Diagram calculus for a type affine C Temperley–Lieb algebra, I

Diagram calculus for a type affine
Temperley–Lieb algebra, I

Dana C. Ernst Department of Mathematics and Statistics, Northern Arizona University PO Box 5717, Flagstaff, AZ 86011-5717, USA Dana.Ernst@nau.edu
July 16, 2019
Abstract.

In this paper, we present an infinite dimensional associative diagram algebra that satisfies the relations of the generalized Temperley–Lieb algebra having a basis indexed by the fully commutative elements (in the sense of Stembridge) of the Coxeter group of type affine . Moreover, we provide an explicit description of a basis for the diagram algebra. In the sequel to this paper, we show that this diagrammatic representation is faithful. The results of this paper and its sequel will be used to construct a Jones-type trace on the Hecke algebra of type affine , allowing us to non-recursively compute leading coefficients of certain Kazhdan–Lusztig polynomials.

2000 Mathematics Subject Classification:
20F55, 20C08, 57M15

1. Introduction

The (type ) Temperley–Lieb algebra , invented by H.N.V. Temperley and E.H. Lieb in 1971 [23], is a finite dimensional associative algebra which first arose in the context of statistical mechanics. R. Penrose and L.H. Kauffman showed that this algebra can be realized as a diagram algebra [18, 21], that is, an associative algebra with a basis given by certain diagrams, in which the multiplication rule in the algebra is given by applying local combinatorial rules to the diagrams.

In 1987, V.F.R. Jones showed that occurs naturally as a quotient of the type Hecke algebra [16]. Given a Coxeter group , the associated Hecke algebra has a basis indexed by the elements of and relations that deform the relations of by a parameter . The realization of the Temperley–Lieb algebra as a Hecke algebra quotient was generalized by J.J. Graham in [6] to the case of an arbitrary Coxeter system. In Section 2.3, we define the generalized Temperley–Lieb algebra of type , denoted , in terms of generators and relations and describe a special basis, called the monomial basis, which is indexed by the fully commutative elements (defined in Section 2.2) of the underlying Coxeter group.

The goal of this paper is to introduce a diagrammatic representation of the Temperley–Lieb algebra (in the sense of Graham) of type . The motivation behind this is that a realization of can be of great value when it comes to understanding the otherwise purely abstract algebraic structure of the algebra. Moreover, studying these generalized Temperley–Lieb algebras often provides a gateway to understanding the Kazhdan–Lusztig theory of the associated Hecke algebra. Loosely speaking, the generalized Temperley–Lieb algebra retains some of the relevant structure of the Hecke algebra, yet is small enough that computation of the leading coefficients of the notoriously difficult to compute Kazhdan–Lusztig polynomials is often much simpler.

In this paper, we construct an infinite dimensional associative diagram algebra that satisfies the relations of . In Sections 3 and 4, we establish our notation and introduce all of the necessary terminology required to define , and once this has been done it is trivial to verify that the relations of are satisfied and that there is a surjective algebra homomorphism from to (Proposition 4.1.3). However, due to length considerations, the injectivity of the homomorphism is resolved in the sequel to this paper [4].

One of the major obstacles to proving that our diagrammatic representation is faithful is having a description of a basis for . In Section 4.2, we define the -admissible diagrams by providing a combinatorial description of the allowable edge configurations involving diagram decorations. Our main result (Theorem 5.4.3) comes at the end of a sequence of technical lemmas and states that the -admissible diagrams form a basis for . Finally, in Section 6, we discuss the implications of our results and future research.

With the exception of type , all other generalized Temperley–Lieb algebras with known diagrammatic representations are finite dimensional. In the finite dimensional case, counting arguments are employed to prove faithfulness, but these techniques are not available in the type case since is infinite dimensional. Instead, we will make use of the author’s classification in [3] of the non-cancellable elements in Coxeter groups of types , , and (also see [2, Chapters 3–5]). The classification of the non-cancellable elements in a Coxeter group of type provides the foundation for inductive arguments used to prove the faithfulness of . Once injectivity has been established, the diagram algebra introduced in this paper will be the first faithful representation of an infinite dimensional non-simply-laced generalized Temperley–Lieb algebra (in the sense of Graham).

This paper is an adaptation of the author’s Ph.D. thesis, titled A diagrammatic representation of an affine Temperley–Lieb algebra [2], which was directed by Richard M. Green at the University of Colorado at Boulder. However, the notation has been improved and some of the arguments have been streamlined. In particular, the author’s thesis describes a framework for constructing a large class of diagram algebras and is more general than what often appears in the literature. For the sake of length, we omit here the general construction and focus on our diagram algebra of interest.

2. Preliminaries

2.1. Coxeter groups

A Coxeter group is a group with a distinguished set of generating involutions having presentation

where is a function and if and only if . It turns out that the elements of are distinct as group elements, and that is the order of . Any minimum length expression for in terms of the generators is called a reduced expression (all reduced expressions for have the same length). The pair is called a Coxeter system.

Given a Coxeter system , the associated Coxeter graph is the graph with vertex set and edges for each . Moreover, each edge is labeled with its corresponding -value, although it is customary to omit the label if . Given a Coxeter graph , we can uniquely reconstruct the corresponding Coxeter system . In this case, we say that the corresponding Coxeter system is of type , and denote the Coxeter group and distinguished generating set by and , respectively.

