aa r X i v : . [ m a t h . R T ] F e b ( ℓ , p ) -JONES-WENZL IDEMPOTENTS ROBERT A. SPENCER AND STUART MARTINA
BSTRACT . The Jones-Wenzl idempotents of the Temperley-Lieb algebra are celebrated ele-ments defined over characteristic zero and for generic loop parameter. Given pointed field ( R , δ ) , we extend the existing results of Burrull, Libedinsky and Sentinelli to determine arecursive form for the idempotents describing the projective cover of the trivial TL Rn ( δ ) -module. I NTRODUCTION
The Temperley-Lieb algebra, TL n , defined over ring R with distinguished element δ bythe generators { u i } n − i = and relations u i = δ u i (0.1) u i u j = u j u i | i − j | ≥ u i u i ± u i = u i ≤ i ± < n ,(0.3)has recently been recast into the limelight.First studied over characteristic zero, as algebras of transfer operators in lattice models,these structures found extensive use in physics and, later, knot theory where VaughanJones famously used them to define the Jones polynomial invariant.More recently, Temperley-Lieb algebras and their variants have become the subject ofstudy by those seeking to understand Soergel bimodule theory [Eli16]. Here they are in-tricately linked to the categorification of two-colour Soergel bimodules. From this cate-gorification arises interesting “canonical” bases of certain Hecke algebras, of which theKazhdan-Lusztig basis is probably the most famed.However, other interesting bases occur, particularly when the underlying ring has pos-itive characteristic. Work by Jensen and Williamson [TW17] develops the so-called p -canonical basis or p -Kazhdan-Lusztig basis for crystallographic Coxeter types. The un-derlying calculations in the two-colour case reduce to results in the representation theoryof the Temperley-Lieb algebra.In the language of Soergel bimodules, Jones-Wenzl idempotents describe the indecom-posable objects. Recently, Burrull, Libedinsky and Sentinelli [BLS19] determined the cor-responding elements of TL n defined over a field of characteristic p > δ =
2. The results of Erdmann and Henke [EH02] implicitly describing the p -canonicalbases are crucial.Throughout these results, the role of the Temperley-Lieb algebra as the centraliser ring,End U q ( sl ) ( V ⊗ n ) , has been key to understanding its modular representation theory (see[And19, AST18], and the reliance on [EH02] in [CGM03, BLS19] for examples). The explicitnature of the tilting theory of U q ( sl ) has underpinned most of the results. However, the algebras themselves admit a pleasing diagrammatic presentation stem-ming from Eqs. (0.1) to (0.3). Most of the theory of the characteristic zero case was de-termined purely “combinatorially” from this [Wes95, RSA14]. In [Spe20] the first authorre-derives many of the results known about the representation theory of TL n over positivecharacteristic without recourse to tilting theory of U q ( sl ) .This paper builds on [Spe20] and [BLS19] to construct ( ℓ , p ) -Jones-Wenzl idempotentsgiving the indecomposable objects in the most general case of any field and any parameter.This answers one of the questions in [BLS19, 1.3.5] by showing that the construction givenextends quite simply to all cases, but our argument is completely “diagrammatic”.This paper is arranged as follows. In Section 1 we recall the known theory on Jones-Wenzlidempotents over characteristic zero and make some observations in positive character-istic. In Section 2 we recount the construction of p -Jones-Wenzl elements due to Burrull,Libedinsky and Sentinelli [BLS19], with a slight modification to allow for generic parame-ter. Section 3 collects the properties of projective modules for TL n that will be needed in theresult and we prove our main theorem in Section 4. Finally, Section 5 examines the actionof the Markov trace on our new elements.1. J ONES -W ENZL I DEMPOTENTS
We will use the notation and conventions for TL Rn ( δ ) set out in [Spe20]. Note that in thisformulation, closed loops resolve to a factor of δ as opposed to − δ as is often found in theliterature. Throughout, we will omit the δ from our notation for the algebra TL Rn , as everyring R discussed will be naturally unambiguously pointed.1.1. Semi-simple case.
The Jones-Wenzl idempotent, denoted JW n , is a celebrated elementof TL Q ( δ ) n (where δ is indeterminate). It is the unique idempotent e such that TL n · e isisomorphic to the trivial module. As such it satisfies the relations(1.1) u i · JW n = ∀ ≤ i < n .It is clear that since JW n is idempotent, the coefficient of the identity diagram n → n is 1. Lemma 1.1.
Let R be any pointed ring. Suppose e is an idempotent of TL Rn such that Eq. (1.1) holds. Then e is invariant under the cellular involution ι , and e · u i = for all ≤ i < n. Thus eis the unique idempotent of TL Rn satisfying Eq. (1.1) .Proof. Since the coefficient of the identity diagram is one and the rest of the diagrams factorthrough m for m < n we see that e = ι e · e = ι e ,whence 0 = ι ( u i · e ) = ι e · ι u i = e · u i . Finally, if e and e are two such idempotents, thenboth have unit coefficient for the identity diagram and so e = e e = e . (cid:3) We now argue that idempotents satisfying Eq. (1.1) exist. Indeed, the algebra TL Q ( δ ) n issemi-simple so the trivial module is projective. Thus the idempotent e such that TL Q ( δ ) n · e is isomorphic to the trivial module suffices.On the other hand, there is also a recursive formulation that explicitly constructs theidempotents: ℓ , p ) -JONES-WENZL IDEMPOTENTS 3 Lemma 1.2.
