Defining an Affine Partition Algebra

We define an affine partition algebra by generators and relations and prove a variety of basic results regarding this new algebra analogous to those of other affine diagram algebras. In particular we show that it extends the Schur-Weyl duality between the symmetric group and the partition algebra. We also relate it to the affine partition category recently defined by J. Brundan and M. Vargas. Moreover, we show that this affine partition category is a full monoidal subcategory of the Heisenberg category.


Introduction
Classical Schur-Weyl duality relates the representations of the symmetric group and the general linear group via their commuting actions on tensor space.The Brauer algebra was introduced in [B37] to play the role of the symmetric group in a corresponding duality for the symplectic and orthogonal groups.The partition algebra was originally defined by P. Martin in [M91] in the context of Statistical Mechanics.V. Jones showed in [J94] that it appears in another version of Schur-Weyl duality.More precisely, if one replaces the general linear group by the finite subgroup of all permutation matrices then the centraliser algebra of its action on tensor space is precisely the partition algebra.The aim of this paper is to define an affine version of the partition algebra.
There are different 'affinization' processes for such algebras.One such process amounts to making the Jucys-Murphy elements of the ordinary algebra into variables, retaining some of the relations between these variables and the standard generators of the ordinary algebra.Starting with the symmetric group algebra, this 'affinization' process gives rise to the much-studied degenerate affine Hecke algebra (see for example [K05]).In the case of the Brauer algebra, M. Nazarov used this process in [Naz96] to define the affine Wenzl algebra (also called the Nazarov-Wenzl algebra or the degenerate affine BMW algebra in the literature).This process was also employed independently in both [RS15] and [Sar13] to define a degenerate affine walled Brauer algebra.R. Orellana and A. Ram introduced a different 'affinization' process in [OR07] focussed on extending the Schur-Weyl dualities, via the affine braid group.They have applied it to the symmetric group, Brauer algebra and their quantum analogues.This process naturally leads to cyclotomic quotients of the affine algebras.
In this paper, we follow the first approach to define an affine partition algebra by turning the Jucys-Murphy elements for the partition algebra into variables and ask them to retain certain relations with the generators.But we also show that this affine partition algebra naturally extends the commuting action on tensor space with the symmetric group.We started this work by trying to use the presentation for the partition algebra given by T. Halverson and A. Ram in [HR05] but were unable to define an algebra with the expected properties in this way.So we instead used the more recent presentation given by J. Enyang in [Eny12] (which uses a new set of generators) to define the affine partition algebra A aff 2k .We prove that it satisfies many properties analogous to those for other affine diagram algebras.
While writing this paper, J. Brundan and M. Vargas produced a preprint [BV21] defining an affine partition category APar as a monoidal subcategory of the Heisenberg category generated by some objects and morphisms.Taking an endomorphism algebra in their category gives an alternative definition of an affine partition algebra, which they denote by AP k .They prove many properties for this category and use it to give a new approach to the representation theory of the partition category.However, as they note in [BV21,Remark 4.12] they have not attempted to give a basis for the morphism spaces in their category, or to give a presentation for it.Inspired by their work, we have explored the connection between our affine partition algebra and the Heisenberg category.We have added a section at the end of our paper where we construct a surjective homomorphism from A aff 2k to an endomorphism algebra in the Heisenberg category.Our argument generalises to show that the affine partition category APar of Brundan and Vargas is in fact the full monoidal subcategory of the Heisenberg category generated by one object.Using work of Khovanov [Kho14], this gives a basis for all morphism spaces in APar and hence also for AP k .We also obtain as a corollary that AP k is a quotient of A aff 2k .We do not know whether these two algebras are in fact isomorphic.If they were, then our definition of A aff 2k would also give a presentation for AP k .The paper is structured as follows.Section 2 deals with the ordinary partition algebra.In Section 2.1 we recall the diagram basis and the original presentation of the partition algebra given by T. Halverson and A. Ram.In Section 2.2, we recall the definition of the Jucys-Murphy elements and the more recent presentation of the algebra given by J. Enyang.We also state some relations which will be needed in defining an affine partition algebra.Section 2.3 introduces a new normalisation of the Jucys-Murphy elements and of Enyang's generators which has the advantage of simplifying many of the relations in our definition.Finally, in Section 2.4, we recall explicitly the Schur-Weyl duality between the partition algebra and the symmetric group.
Section 3 gives the definition of the affine partition algebra A aff 2k in terms of generators and relations and proves some properties.In particular, we show in Section 3.1 that the ordinary partition algebra appears both as a subalgebra and as a quotient of A aff 2k .In Section 3.2, we describe a family of central elements in A aff 2k and formulate a conjecture about its centre.Finally, in Section 3.3 we show that A aff 2k extends the action of the partition algebra on tensor space as desired.
Section 4 deals with the connections with the Heisenberg category and the work of J. Brundan and M. Vargas on their affine partition category.In Section 4.1, we recall the definition of the Heisenberg category including the basis of the morphism spaces given by M. Khovanov in [Kho14].In Section 4.2 we define a homomorphism from A aff 2k to the endomorphism space of a particular object in the Heisenberg category and prove that it is surjective.In Section 4.3, we generalise the arguments from Section 4.2 to show that APar is the full monoidal subcategory of the Heisenberg category generated by one object and deduce that AP k is a quotient of A aff 2k .
We view [k] ∪ [k ′ ] as a formal set on 2k elements, and let Π 2k denote the set of all set partitions of [k] ∪ [k ′ ].Given any α ∈ Π 2k , we say a partition diagram of α is any graph with vertex set [k] ∪ [k ′ ] whose connected components partition the vertices according to the blocks of α.We do not distinguish between α and any partition diagram of α, in particular we will only care about the connected components of such graphs, not the edges forming the components.When drawing such a diagram, we will arrange the vertices in two rows with the top row going from 1 to k, and the bottom row from 1 ′ to k ′ .For example, in Π 10 we have the identification We define a product • on Π 2k as follows: Given α, β ∈ Π 2k , we let α • β ∈ Π 2k be the set partition obtained by first stacking the diagram of α on top of that of β, identifying the bottom row of α with the top row of β, removing any connected components lying entirely within the middle row, and then reading off the connected components formed between the top row of α and the bottom row of β.For example consider α = 1 2 3 4 5 and β = 1 2 3 4 5 in Π 10 .Then we have Clearly this product is associative and independent of the choice of graphs used to represent the set partitions.