The main focus of this paper will be the Coxeter systems of types and , which are defined by the Coxeter graphs in Figures 0(a) and 0(b), respectively, where .

(a) Coxeter graph of type .
(b) Coxeter graph of type .
Figure 1. Coxeter graphs.

We can obtain from by removing the generator and the corresponding relations [15, Chapter 5]. We also obtain a Coxeter group of type if we remove the generator and the corresponding relations. To distinguish these two cases, we let denote the subgroup of generated by and we let denote the subgroup of generated by . It is well-known that is an infinite Coxeter group while and are both finite [15, Chapters 2 and 6].

2.2. Fully commutative elements

Let be a Coxeter system of type and let . According to Stembridge [22], is fully commutative (FC) if and only if no reduced expression for contains a subword of the form of length . We will denote the set of all FC elements of by or .

The elements of are precisely those whose reduced expressions avoid subwords of the following types:

  1. for and ;

  2. for or .

The FC elements of and avoid the respective subwords above.

By [22, Theorem 5.1], contains an infinite number of FC elements, while (and hence ) contains finitely many. There are examples of infinite Coxeter groups that contain a finite number of FC elements (e.g., ) is infinite for , but contains only finitely many FC elements [22, Theorem 5.1]).

2.3. Generalized Temperley–Lieb algebras

Given a Coxeter graph , we can form an associative algebra, (in the sense of Graham [6]), which we call the Temperley–Lieb algebra of type . For a complete description of the construction of , see [2, 6, 10]. For our purposes it suffices to define in terms of generators and relations. We are using [10, Proposition 2.6] (also see [6, Proposition 9.5]) as our definition.

Definition 2.3.1.

The Temperley–Lieb algebra of type , denoted , is the unital algebra generated by with defining relations

  1. for all , where is an indeterminate;

  2. if ;

  3. if and ;

  4. if or .

In addition, (respectively, ) is generated as a unital algebra by (respectively, ) with the relations above.

It is known that we can consider and as subalgebras of in the obvious way.

Note that when is considered as a quotient of the Hecke algebra of type with indeterminate , the indeterminate is defined to be the Laurent polynomial .

Let be a reduced expression for , where each . Define the element via

It is well-known (and follows from [10, Proposition 2.4]) that the set forms a -basis for . This basis is referred to as the monomial basis or “-basis.”

If is a Coxeter system of type , the associated Hecke algebra is an algebra with a basis indexed by the elements of and relations that deform the relations of by a parameter . In general, is a quotient of , having several bases indexed by the FC elements of  [6, Theorem 6.2]. Except for in the case of type , there are many Temperley–Lieb type quotients that appear in the literature. That is, some authors define a Temperley–Lieb algebra to be a different quotient of than the one we are interested in. In particular, the blob algebra of [20] is a smaller Temperley–Lieb type quotient of than . Also, the symplectic blob algebra of [14] and  [19] is a finite rank quotient of , whereas, is of infinite rank. Furthermore, despite being infinite dimensional, the two-boundary Temperley–Lieb algebra of [5] is a quotient of different from . Typically, authors that study these usually smaller Temperley–Lieb type quotients are interested in representation theory, whereas our motivation is Kazhdan–Lusztig theory.

3. Diagram algebras

The goal of this section is to familiarize the reader with the necessary background on diagram algebras. It is important to note that there is currently no rigorous definition of the term “diagram algebra.” Our diagram algebras possess many of the same features as those already appearing in the literature, however the typical developments are too restrictive to accomplish the task of finding a faithful diagrammatic representation of the infinite dimensional Temperley–Lieb algebra (in the sense of Graham) of type . Yet, our approach is modeled after [9][14][17], and [19].

3.1. Undecorated diagrams

First, we discuss undecorated diagrams and their corresponding diagram algebras.

Definition 3.1.1.

Let be a nonnegative integer. The standard -box is a rectangle with marks points, called nodes (or vertices) labeled as in Figure 2. We will refer to the top of the rectangle as the north face and the bottom as the south face.

Sometimes, it will be useful for us to think of the standard -box as being embedded in the plane. In this case, we put the lower left corner of the rectangle at the origin such that each node (respectively, ) is located at the point (respectively, ).

Figure 2. The standard -box.

The next definition summarizes the construction of the ordinary Temperley–Lieb pseudo diagrams.

Definition 3.1.2.

A concrete pseudo -diagram consists of a finite number of disjoint curves (planar), called edges, embedded in the standard -box with the following restrictions. The nodes of the box are the endpoints of edges, which meet the box transversely. All other edges must be closed (isotopic to circles) and disjoint from the box. We define an equivalence relation on the set of concrete pseudo -diagrams. Two concrete pseudo -diagrams are (isotopically) equivalent if one concrete diagram can be obtained from the other by isotopically deforming the edges such that any intermediate diagram is also a concrete pseudo -diagram. A pseudo -diagram (or an ordinary Temperley-Lieb pseudo-diagram) is defined to be an equivalence class of equivalent concrete pseudo -diagrams. We denote the set of pseudo -diagrams by .

Example 3.1.3.

The diagram in Figure 3 is an example of a concrete pseudo 5-diagram.

Figure 3. A concrete pseudo 5-diagram.
Remark 3.1.4.

When representing a pseudo -diagram with a drawing, we pick an arbitrary concrete representative among a continuum of equivalent choices. When no confusion can arise, we will not make a distinction between a concrete pseudo -diagram and the equivalence class that it represents.

