On geometrically defined extensions of the Temperley-Lieb category in the Brauer category

We define an infinite chain of subcategories of the partition category by introducing the left-height ($l$) of a partition. For the Brauer case, the chain starts with the Temperley-Lieb ($l=-1$) and ends with the Brauer ($l=\infty$) category. The End sets are algebras, i.e., an infinite tower thereof for each $l$, whose representation theory is studied in the paper.


Introduction
The partition algebra and its Brauer and Temperley-Lieb (TL) subalgebras [8,58,42] have many applications and a rich representation theory (see e.g.[8,10,60,43,51] and references therein).In particular each representation theory has an intriguing geometrical characterisation [30,16,40,50] (in the Brauer case also embracing the Birman-Murakami-Wenzl (BMW) algebra [54,6,33,53,61]).The TL case can be understood in terms of Lie Theoretic notions of alcove geometry and geometric linkage, via generalised Schur-Weyl duality [41,22,30], but the Brauer case is much richer [47] and, although its complex representation theory is now intrinsically wellunderstood, the paradigm for the corresponding notions is more mysterious.Here we introduce a sequence of (towers of) algebras J l,n which interpolate between the TL algebra and the Brauer algebra as algebras.A particular aim is to study the geometry in their representation theory by lifting this new connection to the representation theory level.To this end we investigate the representation theory of the new algebras using their amenability to tower of recollement (ToR) [15] and monoid methods [56].The representation theory for large l, n eventually becomes very hard, but we are able to prove a number of useful general results, and results over the complex field.For example we obtain the 'generic' semisimple structure in the sense of [14].
By way of further motivation (although we will not develop the point here) we note that both the Brauer and TL algebras provide solutions to the Yang-Baxter (YB) equations [3,57].In addition to their interest from a representation theory perspective, our new algebras can be seen as ways to address the problem of contruction of natural solutions to the boundary YB equations in the TL setting (generalising the blob approach and so on -see e.g.[19] and references therein).
A paradigm here is the XXZ spin chain -a 'toy' model of quantum mechanical interacting spins on a 1-dimensional spacial lattice derived from the Heisenberg model [5]; see e.g.[37,Ch.6].
An outline of the paper is as follows.The partition category has a basis of set partitions, and the Brauer and Temperley-Lieb categories are subcategories with bases of certain restricted partitions.In particular the Temperley-Lieb category has a basis of non-crossing partitions.Here we provide a classification of partitions generalising the geometrical notion of non-crossing.Many such games are possible in principle (see e.g.[12]), but we show that our classification (like non-crossing) is preserved under the partition category composition.This closure theorem allows us to define a sequence of new 'geometrical' subcategories.Next we turn to our motiviating aim: investigation of geometric features in the algebraic representation theory contained in these categories.We focus in particular here on the extensions of the Temperley-Lieb category in the Brauer category.In this paper we establish a framework for modular representation theory of the corresponding towers of algebras.In the case that is modular over C in the sense of [9] we prove that the algebras are generically semisimple.We observe an intriguing subset of parameter values for which they are not semisimple, distinct from both TL and Brauer cases.We conclude by determining branching rules, and hence give combinatorial constructions for the ranks of these algebras.
The TL algebra has a sequence of known generalisations using its characterisation via an embedding of pair partitions in the plane -the blob algebras and the contour algebras [48]; as well as various beautiful generalisations due to R. Green et al [26,23], tom Dieck [59] and others.The blob algebra also has a rich geometricallycharacterised representation theory [50].However none of the previously known cases serve to interpolate between the TL algebra and the Brauer algebra.

Notations and pictures for set partitions
We need to recall a pictorial (and so en passant geometric) realisation of the partition algebra (i.e. of set partitions).This realisation is in common use (see e.g.[43]), but we will need to develop it more formally.
(1.1)If T is a set then P (T ) denotes the set of partitions of T .Noting the standard bijection between P (T ) and the set of equivalence relations on T we write a ∼ p b when a, b in the same part in p ∈ P (T ).
Suppose that p ∈ P (T ) and S ⊂ T .Write p| S for the restriction of p to S. Write f S (p) := #{π ∈ p | π ∩ S = ∅}, the number of parts of p that do not intersect S. (Here we follow [42,Def.20].See also e.g.[45].)(1.2) Let G denote the class of graphs; G(V ) the subclass of graphs on vertex set V ; and G[S] the subclass of graphs whose vertex set contains set S. Define (1.3) We shall use drawings to represent graphs in a conventional way: vertices by points and edges by lines (polygonal arcs between vertex points), as in Fig. 1 or 2.
A picture d of a graph is thus (i) a rectangular region R of the plane; (ii) an injective map from a finite set into R (hence a finite subset of points identified with vertices); and (iii) a subset of R that is the union of lines.Line (polygonal arc) crossings are not generally avoidable (in representing a given graph in this way), but we stipulate 'line regularity': that, endpoints apart, lines touch only at points in the interior of straight segments; and that a line does not touch any vertex point except its endpoints, or the boundary of R except possibly at its endpoints.
Note: (I) Regularity ensures that no two graphs have the same picture, and hence gives us a map 'back' from pictures to graphs.(II) Any finite graph can be represented this way (indeed with the vertices in any position, see e.g.[17]).
For g ∈ G[S] one thinks of S as a set of 'external' vertices, and draws them on the horizontal part of the rectangle boundary.Interior vertices (v ∈ S) will generally not need to be explicitly labelled here (the choice of label will be unimportant).
(1.4) A vacuum bubble in g ∈ G[S] is a purely interior connected component [7] (cf.Fig. 2).The vacuum bubble number is , ..., n} and n ′ := {1 ′ , 2 ′ , ..., n ′ }, and so on.Let We draw them as in Figures 1 and 2. We define Π n,m = Π n∪m ′ , so 2' 3' (1.7) Next we recall the partition category P, as defined in [42, §7].We first fix a commutative ring k say, and δ ∈ k.The object set in P is N 0 .The arrow set P(n, m) is the free k-module with basis P (n, m).Noting (1) this means that elements of P(n, m) could be represented as formal k-linear combinations of (n, m)graphs.In fact one generalises this slightly.In P an (n, m)-graph (as in (1.6)) maps to an element of kP (n, m) via: The composition p * q in P can be defined and computed in naive set theory [42].But it can also be computed by representing composed partitions as stacks of corresponding pictures of graphs, as in Fig. 2. First a composition • : G(n, m) × G(m, l) → G(n, l) is defined: a • b is given by stacking pictures of a and b so that the m vertex sets in each meet and are identified as in the Figure .Then p * q = Π P (a • b) for suitable a, b.For example, in case p = {{1, 2 ′ }, {1 ′ , 3 ′ }} in P (1, 3) and q = {{1, 5 ′ }, {2, 4 ′ }, {3, 1 ′ }, {2 ′ , 3 ′ }} in P (3, 5) then the composition δp * q, or more explicitly can be verified via Fig. 2. In general, denoting the stack of pictures by d|d ′ , then for any d, d ′ such that p = Π P (d) and q = Π P (d ′ ).(Given the set theoretic definition of P [42] the identity (2) would be a Theorem.Here we can take it as the definition, and one must check well-definedness.)Remark: From this perspective the pictures constitute a mild modification of the plane projection of arrows in the tangle category, in which arrows are certain collections of non-intersecting polygonal arcs in a 3D box (see later, or e.g.[32]).

