the tensor diagram is a morphism inHom p , q . We now present and equivalent, but more abstract definition of a rep-resentation of a tensor diagram a.k.a. a tensor network. Definition 2.3.
Let F be a finitely generated free compact closed monoidal cat-egory and Ψ be a strict monoidal functor from F into the category Vect k (thecategory of finite dimensional vector spaces over k ) for some field k . A tensornetwork is the image of a morphism of F under the map Ψ. Organization of the paper . In Section 3, we identity the space of representationsof fixed dimension of a tensor diagram and show that it forms an abelian category.We also discuss the natural group action that arises from isomorphism classes ofrepresentations. Then in Section 4, we formulate the notions of a finite, tame,and wild tensor diagrams in terms of their category of representations. We thenexplicitly show that certain tensor diagrams are of wild type. In Section 5, weshow that the category of representations of a tensor diagram generally containsthe category of representations of any subdiagram. Lastly, in Section 6, we classifythose tensor diagrams which are tame and finite and classify their indecomposablerepresentations.3.
The Space of Representations of a Tensor Diagram
Given a tensor diagram T with wires E and vertices V , a dimension vector isa tuple of non-negative integers d “ p d , . . . , d | E | q . From here on out, we assumethat our wires are labeled by natural numbers. We say that representation of T has dimension vector d if the vector space associated to the wire i has dimension d i , for all i P r| E |s . Given such a representation, we can consider the wire i to havethe vector space k d i associated to it. If we do this, the set of all representation of T with dimension vector d form a vector space, namely R d p T q : “ à v P V Hom ˆ â i P á N p v q k d i , â j P à N p v q k d j ˙ . FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 5
However, by choosing the vector space on the i th wire to be k d i , we have madean implicit choice of basis unnecessarily. We note that we could change this basisand have an isomorphic representation. This induces an action of the followinggroup GL d p T q : “ à i P E GL p d i , k q on R d p T q acting by a change of basis on each wire. To be very concrete, given anelement g “ p g , . . . , g | E | q P GL d p T q , the induced action on the matrix associatedto a vertex v , call it M v , is as follows: g.M v “ ˆ â j P à N p v q g j ˙ M v ˆ â i P á N p v q g ´ i ˙ . The orbits of GL d p T q on R d p T q are precisely the isomorphism classes of represen-tations of T with dimension vector d .We denote the space of all representations of T by R p T q . We will see that thisspace has the structure of an abelian category in addition to being a closed compactmonoidal category. This category will be naturally graded by dimension vectors d .We now define the morphisms between two representations of a tensor diagram.Let R p T q and R p T q be two representations of T “ p V, E q . A morphism Φ : R p T q Ñ R p T q is a collection of linear maps ϕ e P Hom p R p e q , R p e qq for e P E such that the following diagram commutes for every v P V : â i P á N p v q R p i q R p v q (cid:47) (cid:47) ϕ á N p v q (cid:15) (cid:15) â j P à N p v q R p j q ϕ à N p v q (cid:15) (cid:15) â i P á N p v q R p i q R p v q (cid:47) (cid:47) â j P à N p v q R p j q where ϕ à N p v q : “ Â j P à N p v q ϕ j and ϕ á N p v q similarly.We note that it is entirely possible for some vertex v to have the property thateither á N p v q or à N p v q is H . In that case, we recall that the empty tensor product isdefined to be C .We see immediately that Hom p R p T q , R p T qq forms a vector space and thatcomposition of morphisms ϕ : R p T q Ñ R p T q and ψ : R p T q Ñ R p T q is bilinear,given by a collection of maps ψ e ϕ e . As such this category is enriched over thecategory of abelian groups.It is clear that a morphism is a monomorphism if and only if each ϕ e is injectiveand likewise an epimorphism if and only if every ϕ e is a surjection. Let us considera monomorphism ϕ : R p T q ã Ñ R p T q given by a collection of maps ϕ e , e P E .Consider a map ϕ e : R p e q ã Ñ R p e q ; let R p e q denote the cokernel of this map andlet ψ e be the projection map R p e q (cid:16) R p e q . Then for every v P V , define R p v q to be R p v q : “ ˆ â j P á N p v q ψ j ˙ R p v q ˆ â i P à N p v q ψ i ˙ . FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 6
So the collection of maps ψ e define a morphism ψ : R p T q Ñ R p T q . Lastly, let usdenote by 0 the representation of T given by assigning t u to every wire and thezero map to every vertex. Then we see that0 Ñ R p T q ϕ ÝÑ R p T q ψ ÝÑ R p T q Ñ R p T q is the cokernel of ϕ . Note that our construction of R p T q did not depend on ϕ being a monomorphism, so every map has a cokernel. We also need to show thatevery epimorphism is normal and every map has a kernel. However, this will followby describing dual representations that reverse arrows in short exact sequences. Wedo this now.Given a representation R p T q , we define its dual representation R ˚ p T q as follows(this necessarily involves a choice of basis). For every e P E , we define R ˚ p e q : “ R p e q ˚ and for every v , R ˚ p v q : “ R p v q T .Now suppose that we have a morphism ϕ : R p T q Ñ R p T q defined by a collec-tion of maps ϕ e . Then we have dual morphism ϕ T : R p T q ˚ Ñ R p T q ˚ given bythe maps ϕ Te . Furthermore, it is easy to check that ϕ is a monomorphism if andonly if ϕ T is an epimorphism and vice versa.Now suppose we have an epimorphism ϕ : R p T q (cid:16) R p T q . We look at themonomorphism ϕ T : R p T q ˚ ã Ñ R p T q ˚ . We know that we can find a representation R p T q and morphism ψ such that the sequence0 Ñ R p T q ˚ ϕ T ÝÝÑ R p T q ˚ ψ ÝÑ R p T q Ñ Ñ R p T q ˚ ψ T ÝÝÑ R p T q ϕ ÝÑ R p T q Ñ ϕ is normal. Furthermore, given a morphism ϕ : R p T q Ñ R p T q , the kernel of this map is isomorphic to the cokernel of themap ϕ T : R ˚ p T q Ñ R ˚ p T q . So every morphism has a kernel as well. We have nowproved the following proposition. Proposition 3.1.
