On traces of tensor representations of diagrams
ON TRACES OF TENSOR REPRESENTATIONS OF DIAGRAMS
Abstract. Let be a set, of types, and let . A -diagram is a locally ordered directed graph equipped with a function such that each vertex of has indegree and outdegree . (A directed graph is locally ordered if at each vertex , linear orders of the edges entering and of the edges leaving are specified.)
Let be a finite-dimensional -linear space, where is an algebraically closed field of characteristic 0. A function on assigning to each a tensor is called a tensor representation of . The trace (or partition function) of is the -valued function on the collection of -diagrams obtained by ‘decorating’ each vertex of a -diagram with the tensor , and contracting tensors along each edge of , while respecting the order of the edges entering and leaving . In this way we obtain a tensor network.
We characterize which functions on -diagrams are traces, and show that each trace comes from a unique ‘strongly nondegenerate’ tensor representation. The theorem applies to virtual knot diagrams, chord diagrams, and group representations.
Keywords: diagram, tensor representation, trace, partition function, virtual link, chord diagram
Mathematics Subject Classification: 05C20, 14L24, 15A72, 81T
Our theorem characterizes traces of tensor networks, more precisely of tensor representations of diagrams, which applies to knot diagrams, group representations, and algebras. Tensor networks and their diagrammatical notation root in work of Penrose , and were applied to knot theory by Kauffman  and to Hopf algebra in ‘Kuperberg’s notation’ . Other applications were found in areas like quantum complexity (cf. , , , ), statistical physics (cf. , ), and neural networks (cf. ). (See Landsberg  for an in-depth survey of the geometry of tensors and its applications.)
Types and -diagrams
Let be a (finite or infinite) set, of types, and let ( set of nonnegative integers). A -diagram is a (finite) locally ordered directed graph equipped with a function such that each vertex of has indegree and outdegree . Here a directed graph is locally ordered if at each vertex , a linear order of the edges entering and a linear order of the edges leaving are specified. Loops and multiple edges are allowed. Moreover, we allow the ‘vertexless directed loop’ — more precisely, components of a -diagram may be vertexless directed loops.
Let denote the collection of all -diagrams. If is clear from the context, we call a -diagram just a diagram, and denote by . The types can be visualized by small pictograms indicating the type of any vertex, as in the following examples.
Virtual link diagrams. . So and for each . (In pictures like this we assume the entering edges are ordered counter-clockwise and the leaving edges are ordered clockwise. We also will occasionally delete the grey circle indicating the vertex.) Then the -diagrams are the virtual link diagrams (cf. , , ).
Multiloop chord diagrams. , with . Then the -diagrams are the multiloop chord diagrams, which play a key role in the Vassiliev knot invariants (cf. ). They can also be described as cubic graphs in which a set of disjoint oriented circuits (‘Wilson loops’) covering all vertices is specified. By contracting each Wilson loop to one point, the -diagrams correspond to graphs cellularly embedded on an oriented surface.
Groups. Let be a group, and let , with for each . Then -diagrams consist of disjoint directed cycles, with each vertex typed by an element of .
Algebra template. , where the types represent the multiplication and the unit , respectively.
Hopf algebra template. , where the types represent the multiplication , the unit , the comultiplication , the counit , and the antipode , respectively (cf. Kuperberg ).
Directed graphs. , with and for .
Tensor representations and their traces
Throughout this paper, fix an algebraically closed field of characteristic 0.
For any finite-dimensional -linear space , let as usual
If is a set of types, call a function a tensor representation of if for each . We call the dimension of . Let denote the collection of tensor representations . ( depends on the linear space , but we will use only when has been set.)
For a tensor representation , the partition function or trace of is defined as follows. Roughly speaking, we ‘decorate’ each vertex of a -diagram with the tensor , and contract tensors along each edge of , consistent with the orders of the edges entering and of those leaving . In this way we have a tensor network.
To give a more precise description of trace, fix a basis of , with dual basis . Represent any element of as a multi-dimensional array , which are the coefficients of when expressed in the basis of . Set . Then
Here and are the ordered sets of edges entering and leaving , respectively. Moreover, for any ordered set of edges, .