Overview of the paper
We start with a heuristic overview and summary.Later, in order to prove the main Theorems, we will give more formal definitions.
Besides the representation of a set partition p by a graph g, the task of constructing a picture d of p contains another layer -the embedding and depiction of graph g in the plane.Both stages of the representation of set partitions are highly nonunique.However, they lead to some remarkable and useful invariants.To describe these invariants we will need a little preparation.
Suppose we have a picture d of a partition of this kind.Then each polygonal arc l of d partitions the rectangle R into various parts: one or more connected components of R \ l; and l itself.Overall, a picture d subdivides R into a number of connected components, called alcoves, of R \ d (regarding d as the union of its polygonal arcs), together with d itself.
(1.8) Given a picture d, the distance d d (x, y) from point x to y is the minimum number of polygonal arc segments crossed in any continuous path from x to y. Examples (the second picture shows distances to y from points in various alcoves): Note that there is a well-defined distance between a point and an alcove; or between alcoves.
(1.9) The (left-)height of a point in d is defined to be the distance from the leftmost alcove.(By symmetry there is a corresponding notion of right-height.) Given a picture d, a crossing point is a point where two polygonal arc segments cross.Note that these points in particular have heights.For example the upper of the two crossing points in the picture above has height 1, and the other has height 0.
The (left-)height ht(d) of a picture d with crossing points is defined to be the maximum (left-)height among the heights of its crossing points.(We shall say that the left-height of a picture without crossings is -1.) (1.10)Although the picture d of a partition p is non-unique, we can ask, for example, if it is possible to draw p without arc crossings -i.e. if among the drawings d of p there is one without crossings.If it is not possible to draw p without crossings, we can ask what is the minimum height of picture needed -that is, among all the pictures d representing p, what is the minimum picture height?We call this minimum the (left-)height of partition p.
(1.11) Returning to the partition category P = (N 0 , kP (n, m), * ), the existence of a Temperley-Lieb subcategory in P (see e.g.[40, §6.2], [45, §5.1]) corresponds to the observation that the product p * p ′ in P of two partitions of height -1 (i.e.noncrossing) gives rise to a partition that is again height -1.Our first main observation is a generalisation of this: The height of p * p ′ in P does not exceed the greater of the heights of p, p ′ .Thus: For each l ∈ N − := {−1, 0, 1, 2, ...} there is a subcategory spanned by the partitions of height at most l.
We first prove this result.This requires formal definitions of 'left-height' and so on, and then some mildly geometrical arguments.Write J(n, m) ⊂ P (n, m) for the subset of partitions of n ∪ m ′ into pairs.The partitions of this form span the Brauer subcategory: B = (N 0 , kJ(n, m), * ); and the construction above induces a sequence of subcategories here too.The rest of the paper is concerned with the representation theory of the tower of algebras associated to each of these categories, that is, the algebras that are the End-sets in each of these categories.