For every tensor diagram T , every monomorphism and epimor-phism in the category R p T q is normal. Furthermore, this every morphism has akernel and cokernel We now wish to describe how to take the direct sum of two representations of T .This will be a map ‘ : R d p T qˆ R d p T q Ñ R d ` d p T q , where d ` d is coordinate-wiseaddition.Let R p T q be a representation of T . For i P E , let R p i q denote the vector spaceassociated to i and for v P V , let R p v q denote the matrix associated to v . Nowconsider two representations R p T q and R p T q of T . For i P E , we define R ‘ R p i q : “ R p i q ‘ R p i q . To define R ‘ R p v q is slightly trickier, however. We recallthe tensor direct sum ‘ t . Definition 3.2.
Let T P V “  ni “ V i and T P W “  ni “ W i . Then we notethat there are natural projections of  ni “ p V i ‘ W i q onto V and W , denote them p V and p W , respectively. Then T ‘ t T is the unique tensor T P  ni “ p V i ‘ W i q such that p V p T q “ T and p W p T q “ T . FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 7
We now define R ‘ R p v q : “ R p v q ‘ t R p v q . We note that if R p v q P Hom ˆ â i P á N p v q k d i , â j P à N p v q k d j ˙ – â i P á N p v q p k d i q ˚ b â j P à N p v q k d j ,R p v q P Hom ˆ â i P á N p v q k d i , â j P à N p v q k d j ˙ – â i P á N p v q p k d i q ˚ b â j P à N p v q k d j , and so R p v q ‘ t R p v q is in the space â i P á N p v q p k d i ‘ k d i q ˚ b â j P à N p v q p k d j ‘ k d j q – Hom ˆ â i P á N p v q k d i ` d i , â j P à N p v q k d j ` d j ˙ which is the correct space given how we defined the direct sum of R p T q and R p T q on the wires.We see that the direct sum of a finite number of representations is both a prod-uct and coproduct in this category. Thus it makes perfect sense to talk aboutirreducible, simple, and semi-simple representations. This completes the proof that R p T q is an abelian category. Theorem 3.3.
Given a tensor diagram T , R p T q is an abelian category. Actually, this category also has the structure of a closed compact monoidalcategory as well. While we do not prove this rigorously, it is easy to see. To define R b R p T q , define R b R p e q : “ R p e q b R p e q and R b R p v q : “ R p v q b R p v q for all e and v . The monoidal unit is given by the representation p T q , definedas p e q “ C for all e and p v q “ v . There is a clear isomorphism R b R p T q and R b R p T q that defines a symmetric braiding. Lastly, we have alreadydemonstrated how to construct dual objects and morphisms.4. Forbidden Subdiagrams
In the representation theory of any object, the most important task is to classifythe indecomposable representations. In some cases, one can hope for a finite numberof indecomposable representations, or that every indecomposable representationbelongs to one of a finite number of one parameter families. As one encounters two(and higher) parameter families of indecomposable representations, classificationbecomes more difficult.On the flip side, there are objects whose representation theory is prohibitivelycomplex. This is because their category of modules contains the category of mod-ules of every finite dimensional algebra as a full subcategory. An unfortunate motifin representation theory is that once an object has a two-parameter family of in-decomposable representations, it become intractable in this way. We make thesenotions precise and show that for tensor diagrams, such a trichotomy holds.
Definition 4.1.
Given a tensor diagram T , we say that T is ‚ finite if there are finitely many indecomposable representations in R d p T q . ‚ tame if for every dimension vector d , all but finitely many of the inde-composable representations in R d p T q belong to one of a finite number ofone-parameter families. ‚ wild if for every finite dimensional algebra A , A -Mod is equivalent to a fullsubcategory of R p T q . FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 8 (a) The open claw. (b) The needle. (c) The figure eight.
Figure 1.
Forbidden subdiagrams of tame tensor diagrams. Allother orientations of these tensor diagrams given forbidden subdi-agrams.We note that in our definition of wild, it is true but not obvious that for every n , there is an n -parameter family of indecomposables. This follows from the firstwork on tame-wild dichotomy theorems by Drozd [11], whose work implied thesedichotomies for quivers and finite dimensional algebras over algebraically closedfields. Gabriel famously classified the quivers of finite type, given by simply lacedDynkin diagrams [14].Our goal in this section is to find a set of forbidden subdiagrams in the sense thatany tensor diagram containing them will be wild. The first forbidden subdiagramis given by the following theorem. We shall make a few modifications which willgive us the other forbidden subdiagrams. Theorem 4.2 ([1]) . The problem of classifying the orbits of GL p m qˆ GL p n qˆ GL p q q acting on C m b C n b C q is wild. Theorem 4.2 states that the tensor diagram with a single vertex and three dan-gling wires is wild. From this, one would expect that any tensor diagram with avertex of degree at least three is wild. Indeed, this will be the case, but will requiresome work to show. There also exist other forbidden subdiagrams that we need toprove are wild. Lastly we need to show that tensor diagrams with degree at mosttwo are tame, and determine which of these are of finite type.We first give a list of forbidden subdiagrams that, if in a tensor diagram, willforce it to be wild. The list is given in Figure 1. Theorem 4.2 states that the openclaw is wild. We shall need to prove that the needle and the figure eight are alsowild. In fact, the underlying semi-graph of a tensor network is all that is neededto determine if it is of finite, tame, or wild type. So just as in the case of quivers,these properties are orientation independent. We prove this now.
Lemma 4.3.