Note that JW = id . For n > , (1.2) JW n = JW n − ⊗ id − [ n − ][ n ] ( JW n − ⊗ id ) ◦ u n − ◦ ( JW n − ⊗ id ) . In diagrams, (1.3) JW n + ... ... = JW n ... ... − [ n − ][ n ] JW n JW n ... ... ... .For example,(1.4) JW = − [ ][ ] (cid:16) + (cid:17) + [ ] (cid:16) + (cid:17) .An equivalent formulation is given by Morrison as follows: Lemma 1.3. [Mor17, 4.1]
Suppose D is a diagram n + → n + . Let ˆ D be the diagram n + → n formed by folding across the lowest target site of D. Let { i } be the set of positions of simple capsin ˆ D and D i ∈ TL n the diagrams obtained by removing those caps. Then (1.5) coeff ∈ JW n + ( D ) = ∑ { i } [ i ][ n + ] coeff ∈ JW n ( D i ) The author is not aware of any formula for the coefficients of diagrams in the Jones-Wenzl idempotents in the semi-simple case that is not inherently recurrent.1.2.
Characteristic Zero.
If we now specialise to a characteristic zero pointed field ( k , δ ) where δ satisfies [ ℓ ] but no [ m ] for 0 < m < ℓ , the Temperley-Lieb algebras are no longersemi-simple for all n . In this case the trivial module is not, in general, projective, whichmeans that the Jones-Wenzl elements “do not exist”.To be precise, let m ( δ ) ∈ Z [ δ ] be the minimal polynomial of δ over the integers and p theprime ideal of Q [ δ ] generated by m ( δ ) . The element JW n lies in TL Q ( δ ) n . We construct boththe “integer form” of TL n over Q [ δ ] p and the algebra of interest which is defined over thecharacteristic zero “target” field Q [ δ ] p / p p ⊆ k . We can summarise these rings as: Q [ δ ] p Q ( δ ) Q [ δ ] p / p p k i If n ≥ ℓ and n ℓ − TL Q ( δ ) n does not lie in the image of i and thus cannot descend to an element of TL kn . Indeed if itdid, then the trivial module would be projective which is not the case [RSA14, 8.1].We can rephrase as follows. If n ≥ ℓ and n ℓ −
1, then the Jones-Wenzl idempotentJW n written in terms of diagrams cannot be put over a denominator not divisible by m ( δ ) .For example, observe that Eq. (1.4) makes no sense if δ = ± [ ] =
0. However,if n ≡ − ℓ = δ = n = ℓ − n . Recall from [Spe20] the R. A. SPENCER AND STUART MARTIN definition of the “candidate morphisms”(1.6) v r , s = ∑ x h F ( x ) x and the subsequent proposition Proposition 1.4. [GL98, 3.6]
If s < r < s + ℓ and s + r ≡ ℓ − then the map S ( n , r ) → S ( n , s ) given by x x ◦ v r , s is a morphism of TL n modules for every n. The immediate corollary (from setting s = r = ℓ −
2) is that the Jones-Wenzlidempotent JW ℓ − exists over k and it’s diagram coefficients can be found by “rotating thetarget sites” and then computing the hook-formula h F ( x ) .1.3. Positive Characteristic.
Let us now focus on the positive characteristic case. As be-fore, if no quantum number vanishes, we are in a semi-simple case and Eq. (1.2) gives usall JW n . Thus assume that we are working over a pointed field ( k , δ ) of characteristic p andthat ℓ is the least non-negative such that [ ℓ ] is satisfied by ¯ δ ∈ k . Thus we say that we areunder ( ℓ , p ) -torsion.Let m ( δ ) ∈ F p [ δ ] be the minimal polynomial satisfied by δ and let m ( δ ) be a preimagein Z [ δ ] . Then m = ( p , m ( δ )) is a maximal ideal in Z [ δ ] . Consider S = Z [ δ ] m , a localnoetherian domain with maximal ideal m S . Now m ( x ) ( p ) so, ( ) ⊂ ( p ) ⊂ m strictlyand S is regular of Krull dimension 2. As such, its completion with respect to m , which wewill call R is regular and hence a domain. Set F to be the field of fractions of R . This is acharacteristic zero field containing Q ( δ ) . R = \ Z [ δ ] m F Q ( δ ) R / m R k
What we have now is a “ ( ℓ , p ) -modular system” in that we have a triple ( F , R , k ) suchthat F is a characteristic zero field, which is the field of fractions of R , a complete localdomain with residue field k . Thus, any idempotent in an algebra defined over k can beraised to an idempotent over R and then injected in to one defined over F .We would like to know if this is reversible. That is, given an idempotent defined over Q ( δ ) , we consider it as an idempotent over F and ask if it lies in the algebra over R . If so,we may reduce it modulo the maximal ideal to find an idempotent over k .In plain terms, we wish to know if the coefficients of the diagrams in JW n can be writtenwithout denominators divisible by p or m ( δ ) . If so, the idempotent “exists” in our field ofpositive characteristic.Should an element e satisfying Eq. (1.1) exist over k , it is clear that TL kn · e is a trivialmodule and so the trivial module is projective. Thus we can raise the idempotent e to anelement of TL Rn where action by u i sends the element to something divisible by m r for every r and hence equal to zero. That is to say, it lifts to JW n .Thus we may consider an alternative defining property of the Jones-Wenzl element thatit is the idempotent that generates the trivial module’s cover, which is equal to the trivialmodule, whenever that is the case.As such, the results of [Spe20] (in particular Theorem 3.4 combined with 8.3) can beinterpreted as follows: ℓ , p ) -JONES-WENZL IDEMPOTENTS 5 Theorem 1.5.