The element 1 = {{i, i ′ } | i ∈ [k]} ∈ Π 2k is an identity element, and thus (Π 2k , •) is in fact a monoid.Given α, β ∈ Π 2k , we let m(α, β) denote the number of middle components removed in evaluating α • β.In the above example, we have m(α, β) = 1.Now let z be a formal variable and C[z] the polynomial ring.The partition algebra A 2k (z) is the C[z]-algebra whose basis as a free C[z]-module is given by the set Π 2k , and whose product is given by αβ := z m(α,β) α • β for all α, β ∈ Π 2k , extended linearly over C [z].
For 0 ≤ l ≤ k, we identify Π 2l as a submonoid of Π 2k given diagrammatically by for any α ∈ Π 2l .Define Π 2k−1 to be the submonoid of Π 2k consisting of all set partitions of where k and k ′ belong to the same block.We have a chain of monoids ∅ = Π 0 ⊂ Π 1 ⊂ Π 2 ⊂ . . . .For any 0 ≤ r ≤ 2k, we let A r (z) denote the subalgebra of A 2k (z) generated by Π r .We obtain an analogous chain , where B r is the r th Bell number.We can view A r (z) as an infinte dimensional algebra over C with basis {z n α | n ∈ Z ≥0 , α ∈ Π r }.When we do so, we use the notation A r instead.For any δ ∈ C, let (z − δ) denote the ideal of A r generated by z − δ.Then we let A r (δ we define the following elements of Π 2k : These elements generate the monoid (Π 2k , •), and in turn the algebra A 2k (z).Moreover, a presentation in terms of these generators, which we display below, was given in [HR05, Theorem 1.11], see also Theorem 36 and Section 6.3 of [East11].
Theorem 2.1.1.The partition algebra A 2k (z) has a presentation with generating set The presentation above extends to one for the C-algebra A 2k by simply adding z as a central generator.The C-algebra A 2k (δ) has a presentation identical to above with the exception of replacing z with δ in relation (HR2)(i).From the symmetry of the above presentation, one can deduce that we have an anti-involution * : A 2k (z) → A 2k (z) given by flipping a partition diagram up-side-down, and extending linearly over C[z].We denote the image of an element a ∈ A 2k (z) under this anti-involution by a * .

Jucys-Murphy elements and Enyang's presentation
In this section we give the definition of the Jucys-Murphy elements of the partition algebra.These elements were originally defined diagrammatically by Halverson and Ram in [HR05].They were later given a recursive definition by Enyang in [Eny12].For this recursive definition, Enyang introduced new elements σ i which resemble the Coxeter generators s i .We recall this recursive definition, and a new presentation of the partition algebra given in [Eny12] in terms of the generators e i and σ i .The following definition is the one given in Section 2.3 of [Eny13].
Definition 2.2.1.Let L 1 = 0, L 2 = e 1 , σ 2 = 1, and σ 3 = s 1 .Then for i = 1, 2, . . ., define where, for i = 2, 3, . . ., we have Also for i = 1, 2, . . ., define where, for i = 2, 3, . . ., we have Example 2.2.2.The first few non-trivial examples are We will refer to the elements L i as the JM-elements, and the elements σ i as Enyang's generators.A simple proof by induction tells us the following: It was shown in [Eny12] that these elements are invariant under the automorphism * .They also commute with smaller partition algebras with respect to the chain described in the previous section: For any i ≤ r, (1) In particular this shows that the JM-elements pairwise commute.We now give the new presentation of A 2k (z) established in [Eny12].This presentation is given in terms of the generators e i and Enyang's generator's σ i .Remarkably, although the definition and diagrammatic description of the σ i is rather complicated, the defining relations in the following presentation are very simple.This is less surprising when one considers how these elements act on tensor space.This will be discussed in Section 2.4.
Theorem 2.2.4.[Eny12, Theorem 4.1] The partition algebra A 2k (z) has a presentation with generating set and relations: Note we only worked with the elements σ i for i ≥ 3, since σ 2 = 1.The elements s j in the above presentation are precisely the Coxeter generators.From the involution relations we have that s i σ 2i = σ 2i s i = σ 2i+1 .From Equation (1) one can deduce that L i and σ j commute whenever j = i − 1, i, i + 1.We end this section by giving relations which tell us how the JM-elements interact with Enyang's generators when they do not commute.We use results established in [Eny12], although we have adopted the notation of [Eny13].
Remark 2.2.5.The change of notation between [Eny12] and [Eny13] is given respectively by p i ∼ e 2i−1 , , and L i+ 1 2 ∼ L 2i+1 .Proposition 2.2.6.The following relations hold: (2) We examining the right hand side term by term.For the first term we have (ii): By definition, Multiplying this equation on the left and right by σ 2i gives From this, and the fact that L i and σ j commute whenever j = i − 1, i, i + 1, one may deduce that

Rearranging gives
We examine the bracketed terms in Equation (4).Rearranging (i) gives the first bracketed term as Multiplying this on the left and right by σ 2i , and then rearranging gives the second bracketed term Substituting these back into equation (4) yields (iii).
(iv): Analogously to the previous case, one can deduce that Rearranging gives We examine the bracketed terms in Equation (5).Rearranging (2)(ii) gives the first bracketed term as Substituting these back into equation (5) yields (iv).

Normalisation
As mentioned in the introduction, we seek to 'affinize' the partition algebra by replacing the Jucys-Murphy elements with commuting variables, and asking them to retain various relations with the generators.In preparation for this construction, this section collects all the relations we seek to retain in one place.However, instead of working with the JM-elements and Enyang's generators, it turns out to be easier to work with the following elements: For each i ∈ N we set For each i ∈ N we set We also call the elements X i the JM-elements and the elements t i Enyang's generators.By definitions we have that t i ∈ A i+1 (z), X i ∈ A i (z), and that t * i = t i and X * i = X i .One can also deduce that s i t 2i = t 2i s i = t 2i+1 .We briefly collect some simple relations to ease the proof of the proceeding proposition.The relation e 2i e 2i−1 t 2i = e 2i X 2i is obtained by acting by * .
The following proposition contains all the relations we seek to retain for our construction of the affine partition algebra, as such some are identical to relations we have already stated.It provides a presentation of the partition algebra A 2k (z) which is simply Enyang's presentation Theorem 2.2.4 except working with the generators t i instead of σ i .For those relations we have adopted the same naming conventions given in Theorem 2.2.4.Proposition 2.3.2.The partition algebra A 2k (z) has a presentation with generating set and relations: (1) (Involutions) (2) (Braid relations) (i) t 2i+1 t 2j = t 2j t 2i+1 for j = i + 1.