Let T and T be two tensor diagrams with the same underlying semi-graph but with the orientation on a single wire reversed. Then R p T q – R p T q .Proof. We construct a map ϕ : R p T q Ñ R p T q sending R p T q P R p T q to R ϕ p T q P R p T q . Let T “ p V, E q and T “ p V, E q where E is obtained from E by reversingthe orientation on a single wire, call it w . Then for e P E zt w u , we define R ϕ p e q “ R p e q . For all v P V not incident to w , we define R ϕ p v q “ R p v q . Now we choosea basis for R p w q , which induces an isomorphism ψ : R p w q – R p w q ˚ . Suppose w P á N p v q for some v P V . Then ψ induces an isomorphism˜ ψ : R p w q ˚ b â i P á N p v qzt w u R p i q ˚ b â j P à N p v q R p j q – FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 9 â i P á N p v qzt w u R p i q ˚ b â j P à N p v q R p j q b R p w q . We then define R ϕ p v q “ ˜ ψ p R p v qq . We construct a similar isomorphism ˆ ψ if w P à N p v q and then define R ϕ p v q “ ˆ ψ p R p v qq . The map ϕ clearly gives an isomorphism R p T q – R p T q . (cid:3) Wildness of the needle and figure eight.
We first focus on showing thewildness of the needle given in Figure 1 (b) noting that the tensor diagram givenby any other orientation on the needle will also be wild by Lemma 4.3. We denotethe needle by N “ p V, E q where V “ t v u and E “ t e , e u where e is the loopand e is the dangling wire. The dimension vector d “ p d , d q for a representation R p N q means that dim p R p e i qq “ d i for i “ , R p N q with dimension vector p d , d q is a single tensor ř d i “ M i b u i for M i P End p C d q and u i P C d . The group GL p N q acts on R p N q as follows: for g “ p g , g q P GL p d q ˆ GL p d q , g. d ÿ i “ M i b u i “ d ÿ i “ g M i g ´ b g u i . Lemma 4.4.
The problem of determining if two representations R p N q , R p N q P R d p N q , d “ p d , d q , are isomorphic is equivalent to determining if two d -tuplesof d ˆ d matrices p A , . . . , A d q „ p B , . . . , B d q under the following equivalencerelation:(a) For g P GL p d q , p M , . . . , M d q „ p gM g ´ , . . . , gM d g ´ q .(b) For g “ t g jk u P GL p d q , p M , . . . , M d q „ p ř d j “ M j g j , . . . , ř d j “ M j g jd q .Proof. Given R p N q “ ř d i “ M i b u i , we associate it to the tuple p M , . . . , M d q .Call this map ψ . The action p g, q P GL p N q takes ř d i “ M i b u i to ř d i “ gM i g ´ b u i which ψ maps to p gM g ´ , . . . , gM d g ´ q . The action p , g q P GL p N q , with g “t g jk u , takes ř d i “ M i b u i to ř d i “ M i b gu i which ψ maps to p d ÿ j “ M j g j , . . . , d ÿ j “ M j g jd q . (cid:3) We slightly modify the proof of Theorem 4.2 given in [1] to prove that for suffi-ciently large d , the problem of simultaneous similarity of two n ˆ n matrices canbe embedded into the problem of determining isomorphism classes of R p N q withdimension vector p d , q . We can do this for every n , and so determining the iso-morphism classes of R p N q is as difficult as determining simultaneous similarity oftwo n ˆ n matrices, which is a wild problem. Theorem 4.5.
The needle is wild.Proof.
Let us first start with a two pairs n ˆ n matrices p A , B q and p A , B q . Wewant to find two representations R p N q and R p N q that are isomorphic if and onlyif p A , A q is equivalent to p B , B q under simultaneous similarity. Given a pair of FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 10 n ˆ n matrices p A, B q , we associate to the the pair of matrices p Y , Y p A, B qq : “p X ‘ C , X ‘ C p A, B qq where X “ ˆ I n
00 0 ˙ , X “ ˆ I n ˙ ,C “ ¨˚˚˝ I n I n
00 0 I n ˛‹‹‚ , C p A, B q “ ¨˚˚˝
00 0 A B ˛‹‹‚ Now suppose that p Y , Y p A , B qq „ p Y , Y p A , B qq under the equivalence de-fined in Lemma 4.4. Then there exists a matrix in g “ t g jk u P GL p q such that p Y g ` Y p A , B q g , Y g ` Y p A , B q g q is simultaneously similar to p Y , Y p A , B qq . Note that this implies that rank p Y g ` Y p A , B q g q is equal to rank p Y q and similarly rank p Y g ` Y p A , B q g q “ rank p Y p A , B qq . But rank p Y g ` Y p A , B q g q “ n ą Y if both g , g ‰ p Y q “ n ą n ě rank p Y p A , B qq , so g “ g ‰ p Y g ` Y p A , B q g q ą rank p Y p A , B qq if both g , g ‰
0. If g “
0, then g was singular, so we conclude that g “ g ‰
0. So we see that g Y is similar to Y , implying that g “
1, and that g Y p A , B q is similar to Y p A , B q , implying that g “ p Y , Y p A , B qq must be simultaneously similar to p Y , Y p A , B qq .This implies that p C , C p A , B qq and p C , C p A , B qq are simultaneously similar.Let g P GL p n q be such that gC “ C g and gC p A , B q “ C p A , B q g . The firstequation implies that g “ ¨˚˚˝ g g g g g g g g g g ˛‹‹‚ The second equation implies that A g “ g A and B g “ g B . (cid:3) We now look to the figure eight. The proof that this diagram is wild is alsovery similar to the needle and the open claw. Let B denote the figure eight. Weonly consider dimension vectors of the form p d , q again and show that there is anembedding of simultaneous similarity of n ˆ n matrices into the representations of B . Given a representation R p B q with dimension vector p d , d q , we note that it isa tensor of the form ř d i “ M i b E i where M i P End p C d q and E i P End p C d q arethe elementary basis matrices. Lemma 4.6.