The Jones-Wenzl idempotent JW n in TL Q ( δ ) n descends to an element of TL kn iff n < ℓ or n < ℓ p and n ≡ ℓ − or n = a ℓ p k − for some k ≥ and ≤ a < p. An equivalent statement is
Corollary 1.6.
The Jones-Wenzl idempotent JW n descends to an element of TL kn ( δ ) iff the quantumbinomials [ nr ] are invertible in k for all ≤ r ≤ n. This has been shown by Webster in the appendix of [EL17]. However the proof there,and the alternative provided in [Web14] both construct linear maps of representations of U q ( sl ) which are shown to be morphisms (and hence elements of TL n ). This is the onlyproof not to use the Schur-Weyl duality of which the author is aware.Regrettably, the author is not aware of any formula in the style of Eq. (1.6) giving theform of these idempotents in generality. However, we are able to illuminate one furthercase constructively.Given ℓ and p , we notate p ( i ) = ℓ p i − for all i > p ( ) =
1. We will explicitlyconstruct JW p ( r ) −
1. Key in the below is that the construction of JW p ( r ) − as an “unfolding”of a trivial submodule of S ( p ( r ) −
2, 0 ) shows that the first trace vanishes (it correspondsby the action of u p ( r ) − on that module). Proposition 1.7.
The element JW p ( r ) − exists in TL k p ( r ) − for each r ≥ .Proof. We construct an element of TL p ( r ) − for which the identity diagram appears withcoefficient 1 and which is killed by all cups and caps. It is clear that this element thensatisfies Eq. (1.1).To do this, consider the “turn up” operator which takes a morphism n → m to a mor-phism n + → m − u m ,1 : x ( id ⊗ x ) ◦ ( ∩ ⊗ id m − ) .We can iterate this to turn multiple strands: set u m , k = u m − k + ◦ u m , k − .Now, let J i = u i ( JW p ( r ) − ) and J − i = ι ( J i ) for 0 ≤ i ≤ p ( r ) −
1. Thus J i is a morphism p ( r ) − + i → p ( r ) − − i . In diagrams,(1.8) J i = JW p ( r ) − ... p ( r ) − i ... p ( r ) − − i ... ≤ i ≤ p ( r ) − J − i = JW p ( r ) − ... p ( r ) − i ... p ( r ) − − i ... ≤ i ≤ p ( r ) − R. A. SPENCER AND STUART MARTIN
Recall that
T L is equipped with an automorphism which flips all diagrams vertically. Let K i be the image of J i under this morphism.The elements J i and K i have been chosen such that composition with any cup or capresults in the zero morphism. Key to this observation is that the idempotent JW p ( r ) − van-ishes under the trace defined in Section 5.Now we may define(1.10) e = p ( r ) − ∑ i = − p ( r ) − ( − ) i ι ( K i ) ⊗ id ⊗ J i .Diagrammatically, a typical term in the sum (where i ≥
0) looks like(1.11) ( − ) i JW p ( r ) − ... p ( r ) − i ... p ( r ) − i − ... JW p ( r ) − ... p ( r ) − i ... p ( r ) − i − ... The sign coefficient of the summand has been chosen such that the term for which i = e is killed by the (left) action of u j for each 1 ≤ j < p ( r ) − ι ( K i ) and J i are killed by all cups on the left and so the only terms inEq. (1.10) that do not vanish are those for which p ( r ) − i = j or p ( r ) − i = j + ℓ , p ) -JONES-WENZL IDEMPOTENTS 7 These terms are identical up to sign (in which they differ) and so cancel. They are bothgiven (up to sign) by the diagram in Eq. (1.12) and can be written as ( − ) i ι ( K j ) ⊗ ∩ ⊗ J j .(1.12) JW p ( r ) − ... p ( r ) − p ( r ) − − j j JW p ( r ) − ... p ( r ) − p ( r ) − jj − Since all the terms in the summand are sent to zero by all u j , and the coefficient of theidentity diagram is an idempotent, we can invoke Lemma 1.1 to show that this is indeedJW p ( r ) − . (cid:3) Readers familiar with the Dihedral Cathedral [Eli16] may be familiar with the “circular”Jones-Wenzl notation. This stems from the observation that rotation of the Jones-Wenzlelement, JW p ( r ) − , by a single strand leaves it invariant. In the general case where JW n isdefined, this is no longer so. The error is believed to be in the assumption that JW n existswhenever n ≡ ℓ −