Note that is independent of the chosen basis of . The function corresponds to a ‘state’ or ‘edge coloring’ of the ‘vertex model’ of de la Harpe and Jones  (cf. ).
For , define by . Then is -invariant, taking the natural action of on . (We will use only when has been set.)
Webs = tangle diagrams
To characterize which functions on the collection of -diagrams are traces, we need the concept of tangle diagrams, also called webs. When has been set, for , a -tangle diagram, briefly a -web, is a locally ordered directed graph equipped with injective functions and such that has outdegree 1 and indegree 0 (for ) and has indegree 1 and outdegree 0 (for ), and equipped moreover with a function such that each vertex has indegree and outdegree , where .
The vertices in are called the roots and the vertices in are called the sinks. For , is called the label of vertex , and for , is called the label of vertex . Again, loops and multiple edges are allowed, and components of may be the vertexless directe loop \scalebox0.06\includegraphicsloop.pdf. We call a web if it is a -web for some . Let be the collection of all -webs, and let be the collection of all webs. So . (We use this notation if has been set.)
By , , and we denote the linear spaces of formal -linear combinations of elements of , , and , respectively. Like in , we call their elements quantum diagrams, quantum -webs, and quantum webs, respectively. We extend any function on , , or to some linear space linearly to a linear function on , , or .
For , let be the disjoint union of and . More generally, if and , let be the diagram arising from the disjoint union of and by, for each , identifying the -labeled root in with the -labeled sink in , and, for each , identifying the -labeled sink in with the -labeled root in . After each identification, we ignore identified points as vertex, joining the entering and leaving edge into one directed edge (that is, \scalebox0.08\includegraphicsbecomes1.pdf becomes \scalebox0.08\includegraphicsbecomes2.pdf). Note that this operation may introduce vertexless directed loops. We extend this product bilinearly to , setting if and with .
Define, for each , an element as follows. For let be the -web consisting of disjoint directed edges , where the tail of is labeled and its head is labeled , for . Then
Call multiplicative if and for all . Here is the diagram with no vertices and edges, and as before, denotes the disjoint union of and . We say that annihilates a quantum web if for each web .
Theorem 1. Let . Then there exists a tensor representation of of dimension with if and only if is multiplicative and annihilates .
This theorem can be seen as a generalization of the following simple statement. Let be any (finite or infinite) group. Then a class function is the character of some representation of of dimension if and only if for all :
(Similarly for higher dimensions.)
The extended trace
Generally, as the examples described above suggest, we want to have a tensor representation that satisfies certain linear relations between webs (for instance, ‘R-matrices’ for the virtual link example). Such relations can be described by a collection of quantum webs.
Given a finite-dimensional -linear space and a tensor representation , we extend the trace function to a function as follows. Let be a -web, with root function and sink function . Fix a basis of , with dual basis . Then
where , and moreover, for , is the edge leaving , and, for , is the edge entering .
Again, is independent of the chosen basis of . Also, . We set for and web . Then for each , is -invariant. Finally, by letting (also) to be the standard bilinear form on , we have for webs and :
Hence for any set of quantum webs, implies that annihilates (meaning that it annihilates each ) — but not conversely. (An easy example is with , , and , .)
However, as we will see, if annihilates , then there exists such that and for all . So we could take for the collection of all quantum webs annihilated by .
Virtual link diagrams. The following set of quantum webs correspond to the Reidemeister moves:
(In pictures representing quantum webs like this we assume that the roots and sinks in the different webs occurring in the quantum web are labeled consistently suggested by their position in the pictures. The precise numbering of the roots and sinks is irrelevant, as long it is consistent over all webs occurring in the quantum web.) Then the functions annihilating are the virtual link invariants (that is, invariant under Reidemeister moves). Moreover, if and only if is an ‘R-matrix’.
