Combinatorics of injective words for Temperley-Lieb algebras
aa r X i v : . [ m a t h . A T ] J un COMBINATORICS OF INJECTIVE WORDS FORTEMPERLEY-LIEB ALGEBRAS
RACHAEL BOYD AND RICHARD HEPWORTH
Abstract.
This paper studies combinatorial properties of the complex of pla-nar injective words , a chain complex of modules over the Temperley-Lieb algebrathat arose in our work on homological stability. Despite being a linear ratherthan a discrete object, our chain complex nevertheless exhibits interesting com-binatorial properties. We show that the Euler characteristic of this complexis the n -th Fine number. We obtain an alternating sum formula for the rep-resentation given by its top-dimensional homology module and, under furtherrestrictions on the ground ring, we decompose this module in terms of certainstandard Young tableau. This trio of results — inspired by results of Reiner andWebb for the complex of injective words — can be viewed as an interpretationof the n -th Fine number as the ‘planar’ or ‘Dyck path’ analogue of the numberderangements of n letters. This interpretation has precursors in the literature,but here emerges naturally from considerations in homological stability. Ourfinal result shows a surprising connection between the boundary maps of ourcomplex and the Jacobsthal numbers. Contents
1. Introduction 12. Temperley-Lieb algebras 63. Injective words and planar injective words 104. Dyck paths, Catalan numbers, and Fine numbers 135. Planar diagrams and Dyck paths 146. Young tableaux 167. Jacobsthal numbers and the boundary maps of W ( n ) 19References 221. Introduction
In this work we study combinatorial properties of a highly connected complexthat arose in our study of the
Temperley-Lieb algebra in [BH20]. Highly connected
Mathematics Subject Classification.
Key words and phrases.
Temperley-Lieb Algebras, Fine numbers, Jacobsthal numbers, chaincomplexes. complexes arise naturally in many areas of mathematics. In combinatorics theyarise as matroid complexes and order complexes of geometric lattices [Bj¨o92], asorder complexes of Cohen-Macaulay posets [BGS82], and in the theory of shella-bility in its various forms [Bj¨o92, BW83, Koz08], to name just a few. For theauthors, highly connected complexes arise in the theory of homological stability .This subject is motivated by the study of homology and cohomology of groups andspaces, and makes extensive use of complexes such as buildings, split buildings,complexes of partial bases (of vector spaces, modules, and free groups), complexesof arcs in surfaces, and many more besides. (Though no standard introductoryreference currently exists for homological stability, we recommend [Wah13]. Theintroduction of [RWW17] may also give a good impression of the theory’s scope.)The complex of injective words is much studied in both combinatorics and topol-ogy. Its high-connectivity has been proved using various methods, by authors in-cluding Farmer [Far79], Maazen [Maa79], Bj¨orner-Wachs [BW83], Kerz [Ker05],and Randal-Williams [RW13], and is an important ingredient in proofs of homo-logical stability for the symmetric groups [Maa79, Ker05, RW13]. Reiner andWebb [RW04] studied the complex of injective words from a combinatorial pointof view. They showed that its Euler characteristic is the number of derangementsof n letters, and they described its top-dimensional homology representation intwo ways: as an alternating sum, and in terms of standard Young tableaux. Afurther decomposition of the top-dimensional homology was given by Hanlon andHersh in [HH04].In our work on homological stability for Temperley-Lieb algebras [BH20], weintroduced and studied the complex of planar injective words , a chain complex ofmodules over the Temperley-Lieb algebra on n strands, closely analogous to the(chain complex of the) complex of injective words. In particular we proved thatthe homology of our complex is concentrated in degree ( n − n -th Fine number . Wewill also expose an unexpected appearance of the
Jacobsthal numbers .1.1.
Temperley-Lieb algebras and planar injective words.
Let n >
0, let R be a commutative ring, and let a ∈ R . The Temperley-Lieb algebra TL n ( a ) isthe R -algebra with basis given by the planar diagrams on n strands, taken up toisotopy, and with multiplication given by pasting diagrams and replacing closedloops with factors of a . The last sentence was intentionally brief, we hope thatits meaning becomes clearer with the following illustration of two elements x, y ∈ NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 3 TL ( a ) x = y =and their product x · y . x · y = = = a · The Temperley-Lieb algebras arose in theoretical physics in the 1970s [TL71]. Theywere later rediscovered by Jones in his work on von Neumann algebras [Jon83],and used in the first definition of the Jones polynomial [Jon85]. Kauffman gavethe diagrammatic interpretation of the algebras in [Kau87] and [Kau90]. The rankof TL n ( a ) as an R -module is the n -th Catalan number C n [Jon87].Now let a = v + v − where v ∈ R × is a unit (the most commonly studiedcase in the literature). The complex of planar injective words W ( n ) is a chaincomplex of TL n ( a )-modules. In degree i it is given by the tensor product mod-ule TL n ( a ) ⊗ TL n − i − ( a ) , where is the trivial module for TL n − i − ( a ). In theoriginal complex of injective words the i -simplices are words ( x , . . . , x i ) on thealphabet { , . . . , n } with no repeated entries. The action of S n on these simplicesis transitive, and the typical stabiliser is S n − i − , so that the i -th chain group is iso-morphic to R S n ⊗ R S n − i − . Thus W ( n ) is an analogue of (the chain complex of)the complex of injective words, in which the role of S n is now played by TL n ( a ).In [BH20] we showed that H d ( W ( n )) = 0 for d n −
2, and since the complex isconcentrated in degrees from − n −
1, it follows that its only homology groupis H n − ( W ( n )). The restriction to the case a = v + v − is necessary for TL n ( a )to receive a homomorphism from the group algebra of the braid group, which isrequired in order to define the differentials of W ( n ).1.2. Results.