We will refer to a closed curve occurring in the pseudo -diagram as a loop edge, or simply a loop. The diagram in Figure 3 has a single loop. Note that we used the word “pseudo” in our definition to emphasize that we allow loops to appear in our diagrams. Most examples of diagram algebras occurring in the literature “scale away” loops that appear. There are loops in the diagram algebra that we are interested in preserving, so as to obtain infinitely many diagrams. The presence of in the definition above is to emphasize that the edges of the diagrams are undecorated. In the next section, we allow for the presence of decorations.

Let be a diagram. If has an edge that joins node in the north face to node in the south face, then is called a propagating edge from to . (Propagating edges are often referred to as “through strings” in the literature.) If a propagating edge joins to , then we will call it a vertical propagating edge. If an edge is not propagating, loop edge or otherwise, it will be called non-propagating.

If a diagram has at least one propagating edge, then we say that is dammed. If, on the other hand, has no propagating edges (which can only happen if is even), then we say that is undammed. Note that the number of non-propagating edges in the north face of a diagram must be equal to the number of non-propagating edges in the south face. We define the function via

There is only one diagram with -value having no loops; namely the diagram that appears in Figure 4. The maximum value that can take is . In particular, if is even, then the maximum value that can take is , i.e., is undammed. On the other hand, if while is odd, then has a unique propagating edge.

Figure 4. The only diagram having -value 0 and no loops.

We wish to define an associative algebra that has the pseudo -diagrams as a basis.

Definition 3.1.5.

Let be a commutative ring with . The associative algebra over is the free -module having as a basis, with multiplication defined as follows. If , the product is the element of obtained by placing on top of , so that node of coincides with node of , rescaling vertically by a factor of and then applying the appropriate translation to recover a standard -box. (For a proof that this procedure does in fact define an associative algebra see [9, §2] and [17].)

We will refer to the multiplication of diagrams as diagram concatenation. The (ordinary) Temperley–Lieb diagram algebra (see [7, 9, 17, 21]) can be easily defined in terms of this formalism.

Definition 3.1.6.

Let be the associative -algebra equal to the quotient of by the relation depicted in Figure 5.

Figure 5. The defining relation of .

It is well-known that is the free -module with basis given by the elements of having no loops. The multiplication is inherited from the multiplication on except we multiply by a factor of for each resulting loop and then discard the loop. We will refer to as the (ordinary) Temperley–Lieb diagram algebra.

Example 3.1.7.

Figure 6 depicts the product of three basis diagrams of .

Figure 6. An example of multiplication in .

As -algebras, the Temperley–Lieb algebra that was briefly discussed in Section 1 is isomorphic to . Moreover, each loop-free diagram from corresponds to a unique monomial basis element of . For more details, see [18] and [21].

3.2. Decorated diagrams

We wish to adorn the edges of a diagram with elements from an associative algebra having a basis containing . First, we need to develop some terminology and lay out a few restrictions on how we decorate our diagrams.

Let and consider the free monoid . We will use the elements of to adorn the edges of a diagram and we will refer to each element of as a decoration. In particular, and are called closed decorations, while and are called open decorations. Let be a finite sequence of decorations in . We say that and are adjacent in if and we will refer to as a block of decorations of width . Note that a block of width is just a single decoration. The string is an example of a block of width 7 from .

We have several restrictions for how we allow the edges of a diagram to be decorated, which we will now outline. Let be a fixed concrete pseudo -diagram and let be an edge of .

  1. If , then is undecorated.

In particular, the unique diagram with -value 0 and no loops is undecorated.

Subject to some restrictions, if , we may adorn with a finite sequence of blocks of decorations such that adjacency of blocks and decorations of each block is preserved as we travel along .

If is a non-loop edge, the convention we adopt is that the decorations of the block are placed so that we can read off the sequence of decorations as we traverse from to if is propagating, or from to (respectively, to ) with (respectively, ) if is non-propagating.

If is a loop edge, reading the corresponding sequence of decorations depends on an arbitrary choice of starting point and direction round the loop. We say two sequences of blocks are loop equivalent if one can be changed to the other or its opposite by any cyclic permutation. Note that loop equivalence is an equivalence relation on the set of sequences of blocks. So, the sequence of blocks on a loop is only defined up to loop equivalence. That is, if we adorn a loop edge with a sequence of blocks of decorations, we only require that adjacency be preserved.

Each decoration on has coordinates in the -plane. In particular, each decoration has an associated -value, which we will call its vertical position.

If , then we also require the following.

  1. All decorated edges can be deformed so as to take closed decorations to the left wall of the diagram and open decorations to the right wall simultaneously without crossing any other edges.

  2. If is non-propagating (loop edge or otherwise), then we allow adjacent blocks on to be conjoined to form larger blocks.

  3. If and is propagating, then as in 2, we allow adjacent blocks on to be conjoined to form larger blocks.

  4. If , then we have the following.

    1. All decorations occurring on propagating edges must have vertical position lower (respectively, higher) than the vertical positions of decorations occurring on the (unique) non-propagating edge in the north face (respectively, south face) of .

    2. If a block on a propagating edge contains decorations occurring at vertical positions and with , then no other propagating edge may contain decorations at vertical positions in the interval .

    3. Two adjacent blocks occurring on a propagating edge may be conjoined to form a larger block as long as (b) is not violated.

We call a block maximal if its width cannot be increased by conjoining it with another block without violating 4.