Schur-Weyl Duality
In this section we recall the Schur-Weyl duality between the partition algebra A 2k (n) and the group algebra of the symmetric group CS(n) via their actions on tensor space.We will also highlight how X i and t i act on this tensor space, and complete the proof of Proposition 2.3.2.Consider the permutation module with action given by πv a = v π(a) for all π ∈ S(n) and a ∈ [n], extended C-linearly to CS(n).For k ≥ 0, the tensor space and the diagonal action is given by the C-linear extension of πv a = v π(a) for all π ∈ S(n) and a ∈ [n] k .We let End(V ⊗k ) be the algebra of all vector space endomorphims V ⊗k → V ⊗k .We identify any g ∈ CS(n) with the corresponding endomorphism in End(V ⊗k ) given by the diagonal action.Then consider the subalgebra of all S(n) commuting endomorphisms.For the following result see [HR05, Section 3].
Theorem 2.4.1.For any n, k ≥ 0, we have a surjective C-algebra homomorphism defined on the generators z, s i , and e j , by letting z → n and where δ a,b is the Kronecker delta.Moreover, we have that For any a, b ∈ [n], we let (a, b) ∈ S(n) denote the transposition exchanging a and b, and let ε a,b := 1−δ a,b .We now recall how the elements X i and t i act under ψ n,k , which was proven in [Eny12].
The following result tells us that a relation holds in A 2k if and only if it holds under ψ n,k for all n.We will use this result to complete Proposition 2.3.2.
Proof.To ease notation, for any tuple a = (a 1 , . . ., a k ) ∈ [n] k , we represent a simple tensor in V ⊗k by a word in the entries of a, that is We will prove these relations by showing that they hold under ψ n,k for all n ≥ 1, and then employ Lemma 2.4.4.For each relation we will have to consider different cases based on the relative values of the entries a i−1 , a i , and a i+1 , although most cases are trivial.Also note that ψ n,k (1 − e 2i )(a) = ε ai,ai+1 a.
3 Affine partition algebra 3.1 Definition of A aff 2k (z) and basic results In this section we give the definition of the affine partition algebra A aff 2k by generators and relations.We prove some basic properties about this algebra including the fact that the partition algebra A 2k is both a quotient and subalgebra of A aff 2k .We also show that the polynomial algebra C[x 1 , . . ., x 2k ] is a subalgebra and that H k ⊗ H k is a quotient, where H k is the degenerate affine Hecke algebra.We will prove a variety of relations in A aff 2k including counterparts to the recursive definition of both the Jucys-Murphy elements and Enyang's generators.Definition 3.1.1.We define the affine partition algebra A aff 2k to be the associative unitial C-algebra with set of generators and defining relations (1) (Involutions) (2) (Braid relations) , where s j := τ 2j τ 2j+1 + e 2j .
(3) (Idempotent relations) (5) (Contractions) (i) e i e i+1 e i = e i and e i+1 e i e i+1 = e i+1 , for i Note we have overloaded the symbols e i and s j as elements in A 2k and A aff 2k , however we will show shortly that the mapping A 2k → A aff 2k via z → z 0 , e i → e i , and s j → s j realises the subalgebra e i , s j , z 0 of A aff 2k as an isomorphic copy of the partition algebra A 2k .The defining relations above are those present in Proposition 2.3.2,except where the Jucys-Murphy elements X i have been replaced with the affine generators x i , Enyang's generators t j have been replaced by new generators τ j , and the polynomials z(z − 1) l have been replaced by central generators z l .It is worth mentioning that the map A 2k → A aff 2k given by z → z 0 , e i → e i , and t j → τ j does not realise an algebra homomorphism.This is since τ 2 is a non-trivial generator in A aff 2k , while t 2 is abscent in the presentation of Theorem 2.2.4 since it equals 1 − e 2 , hence the braid relation (E2)(iv) is not respected under such a map.The subalgebra e i , τ j , z 0 of A aff 2k is not isomorphic to the partition algebra, and in fact one can show that this subalgebra is infinite dimensional as an C[z 0 ]-module (see Corollary 3.3.3below).
Replacing the Jucys-Murphy elements with commuting variables, and introducing new central generators is very much analogous to the 'affinization' process employed on other diagram algebras.In particular relations (6) to (10) (except (7)) are comparable to the relations in [Naz96, Section 4] which were chosen as the defining relations for the affine Wenzl algebra.The Skein-like relations (8) tell us how the affine generators x i interact with the generators τ j when they do not commute.These relations are to A aff 2k what the defining relation y i+1 = s i y i s i + s i is to the degenerate affine Hecke algebra H k .In the next section we provide a projection of A aff 2k onto a diagram algebra living within the Heisenberg category.Under this projection the Skein-like relations will correspond to moving a decoration over crossings.
We have also chosen to replace the generators t j with new generators τ j , which appears to be a departure from the 'affinization' process.However, we will show that these elements are not needed to generate the algebra, that is A aff 2k = e i , s i , x i , z l .Hence to go from A 2k to A aff 2k we have indeed just adjoined new affine and central genrators.The reason for letting the elements τ j play the role of generators is to allow us to give a cleaner presentation which is more comparable to its counterparts within the literature.We have chosen to include the Braid-like relations (7) as they tell us how none commuting τ j generators interact in a manner which resembles the braid relations of the Coxeter generators s i .These relations will allow us to give counterparts to the recursive definition of Enyang's generators (see Lemma 3.1.11below).
We begin by showing that the partition algebra is a quotient of the affine partition algebra.This follows naturally from its construction.Lemma 3.1.2.We have a surjective C-algebra homomorphism pr : A aff 2k → A 2k , given on the generators by Proof.This follows by comparing the defining relations with those of the same numbering in Proposition 2.3.2, and surjectivity follows since t i , e j , z = A 2k .
Similar to the partition algebra, the affine partition algebra has a corresponding anti-automorphism which fixes the generators.
We now seek to show that A 2k is the subalgebra s i , e j , z 0 of A aff 2k .We first prove a few helpful relations.Proposition 3.1.5.We have a injective C-algebra homomorphism ι : A 2k → A aff 2k given on the generators by ι(z) = z 0 , ι(s i ) = τ 2i τ 2i+1 + e 2i , and ι(e i ) = e i .