The n -th Fine number F n is the number of Dyck paths of length 2 n whose first peak has even height. This is the second of 11 descriptions of theFine numbers given by Deutsch and Shapiro in their survey [DS01]. Deutsch andShapiro also state the following alternating sum formula for F n : F n = 1 n + 1 (cid:20)(cid:18) nn (cid:19) − (cid:18) n − n (cid:19) + 3 (cid:18) n − n (cid:19) − · · · + ( − n ( n + 1) (cid:18) nn (cid:19)(cid:21) (1)(See [DS01, Section 4] and also [Deu99, Moo79, Rob04].) We show that thisalternating sum has a very simple interpretation: its m -th term counts the Dyckpaths whose first peak has height at least m .Remarkably, the complex of planar injective words W ( n ) embodies a represen-tation theoretical ‘lifting’ of the Fine numbers and of this alternating sum formula. NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 4
Theorem A.
Let R be a commutative ring, let v ∈ R × , and let a = v + v − . Thenthe Euler characteristic of W ( n ) is the n -th Fine number, up to sign: χ ( W ( n )) = ( − n − F n . The TL n ( a )-module H n − ( W ( n )) therefore has rank equal to the Fine num-ber F n . We call it the Fineberg module , and we denote it by F n ( a ). (Such top-dimensional homology groups are often called Steinberg modules, after the top-dimensional homology of the Tits building of a vector space.) As a consequence ofTheorem A we obtain the following representation-theoretic lifting of (1) in termsof induced modules. Corollary B.
Under the assumptions of Theorem A, the alternating sum formula [ F n ( a )] = n X m =0 ( − m h ↑ TL n ( a )TL m ( a ) i holds in the Grothendieck group K (TL n ( a )) . (If TL n ( a ) is semisimple, then K (TL n ( a )) is the module of virtual representations of TL n ( a ) .) We now consider the case R = C , so that a = v + v − with v ∈ C × . Then TL n ( a )is semisimple unless q = v is an ℓ -th root of unity for 2 ℓ n . In thecase of semisimplicity the irreducible representations V λ of TL n ( a ) are indexed bypartitions λ ⊢ n with at most two columns. We prove the following descriptionof F n ( a ) in terms of counts of standard Young tableaux (SYT). Theorem C.
Let R = C , let v ∈ C × be such that v is not an ℓ -th root of unityfor ℓ n , and let a = v + v − . Then F n ( a ) ∼ = M λ ⊢ n columns |{ SYT Q of shape λ with top entry of second column odd }| · V λ . (In the case λ = 1 n , the unique SYT of shape λ has no second column, and so wedeclare that the top entry of its second column is ( n + 1) .) The three results listed above are the direct analogues of Reiner and Webb’sresults relating the complex of injective words to the number of derangements of n letters [RW04, Propositions 2.1–2.3]. This suggests an interpretation of the n -thFine number as the number of ‘planar derangements’ of n letters. This interpreta-tion has several precursors in the literature: One precursor is the fact that the n -thFine number is equal to the number of Dyck paths of length 2 n whose first peak haseven height, while D´esarm´enien [D´es83] showed that the number of derangementsof n is equal to the number of permutations whose first ascent π ( i ) < π ( i + 1)occurs for i even. Another precursor is that Dyck paths can be interpreted as per-mutations that avoid the pattern 321 [Sta99, p.224], and the Fine number is thenumber of derangements that avoid 321 [DS01, Section 8]. It is striking that the NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 5 same interpretation has arisen naturally through our work on homological stabilityand injective words.We now turn to a feature of planar injective words that does not have a precursorin the case of injective words. The n -th Jacobsthal number J n is the number ofcompositions of n that end with an odd number. It is equal to the number ofsequences n > a > a > · · · > a r > n . The l th Jacobsthal element in TL n ( a ) is defined to be J nl = X l>a > ··· >a r > l − a odd ( − ( r − l (cid:16) µλ (cid:17) r U a + n − l · · · U a r + n − l . (Here λ and µ are constants involved in the definition of W ( n ), and µλ is equal toeither − v or − v − .) Observe that the number of irreducible terms in J nl is J l . Weprove the following identification of the boundary maps of W ( n ). Theorem D.
Under the assumptions of Theorem A, for i n − theboundary map d i : W ( n ) i → W ( n ) i − acts as right multiplication by J ni +1 in thefollowing sense: d i ( x ⊗ r ) = x · J ni +1 ⊗ r. In particular, the image of the boundary map has number of irreducible terms givenby J i +1 . The formula for the differentials in Theorem D is convenient for explicit compu-tations, and is used in [BH20] to describe (aspects of) the Fineberg module F n ( a )in the case of n even.1.3. Complexes from algebras.
We hope that the results of this paper willencourage others to consider constructing and studying chain complexes of algebramodules from a combinatorial point of view.The general idea is that one can combine combinatorial complexes with S n -action (such as the complex of injective words) with finite-dimensional algebras(such as the Temperley-Lieb algebras) and construct algebraic analogues of thecomplexes. Examples of possible complexes with S n -action include the complexof injective words, the realisation of the poset of ordered partitions of { , . . . , n } with k > partition poset , which consists of partitions of { , . . . , n } with 1 < k < n parts. Examples of possible algebras include the Temperley-Liebalgebras studied here, Temperley-Lieb algebras of types B and D , variants such asthe dilute and periodic Temperley-Lieb algebras, and cousins such as the Brauer,blob and partition algebras. (Here we only list examples that are somewhat closeto the TL n ( a ); there will be many other candidates besides.)In all cases, one can undertake the following ‘process’ that has as input a com-plex C with S n -action and a family of algebras A n , and as output a chain complexof A n -modules. The process takes a S n -orbit of simplices, and replaces it with NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 6 the A n -module induced from the trivial module modulo the subalgebra corre-sponding to the stabiliser of the ‘original’ orbit. (This is of course only a vagueprocess, and its success depends on the nature of C and the A n .)1.4. Outline.
In Section 2 we recall the basics of Temperley-Lieb algebras thatwe require in the rest of the paper. Section 3 recalls the definition of the complexof planar injective words W ( n ) from [BH20]. Section 4 recalls Dyck paths, Dyckwords, Catalan numbers and Fine numbers, and refines the usual relationshipbetween them to take into account the height of the first peak, ending with anew account of the alternating sum in Equation (1). In Section 5 we recall therelationship between planar diagrams and Dyck words, and prove Theorem A andCorollary B. In Section 6 we prove Theorem C. And in Section 7 we recall theJacobsthal numbers and prove Theorem D.1.5. Acknowledgements.