1. 2. p -J ONES -W ENZL I DEMPOTENTS
We now turn to extending the work of Burrull, Libedinsky and Sentinelli [BLS19] in gen-eralising the definition of the Jones-Wenzl idempotent to a sensible element of TL kn for all n .The element constructed in [BLS19] is in fact the idempotent defining the projective coverof the trivial module, although it is not explicitly stated as such. However, the constructionpresent only covers the δ = δ . Our results will specialise in the case δ =
2, whichcorresponds to ℓ = p .We briefly recount their methodology here for completeness and introduce terminologyto indicate dependence on δ . The definition that follows is largely a rewrite of section 2.3in [BLS19] and the reader is encouraged to peruse that paper for further information.Recall the definition of I n in [Spe20]. If n + = ∑ bi = a n i p ( i ) is the ( ℓ , p ) -expansion of n + I n = { n b p ( b ) ± n b − p ( b − ) ± · · · ± n a p ( a ) } Note that this is different to that defined in [BLS19], where it is named supp p ( n ) .A number n is called ( ℓ , p ) -Adam if I n = { n + } so a = b . Equivalently, n < ℓ , or n < ℓ p and is congruent to -1 modulo ℓ or of the form cp ( r ) − ≤ c < p and r ≥ n is ( ℓ , p ) -Adam iff JW n can be lifted to TL kn . R. A. SPENCER AND STUART MARTIN
If a number is not ( ℓ , p ) -Adam, and a is such that n a =
0, define f [ n ] = ∑ bi = a + n i p ( i ) − ( ℓ , p ) -“father” of n . It is then the case that(2.2) I n = (cid:16) I f [ n ] + n a p ( a ) (cid:17) ⊔ (cid:16) I f [ n ] − n a p ( a ) (cid:17) .We inductively define elements p ℓ JW Q ( δ ) n ∈ TL Q ( δ ) n as(2.3) p ℓ JW Q ( δ ) n = ∑ j + ∈ I n λ jn p jn · JW j · ι p jn where p jn are elements of Hom T L ( n , j ) and λ jn are scalars in Q ( δ ) . In diagrams,(2.4) p ℓ JW Q ( δ ) n ... ... = ∑ j + ∈ I n λ jn JW j p jn p jn ... ...These elements will, in fact, be defined over Z [ δ ] m which is to say that neither m ( δ ) nor p will divide any denominators in the coefficients of the diagrams. As such they willdescend to TL kn and will be the idempotents of the projective cover of the trivial.To define p ℓ JW Q ( δ ) n , we induct on the cardinality of I n . When I n = { n + } we set p ℓ JW Q ( δ ) n = JW n so that λ nn = p nn = id n .Now suppose that n is not ( ℓ , p ) -Adam, but all λ jn ′ and p jn ′ are known for n ′ with smallercardinality I n ′ . In particular, λ j f [ n ] and p j f [ n ] are all known. Let m = n − f [ n ] = n a p ( a ) . Thenfor each i + ∈ I f [ n ] , set(2.5) λ i − mn = [ i + − m ][ i + ] λ i f [ n ] , λ i + mn = λ i f [ n ] and(2.6) p i − mn ... ... = p i f [ n ] JW i ...... ...... , p i + mn ... ... = p i f [ n ] ... ......This concludes the definition of p ℓ JW Q ( δ ) n .It is clear that in the case δ =
2, which implies ℓ = p , this coincides with the definitiongiven by Burrull, Libedinsky and Sentinelli. The only changes are to introduce quantumnumbers in the fractions in Eq. (2.5) and to use our generalised sets I n .Letting U in = p in · JW i · ι p jn so that p ℓ JW Q ( δ ) n = ∑ i + ∈ I n λ in U in , we have the followingproposition. The proof carries over exactly from [BLS19], but with the use of quantumnumbers in all the fractions. Proposition 2.1. [BLS19, 3.2]
The element p ℓ JW Q ( δ ) n ∈ TL Q ( δ ) n is an idempotent. Moreover, { λ in U in } i + ∈ I n is a set of mutually orthogonal idempotents. Readers concerned with the lack of sign in Eq. (2.5) should recall that our convention is that loops resolve to δ , instead of − δ . ℓ , p ) -JONES-WENZL IDEMPOTENTS 9 The result hinges on the equation (for each i , j ∈ I n )(2.7) JW i · ι p in · p jn · JW j = ( λ in JW i i = j i = j which we may read as h p in , p in i = λ in in S ( n , i ) . Lemma 2.2.
