Spin Chains as Modules over the Affine Temperley–Lieb Algebra

Abstract

The affine Temperley–Lieb algebra a TL_N(β) is an infinite-dimensional algebra over $\mathbb {C}$ parametrized by a number $\beta \in \mathbb {C}$ and an integer $N\in \mathbb {N}$. It naturally acts on $(\mathbb {C}^{2})^{\otimes N}$ to produce a family of representations labeled by an additional parameter $z\in \mathbb C^{\times }$. The structure of these representations, which were first introduced by Pasquier and Saleur (Nucl. Phys., 330, 523 1990) in their study of spin chains, is here made explicit. They share their composition factors with the cellular aTL_N(β)-modules of Graham and Lehrer (Enseign. Math., 44, 173 1998), but differ from the latter by the direction of about half of the arrows of their Loewy diagrams. The proof of this statement uses a morphism introduced by Morin-Duchesne and Saint-Aubin (J. Phys. A, 46, 285207 2013) as well as new maps that intertwine various aTL_N(β)-actions on the periodic chain and generalize applications studied by Deguchi et al. (J. Stat. Phys., 102, 701 2001) and after by Morin-Duchesne and Saint-Aubin (J. Phys. A, 46, 494013 2013).

Appendices

Appendix A: Some Properties of q-numbers

This appendix recalls the definitions of q-numbers, factorials and binomials while stating some of their properties.

The q-numbers [n]_q and q-factorial [n]_q! are, for $n\in \mathbb Z$,

$$[n]_{q}=\frac{q^{n}-q^{-n}}{q-q^{-1}}\quad \text{and}\quad [n]_{q}!=\begin{cases}[n]_{q}\cdot[n-1]_{q}!&\text{if }n\geq 1,\\ 1& \text{if }n=0,\\ 0&\text{otherwise.} \end{cases}$$

The q-binomial coefficient, for $m,n\in \mathbb Z_{\geq 0}$, is defined by

$$\left[\begin{matrix}m\\n\end{matrix}\right]=\frac{[m]_{q}!}{[n]_{q}![m-n]_{q}!}$$

if m ≥ n and is 0 otherwise. Here are two propositions that generalise properties of the (usual) binomial coefficients.

Proposition A.1 (Pascal identity, see e.g. [20, p. 6])

Let $q\in \mathbb C^{\times }$ and $m,n\in \mathbb N$ with n ≤ m − 1. Then,

$$\left[\begin{matrix}m\\ n\end{matrix}\right]_{q}= q^{\mp n}\left[\begin{matrix}m-1\\ n\end{matrix}\right]_{q} + q^{\pm(m-n) }\left[\begin{matrix}m-1\\ n-1\end{matrix}\right]_{q}. $$

Proposition A.2 (q-binomial theorem, see e.g. [27, p. 9])

Let $q\in \mathbb C^{\times }$ and $m\in \mathbb Z_{\geq 0}$. Then,

$$\sum\limits_{n=0}^{m} (-1)^{n}q^{n(m+1)}\left[\begin{matrix}m\\ n\end{matrix}\right]_{q}= \prod\limits_{n=1}^{m}(1-q^{2n}). $$

The remaining results describe the behavior of q-numbers when q is a root of unity. Strictly speaking, the left-hand side of the next identity should be understood within a limit $\lim _{q\to q_{c}}$ where q_c is the root of unity considered.

Proposition A.3 (Lucas q-theorem, see e.g. [36, Proposition 2.13])

Let $q\in \mathbb C^{\times }$ be such that q² be a primitive ℓ-th root of unity. Write $m,n\in \mathbb Z_{\geq 0}$ as m = m₁ℓ + m₂ and n = n₁ℓ + n₂ with m₁,n₁ ≥ 0 and 0 ≤ m₂,n₂ < ℓ. Then,

$$\left[\begin{matrix}m\\ n\end{matrix}\right]_{q}=q^{\ell(n_{1} \ell(n_{1}-m_{1})-m_{2}n_{1}-m_{1}n_{2})} \left( \begin{matrix}m_{1}\\ n_{1}\end{matrix}\right)\left[\begin{matrix}m_{2}\\ n_{2}\end{matrix}\right]_{q}$$

with $\big (\begin {smallmatrix}m_{1}\\ n_{1}\end {smallmatrix}\big )=0$ if m₁ < n₁. In particular $\big [\begin {smallmatrix}m\\ n\end {smallmatrix}\big ]_{q}=0$ if and only if m₁ < n₁ or m₂ < n₂.

As above, the next (easily shown) identities are to be understood multiplied by other terms and within limit signs.

Proposition A.4

If $q\in \mathbb C^{\times }$ is such that q² is a primitive ℓ-th root of unity and if $m,n\in \mathbb Z$, then $q^{\ell ^{2}} = (-1)^{\ell +1}q^{\ell }$,

$$ [m\ell\pm n]_{q}=\pm q^{m\ell}[n]_{q}\quad\text{and}\quad [m\ell]_{q}=mq^{(1-m)\ell}[\ell]_{q}. $$

Appendix B: Representation Theory of Lusztig’s Extension $\mathcal LU_q$

This appendix is devoted to the proof of Theorem 4.6. This theorem is purely representation-theoretic in nature and is of interest in its own right. Throughout $q \in \mathbb {C}^{\times }$ is such that q² is a ℓ th-primitive root of unity with ℓ ≥ 2. We start by stating some basic facts. The first one can be easily deduced from the definition of the divided powers of U_res. The last one is well-known (see e.g. [10, Proposition 9.3.5]) but a proof is given for completeness.

Lemma B.1

Let $k,n\in \mathbb {Z}_{\geq 0}$. Then, in U_res,

$$ E^{(k)}E^{(n)} = {\begin{bmatrix}{k+n}\\{n}\end{bmatrix}}_{t}E^{(k+n)}\quad\text{with}\quad K E^{(n)} K^{-1} = t^{2n} E^{(n)}\quad\text{and}\quad K F^{(n)} K^{-1} = t^{-2n} F^{(n)}. $$

Lemma B.2

[10, p. 297-299] Let $n,c \in \mathbb {Z}$ with n ≥ 0. Then, the element ${\begin {bmatrix}{K;c}\\{n}\end {bmatrix}}_{t}$ of the rational form $U_{t}\mathfrak {sl}_{2}$ defined by

$$ {\begin{bmatrix}{K;c}\\{n}\end{bmatrix}}_{t} = \prod\limits_{m=1}^{n}\left( \frac{K t^{c+1-m}-K^{-1}t^{m-1-c}}{t^{m}-t^{-m}}\right) $$

(B.1)

also belongs to U_res. Moreover the subalgebra generated by $\{K^{\pm 1}, {\begin {bmatrix}{K;c}\\{n}\end {bmatrix}}_{t} | n,c\in \mathbb {Z}$ $\text {with } n\geq 0\}\subseteq U_{\text {res}}$ is commutative.

Proposition B.3

In $\mathcal LU_q$, the following identities hold: E^ℓ = F^ℓ = K^2ℓ −id = 0.