The authors would like to thank the Max Planck In-stitute for Mathematics in Bonn for its support and hospitality.2.
Temperley-Lieb algebras
In this section we will cover the basic facts about Temperley-Lieb algebras thatwe require in the rest of the paper. There is some overlap between the materialrecalled here and in [BH20]. General references for readers new to the TL n ( a ) areSection 5.7 of Kassel and Turaev’s book [KT08] on the braid groups, and especiallySections 1 and 2 of Ridout and Saint-Aubin’s survey on the representation theoryof the TL n ( a ) [RSA14].2.1. Definitions. A planar diagram on n strands consists of two vertical lines inthe plane, decorated with n dots labelled 1 , . . . , n from bottom to top, togetherwith a collection of n arcs joining the dots in pairs. The arcs must lie betweenthe vertical lines, they must be disjoint, and the diagrams are taken up to isotopy.For example, here are two planar diagrams in the case n = 5: x = y = We will often omit the labels on the dots.
Definition 2.1 (The Temperley-Lieb algebra TL n ( a )) . Let R be a commutativering and let a ∈ R . The Temperley-Lieb algebra TL n ( a ) is the R -module withbasis given by the planar diagrams on n strands, and with multiplication definedby placing diagrams side-by-side and joining the ends. Any closed loops createdby this process are then erased and replaced with a factor of a . NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 7
For example, the product xy of the elements x and y above is:= = a · We have subscribed to the heresy of [RSA14] by drawing planar diagrams that gofrom left to right rather than top to bottom. The identity element of TL n ( a ) isthe planar diagram in which each dot on the left is joined to the correspondingdot on the right by a straight horizontal line.For 1 i n −
1, we define U i ∈ TL n ( a ) to be the planar diagram shown below. U i = ...... ii +11 n We refer to an arc joining adjacent dots as a cup . Thus U i has a single cup on leftand right joining dots i and i + 1. The elements U i satisfy the following relations:(1) U i U j = U j U i for j = i ± U i U j U i = U i for j = i ± U i = aU i for all i .The reader can easily verify these relations for themselves; two of them are shownin Figure 1. In fact, the generators U i together with the three relations aboveform a presentation of TL n ( a ) as an R -algebra: Elements of the Temperley-Liebalgebra are formal sums of monomials in the U i , with coefficients in the groundring R , modulo the relations above. This is proved in [RSA14, Theorem 2.4],[KT08, Theorem 5.34], and [Kau05, Section 6]. We often write TL n ( a ) as TL n .We note here that TL = TL = R .2.2. Induced modules.Definition 2.2 (The trivial module ) . The trivial module of the Temperley-Lieb algebra TL n ( a ) is the module consisting of R with the action of TL n ( a ) inwhich every diagram acts as multiplication by 0, except for the identity diagram.Equivalently, it is the module on which all of the generators U , . . . , U n − act as 0.We can regard as either a left or right module, and we will usually do thatwithout indicating so in the notation. Definition 2.3 (Sub-algebra convention) . For m n , we will regard TL m asthe sub-algebra of TL n generated by the elements U , . . . , U m − , or equivalently,the subalgebra in which dots m + 1 , . . . , n on the left are joined to the corre-sponding dots on the right by horizontal straight lines. We will often regard TL n NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 8 ...... ...... ...... ii +1 n = (a) The relation U i = aU i . ...... ...... ...... ...... ii +1 i +2 n = (b) The relation U i U i +1 U i = U i . Figure 1.
Diagrammatic relations in TL n ( a ).as a left TL n -module and a right TL m -module, so that we obtain the left TL n -module TL n ⊗ TL m .The induced modules TL n ⊗ TL m will be the building blocks of the com-plex W ( n ).A planar diagram on n strands with black box of size m is a planar diagram on n strands with the dots 1 , . . . , m on the right encapsulated within a black box , suchthat there are no cups with endpoints in the black box. For example, the planardiagrams with 4 strands and black box of size 3 are shown below. The R -linear span of the planar diagrams on n strands with black box of size m hasthe structure of a left TL n ( a )-module. If x is a planar diagram on n strands, and y is a planar diagram on n strands with black box of size m , then the product x · y isdefined by pasting the diagrams in the usual way, subject to the condition that ifthe pasting produces a cup attached to the black box, then the diagram is identified NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 9 with 0. For example: U U · = = = 0 . Proposition 2.4.
Given m n , TL n ( a ) ⊗ TL m ( a ) is isomorphic to the moduleof planar diagrams on n strands with black box of size m .Proof. Let I m denote the left ideal of TL n generated by the elements U , . . . , U m − .In other words, I m is the span of all diagrams which have a cup on the right amongdots 1 , . . . , m . It is shown in [BH20] that TL n ⊗ TL m is isomorphic to TL n /I m under the isomorphism that sends x ⊗ ∈ TL n ⊗ TL m to x + I m ∈ TL n /I m .Since TL n has basis given by planar diagrams, and I m has basis given by planardiagrams with cup among dots 1 , . . . , m on the right, the quotient TL n /I m has basisgiven by the diagrams with no cups among dots 1 , . . . , m on the right, which wecan identify with the n -planar diagrams with black box of size m . This determinesan R -linear isomorphism between TL n ⊗ TL m and the module of planar diagramson n strands with black box of size m , and it is simple to see that this respectsthe module structures. (cid:3) The braiding elements.
Now we suppose that a = v + v − where v ∈ R isa unit. Definition 2.5 (The braiding elements) . Define s , . . . , s n − ∈ TL n ( v + v − ) bysetting s i = λ + µU i where λ, µ ∈ R are defined by one of the following two options:(1) λ = − µ = v , so that s i = vU i − λ = v and µ = − v , so that s i = v − vU i .It is now easy to verify that the elements s i satisfy the braid relations : • s i s j = s j s i for i = j ± • s i s j s i = s j s i s j for i = j ± s i are invertible and satisfy the rule: s − i = λ − + µ − U i . It is also immediate to verify that s i acts on as multiplication by λ .The s i in fact form the generators in a presentation of TL n ( v + v − ) as a quotientof the Iwahori-Hecke algebra of type A n − . In particular, they satisfy furtherrelations of degree 2 and 3, that we will not list here. See [BH20] for more details. Remark 2.6.