Proof.We first prove that ι is a homomorphism.To do this we show that each of the defining relations of A 2k given in Theorem 2.1.1 is respected under ι.We only check the relations involving s i since the others are accounted for in the definition of A aff 2k .(HR1)(i): where we used (1), (2 where the third equality follows from (3)(v) and the forth from (1)(i).Similarly we have ι(e 2i−1 e 2i+1 s i ) = ι(e 2i−1 e 2i+1 ).
Hence we have shown that ι is indeed an algebra homomorphism.For injectivity, note that pr • ι = id where id : A 2k → A 2k is the identity morphism.Thus ι has a left inverse, and so is injective.
Therefore the partition algebra A 2k is both a subalgebra and quotient of the affine partition algebra A aff 2k .Also note that restricting * down to the partition algebra coincides with the anti-automorphism of flipping a diagram.We now seek to give affine counterparts to the recursive definition of the Jucys-Murphy elements given in Definition 2.2.1.Lemma 3.1.6.The following relations hold in A aff 2k : (i) where, in the first equality we used the fact that τ 2i+1 e 2i = e 2i τ 2i+1 = 0, the second equaltiy we used the substitution τ 2i τ 2i+1 = τ 2i+1 τ 2i = s i − e 2i , and the last equality we used the fact that e 2i and x 2i−1 commute.Now applying (8)(iv) from Definition 3.1.1 to the left hand side of above, we obtain By applying Lemma 3.1.4(ii), and rearranging, we arrive at (i).Item (ii) is proved in an analogous manner were we instead employ relations (8)(ii) and (8)(iii) from Definition 3.1.1.
By rearranging the relations in the above Lemma in terms of the generators τ 2i and τ 2i+1 , we immediately obtain the following: Corollary 3.1.7.We have that A aff 2k = e i , s j , x k , z l i,j,k,l .
Recall that the degenerate affine Hecke algebra H k is the C-algebra given as a vector space by the tensor prooduct C[y 1 , . . ., y k ] ⊗ CS(k), where C[y 1 , . . ., y k ] is the polynomial algebra in commuting variables y 1 , . . ., y k .The defining relations of H k are such that C[y 1 , . . ., y k ] and CS(k) are subalgebras, and l=0 be any sequence of constants in C. Then we have a surjective C-algebra homomorphism f λ : A aff 2k → H k ⊗ H k given on the generators by Proof.We show that each of the defining relations of A aff 2k are upheld under f λ .Since f λ (e i ) = 0, one may observe that most of the defining relations involving generators e i are trivially upheld. (1)(i): (1)(ii): Similar to (1)(i) above.
(6)(ii): Follows since s i y j = y j s i whenever j = i, i + 1.
Thus f λ is a homomorphism.Surjectivity follows as f λ (τ i ), f λ (x j ) i,j = H k ⊗ H k .
Corollary 3.1.9.The polynomial algebra C[x 1 , . . ., x 2k ] is a subalgebra of A aff 2k .Proof.This is the same as asking that all monomials in the generators of the subalgebra x 1 , . . ., x 2k of A aff 2k are linearly independent, which follows since their images under f λ are.
To end this section we establish a counterpart to the recursive relations of Enyang's generators.To do so, we collect the more technical relations needed into the following lemma: Lemma 3.1.10.The following relations hold in A aff 2k : (i) e 2i x 2i e 2i = 0
(iv): We have where the first equality follows from (9)(i) of Definition 3.1.1,and the second from Lemma 3.1.6(i).We examine the five terms above: (1) Substituting back into the above equation gives e 2i−2 x 2i−2 s i e 2i−2 s i = e 2i−2 τ 2i as desired.
Lemma 3.1.11.The following relations hold in A aff 2k : Proof.We prove the first relation, the second follows from by multiplying on the left by s i .We have that where the second equality follows since τ 2i−2 commutes with τ 2i+1 and e 2i τ 2i = τ 2i e 2i = 0. Substituting the above we get For the first term in equation (10) we have where the first equality follows by (7)(i) of Definition 3.1.1,the second from s i−1 τ 2i−2 = τ 2i−2 s i−1 = τ 2i−1 , the third from Lemma 3.1.10(viii), and the forth from (7)(iii) of Definition 3.1.1.Substituting this back into equation (10), and rearranging yields The desired relation is obtained by applying relations (iv) to (vii) of Lemma 3.1.10.

Central elements in A aff 2k
In this section we describe a central subalgebra of A aff 2k consisting of certain polynomials in the affine generators.We end the section with a conjecture describing the center of A aff 2k .
Proof.This follows from Lemma 3.2.1 by induction on n.
For the generators e 2i we have where the first equality follows from the commuting relation (6)(iii) of Definition 3.1.1,and the second equality follows since x 2i+1 e 2i = x 2i e 2i and e 2i x 2i+1 = e 2i x 2i by (9)(ii) and (9)(i) of Definition 3.1.1.
Employing this and relations (i) and (ii) of Lemma 3.2.2,we have Hence showing that [τ 2i , p n ] = 0 is equivalent to showing that the three summations above sum to zero.This follows by changing the second summation accordingly: Under the projection pr : showing that such a subalgebra is central in A 2k .It was shown in [Cre21, Thm 4.2.6] that this is in fact the whole center of A 2k .Note that in [Cre21], the centre is given as SSym[N 1 , . . ., N 2k ] where

Extending the Action on Tensor Spaces
We now seek to extend the action of A 2k on V ⊗k to one of A aff 2k on M ⊗ V ⊗k , where M is any CS(n)-module.The tensor space M ⊗ V ⊗k is also viewed as an CS(n)-module by the diagonal action.Before extending the action, we briefly define some central elements in CS(n).
defined on the generators by Proof.This can been shown by direct computations, much of which are fairly simple but lengthy.To ease notation, for any tuple a = (a 0 , a 1 , . . ., a k ) n,k is well-defined, that is to confirm that these endomorphisms do indeed commute with the diagonal action of S(n).We do this by showing for any π ∈ S(n), that πψ n,k (g) for each generator g of A aff 2k .
One can deduce that πψ n,k (e i ) since the action of the generators e i ignores the M component, and hence this follows from Theorem 2.4.1.
For the generators τ 2i , by the substitution b ′ = π(b).One can show πψ One now needs to confirm that the defining relations of A aff 2k in Definition 3.1.1are upheld under ψ n,k .As mentioned, these can be shown by direct, but lengthy computations.With this in mind, we will only give details of some of the more difficult relations, namely relations (8) through (10).Note that the Braid-like relations (7) follow in a analogous manner to the proof of Lemma 2.4.5.
