Graham’s and Lehrer’s morphisms





0, if[d+1]q=0;

dimR^d_n−1,k⁻¹ ₊dimS^d_n−1,k⁺¹ , if[d+1]q6=0and[d+2]q=0;

dimR^d−_n−¹_1,k+dimR^d+_n−1,k¹ , if[d+1]_q6=0and[d+2]_q6=0.

(4.1)

Proof. The case[d+1]q=0 has been dealt with in the previous proposition. The dimension of the radical is the dimension of the kernel of the Gram matrix. Lemma 3.8 has put the Gram matrix in block-diagonal form and the dimension of its kernel is the sum of those of the kernels of the two diagonal blocks. If[d+2]q6=0, it is simply the sum dimR^d_n−⁻_1,k¹ ₊dimR^d+1_n−1,k. If [d+2]q = 0, then the lower block [d+2]q/[d+1]qG^d+_n−¹_1,k is zero and the dimension is dimR^d_n−1,k⁻¹ ₊dimS^d_n−1,k⁺¹ .

Lemma3.8has shown (somewhat implicitly) that dimS^d_n,k=dimS^d_n⁻₋¹_1,k+dimS^d_n⁺_−1,k¹ since these are the sizes of the diagonal blocks appearing in (3.9). Because I^d_{n,k =} S^d_n,k/R^d_n,k, and thus dimI^d_n,k=dimS^d_n,k−dimR^d_n,k, the previous proposition has an immediate consequence.

Corollary 4.3. The dimension of the irreducible moduleI^d_n,k, for d ∈∆⁰_n,k, is given by

dimI^d_n,k=







dimS^d_n,k, if[d+1]_q=0;

dimI^d_n−1,k⁻¹ , if[d+1]q6=0and[d+2]q=0;

dimI^d−_n−1,k¹ ₊dimI^d+_n−1,k¹ , if[d+1]_q6=0and[d+2]_q6=0.

(4.2)

These results parallel those for the Temperley-Lieb algebras (proposition 5.1 and corollary 5.2 of[15]). They will play a similar role in the characterization of non-trivial submodules of cellular modules whenqis a root of unity.

4.2 Graham’s and Lehrer’s morphisms

The previous section has shown that, whenq is a root of unity, some cellular modules S^d_n,k are reducible. This raises the question of characterizing their structure. In particular, is the radicalR^d_n,k itself reducible? In their study of the affine Temperley-Lieb algebras, Graham and Lehrer[9]constructed a non-trivial morphisms between cellularTL_n-modules atqa root of unity. Because of the relationship betweenb_n,k andTL_n+k, this family of morphisms will answer the question of the structure of the radicalR^d_n,k.

The morphism of Graham and Lehrer is now recalled. The first definition describes a partial order on the links of a(t,s)-diagram.

Definition 4.4. Let D be a(t,s)-diagram and F(D)the set of links of D. Two elements x,y ∈F(D) are ordered x≤ y if x lies in the convex hull of y, namely if y, as an arc, contains x or if y, as a through line, is below x.

The definition is easier to understand through an example. LetDbe the(5, 3)-diagram:

D= x y

z w

−→ z y x w

.

The diagram on the right was obtained by turning the left side ofDclockwise by 90^◦and its right one counterclockwise by the same angle. Here F(D) = {x,y,z,w} and an element is smaller or equal to another if the former is contained in the latter. So, for thisD, the setF(D) is partially ordered by

x≤x, x≤ y, x≤z, y≤ y, y ≤z, z≤z, w≤w.

The setF(D)endowed with such a partial order is an example of a forest, that is, ifx ≤ y and x≤z, then y ≤zorz≤ y. The following proposition due to Stanley[24]states a remarkable property of forests.

Proposition 4.5. Let P be a forest of cardinality n; for y ∈P, denote the number of elements of P which are less than or equal to y by h_y. The rational function

H_P(q):= [n]_q!

∏y∈P

h_y

q

(4.3)

is a Laurent polynomial with integer coefficients.

Recall that[n]_q is theq-number (qⁿ−q⁻ⁿ)/(q−q⁻¹), and [n]_q!, theq-factorial∏1≤j≤n[j]_q. With these definitions and results, Graham’s and Lehrer’s morphism can be described.

Theorem 4.6(Graham and Lehrer, Coro. 3.6,[9]). Let q∈C^× be a root of unity,`the smallest positive integer such that q<sup>2`</sup> =1, and TL_n(β) be the Temperley-Lieb algebra. If t,s∈Λn are such that t<s<t+2`and s+t ≡ −2 mod 2`, then there exists a morphismθ:S^s_n→S_n^t given, for v∈S^s_n, by

θ(v) =

∑

w:s←t monic

sg(w)HF(w)(q)vw, (4.4)

wheresg(w)is a sign^k. Moreoverθ(S^s_{n) =}radS_n^t, with the exception that, if t =0andβ =0, thenθ(S^s_n) =S^t_n.