Proof

The first two assertions follow from the specialization of the relations [ℓ]_t!E^(ℓ) = E^ℓ and [ℓ]_t!F^(ℓ) = F^ℓ of U_res. For the last one, note that K^2ℓ −id can be factorised as ${\prod }_{m=0}^{\ell -1} (K^{2}-q^{2m}\text {id})$. The definition of ${\begin {bmatrix}{K;0}\\{\ell }\end {bmatrix}}_{t}$ also gives

$$\prod\limits_{m=1}^{\ell}(Kt^{1-m}-K^{-1}t^{m-1})={\begin{bmatrix}{K;0}\\{\ell}\end{bmatrix}}_{t}\cdot \prod\limits_{m=1}^{\ell}(t^{m}-t^{-m})$$

whose right-hand side contains a factor (t^ℓ − t^−ℓ) that is zero once specialized at q. The left-hand side of the definition becomes ${\prod }_{m=1}^{\ell }(Kq^{1-m}-K^{-1}q^{m-1})=K^{-\ell }q^{\ell (1-\ell )/2}{\prod }_{m=0}^{\ell -1}(K^{2}-q^{2m}\text {id})$ upon specialization and, since K is invertible in $\mathcal LU_q$, the product ${\prod }_{m=0}^{\ell -1}(K^{2}-q^{2m}\text {id})$, and thus K^2ℓ −id, must vanish. □

Fix now integers $i,r,s \in \mathbb {Z}_{\geq 0}$ such that i = rℓ + s and s < ℓ − 1. Let also j = i + 2(ℓ − s − 1) as in Eq. ??. The following result is well-known (see e.g. [11]), but we still provide a proof.

Lemma B.4

As vector spaces, $\text {Ext}^{1}_{\mathcal LU_q}({\Delta }_{q}(i),{\Delta }_{q}(j))\simeq \mathbb {C}$.

Proof

The short exact sequence of Proposition 4.4 yields the long exact sequence in cohomology

$$ 0 \!\rightarrow\! \text{Hom}({\Delta}_{q}(i),{\Delta}_{q}(j)) \!\rightarrow\! \text{Hom}(P_{q}(i),{\Delta}_{q}(j)) \!\rightarrow\! \text{End} {\Delta}_{q}(j) \!\rightarrow\! \text{Ext}^{1}({\Delta}_{q}(i),{\Delta}_{q}(j)) \!\rightarrow\! \text{Ext}^{1}(P_{q}(i),{\Delta}_{q}(j)) \!\rightarrow\! \dots $$

where the last written term is zero by projectivity of P_q(i) in the category of finite-dimensional (type I) $\mathcal LU_q$-modules. (Every object in the above sequence and in the rest of the proof should carry a subscript $\mathcal LU_q$ which is dropped for brevity.) Hence, the extension group $\text {Ext}^{1}({\Delta }_{q}(i),{\Delta }_{q}(j))$ is a quotient of EndΔ_q(j) and, as the exact sequence of Proposition 4.4 is non-split, it is enough to show that EndΔ_q(j) has dimension at most one to conclude the proof of the lemma. For that goal, apply the left-exact covariant functor Hom(Δ_q(j),−) on the sequence given in Proposition 4.3 to get the exact sequence

$$ 0 \rightarrow \text{Hom}({\Delta}_{q}(j),L_{q}(i))\rightarrow \text{End} {\Delta}_{q}(j) \rightarrow \text{Hom}({\Delta}_{q}(j),L_{q}(j)) $$

(B.2)

and note that Hom(Δ_q(j),L_q(i)) = 0 as the simple module L_q(j) = top Δ_q(j) is not a subquotient of L_q(i). Also, Schur’s lemma gives $\text {Hom}({\Delta }_{q}(j),L_{q}(j)) \simeq \text {End} L_{q}(j) \simeq \mathbb {C}$ so Eq. B.2 forces $\dim \text {End}{\Delta }_{q}(j) \leq 1$ as desired. □

To provide an explicit realization for the indecomposable $\mathcal LU_q$-module P_q(i), it thus suffices to construct a non-trivial extension of the module Δ_q(j) by Δ_q(i). This extension will be obtained as the specialization (at t = q) of the module over the rational form $U_{t}\mathfrak {sl}_{2}$ given in the next proposition.

Proposition B.5

The $\mathbb {Q}(t)$-vector space $\mathcal {T}_{t}(i)$ with basis $\{m_{0},\dots ,m_{j},n_{0},\dots ,n_{i}\}$ is a $U_{t}\mathfrak {sl}_{2}$-module for the action

$$ \begin{gathered} Km_{k} = t^{j-2k}m_{k}, \qquad Kn_{p} = t^{i-2p}n_{p}, \qquad E^{(v)}m_{k} = {\begin{bmatrix}{j-k+v}\\{v}\end{bmatrix}}_{t}m_{k-v}, \qquad F^{(v)}m_{k} = {\begin{bmatrix}{k+v}\\{v}\end{bmatrix}}_{t}m_{k+v},\\ E^{(v)}n_{p} = {\begin{bmatrix}{i-p+v}\\{v}\end{bmatrix}}_{t}n_{p-v}+\gamma_{p,v}(t)m_{\ell-s+p-v-1}\quad\text{and}\quad F^{(v)}n_{p} = {\begin{bmatrix}{p+v}\\{v}\end{bmatrix}}_{t}n_{p+v}+\omega_{p,v}(t)m_{\ell-s+p+v-1} \end{gathered} $$

where the coefficients γ_p,v(t) and ω_p,v(t) are defined by

$$ \begin{array}{@{}rcl@{}} \gamma_{p,v}(t)&=& \frac{1}{[v]_{t}!}\sum\limits_{u=0}^{v-1} {\begin{bmatrix}{\ell-s+p-u-2}\\{p-u}\end{bmatrix}}_{t}\cdot \prod\limits_{a=1}^{u}[i-p+a]_{t}\cdot \prod\limits_{b=u+1}^{v-1}[i+\ell-s-p+b]_{t}\quad\text{and}\\ \omega_{p,v}(t) &=& \begin{cases} {\displaystyle \frac{1}{[\ell-s+i]_{t}}{\begin{bmatrix}{\ell-s+p+v-1}\\{p+v}\end{bmatrix}}_{t}{\begin{bmatrix}{p+v}\\{v}\end{bmatrix}}_{t}} & \text{if } v > i-p,\\ 0 & \text{if } v \leq i-p,\end{cases} \end{array} $$

with γ_p,v(t) = ω_p,v(t) = 0 if p < 0 or p > i.