(8)(i): We seek to show that To show this we examine how each term on the hand right side acts on the simple tensor a, and show that the sum recovers the action of x 2i+1 .It proves easier to do this by tackling two cases, when a i = a i+1 and when a i = a i+1 .
(Case 1): Assume a i = a i+1 , then for the first term we have where we employed the substitution c = (a i , a i+1 )(b).For the second term, For the third term ψ n,k (τ 2i )(a) = 0, and so Hence we just need to confirm that ψ The remaining Skein-like relations follow by employing similar arguments.
(9)(i): We seek to show ψ ).We show this first when working with e 2i , then with e 2i−1 .Assume a i = a i+1 , then For odd indices we have Thus ψ ). Relation (9)(ii) may be shown in a similar manner.(10)(i): Lastly relation (10)(ii) is simple to check since Z n,l belongs to the center of CS(n).
Corollary 3.3.3.The subalgebra τ i , e j i,j of 1 e 2 e 1 for all m ∈ N. We first show that d m ∈ τ i , e j by induction on m.By Lemma 3.1.4(i) we see that e 2 x 2 ∈ τ i , e j .Then multiplying on the right by e 1 yeilds e 2 x 2 e 1 = e 2 x 1 e 1 = x 1 e 2 e 1 = d 1 , where the first equality follows from (9)(ii) of Definition 3.1.1,and the second from (6)(iii).Thus we have the base case d 1 ∈ τ i , e j .Assume d m ′ ∈ τ i , e j for all m ′ < m with m ≥ 2, we seek to show that d m ∈ τ i , e j .Well where the second equality follows from Lemma 3.1.4(i), and the remaining equalities follow in the same manner as the base case.Hence d m ∈ τ i , e j completing induction.We now seek to show that the set {d m | m ∈ N} is C[z 0 ]-linearly independent in A aff 2k , which will complete the proof.Let I ⊂ N be finite and assume where h m (z 0 ) are polynomials in C[z 0 ].We seek to show that h m (z 0 ) = 0 for each m ∈ I. Let M ∈ I be the maximal element, and let R be the set of roots for each h m (z 0 ).Pick an n ∈ N such that n > M + 1 and n / ∈ R. Let F be any free CS(n)-module.For any f ∈ F and (a 1 , . . ., a k ) ∈ [n] k , we have Since F is free, it will follow that the set {ψ This follows since n > M + 1, and hence T m n,a2 contains a permutation consisting of a single cycle of size m + 1, while all permutations in T m ′ n,a2 must have smaller support whenever m ′ < m. now consider the equation Since n is not a root of any h m (z 0 ), and the set {ψ 4 Connections with the Heisenberg category J. Brundan and M. Vargas recently defined in [BV21] an affine partition category APar as a monoidal subcategory of the Heisenberg category introduced by Khovanov in [Kho14] generated by certain objects and morphisms.This was based on the observation made by S. Likeng and A. Savage in [LS21] that the partition category can be realised inside the Heisenberg category.This affine partition category naturally gives rise to another definition of an affine partition algebra, which they denote by AP k by taking the endomorphism algebra End APar ((↑↓) k ) for the object (↑↓) k in APar (see section 4.1.below).
Inspired by the work of Brundan and Vargas, we construct a surjective homomorphism ϕ from A aff 2k to End Heis ((↑↓) k ).In fact, our argument generalises to show that Brundan and Vargas' affine partition category APar is the full monoidal subcategory in Heis generated by the object ↑↓.As a corollary we obtain that AP k is a quotient of A aff 2k .We start by recalling the definition of the Heisenberg category.

Heisenberg Category
The Heisenberg Category Heis is a C-linear monoidal category originally defined by M. Khovanov in [Kho14].The objects of Heis are generated, as a monoidal category, by the two objects ↑ and ↓.We use juxtaposition to denote the tensor product of objects, and the monoidal identity object is the empty word ∅.Hence we view the free monoid ↑, ↓ as the set of objects in Heis.
The space of morphisms Hom Heis (a, b) is the C-vector space generated by certain diagrams modulo local relations.We call such diagrams (a, b)-diagrams and define them as follows: Firstly, we work in the strip R × [0, 1] with boundary B := R × {1} ∪ R × {0}.We call an orientated immersion of the interval [0, 1] and circle S 1 a string and loop respectively.We denote orientations by drawing an arrow on the curve.

Now consider the set of points
with the symbols a i and b j respectively.We say that a set partition of E into pairs is an (a, b)-matching if pairs of points in the same row are coloured by opposite arrows, while pairs of points in different rows are coloured by the same arrow.Then an (a, b)-diagram is a finite collection of strings and loops, modulo rel boundary isotopies, such that: (D1) The endpoints of the strings induce an (a, b)-matching on E (D2) There are only finitely many points of intersection, and no triple or tangential intersections occur (D3) The boundary B doesn't intersect any loops, and only intersects strings at the endpoints E For example let a =↓↓↑ and b =↑↓↓↓↑, then Isotopic deformation of the interior of R × [0, 1] is allowed, and will preserve the relative structure of the points of intersection.If a loop contains no intersections we call it a bubble.Bubbles can have clockwise or anticlockwise orientation.If the endpoints of a string occur in different rows we call it a vertical string, and it has either a down or up orientation.If the endpoints belong to the same row then we call it an arc.Non self-intersecting arcs have either a clockwise or anti-clockwise orientation.In the above example there are two loops, one of which is a bubble, and four strings, three of which are vertical and one an arc.We call an endpoint of a string a source if the arrow of orientation points away from it, and a target otherwise.We consider (a, b)-diagrams modulo the following local relations: Relation (H1) holds regardless of orientations.To apply such a local relation to an (a, b)-diagram one locates a disk which is isotopic to one of the disks above, then replace such a disk according to the corresponding equation.Note that any of the local relations may be rotated in any way to give an equivalent relation.Relation (H1) tells us that any curve may past over a crossing, and relations (H2) and (H3) tells us how to pull part orientated curves, where (H3) shows that this can not always be done for free.Relation (H4) tells us that left curls kill (a, b)-diagrams, and that any anti-clockwise bubble may be removed for free.The composition of morphisms is given by vertical concatenation of diagrams, and rescaling (and extending C-linearly).We denote composition by juxtaposition of symbols.When a = b we write a-diagram instead of (a, a)-diagram.The morphism space End Heis (a) is a C-algebra with identity given by the diagram of non-intersecting vertical strings.Now for later use, we collect some relations regarding arbitrary (a, b)-diagrams.The following local relation follows from (H2) and (H3), see also [LS21, (3.5)]: We now recall a basis for the morphism spaces Hom Heis (a, b) presented in [Kho14].We first introduce a few definitions to help us describe this basis in a manner which will lend itself better for later results.The basis for Hom Heis (a, b) we describe below is obtained by adding decorations (right curls) and decorated clockwise loops to all the simple (a, b)-diagrams in a partiuclar manner.We describe this by introducing some basic diagrams and using the composition of diagrams.