Multiloop chord diagrams. To describe the ‘undirectedness’ of the chords and the ‘4T-relations’, set
(Note the difference between a vertex, indicated by a dot, and a crossing of edges as an effect of the planarity of the drawing.) Then the functions annihilating are the ‘weight systems’. Moreover, if and only if comes from a representation of a Lie algebra (cf. ).
Groups. Define for a group :
where is the unit of . Then if and only if is a representation of .
Algebra template. Define
Then if and only if and are the multiplication tensor and the unit of a finite-dimensional unital associative -algebra.
Hopf algebra template. The Hopf algebra axioms can similarly be translated into quantum diagrams (cf. ).
Directed graphs. Let be the collection of all quantum webs
with and , . Then amounts to requiring that is symmetric under permutations of entering edges and under permutations of leaving edges. Thus we deal with invariants of ordinary directed graphs, with no ordering of edges. This case was considered in , and Theorem Webs = tangle diagrams forms a generalization of its result.
In these examples we were considering tensor representations with and , where is some given collection of quantum webs annihilated by . In fact, for each trace there exists a tensor representation with such that for each quantum web annihilated by . To describe this more precisely, we define nondegeneracy of tensor representations.
Call a tensor representation nondegenerate if for each quantum web with , there exists with . In other words, in view of (The extended trace ), being nondegenerate means that the subspace of is nondegenerate with respect to the standard bilinear form on . Or: for each quantum web annihilated by .
Call strongly nondegenerate if each finite is contained in some finite with the restriction of to being nondegenerate. The proof of the theorem below implies that this is equivalent to: there is a finite subset such that is nondegenerate for each finite with . So strong nondegeneracy implies nondegeneracy, and if is finite, the two concepts coincide. (Actually, we have no example of a nondegenerate which is not strongly nondegenerate.)
In the following theorem, ‘unique’ means: up to the natural action of on the set of tensor representations .
Theorem 2. For each trace there exists a unique strongly nondegenerate tensor representation with .
For given sets of types and of quantum webs, it is a fundamental question to determine the collection of quantum webs that are annihilated by each function annihilating . For the virtual link diagram example this contains the question which virtual link diagrams are equivalent under Reidemeister moves.
A related question is whether for each annihilating and each quantum diagram with , there exists a trace annihilating with (’detecting ’). For instance, for the multiloop chord diagram example, this question was answered negatively by Vogel .
2. Some applications of invariant theory
We give a few consequences of invariant theory, as preparation to the proof of Theorems Webs = tangle diagrams and Nondegeneracy in Section 3. Proof of Theorems and. In this section, fix a finite-dimensional linear space .
If is finite, then is a finite-dimensional linear space, which can be described as:
Then the following is a direct application of the first fundamental theorem (FFT) of invariant theory for (cf.  Corollary 5.3.2), where as usual denotes the set of regular -valued functions on a variety , while if acts on a set , then is the set of -invariant elements of :
Proposition 0.Let be finite and . Then is nondegenerate if and only if the orbit is closed.
Proof. Sufficiency. Let be closed and let be such that for each . Suppose that . As the function is -equivariant, for all . Hence, since is closed, by the Nullstellensatz there exists with for each . Applying the Reynolds operator, we can assume that is -equivariant (as is -equivariant). So by (2. Some applications of invariant theory), for some quantum web . Then , a contradiction.
Necessity. Let . So is the set of all with for each -invariant regular function on (by (2. Some applications of invariant theory)). Hence is a fiber of the projection . So contains a unique closed -orbit ().
Suppose . Then there exists with and . Let be the -module spanned by . The morphism with (for and ) is -equivariant.
Let be an embedding of as -submodule of . (This exists, as is spanned by a -orbit, so that each irreducible -module occurs with multiplicity at most 1 in .) So is a -equivariant morphism . In other words, belongs to , which is by (2. Some applications of invariant theory) equal to . Hence for some . As (since ), we have . As is nondegenerate, there is a web with . So . However, for any , , since , as . So while , contradicting the fact that for each diagram .