There is a homomorphism from (the group algebra of) the braidgroup into TL n ( v + v − ) given on generators by s i s i . This can be regarded as a NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 10 ...... ...... ...... ii +1 n + µ = λs i = λ + µ U i Figure 2.
Smoothings of s i .Kauffman bracket-style smoothing operation from braids to planar diagrams: Theformula for s i tells us to smooth a positive crossing in the two possible ways withweights λ and µ as in Figure 2, and the formula for s − i tells us to smooth a negativecrossing in the two possible ways with weights λ − and µ − . In general, given abraid diagram with p crossings, each crossing is smoothed in the 2 possible ways,with appropriate weights, to obtain a linear combination of 2 p planar diagrams.We may also consider hybrid diagrams obtained by concatenating planar and braiddiagrams, or obtained by partially smoothing braid diagrams, though this will notbe important in the rest of the paper.3. Injective words and planar injective words
Throughout this section we will consider the Temperley-Lieb algebra TL n ( a ),where a = v + v − for v ∈ R a unit. We will make use of the elements s , . . . , s n − of Definition 2.5.An injective word on the letters { , . . . , n } is a tuple ( x , . . . , x i ) whose entriescome from the set { , . . . , n } , with no repeated entries in the tuple. Injectivewords form a poset under the subword relation: w > v if v is a subword of w .The complex of injective words is most commonly defined as the realisation of thisposet, as in [Far79] or [BW83]. However, note that for any injective word w , theposet of elements v w is Boolean. It follows that the poset of injective words isa simplicial poset, and its realisation admits a cell structure in which the cells aresimplices, with an i -simplex for each word ( x , . . . , x i ). This cell complex can beobtained from a semi-simplicial set, as in [RW13]. For us, the complex of injectivewords will be the augmented cellular chains of the cell complex described above,studied for example in [Ker05]. We define it explicitly now. Definition 3.1 (The complex of injective words) . The complex of injective words is the chain complex C ( n ) of S n -modules, concentrated in degrees − n − i is the free R -module with basis given by tuples ( x , . . . , x i )where x , . . . , x i ∈ { , . . . , n } and no letter appears more than once. We allow NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 11 the empty word (), which lies in degree −
1. The differential of C ( n ) sends aword ( x , . . . , x i ) to the alternating sum P ij =0 ( − j ( x , . . . , b x j , . . . , x i ).We can rewrite C ( n ) in terms of the group algebra R S n . Denote by s , . . . , s n − ∈ S n the adjacent transpositions s i = ( i i + 1). These elements satisfy the braidrelations listed beneath Definition 2.5. There is an isomorphism C ( n ) i ∼ = R S n ⊗ R S n − i − , where is the trivial module of R S n − i − . Under this isomorphism the word( x , . . . , x i ) is sent to σ ⊗ σ ∈ S n is a permutation such that σ ( n − i + j ) = x j . Furthermore, the differential d : C ( n ) i → C ( n ) i − becomes the map d : R S n ⊗ R S n − i − −→ R S n ⊗ R S n − i defined by d ( x ⊗
1) = P ij =0 ( − j x · ( s n − i + j − · · · s n − i ) ⊗
1. (See [Hep20].) Thisdescription inspires the following definition of the planar analogue.
Definition 3.2 (The complex of planar injective words [BH20]) . Let R be a com-mutative ring, let v ∈ R × , let a = v + v − , and let n >
0. The complex of planarinjective words is the chain complex W ( n ) ∗ of TL n ( a )-modules defined as follows.For i in the range − i n −
1, the degree- i part of W ( n ) ∗ is defined by W ( n ) i = TL n ( a ) ⊗ TL n − i − ( a ) and in all other degrees we set W ( n ) i = 0. Note that W ( n ) − = TL n ( a ) ⊗ TL n ( a ) = . For i > d i : W ( n ) i → W ( n ) i − is defined to be the alternatingsum P ij =0 ( − j d ij , where d ij : TL n ⊗ TL n − i − ( a ) → TL n ⊗ TL n − i ( a ) x ⊗ r ( x · s n − i + j − · · · s n − i ) ⊗ λ − j r. In the expression s n − i + j − · · · s n − i , the indices decrease from left to right. Observethat d j is well-defined because the elements s n − i , . . . , s n − i + j − all commute withall generators of TL n − i − ( a ). We have depicted W ( n ) ∗ in Figure 3. For notationalpurposes we will write W ( n ) and only use a subscript when identifying a particulardegree. Remark 3.3 (Visualising W ( n )) . Recall from the diagrammatic description of theinduced module TL n ( a ) ⊗ TL m ( a ) when m n given in Section 2.2 that elementsof W ( n ) i can be regarded as diagrams where the first n − i − d : W ( n ) i → W ( n ) i +1 is then given by pasting special elements onto the right of a diagram, followedby taking their signed and weighted sum. These special elements each enlarge theblack box by an extra strand, and plumb one of the free strands into the new space NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 12 TL n ⊗ TL d n − (cid:15) (cid:15) n − n ⊗ TL d n − (cid:15) (cid:15) n − d (cid:15) (cid:15) TL n ⊗ TL n − d (cid:15) (cid:15) n ⊗ TL n − d (cid:15) (cid:15) − Figure 3.
The complex W ( n )in the black box, see Figure 4 The resulting diagrams can be simplified using the d : − λ − + λ − Figure 4.
Example: D : W (4) → W (4) smoothing rules for diagrams with crossings described in Remark 2.6. We leaveit to the reader to make this description as precise as they wish, and note herethat this is where the notion of braiding , so often seen in homological stabilityarguments, fits into our set up.In [BH20] we showed the following analogue of the high-connectivity of thecomplex of injective words. It was the main technical underpinning of our proofof homological stability for Temperley-Lieb algebras. Theorem 3.4 ([BH20]) . H d ( W ( n )) = 0 for d ( n − . The top homology of the Tits building is known as the
Steinberg module . Thisinspires the name in the following definition.
Definition 3.5.