Proof

It is not difficult to verify that the given action is compatible with the defining relations of $U_{t}\mathfrak {sl}_{2}$. The only slightly difficult check, which is of the relation (t − t^− 1)[E,F]n_p = (K − K^− 1)n_p, is now detailed:

$$ \begin{array}{@{}rcl@{}} [E,F]n_{p} &=& E([p+1]_{t}n_{p+1}+\omega_{p,1}(t)m_{\ell-s+p})-F([i-p+1]_{t}n_{p-1}+\gamma_{p,1}(t)m_{\ell-s+p-2})\\ &=& ([i-p]_{t}[p+1]_{t}-[i-p+1]_{t}[p]_{t})n_{p}+{\varsigma} m_{\ell-s+p-1} = [i-2p]_{t} n_{p} + {\varsigma} m_{\ell-s+p-1} \end{array} $$

where the coefficient of m_{ℓ−s+p− 1} is

$$ \begin{array}{@{}rcl@{}} {\varsigma} &= [p+1]_{t} \gamma_{p+1,1}(t)-\gamma_{p,1}(t)[\ell-s+p-1]_{t}+\omega_{p,1}(t)[j-\ell+s-p+1]_{t}-[i-p+1]_{t}\omega_{p-1,1}(t).\end{array} $$

(Note that γ_p+ 1,1(t) and ω_p− 1,1(t) respectively vanish if p = i and p = 0.) Furthermore, if v = 1, ω_p,v(t) takes a simpler form, namely $\omega _{p,1}(t)=\delta _{p,i}{\begin {bmatrix}{1}{i+\ell -s-1}\\{i}\end {bmatrix}}_{t}$, as p must satisfy 1 = v > i − p while remaining in the range 0 ≤ p ≤ i. In particular, the term [i − p + 1]_tω_p− 1,1(t) vanishes. The expressions for the γ’s and ω’s then give

$$ \begin{array}{@{}rcl@{}} &= [p+1]_{t} (1-\delta_{p,i}) {\begin{bmatrix}{\ell-s+p-1}\\{p+1}\end{bmatrix}}_{t}-{\begin{bmatrix}{\ell-s+p-2}\\{p}\end{bmatrix}}_{t}[\ell-s+p-1]_{t} + \delta_{p,i}{\begin{bmatrix}{i+\ell-s-1}\\{i}\end{bmatrix}}_{t}[\ell-s-1]_{t} = 0 \end{array} $$

where the factor (1 − δ_p,i) was added to handle the vanishing of γ_p+ 1,1(t) when p = i. This forces as claimed

$$(t-t^{-1})[E,F]n_{p} = (t-t^{-1})[i-2p]_{t}n_{p} = (K-K^{-1})n_{p}.$$

The only remaining thing to check is that the action of the divided powers agrees with $E^{(v)} = \frac {1}{[v]_{t}!}E^{v}$ and $F^{(v)} = \frac {1}{[v]_{t}!}F^{v}$ for any $v\in \mathbb {N}$. This is done by induction on v. There is nothing to prove when v = 1. In preparation for v ≥ 2, note that $\gamma _{p,1}(t)={\begin {bmatrix}{1}{l-s+p-2}\\{p}\end {bmatrix}}_{t}$ so that the following recursive formula for the γ’s is easily obtained:

$$ \begin{array}{@{}rcl@{}} &&{\begin{bmatrix}{i-p+v-1}\\{v-1}\end{bmatrix}}_{t} \gamma_{p-v+1,1}(t)+[i+\ell-s-p+v-1]_{t}\gamma_{p,v-1}(t) \\ &&\qquad\qquad\qquad\quad=\frac1{[v-1]_{t}!}\gamma_{p-v+1,1}(t)\prod\limits_{a=1}^{v-1}[i-p+a]_{t}\\ &&\qquad\qquad\qquad\quad\quad+\frac1{[v-1]_{t}!}\sum\limits_{u=0}^{v-2}\gamma_{p-u,1}(t)\prod\limits_{a=1}^{u} [i-p+a]_{t}\cdot\prod\limits_{b=u+1}^{v-1}[i+l-s-p+b]\\ &&\qquad\qquad\qquad\quad=\frac1{[v-1]_{t}!}\sum\limits_{u=0}^{v-1}\gamma_{p-u,1}(t)\prod\limits_{a=1}^{u} [i-p+a]_{t}\cdot\prod\limits_{b=u+1}^{v-1}[i+l-s-p+b]\\ &&\qquad\qquad\qquad\quad=[v]_{t}\gamma_{p,v}(t). \end{array} $$

The induction hypothesis thus gives as claimed (all γ’s are evaluated at t)

$$ \begin{array}{@{}rcl@{}} \frac1{[v-1]_{t}!} E^{v}n_{p}&=&E\left( {\begin{bmatrix}{i-p+v-1}\\{v-1}\end{bmatrix}}_{t} n_{p-v+1}+\gamma_{p,v-1}m_{\ell-s+p-v}\right)\\ &=&{\begin{bmatrix}{i-p+v-1}\\{v-1}\end{bmatrix}}_{t}\big([i-p+v]_{t}n_{p-v}+\gamma_{p-v+1,1}m_{\ell-s+p-v-1}\big)\\ &&\qquad\qquad\qquad\quad+\gamma_{p,v-1}[j-\ell+s-p+v+1]_{t} m_{\ell-s+p-v-1}\\ &=&[v]_{t}\left( {\begin{bmatrix}{i-p+v}\\v\end{bmatrix}}_{t} n_{p-v}+ \gamma_{p,v}m_{\ell-s+p-v-1}\right)\\ & =&[v]_{t}E^{(v)}n_{p}. \end{array} $$

This hypothesis also expresses $\frac 1{[v-1]_{t}!} F^{v}n_{p}$ as a linear combination of n_p+v and m_{ℓ−s+p+v− 1}. A quick computation brings the coefficient of n_p+v to be the expected one for the action of [v]_tF^(v). The coefficient of m_{ℓ−s+p+v− 1} is in turn

$$ {\begin{bmatrix}{p+v-1}\\{v-1}\end{bmatrix}}_{t} \omega_{p+v-1,1}(t)+[\ell-s+p+v-1]_{t}\omega_{p,v-1}(t). $$

The vanishing of ω_p+v− 1,1(t) and ω_p,v− 1(t) depends on whether i − p is strictly smaller than v − 1 (then ω_p+v− 1,1(t) = 0), equal to v − 1 (then ω_p,v− 1(t) = 0) or strictly larger than v − 1 (and then both ω’s are zero and so is ω_p,v(t)). A direct computation in the first two cases (< and = v − 1) shows that the non-vanishing term equals [v]_tω_p,v(t), as required. □

From now on, we view $\mathcal {T}_{t}(i)$ as a $\mathbb {Z}[t,t^{-1}]$-module and we restrict the above $U_{t}\mathfrak {sl}_{2}$-action to a U_res-action. The U_res-module thus obtained is not finitely-generated but will still give rise to a finitely-generated $\mathcal LU_q$-module by specialization. In order to specialize this U_res-module, we need to show that the functions γ_p,v(t) and ω_p,v(t) of Proposition B.5 have a well-defined limit as t tends to q. This is the purpose of the next two lemmas. As before, $q\in \mathbb C^{\times }$ is such that q² is a ℓ-th primitive root of unity. Note that a t-number [n]_t can be written as t^{− 2(n− 1)}(t²ⁿ − 1)/(t² − 1) and that the polynomial (t²ⁿ − 1)/(t² − 1) has order 2(n − 1) with all its roots distinct. In other words, the poles of 1/[n]_t are all simple. Moreover, by Lucas q-theorem A.3, all q-binomial coefficients have a limit in $\mathbb C$ as t → q.