Note that the condition on the integerssandtis precisely that they be immediate neighbors in an orbit under reflection through vertical critical lines (see section2.3). Indeed the midpoint betweensandt sits at(s+t)/2= (−2+2m`)/2=m`−1, for somem, and thus on a critical vertical line. In the notation of section2.3,s⁻=tor t⁺=s.

The next step is to construct a similar morphism between b_n,k-modules. The following result, characterizing therestriction functor, a standard tool in the representation theory of associative algebras, will be the key element. (See[25]for a standard treatment of the theory.) Proposition 4.7. LetAbe an algebra, e∈Abe an idempotent andB=eAe. The functorres_e: mod-A→mod-Bthat sends a (finitely-generated) module V to eV and a morphism f :V →W tores_e(f):eV→eW defined by ev7→e f(v)is exact.

The importance of this functor in the present case is partially revealed by the following result.

Lemma 4.8. The restriction functorres_P

k establishes an isomorphism between the restriction of the radical of theTL_n+k-moduleS^d_n+k and theb_n,k-moduleR^d_n,k:

R^d_n,k'res_P

k(R^d_n+k), for all d∈∆n,k.

kThe sign sg(w)will not play any role in the following. It is sufficient to know that it can be recovered from theC-algebras isomorphism betweenTL_n(q+q⁻¹)(used here) andTL_n(−q−q⁻¹)(used by Graham and Lehrer) knowing the sign is always+1 in the−q−q⁻¹case.

Proof. If the bilinear form 〈 ·,· 〉^d_n+k on theTL_n+k-moduleS^d_n+k is identically zero, then so is

〈 ·,· 〉^d_n,k on the b_n,k-moduleS^d_n,k. ThenR^d_{n,k =}S^d_{n,k =}res_P

kS^d_{n+k =}res_P

kR^d_n+k. Suppose then that the bilinear form onS^d_n+kis not identically zero. Let v∈R^d_n+k. Then

〈P_kv,P_kw〉^dn,k=〈P_kv,P_kw〉^d_n+k=〈v,P_kw〉^d_n+k=0

because of the invariance property in proposition3.4and the fact that the bilinear form on theb_n,k-module is the restriction of the one on theTL_n+k-module. HenceR^d_n,k⊃res_P

k(R^d_n+k). Now ifP_kv∈R^d_n,k, then for anyw∈R^d_n+k

〈P_kv,w〉^d_n+k=〈P_kv,P_kw〉^d_n,k=0 andR^d_n,k⊂res_Pk(R^d_n+k).

If radMdenotes the Jacobson radical of the moduleM, then the previous lemma can be reformulated as

rad(res_PkS^d_n+k)'res_Pk(radS^d_n+k).

The restriction functor of proposition 4.7 carries the morphism θ of theorem 4.6 into one betweenb_n,k-modules. But is it a non-trivial morphism? This will be the difficult part of the proof ahead.

Proposition 4.9. Let q ∈C^× be a root of unity, ` > 1the smallest positive integer such that q<sup>2`</sup> = 1, and b_n,k(β) be the seam algebra. If t,s ∈ Λn,k are such that t < s < t +2` and s+t≡ −2 mod 2`, then Graham’s and Lehrer’s morphismθgives rise to a non-trivial morphism res_P

k(θ):S^s_n,k→S^t_n,kgiven, for v∈S^s_n,k, by res_P

k(θ)(v) =

∑

w:s←t monic

sg(w)HF(w)(q)Pkvw. (4.5)

Proof. The restriction functor res_P

kis applied to Graham’s and Lehrer’s morphismθ:S^s_n+k→S_n^t_+k. Functoriality gives the existence of a morphism betweenS^s_n,kandS^t_n,ksuch that the following diagram commutes:

S^s_n+k S_n+k^t

S^s_n,k S^t_n,k

θ

res_Pk res_Pk

res_Pkθ

.

Hence

R_n,k^t _'res_Pk(R^t_n+k), by lemma4.8

=res_P

k(radS_n^t_+k), as` >1 'res_Pk(θ(S^s_n+k)), by theorem4.6 'res_P

k(θ)(res_P

kS^s_n+k), by functoriality

=res_P

k(θ)(S^s_{n,k) =}im(res_P

k(θ)) and there is an isomorphism between the image of res_P

kθ andR^t_n,k; it will thus be sufficient to prove that dimR_n,k^t is not zero to prove that res_P

k(θ)is non-trivial.