The problem of determining if two representations R p B q , R p B q P R d p N q , d “ p d , d q , are isomorphic is equivalent to determining if two d -tuplesof d ˆ d matrices p A , . . . , A d q „ p B , . . . , B d q under the following equivalencerelation:(a) For g P GL p d q , p M , . . . , M d q „ p gM g ´ , . . . , gM d g ´ q .(b) For h “ g b p g ´ q T “ t h jk u , g P GL p d q , p M , . . . , M d q „ p d ÿ j “ M j h j , . . . , d ÿ j “ M j h jd q . FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 11
Proof.
We construct a map ψ that takes ř d i “ M i b E i to the tuple p M , . . . , M d q .Part (a) follows the exact same lines as in Lemma 4.4. For part (b), we note thatEnd p C d q – C d b C d and that the induced action of GL p d q on this space is by g b p g ´ q T . Then part (b) follows along the same lines as in Lemma 4.4. (cid:3) Proposition 4.7.
The figure eight is wild.Proof.
Given two pairs of n ˆ n matrices p A , B q and p A , B q , we construct thefour tuples p Y , Y p A , B q , , q and p Y , Y p A , B q , , q , where Y and Y p A, B q are defined in exactly the same way as in Theorem 4.5. Following the same linesas in the proof of Theorem 4.5, we can conclude that p Y , Y p A , B q , , q and p Y , Y p A , B q , , q are related by the equivalence relation defined in Lemma 4.6 ifand only if p Y , Y p A , B qq and p Y , Y p A , B qq are simultaneously similar. Fromhere, we conclude in the same fashion as in Theorem 4.5 that this happens if andonly if p A , B q and p A , B q are simultaneously similar. (cid:3) Tensor diagrams containing wild subdiagrams are wild
Our next goal is to show that if T is a tensor diagram and S a subdiagram, thenunder certain conditions R p S q is a equivalent to a full subcategory of R p T q . Forthis, we first reduce the problem to the case where S is an induced subdiagram of T . Definition 5.1.
Given a tensor diagram T “ p V, E q , a vertex v P V , and a parti-tioning of the wires incident to v , W \ W “ N p v q , a splitting of T with respectto p v, W , W q is the tensor diagram formed by(1) Replacing v with two distinct vertices v and v (2) The wires W become the wires incident to v and likewise W are the wiresincident to v .(3) A wire from v to v is added.The operation of splitting is the reverse operation of contraction along an edge.Given a tensor diagram T “ p V, E q , a wire e P E , and c P N , we denote by R p T q e “ c the subcategory of R p T q of all representations of T where the dimension of thevector space associated to e is c . This category is not abelian as it is not closedunder direct sums. If c “
1, however, it forms a closed compact monoidal categoryas C b C – C . More generally, given a set r “ tp e i , c i q| e i P E, c i P N u , we canconsider the restricted category R p T q r given by restriction to representations wherethe vector space associated to e i has dimension c i . Again, if all c i “ r , R p T q r forms a closed compact monoidal category. Lemma 5.2.
Given a tensor diagram T “ p V, E q , let T be the tensor diagramformed by a splitting of T with respect to p v, W , W q and e be the added wirebetween v and v . Then R p T q e “ is equivalent to a full subcategory of R p T q .Proof. Given a representation R p T q P R p T q e “ , we define a map φ : R p T q ÞÑ R ϕ p T q P R p T q as follows. Note that T “ p V , E q where V “ p V zt v uq Y t v , v u and E “ E Y t e u . For u P V , u ‰ v , v we define R ϕ p u q : “ R p u q , and for every w P E , w ‰ e , R ϕ p e q : “ R p w q . Then we define R ϕ p v q : “ R p v q b R p v q .We note that the non-injectivity of the map ϕ comes from the fact that αR p v qb α ´ R p v q “ R p v q b R p v q . Let R p T q and R p T q be two representations in R p T q e “ . This implies that for all u P V , u ‰ v , v , R p u q “ R p u q . Furthermore, FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 12 there exists a non-zero α P C such that R p v q “ αR p v q and R p v q “ α ´ R p v q .But this implies that R p T q – R p T q by applying the change of basis α on thewire e . Therefore, the map ϕ is injective on isomorphism classes of representationsin R p T q e “ , showing that this category is equivalent to a subcategory of R p T q .The fullness of the subcategory is obvious. (cid:3) We note that Lemma 5.2 can be applied inductively to show that R p T q r isequivalent to a full subcategory of R p T q , where T is formed from T by a sequenceof splittings and r “ tp e i , qu , where e i is the edge added by the i th splitting.Let T “ p V, E q be a tensor diagram and S “ p U, F q be a subdiagram. We definethe following sequence of splittings of T :(1) For every v P U , partition the wires incident to v into W \ W , where W is the set of the wires whose other endpoint lies in U . Split T with respectto p v, W , W q . After all such splittings we get a new diagram T . We nowdefine U to be all the vertices v formed at each splitting of v P U . Notethat S is a subdiagram of the subdiagram induced by U in T .(2) For every v P U , we partition the wires incident to v into W \ W where W consists of the wire from v Ñ v formed by the splitting in theprevious step, as well as all wires in F . Then we split T with respect to p v , W , W q . We get two new vertices v and v . We let U S be the set ofall vertices of type v . After all such splittings, we get a new diagram T S .We call the above construction isolating S in T . We note that the subdiagramof T S induced by U S , ˜ S , is isomorphic to S . Let I S be the wires added by thesplittings transforming T to T S . Let r S “ tp e, q| e P I S u , then by Lemma 5.2, R p T S q r S is equivalent to a full subcategory of R p T q .We now want to show that R p ˜ S q is equivalent to a full subcategory of R p T S q r S .Since ˜ S is isomorphic to S , this will imply that R p S q is equivalent to a full subcat-egory of R p T q . Since ˜ S is an induced subdiagram of T S , it suffices to show that if T is a tensor diagram and S an induced subdiagram then R p S q is equivalent to afull subcategory of R p T q . To show this, we will first need to consider abelian flowson tensor diagrams. We first recall basic definitions. Definition 5.3.