The n -th Fineberg module is the TL n ( a )-module F n ( a ) = H n − ( W ( n )) . NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 13
We often suppress the a and simply write F n . The rank of F n is the n -th Finenumber F n .4. Dyck paths, Catalan numbers, and Fine numbers
We now recall Dyck paths, Catalan numbers and Fine numbers. We also recallthe familiar formula for the Catalan numbers and extend it to take the heightof the first peak into account, leading to a new proof of Equation (1) from theintroduction.A
Dyck path is a path starting and ending on the horizontal axis, built using thesteps (1 ,
1) and (1 , − ,
1) and (1 , −
1) by u and d respectively. Thus Dyck paths are incorrespondence with Dyck words , i.e. words in the letters u and d containing equalnumbers of u s and d s, and such that no initial segment contains more d s than u s.The following figure shows a Dyck path and its corresponding Dyck word. uuduuddd The n -th Catalan number C n is the number of Dyck paths of length 2 n . Forexample, C = 5:See Corollary 6.2 of [Sta99], and the paragraphs before and after it, for a discussionof the Catalan numbers. A peak in a Dyck path is a sequence of consecutive stepsup, followed by a step down. The n -th Fine number F n is the number of Dyckpaths of length 2 n in which the first peak has even height. For example, F = 2as the previous set of diagrams demonstrates. See [DS01] for a nice discussion ofthe Fine numbers. Proposition 4.1.
The number of Dyck paths of length n whose first peak hasheight m or greater is (cid:18) n − mn − m (cid:19) − (cid:18) n − mn − m − (cid:19) = m + 1 n + 1 (cid:18) n − mn (cid:19) In particular, taking m = 0 gives the familiar result C n = (cid:18) nn (cid:19) − (cid:18) nn + 1 (cid:19) = 1 n + 1 (cid:18) nn (cid:19) . NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 14
Proof.
We begin by recalling the proof of the case m = 0 by the ‘reflection trick’.See Lemma 5.27 of [KT08]. We will then adapt this to the general case.Consider the set of all paths built from the steps (1 ,
1) and (1 , − ,
0) and ending at (2 n, x -axis, and the rest we call bad paths. Given a bad path, we locate the firstpoint at which it meets the line y = −
1, and reflect the remainder of the paththrough that line. The result is a path from (0 ,
0) to (2 n, − , n, − ,
0) to (2 n,
0) has n ups and n downs, so that thereare (cid:0) nn (cid:1) in total. And a path from (0 ,
0) to (2 n, −
2) has n − n + 1downs, so there are (cid:0) nn +1 (cid:1) in total. Therefore the total number of Dyck paths ( C n )is (cid:0) nn (cid:1) − (cid:0) nn +1 (cid:1) .For general m we now repeat the procedure, but only consider paths that beginwith at least m up steps. Then the number of paths from (0 ,
0) to (2 n,
0) is (cid:0) n − mn − m (cid:1) ,and the number from (0 ,
0) to (2 n, −
2) is (cid:0) n − mn − m − (cid:1) , as we see by considering thedistribution of the up moves after the first m . (cid:3) Now let us fix n . Given 0 m , we write B m for the number of Dyck paths whosefirst peak occurs at height m or greater. Thus B m = 0 for m > n . Then ( B m − B m +1 ) is the number of Dyck paths whose first peak has height exactly m , and sothe Fine number F n is nothing other than F n = ( B − B ) + ( B − B ) + · · · = n X m =0 ( − m B m . In particular, using Proposition 4.1 above we recover the formula in Equation (1): F n = n X m =0 ( − m m + 1 n + 1 (cid:18) n − mn (cid:19) = 1 n + 1 (cid:20)(cid:18) nn (cid:19) − (cid:18) n − n (cid:19) + 3 (cid:18) n − n (cid:19) − · · · + ( − n ( n + 1) (cid:18) nn (cid:19)(cid:21) Planar diagrams and Dyck paths
We now recall the familiar relationship between planar diagrams and Dyck paths,and we extend it to take the height of the first peak into account.
Proposition 5.1.
The set of planar diagrams on n strands is in bijection with theset of Dyck paths (or words) of length n . Corollary 5.2.
The rank of TL n ( a ) as an R -module is the Catalan number C n . There are several choices for such a bijection; the one that is relevant to us is asfollows: Take a planar diagram on n strands, and work through the dots in order,starting with 1 , . . . , n on the right, followed by n, . . . , NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 15 encounter an arc, either for the first time or for the second time: if it is the firsttime, record a u , and if it is the second time, record a d . For example, here is aplanar diagram, the corresponding Dyck word, and the corresponding Dyck path. uuuddudd See [RSA14, pp.966-967] or [KT08, Lemma 5.33] for details.
Proposition 5.3.
The rank of TL n ( a ) ⊗ TL m ( a ) is equal to the number of Dyckpaths of length n whose first peak occurs at height m or greater.Proof. Proposition 2.4 shows that TL n ⊗ TL m has basis given by the n -planardiagrams with black box of size m , i.e. the diagrams that have no cups amongdots 1 , . . . , m on the right. These are precisely the diagrams which have no arcsthat start and end among dots 1 , . . . , m on the right. Therefore, under the bijectionbetween planar diagrams and Dyck paths, these diagrams correspond exactly tothe paths that start with m up steps, i.e. the paths whose first peak has height m or greater. (cid:3) We are now in a position to prove Theorem A, which states that the Eulercharacteristic of W ( n ) is ( − n − F n , where F n is the n -th Fine number. Proof of Theorem A.