Formal definitions and notations
We start with a formal definition of a picture, a drawing as in (1.3).Notation is taken largely from Moise [52] and Crowell-Fox [17].
(2.2) A polygonal arc is an embedding l of [0, 1] in R 3 consisting of finitely many straight-line segments.The open arc (l) of l is the corresponding embedding of (0, 1).An arc-vertex in l is the meeting point of two maximal straight segments.
(2.3) A polygonal graph is (i) an embedding ǫ of the vertex set V of some g ∈ G(V ) as points in R 3 ; and (ii) for graph edges E a polygonal embedding ǫ : ⊔ e∈E (0, 1) ֒→ R 3 \ ǫ(V ) such that the closure points of (0, 1) e agree with the endpoints of e.
(2.4) Note that every g has an embedding for every choice of ǫ : V ֒→ R 3 .
(2.5) Note that if the edge labels are unimportant then we can recover the original graph from the map ǫ : V (labeling graph vertex points) and the image ǫ(g).Note well the distinction between graph vertices ǫ(v ∈ V ) and polygonal arc-vertices.
(2.6) Fix a coordinate system on R (2.7) A picture is a triple d = (V, λ, L) consisting of a set V , an injective map λ : V ֒→ R 2 and a subset L ⊂ R 2 such that λ = p • ǫ| V for some regular polygonal graph with g ∈ G(V ); and L is the image L = p(ǫ(g)).(The datum also includes the containing rectangle R, but notationally we leave this implicit.) The point here is that such a d, consisting only of labeled points and a subset of R, determines a graph g; and every graph has a picture.Note that d also determines the set of points where |p −1 (p(k))| = 2 in (2.6)(ii) -the set χ(d) of crossing points.
(2.8) Let us consider pictures with R oriented so that its edges lie in the x and y directions.If the vertex points on the northern (respectively southern) edge of R are not labelled explicitly then they may be understood to be labeled 1, 2, ... (respectively 1 ′ , 2 ′ , ...) in the natural order from left to right.
In particular such a frame-drawn picture without any vertex point labels determines a graph in some G[n ∪ m ′ ] up to labelling of the other 'interior' vertices.
We identify pictures differing only by an overall vertical shift.Given our vertex labelling convention above we could also identify under horizontal shifts, but the horizontal coordinate will be a useful tool in proofs later, so we keep it for now.