Lemma B.6

The limit $\gamma _{p,v} = \lim _{t\rightarrow q}\gamma _{p,v}(t)$ exists and belongs to $\mathbb C$ for all $p\in \{0,\dots ,i\}$ and $v \in \mathbb {N}$.

Proof

By the comments made before the statement, the denominator 1/[v]_t! is the only possible source of singularities in γ_p,v(t). Also, the result holds trivially if v < ℓ as, in this case, [v]_t! does not vanish when t → q. If v = ℓ, then the zero in [v]_t is simple and the limit $\lim _{t\to q}\gamma _{\ell ,v}(t)$ needs to be studied. It turns out that this is the only case to consider. Indeed, for v > ℓ, Lemma B.1 ties the two divided powers E^(v) and E^(ℓ) by $\kappa E^{(v)}=(E^{(\ell )})^{v_{1}}E^{(v_{2})}$ where

$$\kappa = {\begin{bmatrix}{v}\\{v_{2}}\end{bmatrix}}_{t} {\begin{bmatrix}{v_{1}\ell}\\{\ell}\end{bmatrix}}_{t} {\begin{bmatrix}{(v_{1}-1)\ell}\\{\ell}\end{bmatrix}}_{t} {\dots} {\begin{bmatrix}{\ell}\\{\ell}\end{bmatrix}}_{t}$$

and where v has been written as v = v₁ℓ + v₂ with $v_{1},v_{2}\in \mathbb Z_{\geq 0}$ and v₂ < ℓ. Note that $\lim _{t\to q}\kappa$ exists and is non-zero by Theorem A.3. Hence, for v > ℓ, the limit $\lim _{t\to q}\gamma _{p,v}(t)$ can be obtained as the limit of a sum of $\gamma _{p^{\prime },\ell }(t)$ and $\gamma _{p^{\prime \prime },v^{\prime }}(t)$ with $v^{\prime }<\ell$ and $p^{\prime }, p^{\prime \prime }\in \{0,\dots , i\}$. The lemma thus rests on the study of $\lim _{t\to q}\gamma _{p,\ell }(t)$. The rest of the proof is devoted to it.

From now on v = ℓ and the goal is to prove that the sum appearing in γ_p,ℓ(t) has a vanishing limit. The difference i − p is written as p₁ℓ + p₂ with $p_{1},p_{2}\in \mathbb Z_{\geq 0}$ and p₂ < ℓ. The summand in γ_p,ℓ(t) is also written as

$$ \lambda_{u}(t) = {\begin{bmatrix}{\ell-s+p-u-2}\\{p-u}\end{bmatrix}}_{t}\ \prod\limits_{a=1}^{u}[i-p+a]_{t} \cdot \prod\limits_{b=u+1}^{\ell-1}[i+\ell-s-p+b]_{t} $$

so that $\gamma _{p,\ell }(t) = \frac {1}{[\ell ]_{t}!}{\sum }_{u=0}^{\ell -1} \lambda _{u}(t)$. We prove first that $\lim _{t\to q} \lambda _{u}(t)=0$ when u is neither ℓ − p₂ − 1 nor s − p₂.

Case 1: ℓ − p₂ ≤ u. Then, [(p₁ + 1)ℓ]_t appears in the first product of λ_u(t) when a = ℓ − p₂, so the result follows.

Case 2: s − p₂ < u < ℓ − p₂ − 1. Then the q-binomial in λ_u(t) can be written as

$$ {\begin{bmatrix}{\ell-s+p-u-2}\\{p-u}\end{bmatrix}}_{t} = {\begin{bmatrix}{(r-p_{1})\ell+\ell-p_{2}-u-2}\\{(r-p_{1}-1)\ell+\ell-p_{2}-u+s}\end{bmatrix}}_{t} $$

whose limit is zero by Theorem A.3 as 0 ≤ ℓ − p₂ − u − 2 < ℓ − p₂ − u + s < ℓ.

Case 3: u ≤ s − (p₂ + 1). Then, [(p₁ + 1)ℓ]_t appears in the second product of λ_u(t) when b = s − p₂.

Two values of u have escaped the above cases, namely u = s − p₂ and u = ℓ − p₂ − 1. If s < p₂, only $\lambda _{\ell -p_{2}-1}(t)$ of these two possible λ’s will contribute to the sum ${\sum }_{u=0}^{\ell -1} \lambda _{u}(t)$ as u takes non-negative values. However, in this case, [(p₁ + 2)ℓ]_t appears in the second product defining $\lambda _{\ell -p_{2}-1}(t)$ when b = ℓ − p₂ + s and the aforementioned sum tends to zero. Hence only the case s ≥ p₂ remains. A direct computation gives

$$ \begin{array}{@{}rcl@{}} \lim_{t\to q}\lambda_{s-p_{2}}(t)\! &=&\! {\begin{bmatrix}{(r\! -\! p_{1})\ell + \ell\! -\! s\! -\! 2}\\{(r-p_{1})\ell}\end{bmatrix}}_{q}\prod\limits_{a=1}^{s-p_{2}}[i-p+a]_{q}\cdot \prod\limits_{b=s-p_{2}+1}^{\ell-1}[i\! +\! \ell\! -\! s\! -\! p+b]_{q},\\ \lim_{t\to q}\lambda_{\ell-p_{2}-1}(t)\! & = & \!{\begin{bmatrix}{(r\! -\! p_{1}-1)\ell + \ell\! -\! 1}\\{(r\! -\! p_{1}\! -\! 1)\ell+s\! +\! 1}\end{bmatrix}}_{q}\prod\limits_{a=1}^{\ell-p_{2}-1}[i\! -\! p\! +\! a]_{q}\cdot \prod\limits_{b=\ell-p_{2}}^{\ell-1}[i\! +\! \ell\! -\! s\! -\! p+b]_{q}. \end{array} $$

The q-binomials in these two limits can be simplified using Theorem A.3 and Proposition A.4:

$$ \begin{array}{@{}rcl@{}} {\begin{bmatrix}{(r-p_{1})\ell+\ell-s-2}\\{(r-p_{1})\ell}\end{bmatrix}}_{q}&=&q^{\ell(r-p_{1})(\ell+s)},\\ {\begin{bmatrix}{(r-p_{1}-1)\ell+\ell-1}\\{(r-p_{1}-1)\ell+s+1}\end{bmatrix}}_{q}&=&{\begin{bmatrix}{\ell-1}\\{s+1}\end{bmatrix}}_{q} q^{-\ell(r-p_{1}-1)(\ell+s)}=(-1)^{\ell+s}q^{\ell(r-p_{1})(\ell+s)}. \end{array} $$