3. Proof of Theorems Webs = tangle diagrams and Nondegeneracy
I. We first show necessity in Theorem Webs = tangle diagrams. Let for some tensor representation , where is a -dimensional linear space with . Clearly, is multiplicative. Moreover, . Indeed, consider an ‘edge coloring’ in the summation (The extended trace ). As , two edges of have the same -value, say edges and (). Let be the permutation in swapping and . Then and cancel each other out for this , as and have opposite signs. So for each fixed , the term in (The extended trace ) is 0. Therefore, , hence for each , by (The extended trace ).
II. We next show that the condition in Theorem Webs = tangle diagrams implies the existence of a strongly nondegenerate tensor representation with .
Let be multiplicative and annihilate . We can assume that is smallest with this property. Then:
Indeed, by the minimality of , there exists with . Let be obtained from by adding one directed edge disjoint from , with both ends labeled . Then . So .
From now on in this proof, fix an -dimensional -linear space . So and are well-defined. Then is an algebra homomorphism, with respect to the product on the space of quantum diagrams (which is for diagrams just the disjoint union).
Proof. Let with . We prove that . By splitting into homogeneous components, we can assume that is a linear combination of diagrams that all have the same number of vertices; and more strongly, that all have the same number of vertices of type , for any . Then we can assume (by renaming and deleting unused types) that for some , , and for all .
In fact, we can assume that for each . To see this, consider a type with , and introduce a new type, named , with and . For any diagram , let be the sum of those diagrams that can be obtained from by changing the type of one vertex of type to type . So is the sum of diagrams.
To describe , let be a basis of term in (2. Some applications of invariant theory). For , let be the corresponding element in the basis dual to , and let be the element corresponding to for the new term in (2. Some applications of invariant theory). Then
Let be obtained by replacing each in by . As , (3. Proof of Theorems and) gives . Moreover, implies , as we can apply a reverse map (where is obtained from by replacing type by and dividing by ).
Repeating this operation we finally obtain that each type occurs precisely once in each diagram occurring in . So finally . Then .
Make the following web , having vertices , where has type , for , and having in addition roots and sinks. The tails of the edges entering are roots, labeled , in order, where . Moreover, the heads of the edges leaving are sinks, labeled , in order, where .
For each , let . Then for each diagram with vertices, of types respectively, there exists a unique with . Moreover, for all and :
We can write for unique (for :
Define the following polynomial :
for . Note that determines . As , (3. Proof of Theorems and) implies that for all and . This implies, by the second fundamental theorem (SFT) of invariant theory for (cf.  Theorem 12.2.12), that belongs to the ideal in generated by the minors of . That is,
where is the submatrix of and belongs
As each term of comes from a permutation, the variables in any row of have total degree
1 in .
Similarly, the variables in any column of have total degree 1 in .
This implies that we can assume that each has total degree 1 in rows of with index not in
and total degree 0 in rows in with index in .
Similarly for columns with respect to .
This implies .
This claim and the condition in Theorem Webs = tangle diagrams imply that . Hence there exists a linear function such that . Then is a unital algebra homomorphism, as , and as for all :
Now first suppose that is finite. Since by (2. Some applications of invariant theory), there exists with for all (by the Nullstellensatz). So
for all , proving Theorem Webs = tangle diagrams for finite . By the closed orbit theorem (), we can assume that the orbit is closed. Then, by Proposition 2. Some applications of invariant theory, is nondegenerate.
Suppose next that is infinite. Consider any finite subset of . Let be the variety of tensor representations such that . We saw above that, as is finite, . In fact, is a fiber of the projection . Hence contains a unique -orbit of minimal (Krull) dimension (cf.  1.11 and 1.24). (It is in fact the unique closed orbit in .)
Since for each finite , there exists a finite with as large as possible. Then for each finite with :
where is the natural projection . Indeed,
To see this, note that . Then the first inequality in (3. Proof of Theorems and) follows from the fact that is a -orbit in , and that has minimal dimension among all -orbits in .
Choose an arbitrary and consider some finite . We extend and if possible as follows. By (3. Proof of Theorems and), there exists at least one with . As is -equivariant, for the stabilizers one has
Suppose that the