Stack composition of pictures
Here we follow the usual construction of 'diagram categories' (e.g. as in [42, §7]), but take care to emphasise specific geometrical features that we will need later.(See also e.g.(2.10)Note that for d ∈ h 0 (a, b) there is an essentially identical picture with R wider.Thus any two such pictures may be taken to have the same (unspecified, finite) interval of R as their northern edges, and southern edges.The juxtaposition of rectangular intervals, R, R ′ say, by vertical stacking of R over R ′ thus produces a rectangular interval, denoted R|R ′ .This is almost a disjoint union, except that the southern edge of R is identified with the northern edge of R ′ .
Given a pair of pictures d and d ′ , stack R|R ′ induces a corresponding pair of subsets λ(V )|λ ′ (V ′ ) and L|L ′ in the obvious way.For example see Fig. 2.
(2.11) Proposition.The stack juxtaposition of a picture d in h 0 (a, b) over a picture d ′ in h 0 (b, c) defines a picture d|d ′ in h 0 (a, c).
Proof: As noted, the stack R|R ′ induces a corresponding pair of subsets λ(V )|λ ′ (V ′ ) and L|L ′ .The former is a union of finite point sets which clearly agrees with a and c on the relevant edges of R|R ′ .The latter is a union of lines, and the only new meetings are at the marked points in b (as it were).These are now interior marked points.Conditions (2.6)(i-v) hold by construction.✷ (2.12) Allowing rectangles of zero vertical extent in any h 0 (a, a) allows for an identity element 1 a of stack composition in h 0 (a, a).Write h 0 for the 'picture category'.
(2.13) Proposition.For finite subsets a and b of R, there is a surjection given by counting the elements of a (resp.b) from left to right and using Π n,m from (1).✷ (2.14) Proposition.Fix a commutative ring k and δ ∈ k, and let π p denote the generalisation of π e corresponding to , where the product on the right is in the partition category P = (N 0 , kP (n, m), * ) [42].
We can take this as a definition of P cf.(1.7).For a proof of equivalence to the original definition see e.g.[42].In outline, one checks that the stack composition implements the transitive closure condition [42, §7].✷ Let p ∈ P (n, m).Write p * for the element of P (m, n) obtained by swapping primed and unprimed elements of the underlying set.
Note that if d is a picture of p ∈ P (n, m) then d * is a picture of p * ∈ P (m, n).Furthermore, this * is a contravariant functor between the corresponding partition categories.
(2.16) Note that for any picture in the category h 0 with distinct northern and southern edge we can vertically rescale to arbitrary finite separation of these edges.Thus we can make any two pictures have the same separation.For two such pictures d, d ′ there is a picture d ⊗ d ′ obtained by side-by-side juxtaposition.
(2.17) We call a picture a chain picture if every exterior marked point (as in (2.9)) is an endpoint of precisely one line, and every interior marked point is an endpoint of precisely two lines (e.g. as in Fig. 2).Write h 2 0 (a, b) ⊂ h 0 (a, b) for the subset of chain pictures.Note that every p ∈ J(n, m) has a chain picture.We have the following.
(2.18) Lemma.The stack composition (2.11) closes on chain pictures.This gives a subcategory of the category in (2.12).The corresponding π p − quotient category (as in (2.14)) is the Brauer category B = (N 0 , kJ(n, m), * ).✷ (2.19)A pair partition is plane if it has a frame drawn picture (as in (2.8)) without crossings of lines.We write T (n, m) for the subset of plane pair partitions (TL partitions) and T = (N 0 , kT (n, m), * ) for the corresponding subcategory of B.
that is an open straight segment.(For example in some picture d = (V, λ, L), with L = p(ǫ(g)), the non-simple points of L \ λ(V ) are the arc-vertices and crossing points.)Given a picture (V, λ, L), a path in (V, λ, L) is a line l in R such that every k ∈ (l) ∩ L is a simple point of (l) and also a simple point of L \ λ(V ).
Thus a path l in (V, λ, L) has a well-defined number of line crossings, d L (l).
(2.23) Lemma.Given a path l in picture d connecting points x, y ∈ R and a distinct point z ∈ (l), there is a path l ′ connecting x, y that does not contain z.
Proof.Since z ∈ (l) it has a neigbourhood either containing only a segment of l; or only a crossing of l with a straight segment of L. If we modify the path inside the neighbourhood by a small polygonal detour then the modification is a path and does not contain z. ✷ (2.24) Lemma.For each picture d and x, y ∈ R there is a path in d from x to y.
Proof.Draw a small straight line l 1 from x to a point x ′ in an adjacent alcove, choosing x ′ so that the tangent of the straight line x ′ − y is not in the finite set of tangents of segments of lines of L; and the line does not contain any element of the finite set of crossing points of d.Then x − x ′ − y is a path.✷ We suppose that L does not intersect the left edge R L of R. The (left)-height ht L (r) = d L (r, x) in case x is any point on R L .(Note that this is well-defined.) The (left)-height of an alcove A is the left-height of a point in A. See Fig. 3 for examples.
(2.26) Given a picture d = (V, λ, L), recall χ(d) is the set of crossing points of lines in d (recall from (1.3) that, vertex points aside, lines in d only meet at crossing points).The left-height ht(d) is the greatest of the left-heights of the points x ∈ χ(d); or is defined to be -1 if there are no crossings.
For example, d 1 in Fig. 3 has left-height 2; and d 2 has left-height -1.
(2.27) Finally we say that a partition p ∈ P (n, m) has left-height ht(p) = l if it has a frame drawn picture of left-height l, and no such picture of lower left-height.For example, both pictures in Fig. 3 give the same partition p, so ht(p) = −1.
A path realising the left-height of a point in a picture is called a low-height path.A picture realising the left-height of a partition is called a low-height picture.
(2.28) Define P l (n, m) as the subset of partitions in P (n, m) of left-height l, and Define J l (n, m) as the corresponding subset of J(n, m), and J ≤l (n, m) analogously.
(2.29) Remark: Observe that for p ∈ J(n, n), ht(p) ≤ n − 2. Hence, in particular, (2.30) Example.Here we give the J l (3, 3) subsets of J(3, 3).Each element is represented by a low-height picture (of course, other pictures could have been chosen instead).Note that it is a Proposition that a given picture is low-height.One should keep in mind that the elements of J l (3, 3) are pair partitions, not pictures!
More generally the Brauer algebra identity element 1 n ∈ J −1 (n, n); the symmetric group Coxeter generator Removing part or all of a line from a picture cannot produce a picture with higher height.
Proof.The number of crossings of a path cannot be increased by removing a line.✷ (2.32)In particular, if a line has a self-crossing then we can 'short-circuit' the path without increasing the height, or changing the partition.The self-crossing point becomes an arc-vertex with a regular neighbourhood, so the regularity of the picture remains to hold.Thus for each low-height picture there is a low-height picture without line self-crossings.
3 Algebraic structures over J ≤l (n, n) (3.1) Recall (e.g. from [8] or (2.18)) that the multiplication in the Brauer k-algebra B k n = kJ(n, n) may be defined via vertical juxtaposition of representative diagrams.(3.2) Define J k l,n as the k-subalgebra of B k n generated by J ≤l (n, n).For k a fixed commutative ring and δ c ∈ k we write J l,n = J l,n (δ c ) for the base-change: (i.e., regarding k as a k-algebra in which δ acts as δ c ).For example J l,n (1) is the monoid k-algebra for the submonoid of the Brauer monoid associated to B n (1) [51].
(3.3) Define J l as the subcategory of B generated by the sets J ≤l (n, m), over all n, m ∈ N 0 (that is, the smallest k-linear subcategory such that the collection of arrows J l (n, m) contains J ≤l (n, m) for each n, m).
Note that the smallest possible height for a crossing is 0, so J −1 (n, m) is the subset with no crossings, that is J −1 (n, m) = T (n, m).It will be clear then that , by Remark 2.29) is the Brauer algebra.Next we will show (our first main Theorem) that the 'interpolation' is proper in the following sense: (3.4) Theorem.(I) The sets J ≤l (n, m) form a basis for the k-linear category J l .That is, J l = (N 0 , kJ ≤l (n, m), * ).(II) The set J ≤l (n, n) is a k-basis for J k l,n .Proof.(I) Recall from definition (3.3) that the n, m-arrow set in J l is generated by J ≤l (n, m).Note that J ≤l (n, m) is linearly independent over k in J l , as it is linearly independent in the Brauer category B. It is thus enough to show that J ≤l (n, m) × J ≤l (m, j) maps into kJ ≤l (n, j) under the Brauer category product, and that the sets of arrows of form J ≤l (n, m) are therefore suitably spanning.For this, we need to show that for every pair of pair-partitions (p 1 , p 2 ) ∈ J ≤l (n, m) × J ≤l (m, j), determining a partition p 3 ∈ J(n, j), we have p 3 ∈ J ≤l (n, j).By (2.27) the pair-partitions p 1 , p 2 have composable minimum-height pictures, denoted by d 1 , d 2 , respectively.By (2.14) their vertical juxtaposition d 1 |d 2 gives p 3 , so it is sufficient to show that ht(d 1 |d 2 ) ≤ l.
Observe that by construction the set of crossing points of d 1 |d 2 is precisely the disjoint union of those of d 1 and d 2 .Now, also by construction a low-height path from any point x in d 1 remains a (not necessarily of low height) path in d 1 |d 2 .See the path from x in the figure below for example.(3.6) Theorem.There is a subcategory P l = (N 0 , kP ≤l (n, m), * ) of P.
Proof.The proof of (3.4) works mutatis mutandis.✷ We will discuss connections with known constructions in §6.2.