Given words a
The orientation of strings is taken to match a.Although both r i and c w depend on a, we surpress this fact as it should be clear from context. (1)(i): which equals ϕ(1 − e 2i ).One can show that relation (1)(ii) is upheld in a similar manner.
Such elements satisfy the braid relation s i s i+1 s i = s i+1 s i s i+1 by [LS21, Theorem 4.1].
By applying (H2) twice and (H1), the second term above straightens out to For the first term we get where the first equality follows by applying (H3), and the second equality by (H1) and (H2).Therefore collectively we have show (7)(iii).Relation (7)(iv) follows in an analogous manner.
(8)(i): We seek to show that One can check that .
(9) and (10): These relations are immediately seen to be upheld diagrammatically.
The remainder of this section seeks to show that the algebra homomorphism ϕ in the above proposition is surjective.Firstly, from Theorem 4.1.8we know that End Heis ((↑↓) k ) has a basis given by where α ∈ Sim((↑↓) k ).Since ϕ(z l ) = c l and ϕ(x i ) = r i , to prove that ϕ is surjective it is enough to show that Sim((↑↓) k ) ⊂ Im(ϕ).We will prove that Sim((↑↓) k ) ⊂ ϕ(e i ), ϕ(τ j ) i,j ⊂ Im(ϕ).We say that a simple diagram is planar if no intersections occur among its strings, for example the diagrams ϕ(e i ) are all planar for each i ∈ [2k − 1].The total number of planar diagrams in Sim((↑↓) k ) is C 2k , the 2k-th Catalan number.These diagrams are precisely oriented versions of the Temperley-Lieb diagrams.The Jones normal form gives a way of writing the Temperley-Lieb diagrams as a product of generators (see [J83], and also [Kau90, Theorem 4.3 and Figure 16]) which does not involve bubbles, and so may be applied here for the elements ϕ(e i ) to show that any planar diagram belongs to ϕ(e i ) i and hence to Im(ϕ).
Example 4.2.3.For k = 3 and π = (1, 2, 3) ∈ S(3), we have For any π ∈ S(k), it is shown in [Stem97] that we have a reduced expression of the form where Strings in α ↓ (w) may intersect one another more than once, but we can resolve such double crossings by pulling strings apart via the local relations.The descending condition on the indices in this reduced expression means we will never need to employ (H3) to pull strings apart, and thus we must have that α ↓ (w) = π ↓ .Hence π ↓ ∈ Im(ϕ).Rotating π ↓ by 180 • yields (ρπρ −1 ) ↑ where ρ is the of transposition (i, k − i + 1) for each i ∈ [k].Thus we also have that π ↑ ∈ Im(ϕ) for all π ∈ S(k).
To aid upcoming proofs we define a collection of diagrams which loosen the conditions on simple diagrams.
Definition 4.2.4.We call an (a, b)-diagram semisimple if the following hold: (1) It contains no loops or self intersections.
(2) No top arc intersects a bottom arc.
Let SSim(a, b) denote the set of semisimple (a, b)-diagrams, and write SSim(a) for SSim(a, a).
From definitions we have that Sim(a, b) ⊂ SSim(a, b).Any diagram α ∈ SSim((↑↓) k , (↑↓) l ) contains precisely k + l strings, and the endpoints of these strings induce an ((↑↓) k , (↑↓) l )-matching of the endpoints E. We let α denote the unique simple diagram corresponding to such a matching (recalling the discussion after Definition 4.1.3).Lemma 4.2.5.Given any simple diagram α ∈ Sim((↑↓) k , (↑↓) l ), there exists π ∈ S(k), σ ∈ S(l), and a planar diagram β ∈ Sim((↑↓) k , (↑↓) l ) such that π ↑ βσ ↓ is semisimple and Proof.Given any simple diagram γ ∈ Sim((↑↓) k , (↑↓) l ) let (2i, 0), (2j, 0) ∈ E (respectively (2i − 1, 1), (2i − j, 1)) be two ↓ (respectively ↑) endpoints in the bottom row (respectively top row) of γ.Let γ ′ be the simple diagram obtained from γ by permuting these two endpoints around.It can be seen that γ(i, j) ↓ (respectively (i, j) ↑ γ) is semisimple as long as the permutation doesn't swap the orianetation of an arc around, since that is the only way a self intersection can occur.In this situation, one can see that Hence to prove this lemma it is enough to show that we can reach a planar diagram β from α by repeatably permuting the endpoints in the bottom row coloured by ↓, and top row coloured by ↑, in such a way that the orientations of arcs are preserved.We focus on the bottom row, where the top row will follow in the same manner by a 180 • rotation of the diagrammatics.Starting with α we remove intersections one at a time by employing a suitable permutation of endpoints.There are a few cases to consider, and in each such case the endpoints of the strings in the following diagrams will be arbitrary: (Case 1): Crossing of two down strings: (Case 2): Crossing of a down string with an clockwise/anti-clockwise arc: , Note in either situation the orientation of the arc is preserved by the permutation.
(Case 3): Crossing of two arcs: There are four cases based on the orientations of the two arcs given by , , , .
noting that in the last case such a down string must exist.Again, the orientations of the arcs are preserved under the permutation of endpoints in all four of the above situations.In all three of the cases above, it can be seen that the new simple diagram we obtain after the permutation of endpoints has strictly less number of intersections.We claim that applying the moves above on the bottom row, and their 180 • counterparts on the top row, until all such intersections are removed will yield a planar diagram.For contradiction, suppose this is not the case.Thus even after removing all such types of intersections, the diagram still contains some other type of intersection.The other such intersections are either between an up string and arc on the bottom row, a down string and arc on the top row, or an up string and a down string.The former two are 180 • counterparts to one another, hence we only need to consider one such type.Firstly, if an up string intersects a clockwise arc on the bottom we have Note that the parity of the number of endpoints on the bottom row strictly between a and b must be different to the partity of endpoints strictly between b and c.Thus on can deduce that such an endpoint must be a target to a string which intersects the arc, and such an intersection would be accounted for by Case 2 or 3, hence a contradiction.The same argument can be used to show that the case of an up string intersecting an anti-clockwise arc on the bottom is also impossible.Note all intersections involving arcs have now been accounted for.Lastly assume an up string intersects a down string.We have two cases, one of which is a b .