Each TL Q ( δ ) n · λ in U in is isomorphic to S ( n , i ) .Proof. Recall that we are in the semi-simple case and omit the superscripts. Consider the TL n -morphism φ : TL n · λ in U in → S ( n , i ) defined by(2.8) a · p in JW i ι p in a · p in This is surjective since any nonzero element of S ( n , i ) generates the entire module and isinjective as if a · p in = ∈ S ( n , i ) , then the morphism a · p in ∈ Hom
T L ( n , i ) factors throughsome j for j < i and hence a · p in JW i = (cid:3) Corollary 2.3.
As TL Q ( δ ) n -modules, TL Q ( δ ) n · p ℓ JW Q ( δ ) n ≃ L i + ∈ I n S ( n , i ) . Similarly to Proposition 2.1, the proof of the following runs identically to that in [BLS19].
Proposition 2.4.
The idempotents p ℓ JW Q ( δ ) n satisfy the “absorption property”, (2.9) p ℓ JW Q ( δ ) np ℓ JW Q ( δ ) f [ n ] id m = p ℓ JW Q ( δ ) n p ℓ JW Q ( δ ) f [ n ] id m = p ℓ JW Q ( δ ) n .Corollary 2.3 and Proposition 2.4, along with knowledge of the composition factors ofthe projective cover of the trivial module will give us all the ingredients to show the fol-lowing: Proposition 2.5.
The element p ℓ JW Q ( δ ) n can be lifted to Z [ δ ] m and therefore when written in thediagram basis, each coefficient can be written as a / b where b does not vanish in k. The proof is deferred until Section 4. This allows us to define the ( ℓ , p ) -Jones-Wenzlprojector on n strands. Definition 2.6.
Let ( R , δ ) be a field with ( ℓ , p ) torsion. The ( ℓ , p ) -Jones-Wenzl idempotent inTL kn ( δ ) , denoted p ℓ JW kn , is that element obtained by replacing each coefficient a / b in p ℓ JW Q ( δ ) n by itsimage in k. To prove Proposition 2.5, we will need a further claim that will be shown simultaneously.
Theorem 2.7.
The idempotent p ℓ JW kn is primitive and TL kn · p ℓ JW kn is isomorphic to the projectivecover of the trivial TL kn -module. Clearly the latter half of Theorem 2.7 implies the former.
3. I
NDUCTION AND P ROJECTIVE C OVERS OF THE T RIVIAL M ODULE
Induction.
In this section, R and δ will be arbitrary and therefore omitted from thenotation.Recall that TL n − ֒ → TL n naturally by the addition of a “through string” at the lowestsites. In this way, we may induce TL n − -modules(3.1) M ↑ = TL n ⊗ TL n − M and restrict TL n -modules to TL n − -modules, which we denote M ↓ .The proof of the following proposition follows exactly as in the characteristic zero case. Proposition 3.1. [RSA14, 6.3] If δ = or ( n , m ) = (
2, 0 ) , (3.2) S ( n − m ) ↑ ∼ = S ( n + m ) ↓ as TL n -modules. The power in this is that restriction is easily understood, and again the characteristiczero proof applies over arbitrary rings.
Proposition 3.2. [RSA14, 4.1]
There is a short exact sequence of TL n − modules, (3.3) 0 → S ( n − m − ) → S ( n , m ) ↓ → S ( n − m + ) → TL m to TL n for m < n , which we will denote M ↑ nm and the corresponding restriction M ↓ nm . This is achieved by n − m iterations of theinduction described above. A trivial consequence of Proposition 3.2 is that Corollary 3.3.
The module S ( n , m ) ↑ Nn has a filtration by standard modules, and the multiplicityof S ( N , i ) is given by (cid:18) N − n ( m + N − n − i ) /2 (cid:19) . In particular, the only S ( N , i ) appearing are those for which m − N + n ≤ i ≤ m + N − n and fori ∈ { m ± ( N − n ) } the factor S ( N , i ) appears exactly once. Projective Covers of the Trivial Module.
Let us suppose that R has ( ℓ , p ) -torsion (andthat p = ∞ if R is characteristic zero and that ℓ = ∞ if δ satisfies no quantum number).Recall that each projective module of TL n has a filtration by standard (cell) modules. For m ≡ n , let P ( n , m ) be the projective cover of the simple head of S ( n , m ) as a TL n -module.A corollary of Theorems 3.4 and 8.4 of [Spe20] is Corollary 3.4.