Representation theory of J l,n
We now begin to examine the representation theory of J l,n .We follow a tower of recollement (ToR) approach [15].A part in p ∈ P (n, m) is propagating if it contains both primed and unprimed elements of n ∪ m ′ (see e.g.[40,42]).Write P (n, l, m) for the subset of P (n, m) of partitions with l propagating parts; and similarly J(n, l, m).Define J l (n, r, m) = J l (n, m) ∩ J(n, r, m).
We write # p (p) for the propagating number -the number of propagating parts.
Define u as the unique element in J(2, 0).Note that P and B are isomorphic to their respective opposite categories via the opposite mapping c → c * .Thus u * is the unique element in J(0, 2).Define U to be the pair partition in J(2, 2) determined by the following picture.
U := uu * = We use ⊗ to denote the monoidal/tensor category composition in P (that is, the image of the side-by-side concatenation of pictures from (2.16), extended k-linearly).

∈ J(n, n)
Given a partition p in P (n, n) we write p| n−2 for the natural restriction to a partition in P (n−2, n−2).(Note that this restriction does not take J(n, n) to J(n−2, n−2).) 4.1 Index sets for simple J l,n -modules Here we assume we have base-changed as in (4) to a field k.We write J l,n for J l,n (δ c ) if we do not need to emphasise δ c .Write k × for the group of units.For simplicity only we generally assume δ = δ c is invertible in k (e.g. as in k).given by Ψ : ede → ede| n−2 .For δ ∈ k × (and n ≥ 2) this Ψ is an algebra isomorphism.
Proof.Note ede = d ′ ⊗ U for some d ′ ∈ J l,n−2 , so Ψ(ede) = d ′ so the map is injective (just as in the ordinary Brauer case).To show surjectivity in case δ ∈ k × consider d ′ in J l,n by the natural inclusion of J l,n−2 ֒→ J l,n (the key point here is that the natural inclusion J n−2 ֒→ J n takes J l,n−2 ֒→ J l,n since the embedding does not change the height of crossings in the d ′ part, as it were, and does not introduce further crossings), so Ψ(ed ′ e) = δd ′ for any d ′ ∈ J l,n−2 .Other cases are similar.✷ (4.2) Corollary.Suppose that Λ(J l,n ) denotes an index set for classes of simple modules of J l,n , for any n.Then the set of classes of simple modules S of J l,n such that eS = 0 may be indexed by Λ(J l,n−2 ).
Proof.Note that δ −1 e is idempotent and apply Green's Theorem in [25, §6.2].✷ (4.3) Corollary.The index set Λ(J l,n ) may be chosen so that where Λ(J l,n /(J l,n eJ l,n )) is an index set for simple modules of the quotient algebra by the relation e = 0.In other words Λ(J l,n ) ∼ = Λ(J l,n−2 ) ⊔ Λ(J l,n /(J l,n eJ l,n )) and the sets {Λ(J l,n )} n are determined iteratively by the sets {Λ(J l,n /(J l,n eJ l,n ))} n .✷ Define J l (n, <m, n) = ∪ r<m J l (n, r, n) and so on (e.g.J ≤l (n, < n, n) includes every pair partition in J ≤l (n, n) with submaximal number of propagating lines).
(4.4) Proposition.The ideal J l,n eJ l,n = kJ ≤l (n, <n, n) as a k-space.
Proof.Let p ∈ J ≤l (n, n) have submaximal number of propagating lines, that is # p (p) ≤ n − 2 (see page 12), so that it has at least one northern and one southern pair.Let d be a low-height picture of p.We will use d to show that p ∈ J l,n eJ l,n .
For X some subset of {N, S}, let d[−X] denote the picture obtained from d by deleting the lines from north to north (N), south to south (S), or both.Thus d[−S] is a picture of some p t ∈ J(n, j, j) where j = # p (p).Similarly, abusing notation slightly by writing u * for some low-height picture of u * , then d[−S] ⊗ u * is a picture of some p ′ ∈ J(n, j, j + 2).Next, observe that one can add some loops on the right of this picture, (red circles in the example Fig. 5) with no change to the height of the picture, or the resulting partition p.Thus, up to an overall factor of a power of δ, p can be expressed in the form p ′ ep ′′ where a picture for a picture of height ≤ l by construction), and p ′′ is d[−N] ⊗ u ⊗(n−j)/2 , and hence p ′ , p ′′ ∈ J l,n .Finally note that the red loops can be replaced by suitable non-crossing deformations of lines from above and below (cf. the rightmost picture in example Fig. 5).✷ (4.5)By (4.4) the quotient algebra J l,n /(J l,n eJ l,n ) has a basis which is the image of J ≤l (n, n, n).Note that these elements of J l,n form a subgroup as well as their image spanning a quotient algebra.
For n < l + 2 this group is isomorphic to S n ; otherwise it is isomorphic to S l+2 (since there can be no crossings after the first l + 2 lines).The quotient itself is then the corresponding group algebra.
That is, Λ(J l,n /(J l,n eJ l,n )) ∼ = Λ(kS min(n,l+2) ).Combining with Prop.4.3, we thus have the following.n are disjoint copies of where ′ p denotes a range including only p congruent to n mod.2.✷ (4.7)Here J || (n, l, m) denotes the subset of J(n, l, m) of elements p having a picture d for which d[−NS] has no crossings.Recall (e.g. from [47]) the polar decomposition of an element of J(n, m, n): the inverse of the map ν : given by the category composition.Note that if p ∈ J ≤l (n, m, n) then l bounds the height of all three factors in the polar decomposition.(The argument is analogous to the argument at (5).Firstly note that if d is a low-height picture of p then d[−S] | d[−NS] * is a picture of the northern polar factor of no higher height.The other factors are similar.)That is, the restriction On the other hand the image of ν on this codomain lies in J ≤l (n, m, n) by Th.3.4,so the restriction as given is a bijection.