Gathering the common factors in

$${\varsigma}=q^{\ell(r-p_{1})(\ell+s)}\prod\limits_{a=1}^{s-p_{2}}[i-p+a]_{q}\cdot \prod\limits_{b=\ell-p_{2}}^{\ell-1}[i+\ell-s-p+b]_{q},$$

the limit $\lim _{t\to q}{\sum }_{u=0}^{\ell -1}\lambda _{u}(t)$ is then equal to

$$ \begin{array}{@{}rcl@{}} \lim_{t\to q}\left( \lambda_{s-p_{2}}(t)+\lambda_{\ell-p_{2}-1}(t)\right) &= {\varsigma}\left( {\prod}_{b=s-p_{2}+1}^{\ell-p_{2}-1}[i+\ell-s-p+b]_{q}+(-1)^{l+s} {\prod}_{a=s-p_{2}+1}^{\ell-p_{2}-1}[i-p+a]_{q}\right)\end{array} $$

that gives, after changing the index in the first product to c = ℓ + s − 2p₂ − b,

$$ \begin{array}{@{}rcl@{}} &={\varsigma}\left( {\prod}_{c=s-p_{2}+1}^{\ell-p_{2}-1}[2(p_{1}+1)\ell-(i-p+c)]_{q}+(-1)^{l+s} {\prod}_{a=s-p_{2}+1}^{\ell-p_{2}-1}[i-p+a]_{q}\right) \end{array} $$

which is zero by Theorem A.4. This closes the argument. □

Lemma B.7

The limit $\omega _{p,v} = \lim _{t\rightarrow q}\omega _{p,v}(t)$ exists and belongs to $\mathbb C$ for all $p\in \{0,\dots ,i\}$ and $v \in \mathbb {N}.$

Proof

By Lucas q-theorem A.3, the only potential singularity of ω_p,v comes from the t-number [ℓ − s + i]_t = [(r + 1)ℓ]_t appearing in its denominator. As the zero of this t-number at t = q is of order one, it suffices to show that the product ${\begin {bmatrix}{\ell -s+p+v-1}\\{p+v}\end {bmatrix}}_{q}{\begin {bmatrix}{p+v}\\{v}\end {bmatrix}}_{q}$ appearing in ω_p,v vanishes when m_{ℓ−s+p+v − 1}≠ 0 and i − p < v. Write v and p in the usual forms (that is v = v₁ℓ + v₂ and p = p₁ℓ + p₂ with $v_{1},v_{2},p_{1},p_{2}\in \mathbb Z_{\geq 0}$ and v₂, p₂ < ℓ) and consider the three following cases.

Case 1: ℓ ≤ p₂ + v₂. Then, ${\begin {bmatrix}{p+v}\\{v}\end {bmatrix}}_{q} = {\begin {bmatrix}{(p_{1}+v_{1}+1)\ell +p_{2}+v_{2}-\ell }\\{v_{1}\ell +v_{2}}\end {bmatrix}}_{q} = 0$ by Theorem A.3 as 0 ≤ p₂ + v₂ − ℓ < v₂ < ℓ.

Case 2: s < p₂ + v₂ < ℓ. Then, ${\begin {bmatrix}{\ell -s+p+v-1}\\{p+v}\end {bmatrix}}_{q} = \begin {bmatrix}{(p_{1}+v_{1}+1)\ell +p_{2}+v_{2}-s-1}\\{(p_{1}+v_{1})\ell + p_{2}+v_{2}}\end {bmatrix}_{q} = 0$ again by Theorem A.3.

Case 3: p₂ + v₂ ≤ s. Then, i < p + v gives r + 1 ≤ p₁ + v₁ so that m_{ℓ−s+p+v− 1} = 0 as

$$ \begin{array}{@{}rcl@{}} \ell-s+p+v-1 &=& (p_{1}+v_{1}+1)\ell-s+p_{2}+v_{2}-1\\ &\geq&(r+2)\ell-s+p_{2}+v_{2}-1> (r+2)\ell-s-2 = j.\end{array} $$

This concludes the proof as every possible case has been studied. □

Corollary B.8

The $\mathbb {C}$-vector space $\mathcal {T}_{q}(i)$ with basis {m₀,..., m_j, n₀,...,n_i} is a $\mathcal LU_q$-module for the action given by

$$ \begin{array}{@{}rcl@{}} \displaystyle K{\kern-.5pt}m_{k}{\kern-.5pt} & = &{} q^{j-2k}m_{k}, \qquad K{\kern-.5pt}n_{p} {\kern-.5pt} = {\kern-.5pt} q^{i-2p}n_{p}, \qquad E^{({\kern-.5pt}v{\kern-.5pt})}m_{k} {\kern-.5pt} = {\kern-.5pt} {\begin{bmatrix}{j{\kern-.5pt} -{\kern-.5pt} k{\kern-.5pt} +{\kern-.5pt} v}\\{v}\end{bmatrix}}_{q}m_{k-v}, \quad F^{({\kern-.5pt}v{\kern-.5pt})}m_{k} {\kern-.5pt} = {\kern-.5pt} {\begin{bmatrix}{k{\kern-.5pt} +{\kern-.5pt} v}\\{v}\end{bmatrix}}_{q}m_{k+v}{\kern-.5pt},\\ \displaystyle E^{(v)}n_{p}{\kern-.5pt} & = & {} {\begin{bmatrix}{i-p+v}\\{v}\end{bmatrix}}_{q}n_{p-v}+\gamma_{p,v}m_{\ell-s+p-v-1}\quad \text{ and } \displaystyle F^{(v)}n_{p} = {\begin{bmatrix}{p+v}\\{v}\end{bmatrix}}_{q}n_{p+v}+\omega_{p,v}m_{\ell-s+p+v-1}. \end{array} $$

Also, the unrolled element H acts on $\mathcal {T}_{q}(i)$ as H m_k = (j − 2k)m_k and Hn_p = (i − 2p)n_p.

Proof

The first statement follows from Proposition B.5 and Lemmas B.6 and B.7. The last one is easily proved using the definition of the unrolled element H and Proposition B.5. □

As $\mathcal {T}_{q}(i)$ is clearly an extension of Δ_q(j) by Δ_q(i), Lemma B.4 shows that it is enough to show the following result in order to finish the proof Theorem 4.6. Recall that the construction of $\mathcal {T}_{q}(i)$ and Theorem 4.6 both assume that s < ℓ − 1.

Lemma B.9

The $\mathcal LU_q$-module $\mathcal {T}_{q}(i)$ is an indecomposable extension of Δ_q(j) by Δ_q(i).

Proof

Suppose ad absurdum that the extension is trivial and fix a $\mathcal LU_q$-linear isomorphism $\varphi :{\Delta }_{q}(i)\oplus {\Delta }_{q}(j)\rightarrow \mathcal {T}_{q}(i)$. Let $\{\overline m_{k}\}_{k=0}^{j}\subseteq {\Delta }_{q}(j)$ and $\{\overline n_{p}\}_{p=0}^{i}\subseteq {\Delta }_{q}(i)$ be the bases given in Definition 4.1 and observe that the restriction of φ to _q(j) must satisfy $\varphi (\overline m_{k}) = {\varsigma } m_{k}$ for some ${\varsigma }\in \mathbb {C}^{\times }$ as $\text {End}_{\mathcal LU_q}({\Delta }_{q}(j)) \simeq \mathbb {C}$ by the proof of Lemma B.4.