The multiplicity of S ( n , m ) in a standard filtration of P ( n , m ′ ) is 1 iff m + ∈ I m ′ ,otherwise it is 0. Let ν ( p ) ( x ) = ℓ ∤ x and ν p ( x / ℓ ) + ν ( p ) ( x ) gives the position ofthe least significant nonzero ( ℓ , p ) -digit of x . A trivial consequence of the above is that if S ( n , m ) appears in a standard filtration of P ( n , m ′ ) , then ν ( p ) ( m ) = ν ( p ) ( m ′ ) .Suppose now that N > n and that m = N − n is such that m < p ( ν ( p ) ( n )) so ν ( p ) ( n ) > ν ( p ) ( m ) = ν ( p ) ( N ) . Consider the module P ( n , n ) ↑ Nn . This is a projective module. If weconsider the filtration of P ( n , n ) ↑ Nn by cell modules, we see by Corollary 3.3 that the S ( N , j ) appearing in a filtration of P ( n , n ) ↑ Nn must all satisfy(3.4) i − N + n ≤ j ≤ i + N − n ℓ , p ) -JONES-WENZL IDEMPOTENTS 11 for some i + ∈ I n . In particular, the trivial module S ( N , N ) appears exactly once andcareful examination of Proposition 3.2 shows that it must appear in the head of P ( n , n ) ↑ Nn .As such, there is a unique summand of P ( n , n ) ↑ Nn isomorphic to P ( N , N ) . Additionally,from Eq. (2.2) and Corollary 3.3, where n a p ( a ) = m = N − n , we have further that everymodule S ( N , i ) appearing in a filtration of both P ( n , n ) ↑ Nn and P ( N , N ) does so in eachexactly once.To make these observations numerical, consider the Grothendieck group of TL n , de-noted here by G ( TL n ) . This is a free abelian group with basis the isomorphism types ofsimple TL n modules. That is to say, its elements are given by (isomorphism classes of)finite-dimensional TL n -modules, modulo the relation that if there is a short exact sequence0 → M → M → M →
0, then [ M ] = [ M ] + [ M ] , where [ M ] is the class of TL n -modulein G ( TL n ) . Similarly, we will consider K ( TL n ) , the free abelian group on (isomorphismclasses of) indecomposable projective modules with the same relation.A consequence of [GL96, 3.6] is that { [ S ( n , i )] } i ∈ Λ form a basis for G ( TL n ) . In G ( TL n ) ,the above discussion can be rephrased as(3.5) h P ( n , n ) ↑ Nn i = [ P ( N , N )] + ∑ j ∈ J [ S ( N , j )] where J is a multiset disjoint from I n −
1. Equation (3.5) should be read as a rephrasingof [BLS19, Lemma 4.11], but without recourse to the p -Kazhdan-Lusztig basis, Soergel bi-module theory or Schur-Weyl duality.4. M AIN R ESULT
Recall the cde -Triangle of [Web16, §9.5]: G ( TL Fn ) K ( TL kn ) G ( TL kn ) de c Here, c simply takes the image of a module [ M ] ∈ K ( TL kn ) to its image [ M ] ∈ G ( TL kn ) .The map d is defined on the basis of G ( TL Fn ) . Recall that this is a semi-simple algebraand the simple modules are exactly S ( n , i ) for i ≤ n and i ≡ n . We define d ([ S ( n , i )]) =[ S ( n , i )] . It is clear that d is the transpose of the decomposition matrix.To define e , one uses idempotent lifting techniques. Let P be a projective module of TL kn and suppose that P ≃ TL kn · e for some idempotent e defined in diagrams over k . Then lift e to an idempotent over F . This defines a projective TL Fn -module ˆ P and the image e [ P ] is [ ˆ P ] .It is classical theory that c = de .We now have all the results required to prove Proposition 2.5 and Theorem 2.7. Proof.