Quasiheredity of J l,n
The proof of the main result of this section (Thm.4.12) follows closely the Brauer algebra case, as for example in [16].We focus mainly on the new features required for the present case.(4.9) Corollary.Provided that δ ∈ k × , the ideal J l,n e n,t J l,n = kJ ≤l (n, ≤ n−2t, n).
Proof.The proof is the same as for Prop.4.4 except that we use more pairs of loops, instead of single loops, in the final stage of the construction of d ∈ J l (n, ≤ n − 2t, n) (the red loops in ( 6)).Note that there is room for enough loops because of the bound on the number of propagating lines in d. ✷ (4.10) Define the quotient algebra J l,n,t = J l,n /J l,n e n,t+1 J l,n By Prop.4.9 this algebra has basis J ≤l (n, ≥ n − 2t, n).
(4.11) Proposition.For each triple n, l, t the following hold when δ ∈ k × .(i) The algebra A = e n,t J l,n,t e n,t is semisimple over C. (ii) The multiplication map J l,n,t e n,t ⊗ A e n,t J l,n,t µ → J l,n,t e n,t J l,n,t is a bijection of J l,n,t , J l,n,t -bimodules.
Proof.(i) The number of propagating lines must be at least n − 2t, but with the e n,t 's present this is also the most it can be, so every propagating line in e n,t is propagating in A, and indeed A is isomorphic to the group algebra of a symmetric group.(ii) The map is clearly surjective.We construct an inverse using the polar decomposition (4.7).Note that J l,n,t e n,t has a basis in bijection with . Subset e n,t J l,n,t can be treated similarly.It then follows from the definition of the tensor product that the left-hand side has a spanning set whose image is independent on the right.✷ (4.12) Theorem.If δ invertible in k = C then J l,n is quasihereditary, with heredity chain (1, e ′ n,1 , e ′ n,2 , ...).Proof.This follows from (4.11).Cf. e.g.[18,13,20,16].✷

Aside: Slick proof of quasiheredity in the monoid case
Note that for a ∈ J(n, n) we have aa * a = a in J l,n .Since the height of a * is the same as a we have the following.(4.13) Proposition.The algebra J l,n (1) is a regular-monoid k-algebra (i.e. a ∈ aJ l,n a for all a ∈ J l (n, n)).✷ (4.14) Corollary.The algebra J l,n (1) is quasihereditary when k = C.

Standard modules of J l,n
We may construct a complete set of 'standard modules' for each J l,n as follows.The modules we construct are 'standard' with respect to a number of different compatible axiomatisations (the general idea of standard modules, when such exist, is that they interpolate between simple and indecomposable projective modules).For example (I) we can construct quasihereditary standard modules by enhancing the heredity chain in Th.4.12 to a maximal chain cf.[20]; (II) we can construct the modular reductions of lifts of 'generic' irreducible modules in a modular system cf.[4]; (III) we can construct globalisations of suitable modules from lower ranks cf.[43,35,47].We are mainly interested in a useful upper-unitriangular property of decomposition matrices that we establish in (4.27).
(4.15)Note that kJ ≤l (n, m), with m ≤ n say, is a left J l,n right J l,m -bimodule, by the category composition.By the bottleneck principle it has a sequence of submodules: For given l, each section thus has basis J ≤l (n, p, m).In particular the top section has basis J ≤l (n, m, m).
(4.16)The above holds in particular for the case m = n, where our sequence is an ideal filtration of the algebra, cf.Corol.(4.9).Define quotient algebra Note that this is the same as J l,n,t with p + 2 = n − 2t (but now without restriction on δ).The index p tells us that partitions with p or fewer propagating lines are congruent to zero in the quotient.

Towards the Cartan decomposition matrices
The Cartan decomposition matrix encodes the fundamental invariants of an algebra [4].Given the difficulty experienced in computing them in case l = −1 and particularly l = ∞ [47] we can anticipate that they will not be easy to determine in general.However, the main tools used in cases l = −1 and l = ∞ can be developed in general, as we show next.We start with a corollary to Prop.4.27.
(4.28) Corollary.Module ∆ p,λ has a contravariant form (with respect to * ) that is unique up to scalars.The rank of this form determines the dimension of the simple module L p,λ .
Proof.The space of contravariant forms is in bijection with the space of module maps from ∆ p,λ to its contravariant dual (the analogous right-module E n (p, λ)J /p−2 l,n treated as a left-module via ordinary duality).But by the upper-unitriangular property (4.27) this space is spanned by any single map from the head to the socle.✷ The contravariant form here is the analogue of the usual form for the Brauer algebra [27,47].That is, a suitable inner product e i , e j is given by e * i e j = e i , e j E n (p, λ) (this is well-defined by (4.24)).The form rank is given by the matrix rank of the gram matrix over a basis.
As noted, when the determinant vanishes (e.g. as a polynomial in δ) then the indecomposable module ∆ contains a proper submodule L. Thus this gives the cases where the algebra is non-semisimple.The roots in the example are given by

Discussion
The construction of J l and P l raises many interesting collateral questions.In this section we assemble some brief general observations on our construction, on further developments and on open problems (we defer full details to a separate note [31]).