Let us fix n and define B m to be the number of Dyck pathsof length 2 n whose first peak occurs at height m or greater, so that rank(TL n ⊗ TL m ) = B m . And let us write A m for the number of Dyck paths of length 2 n whosefirst peak occurs at height exactly m . Then χ ( W ( n )) = − rank(TL n ⊗ TL n ) + rank(TL n ⊗ TL n − ) − rank(TL n ⊗ TL n − ) + · · ·· · · + ( − n − rank(TL n ⊗ TL ) + ( − n − rank(TL n ⊗ TL )= − B n + B n − − B n − + · · · + ( − n − B = ( − n − [( B − B ) + ( B − B ) + · · · ]with final term in the bracket either B n if n is even, or ( B n − − B n ) if n is odd. Butthis is precisely ( − n − [ A + A + A + · · · + A n ] if n is even, and ( − n − [ A + A + A + · · · + A n − ] if n is odd. In either case, we obtain ( − n − F n . (cid:3) Combined with Proposition 4.1, the proof above gives us Equation (1): F n = ( − n − χ ( W ( n ))= B − B + · · · + ( − n B n = 1 n + 1 (cid:20)(cid:18) nn (cid:19) − (cid:18) n − n (cid:19) + 3 (cid:18) n − n (cid:19) − · · · + ( − n ( n + 1) (cid:18) nn (cid:19)(cid:21) NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 16
We also obtain the representation-theoretic analogue Corollary B. Indeed, for abounded chain complex C of finitely generated modules over any ring or algebra A ,the relation X m ( − m [ C m ] = X m ( − m [ H m ( C )] (2)holds in K ( A ). In particular, if A is semisimple then K ( A ) coincides with therepresentation ring of A , and the same relation holds there. We therefore obtain:[ F n ] = ( − n − n − X d = − ( − d [ H d ( W ( n ))]= ( − n − n − X d = − ( − d [ W ( n ) d ]= ( − n − n − X d = − ( − d [ ↑ TL n TL n − d − ]= n X m =0 ( − m [ ↑ TL n TL m ] . Here the first equation is a consequence of Theorem 3.4, the second is an instanceof (2), and the third follows from the definition W ( n ) d = TL n ⊗ TL n − d − = ↑ TL n TL n − d − . Young tableaux
In this section we will describe the top-dimensional homology F n = H n − ( W ( n ))as a module over TL n when our ground ring R is the complex numbers and thealgebra TL n is semisimple. In this case the irreducible representations of TL n areindexed by certain Young diagrams, and we are able to identify the multiplicity ofeach irreducible in F n . A nice account of the theory used here is given in chapters 4and 5 of [KT08], and see also the brief account in section 11 of Jones’ paper [Jon87].In particular we will use the language of partitions, Young diagrams and Youngtableaux, for which one can refer to Sections 5.1 and 5.2 of [KT08]. More detailedreferences are recalled in Remark 6.1.For this section we will fix n > R is the fieldof complex numbers C , that v and a = v + v − are non-zero complex numbers,and that q = v is not a d -th root of unity for 2 d n . The latter conditionguarantees that TL n ( a ) is semisimple. We also assume that ( λ, µ ) = ( − , v ) inorder to accord with the conventions of [KT08].Under these assumptions, the Temperley-Lieb algebra TL p ( a ) is semisimple foreach 0 p n , with one irreducible representation V λ for each partition λ ⊢ p whose Young diagram has at most two columns. The representation corresponding NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 17 to the partition 1 p = (1 , . . . , ⊢ p , whose Young diagram is a single column of p boxes, is V p = . The operations of restriction and induction on the modules V λ are now determinedby the rules V λ ↓ TL p TL p − ∼ = M µ֒ → λ V µ , λ ⊢ pV λ ↑ TL p TL p − ∼ = M λ֒ → µ V µ , λ ⊢ ( p − λ and µ are assumed to have diagrams with at most two columns. Recallthat the notation µ ֒ → λ means that the diagram of µ is obtained from that of λ by deleting a single corner box, and that λ ֒ → µ means that the diagram of µ isobtained from that of λ by adding a single corner box. See Remark 6.1 below forreferences. Proof of Theorem C.
For this proof, all partitions are assumed to have diagramswith at most two columns.To prove the claim it suffices to identify the isomorphism class [ F n ] within thering of isomorphism classes of TL n -modules. We have( − n − [ F n ] = n − X j = − ( − j [ W ( n ) j ] . (This is analogous to the usual relationship between the Euler characteristic of achain complex and the Euler characteristic of its homology, and is proved in thesame way, using the fact that because TL n is semisimple, any short exact sequenceof TL n -modules splits.) Observe that W ( n ) j = TL n ⊗ TL n − j − = V n − j − ↑ TL n TL n − j − so that altogether we have[ F n ] = ( − n − n − X j = − ( − j [ V n − j − ↑ TL n TL n − j − ] . We now identify the induced modules appearing above. Given λ ⊢ n and p n ,let N λ,p denote the number of SYT for which the labels in the first column begin,starting from the top, with 1 , . . . , p . Observe that if i is in the range 0 i n ,then N λ,i − − N λ,i is precisely the number of SYT of shape λ whose second columnhas top entry i , and whose first column necessarily has top entries 1 , . . . , i − V p ↑ TL n TL p ∼ = M λ ⊢ n V ⊕ N λ,p λ NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 18