Given a tensor diagram T “ p V, E q and a subset U Ď V , let T r U s “ p U, F q be the induced subdiagram. Let U : “ V z U and F : “ E z F . A partial C ˆ -flow on T with respect to U Ď V is a map f : F Ñ C ˆ such that forevery v P U , ź e P á N p v qX F f p e q ´ ˆ ź e P à N p v qX F f p e q “ U Ď V . If U “ V , then we simply call f a C ˆ -flow on T . Definition 5.4.
Let T “ p V, E q be a tensor diagram. Suppose T is a directedgraph and let ˜ T “ p ˜ V , ˜ E q be the graph formed from T by ignoring the orientation.Then we form a new graph ˆ T as follows.(a) For every pair of adjacent vertices u, v P ˜ V , let us merge all wires between theminto a single wire.(b) Remove all loops from ˜ T .The resulting graph ˆ T is simple. If it is a tree, we call T a multi-tree . FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 13
We first consider the case when T is closed. Suppose we have a closed tensordiagram T “ p V, E q and a partial C ˆ -flow f defined with respect to U Ď V . Then T minus the induced subgraph T r U s is a tensor diagram whose dangling wires areprecisely the wires that represent a cut set between U and U . We denote this setof wires by D . We may assume without loss of generality that every wire in D isoriented towards U by Lemma 4.3. The partial flow condition then implies that ś e P D f p e q “ Lemma 5.5.
Let T “ p E, V q be a closed tensor diagram and U Ď V . Let f be apartial C ˆ -flow defined with respect to U Ď V , then f can be extended to a C ˆ -flowon T .Proof. As before, let D be the cut set separating T r U s “ p U, F q from the comple-mentary subgraph in T . We may assume that T r U s has no loops as they contributetrivially to flows. We assume without loss of generality that every e P D is orientedtowards U using Lemma 4.3. Let us first assume that ˆ T r U s is a star. We mayassume that all wires in T r U s are oriented towards the central vertex c by Lemma4.3. Let v P U zt c u be incident to the wires e v , . . . , e kv P F . Let W be the wiresincident to v in D . Then to we define the extension of f , ˆ f , to send every e iv toone of the k th roots of ś w P W f p w q . We need to show this defines C ˆ -flow on T .First of all, it is clear that the flow condition is satisfied for all of the vertices of T r U s except for c . However, the flow condition is satisfied for c by the fact that ś e P D f p e q “
1. So ˆ f is a C ˆ -flow on T .Now let ˆ T r U s be a tree. Let L be the leaves of ˆ T r U s . Let e v , . . . , e kv be a wiresincident to v P L , which we may assume are directed away from the leaf by Lemma4.3. Let W be the wires incident to v in D , which we may assume are orientedtowards v , again by Lemma 4.3. Then for all i P r k s , we define ˆ f p e iv q to be one ofthe k th roots of ś w P W f p w q . We have now extended the partial C ˆ -flow and wemust extend it again to a submulti-tree of T r U s . We repeat this procedure untilˆ T r U s is a star and then we are done.We now consider the case where S “ p U, F q is an arbitrary subdiagram of T .We first choose a spanning tree of S , G “ p U, F q . We may extend f to a functionˆ f onto the edges of G , viewed as a subdiagram of T r U s , using the argument above.Then for every e P F z F , we define ˆ f p e q “
1. This clearly defines a C ˆ -flow on T . (cid:3) Let p U, F q be the induced subgraph of T S “ p V, E q , which by an abuse of notationwe call S . Let r : “ tp e, q| e P E z F u and note that r S Ď r . This implies that R p T S q r is a full subcategory of R p T S q r S . We now consider a map ϕ : R p S q Ñ R p T S q r .Given R p S q P R p S q , the map ϕ takes R p S q to a representation R ϕ p T S q in R p T S q r as follows. For each e P F , R ϕ p e q “ R p e q and for each v P U , R ϕ p v q “ R p v q (possibly using an isomorphism  V i –  V i b C b n b p C ˚ q b m sending  v i ÞÑ Â v i b b n b p ˚ q b m ).For each e P F , R ϕ p e q “ C and for each v P U , R ϕ p v q “ ˆ â e P á N p v q ˚ ˙ b ˆ â e P à N p v q ˙ . Lemma 5.6.
The map ϕ : R p S q Ñ R p T S q r realizes the equivalence of R p S q to afull subcategory of R p T S q r . FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 14 v v ÞÑ v v v v ÞÑ v v Figure 2.
Two splittings of degree three vertices showing the clawas an induced subgraph.
Proof.
We simply need to show that if R ϕ p T S q and R ϕ p T S q are isomorphic for R p S q , R p S q P R p S q , then R p S q – R p S q . So suppose that R ϕ p T S q and R ϕ p T S q are related by an element G P GL p T S q . We may assume that the action of G restricted to S is the identity (again by abuse of notation, by S we mean theinduced subdiagram of T S , p U, F q , isomorphic to S ). So the action of G on every v P U sends the tensor ˆ â e P á N p v q ˚ ˙ b ˆ â e P à N p v q ˙ to itself. We see that the action of G describes a partial C ˆ -flow on T with respectto U . Let D be the set of wires from T r U s to the complementary subdiagram asbefore. The action of G on the wires in D acts non-trivially on the tensors associatedto some of the vertices in U . We need to show that there is a representationof S , R p S q – R p S q such that the induced action of G on R ϕ p T S q restricted tothe vertices U acts trivially. By Lemma 5.5, we may extend the partial C ˆ -flowinduced by G to a C ˆ -flow on all of T S . This defines an element of H P GL p S q .Let R p S q : “ H.R p S q , which is by definition an isomorphic representation of S .Since GL p S q is a subgroup of GL p T S q in a natural way, H lifts to a an element˜ H P GL p T S q . Furthermore, it commutes with G . Then if we look at ˜ HG.R ϕ p T S q ,we see that this gives the flow on T S guaranteed by Lemma 5.5 and that this groupelement is in the isotropy group of GL p T S q acting on T S . So we see that if R ϕ p T S q and R ϕ p T S q are isomorphic by an element G P GL p T S q , then R p S q and R p S q areisomorphic by an element H P GL p S q . (cid:3) With an eye towards Theorem 4.2, we are particularly interested in closed tensordiagrams with a vertex of degree ě
3. If none of the incident edges of this vertexis a loop, then there exists a splitting such that the resulting graph has a claw asan induced subgraph and there are no dimension restrictions on the dimensions ofthe wires in the claw. This is obvious if the tensor diagram is simple; in the non-simple scenario, two other situations arise. Figure 2 shows both situations and theresulting splittings realizing the claw as an induced subgraph. Solid wires have nodimension restrictions while the dotted edges are the edges added by the splittingand thus can only have representations of C . Lemma 5.6 implies that determiningthe indecomposable representations of a closed tensor diagram with such vertexof degree ě
3, not counting loops, is as hard as determining the indecomposablerepresentations of the claw diagram. The next lemma states that this is as hard asdetermining the indecomposable representations of trivalent tensors.
FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 15
Lemma 5.7.
Let C be a claw diagram (with any orientation on the wires) and C the open claw diagram. Then R p C q is a full subcategory of R p C q . Similarly, let ˆ N be the needle diagram with a vertex added to its dangling wire and N be the needlediagram. Then R p N q is a full subcategory of R p ˆ N q .Proof. We may assume without loss of generality that all wires are directed awayfrom the leaves by Lemma 4.3. Now let us construct a map ϕ : R p C q Ñ R p C q sending R p C q P R p C q to R ϕ p C q P R p C q . Let c denote the only vertex in C .Then R ϕ p c q : “ R p c q and for all other vertices v in C , define R ϕ p v q “
0. It isclear that R ϕ p C q – R ϕ p C q if and only if R p C q – R p C q . The fullness of thesubcategory is also clear.Similarly for R p N q , we define a map ϕ : R p N q Ñ R p ˆ N q . Given R p N q P R p N q ,we define R ϕ p ˆ N q P R p ˆ N q as follows: R ϕ p e q “ R p e q for all wires, R ϕ p c q “ R p c q forthe single vertex c in N , and R ϕ p v q “
0, where v is the vertex in ˆ N that is not in N . It is clear that that R ϕ p ˆ N q – R ϕ p ˆ N q if and only if R p N q – R p N q . The fullnessof the subcategory is also clear. (cid:3) Corollary 5.8.
Every closed tensor diagram with a vertex of degree ě (notcounting loops) is wild. Any closed tensor diagram with a vertex with a loop anda non-empty neighborhood is wild. Any tensor diagram which has a connectedcomponent consisting of a single vertex with at least two loops is wild.Proof. This follows directly from Lemmas 5.2, 5.6 and 5.7, Theorem 4.2, Theorem4.5, and Proposition 4.7 and the observation that given a disconnected tensor dia-gram T “ T \ T , the categories R p T q and R p T q are clearly full subcategories of R p T q . (cid:3) We now turn out attention to tensor diagrams with dangling wires. We first ob-serve that we may assume that a tensor diagram has no wire with no endpoints. Thereason is it shares the same vertices as the tensor diagram with this wire removed.As such, these two diagrams have precisely the same category of representations.So we only need look at tensor diagrams where there are wires with one vertex.Let T “ p V, E q be a tensor diagram and let H Ď E be the set of dangling wires,i.e. wires with one endpoint. Now let us consider any representation R p T q . Let v P V be incident to some e P H . Then applying either αI or α ´ I to the wire e (depending on the orientation of e ), we get that the representation R p T q givenby R p w q “ R p w q for w ‰ v and R p v q “ αR p v q is an isomorphic representation to R p T q .Given R p T q , we may now apply either αI or α ´ I to another wire incident to v , say f , depending on the orientation of f , such that R p v q gets replaced with α ´ R p v q “ R p v q . If f P H , then we see that multiplying by a scalar on oneof the dangling wires incident to v is isomorphic by multiplying by a (potentiallydifferent) scalar on a different dangling wire incident to v . If f R H , then let u beits other endpoint, which multiplies R p y q by α . Then we get a new representation R p T q , where R p u q “ αR p u q and for w ‰ u , R p w q “ R p w q . We see that R p T q is isomorphic to R p T q . In this way we get the following fact: Fact 1.
Let T “ p V, E q be a connected tensor diagram with dangling wires H .Given a representation R p T q , any representation formed from R p T q by multiplying R p v q , v P V , by a non-zero scalar α is an isomorphism of representations. Further-more, the representation induced by multiplying every w P H by different scalars FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 16 is isomorphic to the representation induced by multiplying a single wire in H by acertain scalar. Theorem 5.9.
If a tensor diagram T “ p V, E q with dangling wires H Ď E containsthe open claw, needle, or figure eight as a subdiagram, then it is wild.Proof. First of all, we note that if the figure eight is a subdiagram, then either it isa connected component, or the the needle is also a subdiagram. If the figure eightis a connected component, we are finished by Theorem 5.8. So it suffices to showthat if T contains the open claw or needle as a subdiagram, T is wild.Let S “ p U, F q be the subdiagram of T that is either the open claw or the needle.We consider the diagram T S as before, recalling that R p T S q is equivalent to fullsubcategory of R p T q by Lemma 5.2. We let r “ tp e, q| e R F u ; we consider themap ϕ : R p S q Ñ R p T S q r constructed for the proof of Lemma 5.6, noting that it iswell defined even if S and T have dangling wires.We know from the proof of Lemma 5.6 that the subgroup GL p H q Ď GL p T S q ,defined by acting by the identity on the wires H , preserves isomorphism classesof representations of S under the image of ϕ . Let GL p H q be the subgroup ofGL p T S q that acts as the identity on all the wires in E z H . We note that GL p T S q – GL p H q ˆ GL p H q . So we need to show that GL p H q preserves isomorphism classesof representations of S under the image of ϕ .If H Ď F , then this is trivial. So let us look at the action of GL p H q on thedangling wires outside of F , which acts by multiplication by scalars on each of thewires. By Fact 1, we need only consider the action on a single dangling wire outsideof F . But as U “ t c u , this is isomorphic to multiplying the tensor associated to c bya scalar α , again by Fact 1. Lastly, we note that if we have a representation R p S q , αR p S q – R p S q for all α P C ˆ as both the needle and open claw have a danglingwire. So GL p H q preserves isomorphism classes of representations of S in the imageof ϕ .Thus we have that the map ϕ realizes R p S q as a category equivalent to a fullsubcategory of R p T q . Since R p S q is wild, this implies that R p T q is wild. (cid:3) As a consequence of Theorem 5.9, we have that a tensor diagram is wild if it hasvertex of degree ě
3, where a loop is counted as a degree two edge. In the nextsection, we will prove that these are all the wild tensor diagrams.6.