Write $\varphi (\overline n_{0}) = {\sum }_{k = 0}^{j} \alpha _{k}^{(m)} m_{k}+{\sum }_{p=0}^{i} \alpha _{p}^{(n)} n_{p}$ where $\alpha _{0}^{(m)},\dots ,\alpha _{j}^{(m)},\alpha _{0}^{(n)},\dots ,\alpha _{i}^{(n)}\in \mathbb {C}$. Then,

$$ 0 = H\varphi(\overline n_{0})-\varphi(H\overline n_{0}) = 2\sum\limits_{k=0}^{j} \alpha_{k}^{(m)}(\ell-s-1-k)m_{k}- 2\sum\limits_{p=0}^{i} p\alpha_{p}^{(n)}n_{p} $$

gives $\varphi (\overline n_{0})=\alpha _{0}^{(n)}n_{0}+\alpha _{\ell -s-1}^{(m)}m_{\ell -s-1}$. However, this implies

$$ \begin{array}{@{}rcl@{}} 0 & = & E \varphi(\overline n_{0}) = (\gamma_{0,1}\alpha_{0}^{(n)} + \alpha_{\ell-s-1}^{(m)}[j - \ell + s + 2]_{q})m_{\ell-s-2}\\ & = & (\alpha_{0}^{(n)} + \alpha_{\ell-s-1}^{(m)}[(r + 1)\ell]_{q}) m_{\ell-s-2} = \alpha_{0}^{(n)}m_{\ell-s-2}, \end{array} $$

and $\alpha _{0}^{(n)}=0$ as s < ℓ − 1. Hence ${\varsigma }\varphi (\overline n_{0}) = {\varsigma }\alpha _{\ell -s-1}^{(m)}m_{\ell -s-1} = \alpha _{\ell -s-1}^{(m)}\varphi (\overline m_{\ell -s-1})$ and we contradict the injectivity of φ. □

Appendix C: Remaining Cases and Problematic Pairs

In this appendix, we finish the analysis done in Section 4 by letting q = ± 1 and by replacing Lusztig’s quantum group $\mathcal LU_q$ by the universal envelopping algebra $U\mathfrak {sl}_{2}$. We also show some results involving the problematic pairs (0,±i) ∈ λ_N (with N even, see Section 2) which were left unproven in Sections 3 and 5.

1.1 C.1 Periodic Chain Morphisms for ℓ = 1

Throughout this subsection, the parameter q is set to + 1 or − 1 and the algebra $U_{q}\mathfrak {sl}_{2}$ is understood to be the universal enveloping algebra $U\mathfrak {sl}_{2}$. This algebra is generated by elements e,f,h satisfying the defining relations

$$[h,e]=2e,\qquad [h,f]=-2f\quad\text{and}\quad[e,f]=h.$$

For $n\in \mathbb {Z}_{\geq 0}$, we denote by L(n) the unique simple $U\mathfrak {sl}_{2}$-module with $\dim L(n) = n+1$. This representation has a basis $\{v_{0},{\dots } ,v_{n}\}$ with explicit action defined by ev_i = i(n − i + 1)v_i− 1, fv_i = v_i+ 1 and hv_i = (n − 2i)v_i (where v_− 1 = v_n+ 1 = 0). Let ${\Delta }:U\mathfrak {sl}_{2}\rightarrow U\mathfrak {sl}_{2}\otimes U\mathfrak {sl}_{2}$ be the algebra morphism given by

$$ {\Delta}(e) = e\otimes 1+q(1\otimes e), \qquad {\Delta}(f) = f\otimes 1+q(1\otimes f) \quad \text{and}\quad {\Delta}(h) = h\otimes 1+1\otimes h. $$

This morphism for q = 1 is the usual coproduct used to define the tensor product of representations of $\mathfrak {sl}_{2}$. A quick check shows that it is also an algebra morphism at q = − 1.

Define recursively a family of maps $\{{\Delta }_{n}:U\mathfrak {sl}_{2}\rightarrow U\mathfrak {sl}_{2}^{\otimes n}\}_{n\geq 2}$ with Δ₂ = Δ and Δ_n+ 1 = (id ⊗Δ) ∘Δ_n where id denotes the identity operator on $U\mathfrak {sl}_{2}^{\otimes (n-1)}$. (Remark that Δ₃≠(Δ ⊗id) ∘Δ when q = − 1 so that Δ is then not coassociative. Therefore the pullback of a threefold tensor product of $U\mathfrak {sl}_{2}$-modules by (Δ ⊗id) ∘Δ and its pullback by Δ₃ may be non-isomorphic and care must be taken when defining a $U\mathfrak {sl}_{2}$-action on the periodic chains with q = − 1.) Identify moreover L(1) with $\mathbb {C}^{2}$ through the change of basis given by v₀↦|+〉 with v₁↦|−〉 and associate the periodic chain $\mathsf {X}_{N;z}^{+}$ ($=\mathsf {X}_{N;z}^{-}$) to the pullback of (L_q(1))^⊗N by Δ_N. Then, an easy computation gives the following formulas for the $U\mathfrak {sl}_{2}$-action on $\mathsf {X}_{N;z}^{+}$

$$ \begin{array}{@{}rcl@{}} e|x_{1}{\dots} x_{N}\rangle_{z}^{+} &=& \sum\limits_{\substack{1\leq j\leq N\\ x_{j} = -}} q^{j-1}|x_{1}{\dots} x_{j-1}(+)x_{j+1}{\dots} x_{N}\rangle_{z}^{+}, \\ f|x_{1}{\dots} x_{N}\rangle_{z}^{+} &=& \sum\limits_{\substack{1\leq j\leq N\\ x_{j}=+}}q^{j-1}|x_{1}{\dots} x_{j-1}(-)x_{j+1}{\dots} x_{N}\rangle_{z}^{+} \end{array} $$

and $h|x_{1}{\dots } x_{N}\rangle _{z}^{+} = 2S^{z} |x_{1}{\dots } x_{N}\rangle _{z}^{+} = d|x_{1}{\dots } x_{N}\rangle _{z}^{+}$ where $d = {\sum }_{j=1}^{N} x_{j}$. Remark that f acts on $\mathsf {X}_{N;z}^{+}$ as the conjugation of e with the spin-flip s of Section 3.2. This will be used without mention later.

Theorem C.1

Let $d \in \mathbb {Z}$ be of the parity of N and fix $z \in \mathbb {C}^{\times }$ such that q^d = z². Set d_m = d + 2m with z_m = zq^m for $m\in \mathbb {Z}$. Then $F_{d_{m},z_{m}}^{(1)}:\mathsf {X}_{N;d_{m},z_{m}}^{+}\rightarrow \mathsf {X}_{N;d_{m-1},z_{m-1}}^{+}$ given by $|x_{1}{\dots } x_{N}\rangle _{z_{m}}^{+}\mapsto f|x_{1}{\dots } x_{N}\rangle _{z_{m-1}}^{+}$ is aTL_N>-linear.