We show the results by mathematical induction on | I n | . When | I n | =
1, we have that p ℓ JW kn = JW n exists in TL kn by Theorem 1.5.Otherwise, assume both Proposition 2.5 and Theorem 2.7 hold for all n ′ with | I n ′ | < | I n | . In particular the results are known for f [ n ] . Thus p ℓ JW Q ( δ ) f [ n ] descends to an idempotent p ℓ JW k f [ n ] in TL kn which describes the projective module P ( f [ n ] , f [ n ]) . Now consider the module P ( f [ n ] , f [ n ]) ↑ n f [ n ] . This is isomorphic to the module TL kn · g where g is the idempotent p ℓ JW k f [ n ] ⊗ id n − f [ n ] in TL kn . The composition factors of this mod-ule are described in Eq. (3.5) and the discussion preceding it. In particular, if e is theunique idempotent of TL kn describing the projective cover of the trivial module, then e · P ( f [ n ] , f [ n ]) ↑ n f [ n ] is a non-zero module isomorphic to P ( n , n ) .As such e · g = g · e = e . If we lift both of these elements to F , say to ˆ e and ˆ g we obtainˆ e · ˆ g = ˆ g · ˆ e = ˆ e , so in particular(4.1) ˆ e ∈ ˆ g · TL Fn · ˆ g ∼ = End TL Fn ( TL Fn · ˆ g ) .But TL Fn · ˆ g is exactly the lift of P ( f [ n ] , f [ n ]) ↑ n f [ n ] to characteristic zero and so ˆ g = p ℓ JW Q ( δ ) f [ n ] ⊗ id n − f [ n ] . Thus Proposition 2.4 claims that p ℓ JW Q ( δ ) n · ˆ g = ˆ g · p ℓ JW Q ( δ ) n = p ℓ JW Q ( δ ) n . Thus too(4.2) p ℓ JW Q ( δ ) n ∈ ˆ g · TL Fn · ˆ g ∼ = End TL Fn ( TL Fn · ˆ g ) .However, recall that TL Fn is semi-simple and that e [ P ( f [ n ] , f [ n ]) ↑ n f [ n ] ] = e [ P ( n , n )] + M where M does not have support intersecting that of e [ P ( n , n )] . This is to say that there isa unique idempotent in End TL Fn (cid:16) P ( f [ n ] , f [ n ]) ↑ n f [ n ] (cid:17) with image having composition factorsgiven by e [ P ( n , n )] . Clearly by construction ˆ e is such an idempotent, and Corollary 2.3shows that p ℓ JW Q ( δ ) n is too. Hence they must be equal. (cid:3)
5. T
RACES
The trace map τ : Hom T L ( n , m ) → Hom
T L ( n − m − ) is defined diagrammaticallyas(5.1) τ f = f ... ...By linear extension, this induces the trace τ : TL n → TL n − for each n ≥ τ n : TL n → TL ≃ k .The action of the trace on Jones-Wenzl elements over characteristic zero is well under-stood. Lemma 5.1. [BLS19, Lemma 3.1]
In TL Q ( δ ) n , (5.2) τ m ( JW n ) = [ n + ][ n + − m ] JW n − m . Proof.
It will suffice to show that τ ( JW n ) = [ n + ] / [ n ] JW n − . Recall Lemma 1.2 whence(5.3) τ ( JW n + ) = (cid:18) δ − [ n − ][ n ] (cid:19) JW n = [ n ][ ] − [ n − ][ n ] JW n = [ n + ][ n ] JW n . (cid:3) However, this breaks down over positive characteristic. For example, the form of JW p ( r ) − given in Proposition 1.7 makes clear that(5.4) τ (cid:16) JW p ( r ) − (cid:17) = J p ( r ) − ⊗ J − p ( r ) + . ℓ , p ) -JONES-WENZL IDEMPOTENTS 13 This can be read as stating that [ p ( r ) − ][ p ( r ) − ] JW p ( r ) − exists over k and has value given byEq. (5.4). Notice that such a morphism has zero through-degree.Recall from [Spe20, Lemma 2.2] that [ ℓ m ] / [ ℓ n ] = m / n in k . The following is a trivialcorollary of Lemma 5.1. Corollary 5.2.
Let ≤ a ≤ p and r ≥ . Then as elements in TL k , (5.5) τ p (cid:16) JW kap ( r ) − (cid:17) = aa − k ( a − ) p ( r ) − .We now ask how the trace acts on elements p ℓ JW Q ( δ ) n . Let m = n − f [ n ] > [ n ] = f [ n − ] and(5.6) I n = (cid:16) I f [ n ] + m (cid:17) ⊔ (cid:16) I f [ n ] − m (cid:17) ; I n − = (cid:16) I f [ n ] + m − (cid:17) ⊔ (cid:16) I f [ n ] − m + (cid:17) Then notice that for i + ∈ I f [ n ] , τ U i − mn = U i − m + n − (5.7) τ U i + mn = [ i + m + ][ i + m ] U i + m − n − (5.8)Recall from Eq. (2.5) that(5.9) p ℓ JW Q ( δ ) n = ∑ i + ∈ I f [ n ] (cid:18) [ i + − m ][ i + ] λ i f [ n ] U i − mn + λ i f [ n ] U i + mn (cid:19) and also(5.10) p ℓ JW Q ( δ ) n − = ∑ i + ∈ I f [ n ] (cid:18) [ i + − m ][ i + ] λ i f [ n ] U i − m + n + λ i f [ n ] U i + m − n (cid:19) Then Eqs. (5.7) and (5.8) with Eq. (5.9) give(5.11) τ (cid:16) p ℓ JW Q ( δ ) n (cid:17) = ∑ i + ∈ I f [ n ] [ i + − m ][ i + − ( m − )] [ i + − m ][ i + ] λ i f [ n ] U i − m + n − + [ i + + m ][ i + + ( m − )] λ i f [ n ] U i + m − n − ! Now, for all i + ∈ I f [ n ] we have that ℓ | i + [ i + + m ] = ± [ m ] (depending onif [ i + ] = ±
1) and similarly for [ i + − m ] . Hence we see that all the factors actually fallout and(5.12) τ (cid:16) p ℓ JW kn (cid:17) = [ m ][ m − ] p ℓ JW kn − ,recovering a form of Eq. (5.3) for ( ℓ , p ) -Jones-Wenzl elements. If, on the other hand, f [ n ] = n − m =
1, then τ (cid:16) p ℓ JW kn (cid:17) = ∑ i + ∈ I f [ n ] [ i ][ i + ] λ i f [ n ] U in − + [ i + ][ i + ] λ i f [ n ] U in − ! = [ ] ∑ i + ∈ I f [ n ] λ i f [ n ] U i f [ n ] = [ ] p ℓ JW kn − .(5.13)A result of the above computation is that τ m (cid:16) p ℓ JW kn (cid:17) = [ ][ m ] p ℓ JW k f [ n ] .The remaining case where n is ( ℓ , p ) -Adam, apart from the cases p ( r ) − p ( r ) − p ℓ JW kn is the image of JW n and hence τ p ℓ JW kn is the image of [ n + ] / [ n ] JW n . As such, since JW n is killed by the action of all u i , so too must τ p ℓ JW kn be. By considering all diagrams of the form | x ih y | (see [Spe20] fornotation) where y is a fixed diagram of maximal degree d , we see that this implies theexistence of a trivial submodule of S ( n , d ) . Knowledge of where these modules could existthen give us restrictions on the valid values of d .R EFERENCES[And19] Henning Haahr Andersen. Simple modules for Temperley-Lieb algebras and related algebras.