Finite and Tame Tensor Diagrams
In the previous subsection, we showed that any tensor diagram containing avertex with degree at least three is wild. We claim that these are all the wild tensordiagrams. The remaining tensor diagrams to consider are those with the followingunderlying semi-graphs:(a) The path P n , which consists of vertices labeled 1 , . . . , n and wires connectingvertex i to i ` i P t , . . . , n ´ u .(b) The open path A n , which is the path P n ` with the two leaves removed.(c) The half-open path A n , which is the path P n ` with one of the leaves removed.(d) The loop J n which is formed by taking the path P n and adding a wire betweenvertices 1 and n .As was mentioned in the Preliminaries, a closed tensor diagram T is a morphismin Hom p , q in a finitely generated free compact closed monoidal category F . As FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 17 such, any representation of a closed tensor diagram is a morphism in Hom p C , C q – C . That is to say, there is a fundamental invariant of the representation. In fact,this invariant is a GL p T q invariant polynomial. This implies that are infinitelymany non-isomorphic representations of T as there is an invariant that can takeany value in C . As such, the only tensor diagrams that are eligible to be finite arethose whose underlying semi-graph is the open or half-open path. Lemma 6.1.
The tensor diagrams with underlying semi-graph A n and A n arefinite.Proof. We shall first show that R p A n q is equivalent to A n ` ´ Mod, where A n ` isthe quiver associated with the Dynkin diagram A n ` , whose orientation we shallspecify shortly. We may assume that all wires in A n are oriented in the samedirection by Lemma 4.3; similarly for the quiver A n ` . Let R p A n q P R p A n q . Weconstruct a map ϕ taking R p A n q to a representation R ϕ p A n ` q .Let the wires in A n be labeled in sequence e , . . . , e n ` . Let q , . . . , q n ` be thevertices of A n ` in sequence. Then we define R ϕ p q i q “ R p e i q . Let p , . . . , p n bethe arcs of A n ` listed in sequence and v , . . . , v n be the vertices of A n listed insequence. Then define R ϕ p p i q “ R p v i q . It is clear that ϕ defines a equivalence ofcategories, in fact an isomorphism. Since A n ` is finite by [14], A n is also finite.Now we show that R p A n q is equivalent to a full subcategory of A n ` -mod byconstructing a map ϕ . Let R p A n q P R p A n q . As before, we assume the orientation ofall the wires in A n and A n ` are in the same direction. We let e , . . . , e n be the wiresof A n in sequence, with e the dangling wire, and p , . . . , p n be the arcs of A n ` insequence. We let v , . . . , v n be the vertices of A n in sequence and q , . . . , q n ` bethe vertices of A n ` in sequence. For i P r n s , we define R ϕ p q i q “ R p e i q and define R ϕ p q n ` q “ C . Then for i P r n s , we define R ϕ p p i q “ R p v i q . This clearly shows that R p A n q is a full subcategory of A n ` -Mod and so A n is finite. (cid:3) We know that the tensor diagrams whose underlying semi-graph is the path P n or loop J n are not finite as they are closed. We know from Lemma 5.6 that itis sufficient to show that J n is tame as any tensor diagram with underlying semi-graph R p P n q is equivalent to a full subcategory of a tensor diagram with underlyingsemi-graph R p J n q . Lemma 6.2.
The category R p T q , where T has underlying semi-graph J n , is equiv-alent to the category ˜ A n ´ -Mod, where ˜ A n ´ is the affine Dynkin diagram.Proof. We construct a map ϕ : R p T q Ñ ˜ A n ´ -Mod as follows. Let R p T q P R p T q .We may assume that the orientation of the wires of J n and ˜ A n ´ are all in thesame direction by Lemma 4.3. Let v , . . . , v n be the vertices of J n in sequenceand q , . . . , q n be the vertices of ˜ A n ´ in sequence. Similarly let e , . . . , e n be thewires of J n in sequence and p , . . . , p n be the arcs of ˜ A n ´ in sequence. Then wedefine R ϕ p p i q “ R p v i q and R ϕ p q i q “ R p e i q for all i P r n s . This clearly defines anequivalence, and indeed an isomorphism of the categories ˜ A n ´ -Mod and R p T q .Since the quiver ˜ A n ´ is tame (cf. [3]), so is T . (cid:3) This concludes our classification, giving us the following theorem:
Theorem 6.3.
A connected tensor diagram is(a) finite if and only if it its underlying semi-graph is either A n or A n ,(b) tame (but not finite) if and only if its underlying semi-graph is either P n or J n , FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 18 (c) wild otherwise.Proof.
This follows directly from Theorem 5.9 and Lemmas 6.1 and 6.2. (cid:3)
Classifying indecomposable representations of finite and tame ten-sor diagrams.