Proof

Fix a basis element $|x_{1}{\dots } x_{N}\rangle _{z_{m}}^{+}$ of $\mathsf {X}_{N;d_{m},z_{m}}^{+}$. Then, a straightforward computation using Eq. 3.6 gives

$$ e_{1}F_{d_{m},z_{m}}^{(1)}|x_{1}{\dots} x_{N}\rangle_{z_{m}}^{+} = {\varsigma}+{\varsigma}_{1}\quad\text{and}\quad F_{d_{m},z_{m}}^{(1)}e_{1}|x_{1}{\dots} x_{N}\rangle_{z_{m}}^{+}={\varsigma}+{\varsigma}_{2} $$

where, as q² = 1,

$$ \begin{array}{@{}rcl@{}} {\varsigma}_{1} &=& \delta_{x_{1},+}\delta_{x_{2},+}((1-q^{2})|(+)(-)x_{3}{\dots} x_{N}\rangle_{z_{m-1}}^{+}+(q-q^{-1})|(-)(+)x_{3}{\dots} x_{N}\rangle_{z_{m-1}}^{+}) = 0, \\ {\varsigma}_{2} &=& \delta_{x_{1}+x_{2},0}((\delta_{x_{2},+}+q\delta_{x_{1},+})|(-)(-)x_{3}{\dots} x_{N}\rangle_{z_{m-1}}^{+}\\ &&- q^{x_{1}}\delta_{x_{1},+}|(-)x_{2}{\dots} x_{N}\rangle_{z_{m-1}}^{+} - q^{x_{1}+1}\delta_{x_{2},+}|x_{1}(-)x_{3}{\dots} x_{N}\rangle_{z_{m-1}}^{+})=0 \text{ and}\\ {\varsigma} &=& \sum\limits_{\substack{3\leq j\leq N \\ x_{j} = +}}q^{j-1} \delta_{x_{1}+x_{2},0}(|x_{2}x_{1}x_{3}{\dots} x_{j-1}(-)x_{j+1}{\dots} x_{N}\rangle_{z_{m-1}}^{+} - q^{x_{1}}|x_{1}{\dots} x_{j-1}(-)x_{j+1}{\dots} x_{N}\rangle_{z_{m-1}}^{+}). \end{array} $$

Thus, $e_{1}F_{d_{m},z_{m}}^{(1)}|x_{1}{\dots } x_{N}\rangle _{z_{m}}^{+} =F_{d_{m},z_{m}}^{(1)}e_{1}|x_{1}{\dots } x_{N}\rangle _{z_{m}}^{+}$ as desired. We also have

$$ \begin{array}{@{}rcl@{}} {\Omega}_{N} F^{(1)}_{d_{m},z_{m}}|x_{1}{\dots} x_{N}\rangle_{z_{m}}^{+} &=& z_{m-1}\delta_{x_{1},+}|x_{2}{\dots} x_{N}(-)\rangle_{z_{m-1}}^{+}\\ &&+z_{m-1}^{-x_{1}}\sum\limits_{\substack{2\leq j \leq N\\x_{j} =+}}q^{j-1}|x_{2}{\dots} x_{j-1}(-)x_{j+1}{\dots} x_{N}x_{1}\rangle_{z_{m-1}}^{+},\\ F^{(1)}_{d_{m},z_{m}}{\Omega}_{N} |x_{1}{\dots} x_{N}\rangle_{z_{m}}^{+}&=& z_{m}^{-1}q^{N-1}\delta_{x_{1},+}|x_{2}{\dots} x_{N}(-)\rangle_{z_{m-1}}^{+}\\ &&+z_{m}^{-x_{1}}\sum\limits_{\substack{1\leq j\leq N-1\\x_{j+1}=+}}q^{j-1}|x_{2}{\dots} x_{j}(-)x_{j+2}{\dots} x_{N}x_{1}\rangle_{z_{m-1}}^{+}, \end{array} $$

so ${\Omega }_{N} F^{(1)}_{d_{m},z_{m}}|x_{1}{\dots } x_{N}\rangle _{z_{m}}^{+} = F^{(1)}_{d_{m},z_{m}}{\Omega }_{N}|x_{1}{\dots } x_{N}\rangle _{z_{m}}^{+}$ as $z_{m-1}^{-x_{1}} = q^{-1}z_{m}^{-x_{1}}$ and $z_{m-1} = q^{d-1}z_{m}^{-1} = q^{N-1}z_{m}^{-1}$ by hypothesis. □

Suppose that q^d = z². Then, q^−d = (z^− 1)² and Theorem C.1 produces a morphism of a TL_N-modules from $\mathsf {X}_{N;-d_{m-1},z_{m-1}^{-1}}^{+}$ to $\mathsf {X}_{N;-d_{m},z_{m}^{-1}}^{+}$. Denote by $E^{(1)}_{d_{m},z_{m}}:\mathsf {X}_{N;d_{m-1},z_{m-1}}^{+}\rightarrow \mathsf {X}_{N;d_{m},z_{m}}^{+}$ the aTL_N-linear morphism obtained by conjugating this last map with the spin flip. (Recall that $\mathsf {X}_{N;d,z}^{+}$ and $\mathsf {X}_{N;d,z}^{-}$ are equal as q² = 1.) Note also that the assumption q^d = z² amounts here to saying that $(d_{m-1},z_{m-1})\unlhd (d_{m},z_{m})$ whenever $m\in \mathbb {Z}$ where $\unlhd$ is the partial order on ${\Delta }=\mathbb {Z}\times \mathbb {C}^{\times }$ given through Eq. 4.3. Fix $m\in \mathbb {N}$ and let $\mu ^{(m)}:\mathsf {X}_{N;d_{m},z_{m}}^{+}\rightarrow \mathsf {X}_{N;d,z}^{+}$ and $\nu ^{(m)}:\mathsf {X}_{N;d,z}^{+}\rightarrow \mathsf {X}_{N;d_{m},z_{m}}^{+}$ be the a TL_N-morphisms defined by

$$ \mu^{(m)} = F^{(1)}_{d_{1},z_{1}}F^{(1)}_{d_{2},z_{2}}{\dots} F^{(1)}_{d_{m},z_{m}}\qquad\text{and}\qquad \nu^{(m)}=E^{(1)}_{d_{m},z_{m}}E^{(1)}_{d_{m-1},z_{m-1}}{\dots} E^{(1)}_{d_{1},z_{1}}. $$

Then $\mu ^{(m)}|x_{1}{\dots } x_{N}\rangle _{z_{m}}^{+} = f^{m}|x_{1}{\dots } x_{N}\rangle _{z}^{+}$ and $\nu ^{(m)}|x_{1}{\dots } x_{N}\rangle _{z}^{+} =e^{m}|x_{1}{\dots } x_{N}\rangle _{z_{m}}^{+}$ for $x_{1},{\dots } ,x_{N} \in \{+,-\}$ since the $U\mathfrak {sl}_{2}$-action on $\mathsf {X}_{N;z}^{+}$ is independent of z. The morphisms μ^(m) and ν^(m) satisfy properties similar to those described in Theorem 4.20. To prove this fact, observe that, since the category of finite-dimensional $U\mathfrak {sl}_{2}$-modules is semisimple, we can decompose the $U\mathfrak {sl}_{2}$-module $\mathsf {X}_{N;z}^{+}$ as a sum of simple modules L(n) with $n\in \mathbb {Z}_{\geq 0}$. Within L(n), the action of e^jf^j on the basis vector v_i gives a non-zero multiple of v_i if i + j ≤ n. This well-known fact and its analogue for f^je^j lead to the following lemma.