The dashed vertical line is simply an aid for arguments to come, and has been draw so that the endpoints a and b are the closest endpoints to its left.The other case is given by rotating the above by 180 • and will follow analgously.The parity of the number of endpoints to the right of a is odd, while the parity of the number of endpoints to the right of b is even.This implies that there exists a string s such that one of its endpoints belongs to the right of the dashed line, while the other belongs to the left.Moreover, since the right-most endpoint on the top and bottom row are coloured by ↓, we can say that the endpoint of s which is to the right of the dashed line is a source while the endpoint to the left is a target.So s must intersect one of the above strings, and must be a vertical string since all intersections with arcs are accounted for.Hence, Note that s may intersect both of the other strings and not just one, but it is always forced to intersect the string depicted.As such each situation exhibits an intersection accounted for in Cases 1 (or its 180 • counterpart), giving the desired contradiction.Thus removing all intersections of the types presented in Cases 1 to 3 (and their 180 • rotated counterparts) will result in a planar diagram, completing the proof.
Let R be an open subspace of R × [0, 1] and let α be an (a, b)-diagram.Examining α locally in R will give a configuration of curve segments, and we refer to such as a region of α.Within a given region we treat distinct curve segments as different curves, even if in α itself the two segments belong to the same curve.In particular, if in R two distinct curve segments intersect one another, and in α these two segments belong to the same curve, we will not call such an intersection a self-intersection in R, but it is a self-intersection in α.
Recall that the local relation (H4) tells us that if a left curl appears in a diagram we may annihilate such a diagram.This relation asks that the region enclosed in the curl is absent of any other strings.The following result shows that even if such a region is non-empty, as long as its contains no loops or self-intersections, we can annihilate the diagram.Lemma 4.2.6.Let α be an (a, b)-diagram containing a left curl where the region bounded by the curl contains no loops or self-intersecting curve segments, then α = 0.
Proof.By assumption α contains a configuration of the form where we let R denote the interior region bounded by the curl, which contains no loops or self-intersecting curve segments, and g 0 , . . ., g m account for all the intersections which occur on the curl.Note we have only drawn the segments of the g i 's which realise the intersection on the curl.We prove the result by induction on the number of intersections occurring in R. Assume that no intersections occur in R, hence R gives a planar configuration of strings.One can deduce that there exists neighbours g i mod(m+1) and g (i+1)mod(m+1) such that either In the former situation, since we are dealing with a left curl, one can check that regardless of the orientation of the depicted string in R, it may be pulled outside the curl by (H2).For the latter situation we may employ (H1) to pull the string out of the curl over the crossing at the top.Continually pulling out such strings one at a time will result in making R empty, and then applying (H4) gives α = 0. Now suppose that the result holds whenever R contains n or less intersections for some n ≥ 0, and assume that R contains n + 1 intersections.It is clear that there must exist an empty region R ′ in R bounded by the curl and various segments.Diagrammatically we have where g i , g i+1 , and the (possibly empty) set of curve segments H = {h 1 , . . ., h l } make up the remainder of the boundary of R ′ .Note such curve segments may not be pairwise distinct in R. In the case when H is empty, we simply have the situation .
Since R ′ is empty we may pull this crossing out of the curl by (H1), which will decrease the number of intersection in R and thus by induction α = 0. Hence we may assume that H is non-empty.The general case H = {h 1 , . . ., h l } is solved by focusing on h 1 , and in fact solving the case H = {h 1 } is sufficient to understand the general case, hence we only prove this case.So we are working with the sitaution There are two cases to consider based on the orientation of h 1 .For the first case we have by (H2).Then we may pull the crossing between either g i and h 1 , or g i+1 and h 1 out of the curl by (H1), which will decrease the number of intersections in R by one and so α = 0 by induction.With the opposite orientation on h 1 we have by (H3).Here denote the first diagram on the right of the above equation by α 1 and the second by α 2 .For α 1 , as was done in the previous case we may pull one of the crossings outside of the curl, and thus decrease the number of intersections in R by one, and hence α 1 = 0 by induction.For α 2 the curve containing h 1 and the original left curl have been turned into the two new curves h (1) 1 and h (2) 1 .Note the original left curl is no longer present, but regardless of how the original curve containing h 1 intersected the curl, at least one of the new curves h (1) 1 and h (2) 1 must form a new, smaller, left curl.The region bounded by this new curl is a subregion of R containing strictly less number of intersections.Hence by induction α 2 = 0, and so collectively α = α 1 + α 2 = 0 completing the proof by induction.Note the general case for H = {h 1 , . . ., h l } is tackled in the exact same manner by pulling h 1 out of the curl, the diagrammatics are just more cluttered, but the remaining segments h 2 , . . ., h l do not interfer with the above argumenets.Proof.Given two distinct strings s and t in α, let n be the number of intersections occurring between the two strings.If n is even set µ({s, t}) = n, while if n is odd set µ({s, t}) = n − 1.Note µ({s, t}) is always even.We let η(α) = s,t µ({s, t}), where the sum runs over all unordered pairs of distinct strings of α.Informally, η(α) is the number of intersections of α which prevent it from being simple, in particular η(α) = 0 if and only if α ∈ Sim(a, b).
We will prove this proposition by induction on η(α), where the base case of η(α) = 0 follows immediately since α = α.Assume the result holds for all α ∈ SSim(a, b) such that η(α) < n for some n > 0. Now let α be such that η(α) = n.Pick two strings s and t in α such that µ({s, t}) ≥ 2. Order the points of intersections between s and t according to when they appear as one travels from the source of s to its target.Under this ordering pick two neighbouring points of intersection p and q.Then diagrammatically we have a configuration of strings of the form R g 1 g m h 1 h x . . . . . .

s t p q
where R is the interior region bounded by the curve segments of s and t between the points of intersection p and q, and the two (possible empty) sets of string segments G = {g 1 , . . ., g m } and H = {h 1 , . . ., h x } account for all the intersections of the boundary of R through t and s respectively.We may assume that we are not in one of the following three situations: (i) : (ii) : or (iii) : since otherwise we may pick the more nested pair of intersections to work with instead.Since situations (i) and (iii) are not present, any string segment g i must connect to a h j (rather than another segment in G).