Requirement 1 is related to the concept of “exposed” that appears in the context of the Temperley–Lieb algebra of type  [7, 8, 9]. The general idea is to mimic what happens in the type case on both the east and west ends of the diagrams. Note that 4 is an unusual requirement for decorated diagrams. We require this feature to ensure faithfulness of our diagrammatic representation on the monomial basis elements of indexed by the type I elements of the Coxeter group of type (see [3]).

Definition 3.2.1.

A concrete LR-decorated pseudo -diagram is any concrete -diagram decorated by elements of that satisfies conditions 14.

Example 3.2.2.

Here are a few examples.

  1. The diagram in Figure 6(a) is an example of a concrete LR-decorated pseudo -diagram. In this diagram, there are no restrictions on the relative vertical position of decorations since the -value is greater than 1. The decorations on the unique propagating edge can be conjoined to form a maximal block of width 4.

  2. The diagram in Figure 6(b) is another example of a concrete LR-decorated pseudo -diagram, but with -value 1. We use the horizontal dotted lines to indicate that the three closed decorations on the leftmost propagating edge are in three distinct blocks. We cannot conjoin these three decorations to form a single block because there are decorations on the rightmost propagating edge occupying vertical positions between them. Similarly, the open decorations on the rightmost propagating edge form two distinct blocks that may not be conjoined.

  3. Finally, the diagram in Figure 6(c) is an example of a concrete LR-decorated pseudo -diagram with maximal -value and no propagating edges.

(a)
(b)
(c)
Figure 7. Examples of concrete LR-decorated pseudo diagrams.

Note that an isotopy of a concrete LR-decorated pseudo -diagram that preserves the faces of the standard -box may not preserve the relative vertical position of the decorations even if it is mapping to an equivalent diagram. The only time equivalence is an issue is when . In this case, we wish to preserve the relative vertical position of the blocks. We define two concrete pseudo LR-decorated -diagrams to be -equivalent if we can isotopically deform one diagram into the other such that any intermediate diagram is also a concrete LR-decorated pseudo -diagram. Note that we do allow decorations from the same maximal block to pass each other’s vertical position (while maintaining adjacency).

Definition 3.2.3.

An LR-decorated pseudo -diagram is defined to be an equivalence class of -equivalent concrete LR-decorated pseudo -diagrams. We denote the set of LR-decorated diagrams by .

As in Remark 3.1.4, when representing an LR-decorated pseudo -diagram with a drawing, we pick an arbitrary concrete representative among a continuum of equivalent choices. When no confusion will arise, we will not make a distinction between a concrete LR-decorated pseudo -diagram and the equivalence class that it represents.

Remark 3.2.4.

We make several observations.

  1. The set of LR-decorated diagrams is infinite since there is no limit to the number of loops that may appear.

  2. If is an undammed LR-decorated diagram, then all closed decorations occurring on an edge connecting nodes in the north face (respectively, south face) of must occur before all of the open decorations occurring on the same edge as we travel the edge from the left node to the right node. Otherwise, we would not be able to simultaneously deform decorated edges to the left and right. Furthermore, if an edge joining nodes in the north face of is adorned with an open (respectively, closed) decoration, then no non-propagating edge occurring to the right (respectively, left) in the north face may be adorned with closed (respectively, open) decorations. We have an analogous statement for non-propagating edges in the south face.

  3. Loops can only be decorated by both types of decorations if is undammed. Again, we would not be able to simultaneously deform decorated edges to the left and right, otherwise.

  4. If is a dammed LR-decorated diagram, then closed decorations (respectively, open decorations) only occur to the left (respectively, right) of and possibly on the leftmost (respectively, rightmost) propagating edge. The only way a propagating edge can have decorations of both types is if there is a single propagating edge, which can only happen if is odd.

Example 3.2.5.

The diagram of Figure 6(c) is an example that illustrates conditions (2) and (3) of Remark 3.2.4, while the diagram of Figure 6(a) illustrates condition (4).

Definition 3.2.6.

We define to be the free -module having the LR-decorated pseudo -diagrams as a basis.

We define multiplication in by defining multiplication in the case where and are basis elements, and then extend bilinearly. To calculate the product , concatenate and (as in Definition 3.1.5). While maintaining -equivalence, conjoin adjacent blocks. We claim that the multiplication just defined turns into a well-defined associative -algebra. To justify this claim we require the following lemma.

Lemma 3.2.7.

Let be diagram with . Suppose that the unique non-propagating edge in the north face of joins to . Let be any other diagram with . Then if and only if and the unique non-propagating edge in the south face of joins either (a) to , (b) to , or (c) to .

Proof.

First, assume that . It is a general fact that , which implies that .

Conversely, assume that and that the unique non-propagating edge in the south face of joins either (a) to , (b) to , or (c) to .

Assume that we are in situation (a). Suppose that the propagating edge leaving node in the south face of is connected to node in the north face. Also, suppose that the propagating edge leaving node in the north face of is connected to node in the south face. Then has a propagating edge joining node to node . Furthermore, the only non-propagating edge in the north (respectively, south) face of is the same as the unique non-propagating edge in the north (respectively, south) face of (respectively, ). It follows that .

Next, assume we are in case (b). Then has one more loop than the sum total of loops from and . Furthermore, the only non-propagating edge in the north (respectively, south) face of is the same as the unique non-propagating edge in the north (respectively, south) face of (respectively, ), and so .