Proposition4.2indicates that three cases are to be considered. First[t+1]q=0, that is, tis critical. This cannot happen as tandsare in a non-critical orbit under reflection. Second [t+2]q=0 and thus, by (4.1),

dimR^t_n,k=dimR^t−1_n−1,k+dimS_n^t+1_−1,k. (4.6)

Ast+1≤s−1≤n+k−1, the dimension dimR_n,k^t _≥S_n−1,k^t+1 is surely not 0. Third[t+2]q6=0.

In this last case, equation (4.1) gives

dimR^t_n,k=dimR_n^t−1_−1,k+dimR_n^t+1_−1,k. (4.7) The upper index ofR^t+1_n−1,k can further increase bym+2 uses of the same relation, such that [t+m+2]q=0:

dimR_n,k^t ₌dimR^t_n⁻₋¹_1,k+dimR^t_{n−2,k+· · ·+}dimR_n^t+₋^m_m⁻₋¹_1,k+dimS^t_n⁺₋^m_m⁺₋¹_1,k. (4.8) (An example of this recursive process follows the proof.) The method to get the term dimS^t+m+_n−m−¹_1,k insures thatt+m+1 belongs to∆_n−m−1,k. To see this, note first that the con-ditionn+d ≥k in the definition (3.8) of∆n,k remains satisfied at each step. Second, since [t+m+2]q=0, the indext+m+1 is critical andsis thus equal tot+2(m+1). Sinces∈∆n,k, the conditions≤n+kimplies t+m+1≤n−m−1+kandt+m+1∈∆_n−m−1,k. Hence, dimR^t_n,k is non-zero asS_n−m−^t+m+1_1,kis non-trivial. The non-triviality of res_P

k(θ)follows.

Figure4provides an example of the recursive process used in the proof. It is drawn forb_4,3 with`=5,s=7 andt =1. The morphism res_P3θ:S⁷_4,3→S¹_4,3is non-trivial as the dimension ofR¹_4,3, isomorphic to its image, is larger or equal to dimS⁴_1,3=1, as can be seen by multiple applications of (4.1):

dimR¹_4,3=dimR⁰_3,3+dimR²_3,3

=dimR⁰_3,3+dimR¹_2,3+dimR³_2,3

=dimR⁰_3,3+dimR¹_2,3+dimR²_1,3+dimS⁴_1,3.

The following Bratteli diagram displays the process with full lines representing applications of proposition4.2and circles the indices of the modules (radicals or cellular) appearing in the last line of the above equation. As before, the dashed vertical line is critical.

0 2

0 2 4

0 2 4 6

1 3

1 3 5

1 3 5 7

1

4

Figure 4: Recursive process forb_4,3with`=5,s=7 andt=1.

The existence of this family of morphims leads to the structure of the cellular modules overb_n,k. Proposition4.1showed thatS^d_n,k is irreducible ifd is critical. The next proposition studies the casednon-critical.

Proposition 4.10. If d∈∆n,kis non-critical, then the short sequence ofb_n,k-modules

0−→I^d_n,k⁺ _−→S^d_n,k−→I^d_n,k−→0 (4.9) is exact and non-split and thusR^d_{n,k '} I^d_n,k⁺. (Note that, if d 6∈∆⁰_n,k, then R^d_n,k = S^d_n,k and the moduleI^d_n,k in the above short sequence is understood to be0.)

Proof. Denote by [d] the orbit of d under the reflection through mirrors at critical integers and byd_r∈∆n,k the rightmost (or the largest) integer in this orbit. By definitiond⁺_r is not in

∆n,k. Letm∈Nbe such that(m−1)`−1<d<sub>r</sub><m`−1. The reflectiond_r⁺ofd_r through the critical integerm`−1 is determined by(dr+d<sub>r</sub><sup>+</sup>)/2= m`−1. Since d⁺_r is not in∆n,k, then n+k<d⁺_r =2m`−2−d_r and thus

n+k+d_r

2 +1<m`.

The second product of detG^d_n,k^r (see (2.4)) is the only one that can vanish. The numerators in this product run from[d_r+2]_qto[(n+k+d_r)/2+1]_q and thus never vanishes. HenceS^d_n,k^r is irreducible. Again, ifd⁺_r is not in∆n,k, it is not in∆_n+k either so that the cellularTL_n+k -moduleS^d_n^r_+kis also irreducible. In this case, the short exact sequence of theorem2.3is simply 0→0→S^d_n+k^r _→I^d_n+k^r _→0 and applying the exact restriction functor res_Pk to it gives

0−→0−→S^d_n,k^r _−→res_P

k(I^d_n^r_+k)−→0, which proves that res_P

k(I^d_n^r_+k)' I^d_n,k^r . If d_r⁻ is not in ∆n,k, then the proof is finished for this orbit.