Hence m = x and R realises a pairing of the string segments G with H. Diagrammatically we have where B is some permutation connecting segments in G with those in H.Moreover, since situation (ii) is not present, this means that no string segments in B can intersect more that once.In other words B is built out of crossings, and so we may pull all of B outside of the region R one crossing at a time by (H1), and thus obtain Lastly we may pull these horizontal strings out of R through the top or bottom crossing by (H1).Hence we have emptied R by employing only local relation (H1), and so the value η(α) has remained the same.Now there are four different cases depending on the orientations of the strings s and t.In three of these cases, since R is empty, we may pull the strings s and t apart by applying (H2), and thus remove the two intersections p and q.This decreases η(α) by two, and so the result follows by induction.The last case is given with orientations as follows Proof.As discussed previously, this will follow by showing that α ∈ ϕ(e i ), ϕ(τ j ) i,j for all α ∈ Sim((↑↓) k ).We prove this by downwards induction on deg(α) where c γ ∈ Z.By induction all the simple terms in the above summation belong to Im(ϕ).Also from previous discussions we know that π ↑ βσ ↓ ∈ Im(ϕ), thus rearranging the above equation shows that α ∈ Im(ϕ), completing the proof by induction.
Remark 4.2.9.Equation ( 14) is the key to Theorem 4.2.8, and follows from Proposition 4.2.7.This proposition applies to all semisimple diagrams which are much more general than those appearing here.Ideally, one would like to prove that Equation ( 14) holds for π ↑ βσ ↓ by some inductive argument without needing to show it for all semisimple diagrams.However it is a very delicate task to check which properties are preserved by an inductive process.So we ended up using this more general approach instead, even though many of the cases considered in proving Proposition 4.2.7 probably won't occur in this case.

The affine partition category of Brundan and Vargas
In this last section we relate our affine partition algebra to the work of J. Brundan and M. Vargas in [BV21] and prove a new result on their category.We start by recalling the definition of their affine partition category APar as a subcategory of Heis generated by certain objects and morphisms, and of their affine partition algebra AP k , which is an endomorphism algebra within APar. , The affine partition algebra is defined to be AP k := End APar ((↑↓) k ).
We can generalise the arguments in the proof of Theorem 4.2.8 to show the following result.
Theorem 4.3.2.The affine partition category APar is the full monoidal subcategory of Heis generated by the object ↑↓.
Using Theorem 4.1.8,we need to show that any element of the form c kw w . . .c k1 1 c k0 0 r s1 1 r s3 3 . . .r where α ∈ Sim((↑↓) k , (↑↓) l ) can be written in terms of the generating morphisms in APar.The morphisms r i can be obtained by tensoring the generators (18) with the appropriate identity morphisms on the left and right (and subtracting the identity).Moreover, the morphisms c i can be obtained by concatenating r i 1 with the generators (17) on top and bottom.Thus, it remains to show that any diagram α ∈ Sim((↑↓) k , (↑↓) l ) can be written in terms of the generating morphism in APar.A generalisation of Jones' normal form shows that any planar α ∈ Sim((↑↓) k , (↑↓) l ) can be written in terms of the generators ( 16) and (17) (see for example [RSA14, Proof of Lemma 2.1] for an explicit construction).Now Lemma 4.2.5 allows us the write any α ∈ Sim((↑↓) k , (↑↓) l ) as α = π ↑ βσ ↓ where π ∈ S(k), σ ∈ S(l) and β is planar.Note that s ↑ i and s ↓ i can be written using the generators (19) and the composition of the generators (16) (and tensoring with the appropriate identity morphism on the left and right).So using the discussion following Example 4.2.3 we know that π ↑ and σ ↓ belong to Hom APar ((↑↓) k , (↑↓) l ).Now we can follow exactly the same proof as for Theorem 4.2.8 noting that in this case the maximum degree is (k + l, k + l) and the only simple diagram with that degree is the one containing k consecutive arcs at the top and l consecutive arcs at the bottom, which is planar.The rest of the proof can be followed verbatum simply replacing Imϕ by Hom APar ((↑↓) k , (↑↓) l ).
We immediately obtain the following consequences.We do not know whether the map ϕ is an isomorphism.If it were, then we would also have a presentation for AP k .
Lemma 4.1.1.Clockwise bubbles satisfy the commuting relation = Although left curls annihilate diagrams, right curls do not, and they play an important role.We will represent right curls by a decoration, and label such decorations with weights to denote multiplicity: is a simple application of the local relations.Lemma 4.1.2.The following two local relations hold: Definition 4.1.3.For a, b ∈ ↑, ↓ , we say an (a, b)-diagram is simple if it contains no loops, no selfintersections, and two strings intersect at most once.Let Sim(a, b) denote the set of simple (a, b)-diagrams, and write Sim(a) for Sim(a, a).
↓}, let b * denote the word obtained from b by replacing up arrows with down arrows, and down arrows with up arrows.Let u equal the number of up arrows appearing in a and b * , and d the number of a down arrows.Then by (D1), one can deduce that Hom Heis (a, b) is non-empty if and only if u = d.In this situation we have that |Sim(a, b)| = u!, since there is one simple (a, b)-diagram for every (a, b)-matching.Such a correspondence is given by reading the pairings of endpoints formed from the strings of a simple diagram.Example 4.1.4.Consider the words a =↑↓ and b =↓↑↑↓.Then the 6 = 3! simple (a, b)-diagrams are These diagrams are in a one-to-one correspondence with the (a, b)-matchings of the set of endpoints E = {(i, 1), (j, 0) | i ∈ [2], j ∈ [4]}.As an example, for the first diagram above we have ↔ {(1, 1), (2, 0)}, {(2, 1), (1, 0)}, {(3, 0), (4, 0)} .

colouring the string s in red we have
Let a = a 1 • • • a k and b = b 1 • • • b l for a i , b i ∈ {↑,↓}, and consider the map deg : SSim(a, b) → Z ≥0 × Z ≥0 be given by deg(α) = (A(α), C(α)) where A(α) is the number of arcs in α, and C(α) is the number of clockwise arcs in α.We order the degrees by using the lexicographical ordering < on Z ≥0 × Z ≥0 .Note that for any α ∈ SSim(a, b) we have deg(α) = deg(α).
for all l ∈ Z ≥0 .