To see this, we induce V p up to TL p +1 , then TL p +2 , and so on. At each stage, themodule will be a direct sum of modules V λ for various λ , and we will consider eachirreducible summand V λ to be labelled by an SYT of the relevant shape λ , accordingto the following rules. At the initial step, the single summand V p is labelled bythe unique SYT of shape 1 p , which is a single column with labels 1 , . . . , p . And asone passes from one step to the next, we interpret the rule V λ ↑ TL m +1 TL m ∼ = M λ֒ → µ V µ as saying that when we induce up the module labelled by an SYT Q , we obtainthe sum of the two modules labelled by the SYT obtained from Q by adding asingle box containing ( m + 1). Beginning with the unique SYT of shape 1 p , andadding single boxes labelled p + 1 , p + 2 , . . . , n so that one has an SYT at eachstage, produces precisely one copy of each SYT with n boxes whose first columnstarts 1 , . . . , p . Compare with Exercise 5 on p.93 of [Ful97]. This proves the claim.The claim above gives us[ F n ] = ( − n − n − X j = − ( − j [ V n − j − ↑ TL n TL n − j − ]= ( − n − n − X j = − ( − j X λ ⊢ n N λ,n − j − [ V λ ]= X λ ⊢ n " n X k =0 ( − k N λ,k [ V λ ] . Thus the multiplicity of V λ in F n is P nk =0 ( − k N λ,k . If λ = 1 n , then this multi-plicity is precisely X i odd , i n ( N λ,i − − N λ,i ) = |{ SYT of shape λ with top entry of second column odd }| as required. (The assumption λ = 1 n guarantees that N λ,n = 0, so that a potentialfinal term in the case of n even does not make a difference.) And if λ = 1 n then N λ,i = 1 for all i , so that the multiplicity is 0 if n is odd and 1 if n is even,which agrees with the special convention outlined in the statement. This completesthe proof. (cid:3) Remark 6.1 (References for the representation theory of Temperley-Lieb alge-bras) . With the assumptions from the start of the section, Theorem 2.2 of [Wen88]shows that the Iwahori-Hecke algebra H n ( q ) is semisimple. Theorem 5.18 of [KT08]shows that the distinct irreducible modules of H n ( q ) are the modules V λ , one foreach partition λ ⊢ n , with no restriction on the shape. Section 5.7.3 of [KT08]then shows that TL n ( a ) is semisimple, with one irreducible representation V λ foreach partition λ ⊢ n of n whose Young diagram has at most two columns, and NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 19 that these V λ pull back to the representations of H n ( q ) with the same names.The fact that V n = can be seen by comparing Examples 5.12(b) and Theo-rem 5.29 of [KT08]. The rules for induction and restriction of the representa-tions V λ of H n ( q ) are V λ ↓ H n ( q ) H n − ( q ) ∼ = M µ֒ → λ V µ , λ ⊢ n,V λ ↑ H n ( q ) H n − ( q ) ∼ = M λ֒ → µ V µ , λ ⊢ ( n − , again with no restriction on the shape of λ and µ . The first of these rules isProposition 5.13 of [KT08], and the second follows by Frobenius reciprocity. Fromthe first of these we can immediately deduce the stated rule for restriction in theTemperley-Lieb case, and the rule for induction then follows, again by Frobeniusreciprocity. The assumptions on n stated at the start of this section imply theanalogous assumptions for all p in the range 0 p n , so that we can replace n with any such p throughout this remark.7. Jacobsthal numbers and the boundary maps of W ( n )In this section we give a combinatorial description of the boundary maps in W ( n )and relate them to the Jacobsthal numbers . We start by recalling the boundarymaps, and the Jacobsthal numbers.Recall from Definition 3.2 that for i > d i : W ( n ) i → W ( n ) i − d i : TL n ⊗ TL n − i − → TL n ⊗ TL n − i x ⊗ r i X j =0 ( − j d ij ( x ⊗ r )= i X j =0 ( − j ( x · s n − i + j − · · · s n − i ) ⊗ λ − j r = i X j =0 ( − j λ − j ( x · ( λ − µU n − i + j − ) · · · ( λ − µU n − i )) ⊗ r. Here there are two possibilities for λ and µ , namely ( λ, µ ) = ( − , v ) or ( λ, µ ) =( v , − v ), and note for future reference that µλ = − v and µλ = − v − respectively.The n -th Jacobsthal number J n [Slo] is (among other things) the number ofcompositions of n that end with an odd number. So for example, taking n = 4the relevant compositions are 31, 13, 211, 121, 1111. The Jacobsthal number J n can also be described as the number of sequences n > a > a > · · · > a r > NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 20 whose initial term has the opposite parity to n . These sequences are in one-to-one correspondence with the compositions. If you label the ‘gaps’ in 1 , . . . , n by 1 , . . . , n − , . . . , n to obtain the composition. For the above examples, when n = 4, the relevantsequences are 3, 1, 3 >
2, 3 > > >
1. (We allow the empty sequence,and say that by convention its initial term is a = 0, and r = 0. Of course this onlyoccurs when n is odd.) More precisely, the correspondence between compositionsand sequences is as follows: Given a composition c c · · · c r , the correspondingsequence is n > a > · · · > a r − > a j = n − ( c r + c r − + · · · + c r − j +1 ).Observe that the initial term is a = n − c r , so that since c r is odd, a has theopposite parity to n .The Jacobsthal numbers are determined by the recursion J n = J n − + 2 J n − for n >
2, and also satisfy the closed form J n = n − ( − n . Thus, the compositionsand sequences counted by the Jacobsthal number are about one-third of the totalpossible sequences and compositions. Definition 7.1.
Let a = v + v − where v ∈ R × is a unit. For every 0 l n , wedefine the l th Jacobsthal element in TL n ( a ) as follows: J nl = X l>a > ··· >a r > l − a odd ( − ( r − l (cid:16) µλ (cid:17) r U a + n − l · · · U a r + n − l The indices of the U j which occur vary from ( n − ( l − n −
1) and hence are non-trivial in TL n ( a ) ⊗ TL n − ( l − ( a ) . Recall that we allow the empty sequence ( a = 0and r = 0) when l is odd. This corresponds to a constant summand 1 in J nl forodd l . Note that the number of irreducible terms in J nl is J l . When l = n , m n = 0and the formula simplifies. We call J nn the Jacobsthal element , and denote it J n . Proof of Theorem D.