Next steps in reductive representation theory of J l,n
The next steps parallel the program for the Brauer algebra used in [47], but now for each l in turn.Essentially we should compute the blocks, and construct 'translation functors'; and then construct corresponding analogues of Kazhdan-Lusztig polynomials.
We write Ind− for the induction functor adjoint to Res− as in (5.3) above.The precursor of translation functors in the Brauer case is the natural isomorphism of functors expressed as Ind ∼ = Res.G.The general setup here is as follows.In the Brauer case we have the 'disk lemma': Consider a partition in J(n+ 1, n+ 1).Replacing the vertex n+1 ′ with a vertex n+2 defines a map η : J(n+1, n+1) → J(n+2, n).One easily checks that this is a bijection; and an isomorphism of n+1, nbimodules.On the other hand B n+2 e has a basis of partitions in J(n + 2, n + 2) with a pair {n + 1 ′ , n + 2 ′ }.There is a natural bijection of this basis with J(n + 2, n) (simply omit the indicated pair).Altogether then, B n+1 and B n+2 e are isomorphic as bimodules.In our case we have the following.(6.2) Lemma.There is a well-defined restriction of η to a map η : J l,n+1 → J l,n+2 e; and this is an isomorphism of left-J l,n+1 right-J l,n -modules.
Proof.One checks that η does not change height.✷ Thus we have the following.This is a powerful result since, for example, the Ind− functor takes projectives to projectives, while Prop.5.3 tells us what Res− does to standard modules.Thus we have an iterative scheme for computing the ∆-content of projective modules, cf.[47].(6.4) Closely related to Prop.6.3, a generalised Jones Basic Construction [38] applies here (cf. the original Jones Basic Construction [24]).It is analogous to the case in [44].(6.5)We can use the graphs R l from (5.4) to give an explicit contruction for the basis states of the standard modules; and indeed of the entire algebra -a generalised Robinson-Schensted correspondence [34].See [31] for details.

Remarks on J l,n construction
The J l construction is amenable to several intriguing generalisations.Here we briefly mention just one particular such generalisation, which case makes a contact with existing studies.(6.6)The first case with crossings, J l=0,n , is connected to the blob algebra [49]: We say a picture is left-simple if the intersection of the 0-alcove (as for example in Fig. 3) with the frame of R is connected.A partition is left-simple if every picture of it is left-simple.(For instance the identity in B n is left-simple, while the example in Fig. 3 is not.)Define J 1 ≤l (n, m) as the subset of J ≤l (n, m) of left-simple partitions.(6.7) Proposition.The subspace kJ 1 ≤0 (n, n) is a subalgebra of J 0,n .This subalgebra is isomorphic to the blob algebra b n−1 (q, q ′ ), with q, q ′ determined by δ as follows.Parameterising (as in [49]) with x = q + q −1 as the undecorated loop parameter; and y = q ′ + q ′ −1 as the decorated loop parameter, we have x = δ and y = δ+1 2 .Proof.(Outline) It is easy to check that the subspace is a subalgebra.It is also easy to show a bijection between J 1 ≤0 (n, m) and the set of (n − 1, m − 1)-blob diagrams.This does not lift to an algebra map, but shows the dimensions are the same.A heuristic for the algebra isomorphism is to note that the intersection of the propagating number zero ideal (4.16) with the subset of left-simple partitions is empty, so there is no ∆ 0,∅ representation.With this node removed, the Rollet diagram from §5 becomes a ('doubly-infinite') chain, which is the same as for the blob algebra.See [31] for an explicit proof.(6.8)On the other hand one can check using §5 that higher l cases such as the algebra generated by J 1 ≤1 (n, n) do not coincide with the higher contour algebras [48] or the constructions in [23,26,59].(6.9) Diagram bases may be used to do graded representation theory (in the sense, for example, of [11]) for graded blob and TL algebras (see e.g.[55]), regarded as quotients of graded cyclotomic Hecke algebras.It would be interesting to try to generalise [55] to J l,n .(6.10)As noted (cf.(1.7), (2.16) and (2.19)), P, B, T are monoidal categories with object monoid (N, +) and monoid composition visualised by lateral (as opposed to vertical) juxtaposition of diagrams.Note however that the categories J l , P l do not directly inherit this structure (except in cases J −1 = T and J ∞ = B).
[57, 2, 1].) (2.9)Given d = (V, λ, L) in rectangle R write n(d) for the subset of R giving the intersection of L with the northern edge of R (thus by (2.6)(v) the collection of x-values of 'northern' exterior vertex points, or 'marked points').Write s(d) for the corresponding southern set.Write h 0 (a, b) for the class of pictures d with n(d) = a and s(d) = b, and L not intersecting the two vertical edges of the containing rectangle.

( 2 . 15 )
If d ∈ h 0 (a, b) is a picture as above, let d * ∈ h 0 (b, a) denote the picture obtained by flipping d top-to-bottom.

2. 2
Paths and the height of a picture/a partition

( 2 . 20 )
Remark.Fix a rectangle R. Each non-self-crossing line l with exterior endpoints in R can be considered to define a separation of R into three parts -the component of R \ l containing the left edge; the other component of R \ l; and l.(2.21)An alcove of picture (V, λ, L) is a connected component of R \ L.