Finally, if we are in situation (c), then the proof that is symmetric to case (a). ∎

It is quickly seen that concatenating two diagrams that satisfy 1 will result in a diagram that satisfies the same conditions. The claim that is a well-defined associative -algebra now follows from arguments in [19, §3] and Lemma 3.2.7 above. The only case that requires serious consideration is when multiplying two diagrams that both have -value . If while , then there are no concerns. However, if , then according to Lemma 3.2.7, if the unique non-propagating edge in the south face of joins to , it must be the case that the unique non-propagating edge in the north face of joins either (a) to , (b) to , or (c) to . If (a) or (c) occurs, then the only blocks that get conjoined are the blocks on and , which presents no problems. If (b) occurs, then we get a loop edge and we conjoin the blocks from and . As a consequence, it is possible that the block occurring on a propagating edge of having the lowest vertical position may be conjoined with the block occurring on a propagating edge of having the highest vertical position. This can only happen if these two edges are joined in , and regardless, presents no problems.

We remark that since the set of LR-decorated diagrams is infinite, is an infinite dimensional algebra.

3.3. Diagrammatic relations

Our immediate goal is to define a quotient of having relations that are determined by applying local combinatorial rules to the diagrams.

Let and define the algebra to be the quotient of by the following relations:

  1. ;

  2. ;

  3. ;

  4. .

The algebra is associative and has a basis consisting of the identity and all finite alternating products of open and closed decorations.

For example, in we have

where the expression on the right is a basis element, while the expression on the left is a block of width 7, but not a basis element. We will refer to as our decoration algebra.

The point is that there is no interaction between open and closed symbols. It turns out that if , the algebra is equal to the free product of two rank 3 Verlinde algebras. For more details, see Chapter 7 of the author’s Ph.D. thesis [2].

Definition 3.3.1.

Let be the associative -algebra equal to the quotient of by the relations depicted in Figure 8, where the decorations on the edges represent adjacent decorations of the same block.

(a)

(b)

(c)

(d)

(e)
Figure 8. The defining relations of .

Note that with the exception of the relations involving loops, multiplication in is inherited from the relations of the decoration algebra . Also, observe that all of the relations are local in the sense that a single reduction only involves a single edge. As a consequence of the relations in Figure 8, we also have the relations of Figure 9.

(a)

(b)
Figure 9. Additional relations of .
Example 3.3.2.

Figure 10 depicts multiplication of three diagrams in and Figure 11 shows an example where each of the diagrams and their product have -value 1. Again, we use the dotted line to emphasize that the two closed decorations on the leftmost propagating edge belong to distinct blocks.

= 2

Figure 10. Example of multiplication in .

=

Figure 11. Example of multiplication in with diagrams having -value 1.

3.4. Irreducible LR-decorated diagrams as a basis

We need to show that a basis for consists of the set of LR-decorated diagrams having maximal blocks corresponding to nonidentity basis elements in . That is, no block may contain adjacent decorations of the same type (open or closed). To accomplish this task, we will make use of a diagram algebra version of Bergman’s Diamond Lemma [1]. For other examples of this type of application of Bergman’s Diamond Lemma, see [14] and [19].

Define the function via

In the literature, if has no loops, then is sometimes referred to as the “shape” of .

Next, define a function via

Define on via if and only if and .

Consider the collection of reductions determined by the relations of given in Definition 3.3.1. If we apply any single reduction (loop removal or any other local reduction) to a diagram from , then we obtain a scalar multiple of a strictly smaller diagram with respect to . Thus, our reduction system (i.e., diagram relations) is compatible with . Now, suppose that and let be any other element from . Then and . Since , multiplying or on the same side by will increase the number of decorations and number of loops by the same amount. So, we have and . Therefore, and . This shows that is a semigroup partial order on . Clearly, satisfies the descending chain condition.

Proposition 3.4.1.

The set of LR-decorated diagrams having no relations to apply forms a basis for .

Proof.

Let be as above. Following the setup of Bergman’s Diamond Lemma, it remains to show that all of the ambiguities are resolvable.

By inspecting the relations of Definition 3.3.1, we see that there are no inclusion ambiguities, so we only need to check that the overlap ambiguities are resolvable.

Let be a diagram from and suppose that there are two competing reductions that we could apply. If both reductions involve the same non-loop edge, then the ambiguity is easily seen to be resolvable since the algebra is associative. In particular, in the -value case, the reductions could involve two distinct blocks on the same edge, in which case, the order that we apply the reductions is immaterial. If the reductions involve distinct edges, loop edges or otherwise, the ambiguity is quickly seen to be resolvable since the reductions commute. Finally, suppose that the two competing reductions involve the same loop edge. There are three possibilities for this loop edge: (a) the loop is undecorated, (b) the loop carries only one type of decoration (open or closed), and (c) the loop carries both types of symbols. Note that (a) cannot happen since then there could not have been two competing reductions involving this edge to apply. If (b) occurs, then any ambiguity involving this loop edge (including removing the loop) is resolvable since multiplication of closed (respectively, open) decorations is commutative and associative. Finally, assume (c) occurs. Note that the nature of our relations prevents the complete elimination of closed (respectively, open) decorations from this loop edge. Since all loop relations involve either undecorated loops or loops decorated with a single type of decoration, this loop edge can never be removed. Since is associative and none of the relations involve both decoration types at the same time, the ambiguity is easily seen to be resolvable since the reductions commute.