If the left neighbor d_r⁻ belongs to ∆n,k, then the short exact sequence forS^d_n+k^r− given in theorem2.3is

0−→I^d_n+k^r _−→S^d_n+k^−r _−→I^d_n+k^r− _−→0,

whereI^d_n+k^r _'R^d_n+k^−r , also by theorem2.3. The restriction functor thus gives 0−→res_P

k(I^d_n+k^r _)−→S^d_n,k^r− _−→res_P

k(I^d_n+k^r− _)−→0.

It has just been proved that res_P

k(I^d_n+k^r _{) '} I^d_n,k^r and lemma 4.8 states that R^d_n,k^r− _' res_P

k(R^d_n+k^−r _{) =} res_P

k(I^d_n+k^r ₎, thus showing that R^d_n,k^r− _' I^d_n,k^r . The first isomorphism the-orem gives

res_P

k(I^d_n^−r_+k)'S^d_n,k^r−_/res_P

k(I^d_n^r_+k)'S^d_n,k^r−_/R^d_n,k^{r− def}₌ I^d_n,k^−r , thus giving

0−→I^d_n,k^r _−→S^d_n,k^r− _−→I^d_n,k^r− _−→0.

If d_r⁻ 6∈∆⁰_n,k, then R^d_n,k^−r is the whole moduleS^d_n,k^r− andI^d_n,k^−r should be understood as 0 in the sequence, therefore givingS^d_n,k^−r _'I^d_n,k^r . Otherwise, the proof of the exactness of this sequence forS^d_n,k^r− has given res_P

k(I^d_n+k^r− _)'I^d_n,k^r−, like the proof for the existence of an exact sequence for S^d_n,k^r had given res_P

k(I^d_n^r_+k)'I^d_n,k^r . So the present paragraph can be repeated for any elementd of the orbit[dr], thus closing the proof of the existence of the exact sequence (4.9).

It remains to show that these short exact sequences do not split. Proposition 3.11has shown that the cellular moduleS^d_n,kis cyclic for alld∈∆n,k. Suppose then thatzis a generator and thatS^d_n,kis a direct sumA_⊕Bof two submodules. The elementzcan be written asz_A+z_B withz_A∈Aandz_B∈B. If bothz_Aandz_Bwere inR^d_n,k, thenb_n,kz=b_n,kz_A⊕b_n,kz_B⊂R^d_n,k, contradicting the fact thezis a generator ofS^d_n,k. Then, one ofz_Aandz_Bis not inR^d_n,kand, by the argument above, is a generator ofS^d_n,k. If it isz_A, thenBmust be zero, and if it isz_B, then it isAthat must be zero. HenceS^d_n,kis indecomposable and the exact sequence (4.9) does not split.

Here is an example. The algebrab_4,2with`=3 has four cellular modules. Its line in the Bratteli diagram reads:

S⁰_4,2 S²_4,2 S⁴_4,2 S⁶_{4,2 .}

The modulesS²_4,2andS⁶_4,2are irreducible, the first because the integer 2 is critical, the second because 6 does not have a right neighbor in its orbit.

Because k+1=`, the bilinear form〈 ·,· 〉⁰_4,2 is identically zero. But the “renormalized” (4, 6)both satisfy the hypotheses of proposition4.9and there are thus two morphisms

θ1:S⁶_4,2−→S⁴_4,2, θ2:S⁴_4,2−→S⁰_4,2.

The action of the morphismθ1 on the unique element v of the basisB⁶_4,2 is given by a sum over the five elementsw_i of the setB_6←4: The coefficients given by proposition4.5are

H_F(w1)(q) = [5]_q!

Note that the contribution ofw₅ will cancel because P₂vw₅ = 0. With the proper signs and replacing the value of theq-numbers at q = e^πi/3 ([2]q = 1,[3]q = 0 and [4]q = −1), the

morphism is

θ1(v) = − + .

The morphism θ2 is given on the four elements x_i of the basis B⁴_4,2 by a sum on the two elementsz₁,z₂of the setB_4←0. AsH_F(z1)= [2]q andH_F(z2)=1, the morphism is defined by

θ2(v1) = [2]q − =− , θ2(v2) =− ,

θ2(v3) =− , θ2(v4) = [2]q − .

Note that the image ofθ1 lies in the kernel ofθ2:

θ2(θ1(v)) =θ2(v1−v₂+v₄) =θ2(v1)−θ2(v2) +θ2(v4) =0.

Dans le document The representation theory of seam algebras (Page 23-29)