Firstly we note that the terms appearing in J ni +1 are non-zero in TL n ⊗ TL n − i : the target of d i . We consider the cases i odd and i even forclarity. For ease of notation, let p = n − i −
1. When i is odd, then d i is a sumover an even number of terms, and acts by right multiplication on the left factor NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 21 of x ⊗ r by the following element: i X j =0 ( − j λ − j ( λ − µU p + j ) · · · ( λ − µU p +1 )= ( i − / X j =0 (cid:16) λ − j ( λ − µU p +2 j ) · · · ( λ − µU p +1 ) − λ − j − ( λ − µU p +(2 j +1) )( λ − µU p +(2 j +1) − ) · · · ( λ − µU p +1 ) (cid:17) = ( i − / X j =0 (cid:16)(cid:2) λ − j ( λ − µU p +2 j ) · · · ( λ − µU p +1 ) − λ − j − ( λ )( λ − µU p +(2 j +1) − ) · · · ( λ − µU p +1 ) (cid:3) + λ − j − ( µU p +(2 j +1) )( λ − µU p +(2 j +1) − ) · · · ( λ − µU p +1 ) (cid:17) = ( i − / X j =0 λ − j − µU p +(2 j +1) ( λ − µU p +(2 j +1) − ) · · · ( λ − µU p +1 )Here the final equality is given by noting that the terms in the square bracketcancel out. Substituting k = 2 j + 1 gives that d i is multiplication by X Under the above identifications, the top differential of W ( n ) isright-multiplication by J n . In particular, there is an exact sequence −→ F n ( a ) −→ TL n ( a ) −·J n −−−→ TL n ( a ) . Proof. This is an application of Theorem D for the case i = n − 1, which shows d n − ( x ⊗ r ) = x · J n ⊗ r . The identifications above send x ⊗ r to x · r and so under thesethe map d n − is left multiplication by J n as described. (cid:3) References [BGS82] Anders Bj¨orner, Adriano M. Garsia, and Richard P. Stanley. An introduction toCohen-Macaulay partially ordered sets. In Ordered sets (Banff, Alta., 1981) , vol-ume 83 of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci. , pages 583–615. Reidel,Dordrecht-Boston, Mass., 1982.[BH20] Rachael Boyd and Richard Hepworth. Homological stability for Temperley-Lieb alge-bras. arXiv: uploaded concurrently with this article, 2020.[Bj¨o92] Anders Bj¨orner. The homology and shellability of matroids and geometric lattices. In Matroid applications , volume 40 of Encyclopedia Math. Appl. , pages 226–283. Cam-bridge Univ. Press, Cambridge, 1992. NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 23 [BW83] Anders Bj¨orner and Michelle Wachs. On lexicographically shellable posets. Trans.Amer. Math. Soc. , 277(1):323–341, 1983.[D´es83] Jacques D´esarm´enien. Une autre interpr´etation du nombre de d´erangements. S´em.Lothar. Combin. , 8(B08b), 1983.[Deu99] Emeric Deutsch. Dyck path enumeration. Discrete Math. , 204(1-3):167–202, 1999.[DS01] Emeric Deutsch and Louis Shapiro. A survey of the Fine numbers. Discrete Math. ,241(1-3):241–265, 2001. Selected papers in honor of Helge Tverberg.[Far79] Frank D. Farmer. Cellular homology for posets. Math. Japon. , 23(6):607–613, 1978/79.[Ful97] William Fulton. Young tableaux , volume 35 of London Mathematical Society StudentTexts . Cambridge University Press, Cambridge, 1997. With applications to represen-tation theory and geometry.[Hep20] Richard Hepworth. Homological stability for Iwahori-Hecke algebras. arXiv: uploadedconcurrently with this article, 2020.[HH04] Phil Hanlon and Patricia Hersh. A Hodge decomposition for the complex of injectivewords. Pacific J. Math. , 214(1):109–125, 2004.[Jon83] Vaughan F. R. Jones. Index for subfactors. Invent. Math. , 72(1):1–25, 1983.[Jon85] Vaughan F. R. Jones. A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. (N.S.) , 12(1):103–111, 1985.[Jon87] Vaughan F. R. Jones. Hecke algebra representations of braid groups and link polyno-mials. Ann. of Math. (2) , 126(2):335–388, 1987.[Kau87] Louis H. Kauffman. State models and the Jones polynomial. Topology , 26(3):395–407,1987.[Kau90] Louis H. Kauffman. An invariant of regular isotopy. Trans. Amer. Math. Soc. ,318(2):417–471, 1990.[Kau05] Louis H. Kauffman. Knot diagrammatics. In Handbook of knot theory , pages 233–318.Elsevier B. V., Amsterdam, 2005.[Ker05] Moritz C. Kerz. The complex of words and Nakaoka stability. Homology HomotopyAppl. , 7(1):77–85, 2005.[Koz08] Dmitry Kozlov. Combinatorial algebraic topology , volume 21 of Algorithms and Com-putation in Mathematics . Springer, Berlin, 2008.[KT08] Christian Kassel and Vladimir Turaev. Braid groups , volume 247 of Graduate Textsin Mathematics . Springer, New York, 2008. With the graphical assistance of OlivierDodane.[Maa79] Hendrik Maazen. Homology stability for the general linear group . Utrecht University,1979. Thesis (Ph.D.).[Moo79] J. W. Moon. Some enumeration problems for similarity relations. Discrete Math. ,26(3):251–260, 1979.[Rob04] Aaron Robertson. Restricted permutations from Catalan to Fine and back. S´em.Lothar. Combin. , 50:Art. B50g, 13, 2003/04.[RSA14] David Ridout and Yvan Saint-Aubin. Standard modules, induction and the structureof the Temperley-Lieb algebra. Adv. Theor. Math. Phys. , 18(5):957–1041, 2014.[RW04] Victor Reiner and Peter Webb. The combinatorics of the bar resolution in groupcohomology. J. Pure Appl. Algebra , 190(1-3):291–327, 2004.[RW13] Oscar Randal-Williams. Homological stability for unordered configuration spaces. Q.J. Math. , 64(1):303–326, 2013.[RWW17] Oscar Randal-Williams and Nathalie Wahl. Homological stability for automorphismgroups. Adv. Math. , 318:534–626, 2017.[Slo] Neil J. A. Sloane. The On-Line Encyclopaedia of Integer Sequences. https://oeis.org/A001045 . NJECTIVE WORDS FOR TEMPERLEY-LIEB ALGEBRAS 24 [Sta99] Richard P. Stanley. Enumerative combinatorics. Vol. 2 , volume 62 of Cambridge Stud-ies in Advanced Mathematics . Cambridge University Press, Cambridge, 1999. With aforeword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.[TL71] H. N. V. Temperley and E. H. Lieb. Relations between the “percolation” and “colour-ing” problem and other graph-theoretical problems associated with regular planarlattices: some exact results for the “percolation” problem. Proc. Roy. Soc. LondonSer. A , 322(1549):251–280, 1971.[Wah13] Nathalie Wahl. The Mumford conjecture, Madsen-Weiss and homological stability formapping class groups of surfaces. In Moduli spaces of Riemann surfaces , volume 20 of IAS/Park City Math. Ser. , pages 109–138. Amer. Math. Soc., Providence, RI, 2013.[Wen88] Hans Wenzl. Hecke algebras of type A n and subfactors. Invent. Math. , 92(2):349–383,1988. Max Planck Institute for Mathematics, Bonn E-mail address : [email protected] URL : https://guests.mpim-bonn.mpg.de/rachaelboyd/ Institute of Mathematics, University of Aberdeen E-mail address : [email protected] URL ::