According to Bergman’s Diamond Lemma [1], we can conclude that the set of LR-decorated diagrams having no relations to apply is a basis, as desired. ∎

4. The simple and admissible diagrams

In this section, we define the diagram algebra as a subalgebra of that turns out to be a faithful diagrammatic representation of . We will be able to quickly conclude that there is a surjective homomorphism from to . In the sequel to this paper [4], we show that this homomorphism is injective, thus showing that the algebras are isomorphic. In the next section of this paper, we define the admissible diagrams and show that they are a basis for . In fact, we will show that the image of each monomial basis element of is admissible.

4.1. Simple diagrams

Define the simple diagrams as in Figure 12. Note that the simple diagrams are elements of the basis for described in Proposition 3.4.1.

Figure 12. The simple diagrams.

It is not immediately obvious, but we shall see that the algebra generated by the simple diagrams is infinite dimensional yet strictly smaller that .

Remark 4.1.1.

Checking that each of the following relations is satisfied for the simple diagrams is easily verified.

  1. for all ;

  2. if ;

  3. if and ;

  4. if or .

This shows that the simple diagrams satisfy the relations of given in Definition 2.3.1.

Finally, we are ready to define the diagram algebra that we are ultimately interested in. Defining the algebra is easy, but having a description of a collection of basis diagrams is not. The issue of the basis will be handled in Section 5.

Definition 4.1.2.

Let be the -subalgebra of generated as a unital algebra by with multiplication inherited from .

Now, define to be the function determined by . The next theorem follows quickly.

Proposition 4.1.3.

The map defined above is a surjective algebra homomorphism.

Proof.

By Remark 4.1.1, the simple diagrams satisfy the relations of . This shows that is an algebra homomorphism, but since the simple diagrams generate , is surjective. ∎

The main result of the sequel to this paper [4] is that is injective.

4.2. Admissible diagrams

The next definition describes the set of -admissible diagrams, which will turn out to form a basis for . Our definition of -admissible is motivated by the definition of -admissible (after an appropriate change of basis) given by R.M. Green in [8, Definition 2.2.4] for diagrams in the context of type . Since the Coxeter graph of type is type at “both ends”, the general idea is to build the axioms of -admissible into our definition of -admissible on the left and right sides of our diagrams.

Definition 4.2.1.

Let be an irreducible LR-decorated diagram. Then we say that is -admissible, or simply admissible, if the following axioms are satisfied.

  1. The only loops that may appear are equivalent to the one in Figure 13.

    Figure 13. The only allowable loop in -admissible diagrams.
  2. Assume and let be the edge connected to node 1. If is not connected to node , then it is decorated and the first decoration is a . If is connected to , then exactly one of the following three conditions are met:

    1. is undecorated.

    2. is decorated by a single .

    3. is decorated by a single block of decorations consisting of an alternating sequence of closed and open decorations such that the first decoration is a .

    We have analogous restrictions for nodes , , and , where we replace first with last for nodes and and closed decorations are replaced with open decorations for nodes and .

  3. Assume . Then the western end of is equal to one of the diagrams in Figure 14, where and the other rectangles represent a sequence of blocks (possibly empty) such that each block is a single . Moreover, if is the diagram in Figure 13(b), then no more decorations occur on . Also, the occurrences of the decorations occurring on the propagating edges have the highest (respectively, lowest) relative vertical position of all decorations occurring on any propagating edge. We have an analogous restrictions for the eastern end of , where the closed decorations are replaced with open decorations.

  4. No other or decorations appear on other than those required in 2 and 3.

Let denote the set of all -admissible -diagrams.

(a)
(b)
(c)
(d)
(e)
Figure 14. The western end of diagrams exhibiting axiom 3.
Remark 4.2.2.

We collect several comments concerning the admissible diagrams.

  1. The only time an admissible diagram can have an edge adorned with both open and closed decorations is if is undammed (which only happens when is even) or if has a single propagating edge (which only happens when is odd). The latter case coincides with part (c) of axiom 2. See parts (a) and (c) of Example 3.2.2 for examples of diagrams having edges adorned with both types of decorations.

  2. If is an admissible diagram with , then the restrictions on the relative vertical position of decorations on propagating edges along with axiom 3 imply that the relative vertical positions of closed decorations on the leftmost propagating edge and open decorations on the rightmost propagating edge must alternate. In particular, the number of closed decorations occurring on the leftmost propagating edge differs from the number of open decorations occurring on the rightmost propagating edge by at most 1. For example, if is the diagram in Figure 15, where the leftmost propagating edge carries decorations, then the rightmost propagating edge must carry decorations, as well.

    Figure 15. Example of a diagram exhibiting axiom 3.
  3. It is clear that is an infinite set. If an admissible diagram is undammed, then there is no limit to the number of loops given in axiom 1 that may occur. Also, if is an admissible diagram with exactly one propagating edge, then there is no limit to the width of the block of decorations that may occur on the lone propagating edge. Furthermore, if is admissible with , then there is no limit to the number of -blocks (respectively, -blocks) that may occur on the leftmost (respectively, rightmost) propagating edge.

  4. Each of the admissible diagrams is a basis element of .

  5. The symbol in the notation is to emphasize that we are constructing a set of diagrams that is intended to correspond to the monomial basis of . In a sequel to this paper, we will construct diagrams that correspond to the “canonical basis” of , which is defined for arbitrary Coxeter groups in [13].