In the proofs of Lemmas 6.1 and 6.2, we saw that R p A n q wasisomorphic to A n ` -Mod, where A n ` is the Dynkin diagram. Similarly R p J n q was isomorphic to ˜ A n -Mod, where ˜ A n is the affine Dynkin diagram. The indecom-posable representations of any quiver whose underlying graphs are these Dynkindiagrams has been well worked out and gives a classification of the indecomposablerepresentations of A n and J n .Let us first consider the the indecomposable representations of the quivers withunderlying graph the Dynkin diagram A n , which is the path on n vertices. Givenany orientation on A n , we call a representation of A n , R p A n q , thin if dim p R p v qq “ , v . We say that the representation of R p A n q is connected if theunderlying graph of A n formed by deleting those vertices v with dim p R p v qq “ e with R p e q “
0, is connected.
Theorem 6.4 ([14]) . A representation of A n is indecomposable if and only if it isthin and connected. This theorem follows from Gabriel’s result that the indecomposable represen-tation of a simply laced Dynkin diagram are in correspondence with the positiveroots of the root system defined by the Dynkin diagram and the connection of thesepositive roots with the Tits form associated to a quiver. A short proof ot Theorem6.4 can be found here [35].In the proof of Lemma 6.1, we showed that the representations of A n were inbijection with the representations of A n ` where one of the leaves of A n ` wasforced to always be associated to C . If q , . . . , q n ` are the vertices of A n ` insequence, let us take the convention that it is q n ` that must be dimension one.This gives us the following corollary. Corollary 6.5.
Up to isomorphism, the indecomposable representations of A n arein bijection with the thin and connected representations of A n ` , R p A n ` q , where dim p R p q n ` qq “ .Proof. This follows directly from the proof of Lemma 6.1 and Theorem 6.4. (cid:3)
We now look to the determining the indecomposable representations of P n . Sinceit is a subdiagram of J n , we know that its indecomposable representations musttake the form of indecomposable representations of the quiver ˜ A n ` . Looking atthe proofs of Lemma 5.2 and Lemma 6.2, we see that the representations of P n correspond to representations of ˜ A n ` where at least one of the matrices associatedto an arc has rank one.We now recall the classification of indecomposable representations of ˜ A n ` . Letus denote the vertices of ˜ A n ` in sequence by q , . . . , q n ` , and its arcs in sequenceby p , . . . , p n ` . We will consider ˜ A n ` with the orientation such that the arc p i leaves q i and enters q i ` . Let us consider a representation R p ˜ A n ` q , and let usdefine the matrix L “ ś n ` i “ R p p i q P End p R p q qq .We may assume that L is in Jordan canonical form by performing the appropriatechange of basis on R p q q . Let λ be an eigenvalue of L with eigenvector v . Thenwe define v i ` “ R p p i q v i for i “ , . . . , n . If λ “
0, then for some i P r n ` s , FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 19 R p p i q v i “
0. Then consider the representation on ˜ A n ` given by R p p j q “ R p q j q “ t u for j ‰ i , and R p p i q “ R p p i q and R p q i q “ C v i . This is a simplesubrepresentation of R p ˜ A n ` q which we denote V ,i .Otherwise, if λ ‰
0, then we get the following induced subrepresentation: R p p i q “ C v i for all i P r n ` s which is isomorphic the representation where R p p i q “ C forall i P r n ` s , R p q i q “ i P r n s , and R p q n ` q “ λ . This is also a simplerepresentation which we denote V λ .Given a representation of P n , we may view it as a representation of ˜ A n ` wherethe matrix L is rank one. If L is diagonalizable, then it has a single eigenvalue c and it is isomorphic to one of the above simple representations. Otherwise, L hasonly zero eigenvalues, and a single Jordan block of size two.Let R p ˜ A n ` q be a representation where L has this form. We know that R p q q is two dimensional with basis vectors v , w such that Lv “ Lw “ v .We define v i and w i as before. The fact that Lv “ i P r n ` s , R p p i q v i “
0. For that same i , R p p i q w i maps to a non-zero vectorin R p q i ` q . We see that the dimension of all vector spaces R p q j q for 1 ď j ď i ,there is a two-dimensional subspace with v j and w j forming a basis. For j ą i , wehave a subspace of R p q j q spanned by w j . So we have a subrepresentation given by R p q j q “ span t v j , w j u for j P r n ` s (noting that v j “ j ą i ) with the inducedmaps sending v j Ñ v j ` w j Ñ w j ` associated to R p p i q . It is clear that this moduleis indecomposable by the fact that the Jordan matrix of size two with eigenvaluezero cannot be decomposed as a non-trivial matrix direct sum. We denote thisrepresentation by W i . This proves the following proposition. Proposition 6.6.
Every indecomposable representation of P n is isomorphic to oneof either V ,i , V λ , or W i , for λ P C zt u , i P r n ` s . Conclusion
Tensor networks are an important and widely used tool in physics, computerscience, and statistics. In many cases, one is faced with determining when twotensor networks can be related by an element of GL p T q , where T is the underlyingtensor diagram. In this paper, we have shown that any classification scheme oforbits is an intractable problem if one considers arbitrary tensor networks on mosttensor diagrams. The tensor diagrams that admit classifications are very basic anduninteresting, reducing to a small subset of the tame cases arising in quiver theory.However, in applications, one often does not consider arbitrary tensor networkson a given diagram. For example, in computer science applications, the dimensionsof the representation are limited so that every wire has a two dimensional vectorspace associated to it. In tensor network states, or the study of entanglement ofdensity operators, one is interested in closed orbits rather than all orbits.While it may be that many of these problems will still be wild, a much largersubset may be tractable than those presented in the current work. It is completelyunknown how a trichotomy theorem would manifest itself with these restrictions andthe current work represents only the first step towards understanding the difficultyof these more common questions. As such, much more work in this direction isnecessary. FINITE-TAME-WILD TRICHOTOMY THEOREM FOR TENSOR DIAGRAMS 20
Acknowledgments.
The author would like to thank Llu´ıs Vena for helpful dis-cussions. The research leading to these results has received funding from the Eu-ropean Research Council under the European Union’s Seventh Framework Pro-gramme (FP7/2007-2013) / ERC grant agreement No 339109.
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