J. Alge-bra , 520:276–308, 2019. doi:10.1016/j.jalgebra.2018.10.035.[AST18] Henning Haahr Andersen, Catharina Stroppel, and Daniel Tubbenhauer. Cellular structures using U q -tilting modules. Pacific J. Math. , 292(1):21–59, 2018. doi:10.2140/pjm.2018.292.21.[BLS19] Gaston Burrull, Nicolas Libedinsky, and Paolo Sentinelli. p -Jones-Wenzl idempotents. Adv. Math. ,352:246–264, 2019. doi:10.1016/j.aim.2019.06.005.[CGM03] Anton Cox, John Graham, and Paul Martin. The blob algebra in positive characteristic.
J. Algebra ,266(2):584–635, 2003. doi:10.1016/S0021-8693(03)00260-6.[EH02] Karin Erdmann and Anne Henke. On Schur algebras, Ringel duality and symmetric groups.
J. PureAppl. Algebra , 169(2-3):175–199, 2002. doi:10.1016/S0022-4049(01)00071-8.[EL17] Ben Elias and Nicolas Libedinsky. Indecomposable Soergel bimodules for universal Coxeter groups.
Trans. Amer. Math. Soc. , 369(6):3883–3910, 2017. doi:10.1090/tran/6754. With an appendix by Ben Web-ster.[Eli16] Ben Elias. The two-color Soergel calculus.
Compos. Math. , 152(2):327–398, 2016.doi:10.1112/S0010437X15007587.[GL96] J. J. Graham and G. I. Lehrer. Cellular algebras.
Invent. Math. , 123(1):1–34, 1996.doi:10.1007/BF01232365.[GL98] J. J. Graham and G. I. Lehrer. The representation theory of affine Temperley-Lieb algebras.
Enseign.Math. (2) , 44(3-4):173–218, 1998.[Mor17] Scott Morrison. A formula for the Jones-Wenzl projections. In
Proceedings of the 2014 Maui and2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones’ 60th birthday , volume 46 of
Proc.Centre Math. Appl. Austral. Nat. Univ. , pages 367–378. Austral. Nat. Univ., Canberra, 2017. URL https://projecteuclid.org/euclid.pcma/1487646031 .[RSA14] David Ridout and Yvan Saint-Aubin. Standard modules, induction and the struc-ture of the Temperley-Lieb algebra.
Adv. Theor. Math. Phys. , 18(5):957–1041, 2014. URL http://projecteuclid.org/euclid.atmp/1416929529 .[Spe20] R. A. Spencer. The modular Temperley-Lieb algebra, 2020, 2011.01328.[TW17] Lars Thorge Jensen and Geordie Williamson. The p -Canonical Basis for Hecke Algebras. Categorificationand Higher Representation Theory , 683:333–361, 2017.[Web14] Ben Webster. When are jones-wenzl projectors defined? MathOverflow, 2014. URL https://mathoverflow.net/q/138270 . URL:https://mathoverflow.net/q/138270 (version: 2014-09-16).[Web16] Peter Webb.
A course in finite group representation theory , volume 161 of
Cambridge Studies in AdvancedMathematics . Cambridge University Press, Cambridge, 2016. doi:10.1017/CBO9781316677216. ℓ , p ) -JONES-WENZL IDEMPOTENTS 15 [Wes95] B. W. Westbury. The representation theory of the Temperley-Lieb algebras. Math. Z. , 219(4):539–565,1995. doi:10.1007/BF02572380.D
EPARTMENT OF P URE M ATHEMATICS AND M ATHEMATICAL S TATISTICS , U
NIVERSITY OF C AMBRIDGE , C AM - BRIDGE
CB3 0WA, UK
Email address : [email protected] Email address ::