Definition 4.2.3.

Let be the -submodule of spanned by the admissible diagrams.

Proposition 4.2.4.

The set of admissible diagrams is a basis for the module .

Proof.

Linear independence follows immediately from Remark 4.2.2(4). ∎

4.3. Temperley–Lieb diagram algebras of type

We will briefly discuss how and are related to .

Definition 4.3.1.

Let and denote the subalgebras of generated by the simple diagrams and , respectively. We refer to (respectively, ) as the Temperley–Lieb diagram algebra of type (respectively, type ).

It is clear that (respectively, ) consists entirely of diagrams that are decorated with only closed (respectively, open) decorations. Also, note that all of the technical requirements about how to decorate a diagram when are irrelevant since only the leftmost (respectively, rightmost) propagating edge can carry decorations in (respectively, ). The following fact is implicit in [8, §2] after the appropriate change of basis involving a change of basis for the decoration set.

Proposition 4.3.2.

As -algebras, and , where each isomorphism is determined by for the appropriate restrictions on .

After making the appropriate change of basis on the decoration set (which involves making a change of basis on the rank 3 Verlinde algebra), the basis diagrams in (respectively, ) become -admissible in the sense of [8, 9]. Moreover, it is easily verified that the axioms for -admissible given in [8, Definition 2.2.4] imply (again, under the appropriate change of basis involving the decoration set) that all of the basis diagrams in and are -admissible.

5. A basis for

Our main objective in the remainder of this paper is to show that the admissible diagrams form a basis for . Before proceeding, we wish to outline our method of attack. We will show the following:

  1. The admissible diagrams are generated by the simple diagrams (Proposition 5.2.4).

  2. The module is closed under multiplication, making it a subalgebra of (Corollary 5.4.2).

  3. The algebras and are equal having the admissible diagrams as a basis (Theorem 5.4.3).

Items (1) and (2) above require numerous technical lemmas. However, once we overcome these difficulties, (3) will yield itself easily.

5.1. Preparatory lemmas

Our first significant obstacle in proving that the -admissible diagrams form a basis for is proving that the admissible diagrams are generated by the simple diagrams (see Proposition 5.2.4). To achieve this end, we require several intermediate results.

If is an admissible diagram, then we say that a non-propagating edge joining to (respectively, to ) is simple if it is identical to the edge joining the same nodes of the simple diagram . Note that a simple edge is undecorated except when one of the vertices is 1 or (respectively, or ), in which case it is decorated by only a single (respectively, ).

The next six lemmas mimic Lemmas 5.1.4–5.1.7 in [8]. The proof of each lemma is immediate and throughout we assume that is admissible. Figure 16 provides visual representations of each lemma, where represents an arbitrary (possibly empty) block of decorations. Each of Lemmas 5.1.15.1.6 have left-right symmetric analogues (perhaps involving closed decorations), as well as versions that involve edges in the south face.

(a) Lemma 5.1.1
(b) Lemma 5.1.2
(c) Lemma 5.1.3
(d) Lemma 5.1.4
(e) Lemma 5.1.5
(f) Lemma 5.1.6
Figure 16. Visual representations of Lemmas 5.1.15.1.6
Lemma 5.1.1.

Assume that in the north face of there is an edge, say , connecting node to node , and assume that there is another, undecorated, edge, say , connecting node to node with and . Then is the admissible diagram that results from by removing , disconnecting from node and reattaching it to node , and adding a simple edge to and (note that edge maintains its original decorations). See Figure 15(a).

Lemma 5.1.2.

Assume that in the north face of there is an edge, say , connecting node to node labeled by a single (this can happen only if is even), and assume that there is a simple edge, say , connecting node to node (which must be labeled by a single ). Then is the admissible diagram that results from by joining the right end of to the left end of , and adding a simple edge that joins to . Note that the new edge formed by joining and connects node to node and is labeled by the block . See Figure 15(b).

Lemma 5.1.3.

Assume that has a propagating edge, say , joining node to node with . Further, assume that there is a simple edge, say , joining nodes and . Then is the admissible diagram that results from by removing , disconnecting from node and reattaching it to node , and adding a simple edge to and (note that retains its original decorations). See Figure 15(c).

Lemma 5.1.4.

Assume that has simple edges joining node to node and node to node . Then is the admissible diagram that results from by adding a to the edge joining to . See Figure 15(d).

Lemma 5.1.5.

Assume that has two edges, say and , joining node to node and node to node , respectively, where and is simple. Then is the admissible diagram that results from by removing the decorations from and adding them to . This procedure has an inverse, since . See Figure 15(e).

Lemma 5.1.6.

Assume that has two edges, say and , joining node to node and node to node , respectively, with . Further, assume that is decorated by a single decoration only and that is decorated by a single decoration only. Then is the admissible diagram that results from by removing the decoration from and adding it to to the right of the decoration. See Figure 15(f).

5.2. The admissible diagrams are generated by the simple diagrams

Next, we state and prove several lemmas that we will use to prove that each admissible diagram can be written as a product of simple diagrams in .

Lemma 5.2.1.

If is an admissible diagram with , then can be written as a product of simple diagrams.

Proof.

Assume that is an admissible diagram with . The proof is an exhaustive case by case check, where we consider all the possible diagrams that are consistent with axiom 3. We consider five cases (see Figure 17); all remaining cases follow by analogous arguments.