Figure 3 :
Figure 3: Example pictures with left-heights of alcoves.(Remark: By [52, §6] piecewise linear and piecewise smooth lines are effectively indistinguishable as far as physically drawn figures are concerned.)

( 2 . 25 )
Given a picture d = (V, λ, L), and points r, x in R, the x-height d d (r, x) = d L (r, x) of r is the minimum value of d L (l) over paths l in d from r to x.

( 6 )Figure 5 :
Figure 5: Writing p ∝ p ′ ep ′′ .By the Deletion Lemma 2.31 the height of d[−N] does not exceed that of d, and similarly for d[−NS] and d[−S].Note that, since d[−NS] is a picture of a permutation (of the propagating lines); and d[−NS] * is a picture of the inverse, we have that in the picture category (Prop.2.11),d ′ = d[−S] | d[−NS] * | d[−N](5)is another picture for p. (An example is provided by Fig.5: the original picture of p is on the left, whereasd[−S] | d[−NS] * | d[−N] is on the center-left.)Next,observe that one can add some loops on the right of this picture, (red circles in the example Fig.5) with no change to the height of the picture, or the resulting partition p.Thus, up to an overall factor of a power of δ, p can be expressed in the form p ′ ep ′′ where a picture forp ′ is (d[−S] | d[−NS] * ) ⊗ (u * ) ⊗(n−j)/2 (a picture of height ≤ l by construction), and p ′′ is d[−N] ⊗ u ⊗(n−j)/2 , and hence p ′ , p ′′ ∈ J l,n .Finally note that the red loops can be replaced by suitable non-crossing deformations of lines from above and below (cf. the rightmost picture in example Fig.5).✷

( 4 . 19 )
With this functor in mind we recall some facts about the symmetric groups.For any symmetric group S m and a partition λ ⊢ m, let S λ denote the corresponding Specht module of S m -see e.g.[28,29].Recall that there is an element ǫ λ in kS m such that S λ = kS m ǫ λ .If k ⊃ Q then ǫ λ may be chosen idempotent.Example: The element ǫ (2) ∈ kS 2 is unique up to scalars: ǫ (2) = 1 2 + σ 1 (in the obvious notation).Thus a basis for S (2) is b (2) = {ǫ (2) }.

( 4 . 22 )
Proposition.Let λ ⊢ p l and let b λ be a basis for S λ = kS p l ǫ λ .Then a basis for the J l,n -module D p,λ is, up to (irrelevant arc ignoring) isomorphism,B p,λ = {x(y ⊗ 1 p−p l ) | x ∈ J ≤l (n, 1 p , p); y ∈ b λ }regarded as a submodule of J p n,p , where J ≤l (n, 1 p , p) = J || ≤l (n, p, p) as in (4.7).Proof.By definition D p,λ = J /p−2 l,n E n (p, λ) is spanned by elements x(y ⊗ 1 p−p l ) as in B p,λ except with x ∈ J ≤l (n, p, p) (again ignoring irrelevant arcs in the bottom-right).

( 4 . 23 )( 4 . 25 )
First note that a basis element must have p propagating lines by the quotient.Any crossing in the first min(p, l + 2) of these can be 'absorbed' by the b λ part of the basis.There cannot be a crossing in any remaining propagating lines by the height restriction, cf.(4.7).✷ Corollary.D p,λ ∼ = ∆ p,λ Proof.Compare our basis above with the construction for ∆ p,λ .The main difference is combination via ⊗ rather than multiplication (up to some subtleties when δ = 0).This gives us a surjective map right to left.One then compares dimensions.✷ (4.24) Proposition.For given l, E n (p, λ) J/p−2 l,n E n (p, λ) = kE n (p, λ).Proof.Pictorially/schematically we have: Finally ǫ λ kS l+2 ǫ λ = kǫ λ by the Specht property (4.19)[29].✷ Corollary.For (p, λ) ∈ Λ(J l,n ), D p,λ is indecomposable projective as a J /p−2 l,n -module; and hence indecomposable with simple head as a J l,n -module.✷In light of (4.1) J l,n+2 e n+2 is a right J l,n -module.Thus we have also the adjoint pair of functors J l,n − mod Ge . .J l,n+2 − mod

( 4 . 27 )
Proposition.(I) The modules {∆ p,λ } are a complete set of standard modules in the quasihereditary algebra cases (δ invertible in k = C).(II) The simple decomposition matrix C ∆ for this set of modules is upper unitriangular (when written out in any order so that (p, λ) > (p ′ , λ ′ ) when p > p ′ ).Proof.(I) Follows from Corol.4.23.(II) There are several ways to prove this.Prop.4.26 implies that the composition factors with the same label in ∆ n p,λ and ∆ n+2 p,λ have the same multiplicity.The only possible new factors in ∆ n+2 p,λ have the property e n+2 S = 0. Applying this iteratively on n gives the claimed result.✷

Figure 6 :
Figure 6: The J l,n ⊂ J l,n+1 standard Rollet diagram in case l = 1 and case l = 2.

( 6 . 1 )
Lemma.Suppose we have a sequence of algebras A ⊃ B ⊃ C and an idempotent e in A such that eAe ∼ = C.If B and Ae are isomorphic as left-B right-C-modules then functors Ind ∼ = Res G (G = G e defined as in (4.26)).Proof.We have Ind B C − = B C ⊗ C − and G A C − = Ae ⊗ C −.

( 6 . 3 )
Proposition.Fix any l.Then for all n we have Ind ∼ = Res G.