(a)
(b)
(c)
(d)
(e)
Figure 17. The five cases of Lemmas 5.2.1 and 5.3.7.

Case (1). First, assume that is the diagram in Figure 16(a), where the leftmost propagating edge carries decorations, and hence, the rightmost propagating edge carries decorations by Remark 4.2.2(2). In this case, it can quickly be verified that we can obtain via

where and . Therefore, can be written as a product of simple diagrams, as desired.

Case (2). For the second case, assume that is the diagram in Figure 16(b), where . Note that does not carry any open decorations. In this case, either or , and regardless can be written as a product of simple diagrams, as expected.

Case (3). For the third case, assume that is the diagram in Figure 16(c), where the leftmost propagating edge carries decorations, so that the rightmost propagating edge carries decorations. Then

where and are as in case (1), and hence can be written as a product of simple diagrams.

Case (4). Next, assume that is the diagram in Figure 16(d), where the simple edge in the south face connects nodes and with , and the leftmost propagating edge carries decorations. Then by Remark 4.2.2(2), the rightmost propagating edge carries decorations, where or . If , then define to be the diagram in Figure 16(a), where the leftmost (respectively, rightmost) propagating edge carries (respectively, ) decorations. By case (1), can be written as a product of simple diagrams. We see that

which implies that can be written as a product of simple diagrams, as desired. If, on the other hand, , then define to be identical to except that the last decoration occurring on the leftmost propagating edge has been removed. Then by the subcase we just completed (where the rightmost propagating edge carried one more decoration than the leftmost propagating edge carried decorations), can be written as a product of simple diagrams. We see that

which implies that can be written as a product of simple diagrams.

Case (5). For the final case, assume that is the diagram in Figure 16(e), where the simple edge in the north face connects nodes and with , the simple edge in the south face connects nodes and with , and the leftmost propagating edge carries decorations. Then again by Remark 4.2.2(2), the rightmost propagating edge carries decorations, where . Without loss of generality, assume that , so that or . If , define to be the diagram in Figure 16(d), where the leftmost propagating edge carries decorations while the rightmost propagating edge carries decorations. By case (4), can be written as a product of simple diagrams. We see that

which implies that can be written as a product of simple diagrams. If, on the other hand, , then without loss of generality, assume that the first decoration occurring on the leftmost propagating edge has the highest relative vertical position of all decorations occurring on propagating edges. Define to be the diagram in Figure 15, where the leftmost (respectively, rightmost) propagating edge carries (respectively, ) decorations. Again, by case (4), can be written as a product of simple diagrams. Also, we see that

which implies that can be written as a product of simple diagrams, as desired. ∎

Lemma 5.2.2.

If is an admissible diagram with such that all non-propagating edges are simple, then can be written as a product of simple diagrams.

Proof.

Let be an admissible diagram with such that all non-propagating edges are simple. (Note that the restrictions on imply that has more than one propagating edge and at least one non-propagating edge.) We consider two cases, where the second case has two subcases.

Case (1). First, assume that has a vertical propagating edge, say , joining to . Now, define the admissible diagrams and to be the diagrams in Figures 17(a) and 17(b), respectively, where each of the shaded regions is identical to the corresponding regions of . Then . Furthermore, since (respectively, ) satisfies requirement 1 for LR-decorated diagrams, the diagram is only decorated with closed (respectively, open) decorations. Since is admissible, while . This implies that both and can be written as a product of simple diagrams. Therefore, can be written as a product of simple diagrams, as desired.

(a)
(b)
Figure 18. Diagrams for case (1) of the proof of Lemma 5.2.1.

Case (2). Next, assume that has no vertical propagating edges. Suppose that the leftmost propagating edge joins node in the north face to node in the south face, and without loss of generality, assume that . (Note that since has more than one propagating edge, .) We wish to make use of case (1), but we must consider two subcases.

(a) For the first subcase, assume that . Since is admissible, must be the diagram in Figure 18(a),where on the propagating edge from to is either trivial (i.e., the edge is undecorated) or equal to a single decoration. Define the admissible diagram to be the diagram in Figure 18(b), where the leftmost propagating edge carries the same decoration as the leftmost propagating edge in and the shaded region is identical to the corresponding region of . By case (1), can be written as a product of simple diagrams. By making repeated applications of Lemma 5.1.3, we can transform into , which shows that can be written as a product of simple diagrams, as desired.

(a)
(b)
Figure 19. Diagrams for case (2)(a) of the proof of Lemma 5.2.1.

(b) For the second subcase, assume that , so that is the diagram in Figure 19(a). Since , there is at least one other propagating edge occurring to the right of the leftmost propagating edge. Furthermore, since the number of non-propagating edges in the north face is equal to the number of non-propagating edges in the south face, there is at least one undecorated non-propagating edge in the south face of . By making repeated applications, if necessary, of the southern version of Lemma 5.1.3, we may assume that is the diagram in Figure 19(b).

(a)
(b)
(c)
(d)
Figure 20. Diagrams for case (2)(b) of the proof of Lemma 5.2.1.

Now, define the admissible diagrams and via the diagrams in Figures 19(c) and 19(d), respectively, where the shaded regions are identical to the corresponding regions of . By case (1), can be written as a product of simple diagrams. Also, we see that , which implies that can be written as a product of simple diagrams, as well. By making repeated applications of Lemma 5.1.3, we must have that can be written as a product of simple diagrams. ∎

Lemma 5.2.3.

If is odd and is an admissible diagram with