Lemma C.2

Let $n,m\in \mathbb {Z}_{\geq 0}$ and $d \in \mathbb {Z}$ with |d|≤ n and d ≡ n modulo 2. Then the action of e^mf^m (or f^me^m) on L(n) induces a $\mathbb {C}$-linear automorphism of the eigenspace L(n)|_h=d if |d − 2m|≤|d| (or |d + 2m|≤|d|, respectively).

The analogue of Theorem 4.20 follows directly from this lemma as the action of $h\in U\mathfrak {sl}_{2}$ on the periodic chain $\mathsf {X}_{N;z}^{+}$ coincide with the action of the diagonal spin operator 2S^z defining the eigenspaces $\mathsf {X}_{N;d,z}^{+}$.

Theorem C.3

With the above notation, the maps ν^(m)μ^(m) and μ^(m)ν^(m) are respectively one-to-one if |d|≤|d + 2m| and |d|≥|d + 2m|. In particular, μ^(m) is injective (or surjective) if |d|≤|d + 2m| (or if |d|≥|d + 2m|, respectively) and ν^(m) is injective (or surjective) if |d|≥|d + 2m| (or if |d|≤|d + 2m|, respectively).

1.2 C.2 Problematic Pairs

This section studies the problematic pairs, that is the pairs (0,q) and (0,q^− 1) in the case where q + q^− 1 = 0 and N is even. Throughout the section q and N will be assumed to satisfy these conditions. Of course, this means that q is either i or − i. Note that $(0,q)\sim (0,q^{-1})$ in Λ and remark that theorem 2.4 of [17] implies that the head of the module W_N;0,q does not provide any new simple module that is not already labeled in Λ_N (see Eq. 2.1). The main purpose of this subsection is thus to determine the content and structure of these cellular modules. Here it is.

Proposition C.4

Let $N\in 2\mathbb N$ and q be either i or − i. The Loewy diagram of the cellular module W_N;0,q is

Proof

The first step here is to compute the direct successors in λ_N of (0,q). These are precisely the nodes of the proposed diagram, with arrows pointing toward the successors. These nodes are labelled by an integer $i\in \mathbb {N}$ which is bounded above by n = N/2 and are given explicitly by (2i,y_i) for the left column and by (2i,−y_i) for the right where y_i = q¹⁻ⁱ. The direct successors of (0,q) through condition A and B are respectively (2,1) and (2,− 1). Proposition 3.2 applies as the pairs (0,q), (2,1) and (2,− 1) all belong to λ_N. There are hence two injective a TL_N-morphisms W_N;2,1 →W_N;0,q and W_N;2,− 1 →W_N;0,q and the Loewy diagram of W_N;0,q must contain the diagrams of W_N;2,1 and W_N;2,− 1. Fortunately, the latter diagrams may be deduced from Theorem 2.2: they are precisely those obtained by removing, from the diagram drawn in the statement, the node (2,− 1) for W_N;2,1 and (2,1) for the other. It is thus sufficient to show that the dimension of W_N;0,q is equal to that of the sum of the composition factors appearing in the statement.

As said above, W_N;2,1 and W_N;2,− 1 share all their composition factors except for their respective simple heads L_N;2,1 and L_N;2,− 1. Since $\dim \mathsf {W}_{N;2,1}=\dim \mathsf {W}_{N;2,-1}$ (the dimension Eq. 3.3 of W_N;d,z is independent of z), the two simple modules L_N;2,1 and L_N;2,− 1 have the same dimension. This argument can be repeated for each pair of simple modules $\mathsf {L}_{N;2i,y_{i}}$ and $\mathsf {L}_{N;2i,-y_{i}}$. We thus want to prove that $\dim \mathsf {W}_{N;0,q}={\sum }_{1\leq i\leq n}2d_{i}$ where $d_{i}=\dim \mathsf {L}_{N;2i,y_{i}}$. The dimensions d_i satisfy

$$ \begin{array}{@{}rcl@{}} d_{i}&=&\dim \mathsf{L}_{N;2i,y_{i}}=\dim\mathsf{W}_{N;2i,y_{i}}-2\sum\limits_{i+1\leq j\leq n}d_{j}\\ &=&\dim \mathsf{W}_{N;2i,y_{i}}-\dim \mathsf{W}_{N;2(i+1),y_{i+1}}-d_{i+1}= \begin{pmatrix}2n\\ n+i\end{pmatrix}-\begin{pmatrix}2n\\ n+i+1\end{pmatrix}-d_{i+1}. \end{array} $$

The third equality was obtained by gathering up all d_j’s in the sum but one of the the two d_i+ 1’s and the last one follows from Eq. 3.3. This recurrence ends at $d_{n}=\dim \mathsf {L}_{N;N,y_{n}}=\dim \mathsf {W}_{N;N,y_{n}}=1$. Its solution is

$$d_{i}=\begin{pmatrix}2n\\ n+i\end{pmatrix}+2{\sum}_{1\leq j\leq n-i}(-1)^{j}\begin{pmatrix}2n\\ n+i+j\end{pmatrix}.$$

Hence,

$$ \begin{array}{@{}rcl@{}} \sum\limits_{1\leq i\leq n} 2d_{i} &=&\dim\mathsf{W}_{N;2,y_{1}}+d_{1} =\begin{pmatrix} 2n\\ n-1\end{pmatrix}+\begin{pmatrix} 2n\\ n+1\end{pmatrix}+2\sum\limits_{1\leq j\leq n-1}(-1)^{j}\begin{pmatrix}2n\\ n+1+j\end{pmatrix}\\ &=&-2\sum\limits_{1\leq i\leq n}(-1)^{i}\begin{pmatrix}2n\\ n+i\end{pmatrix} =2(-1)^{n+1}\sum\limits_{0\leq i\leq n-1}(-1)^{i}\begin{pmatrix}2n\\ i\end{pmatrix}.\end{array} $$

Note that the alterning sum contains the n first terms of the binomial expansion of (1 − 1)²ⁿ. Thus,

$$ \begin{array}{@{}rcl@{}} &=&\begin{pmatrix} 2n\\ n\end{pmatrix}+(-1)^{n+1}\sum\limits_{0\leq i\leq 2n}(-1)^{i}\begin{pmatrix}2n\\ i\end{pmatrix}=\begin{pmatrix} 2n\\ n\end{pmatrix}+(-1)^{n+1}(1-1)^{2n}\\ &=&\dim\mathsf{W}_{N;0,q} \end{array} $$

as desired. This ends the demonstration. □

We can now adapt the proofs of Theorem 2.4 and of Corollary 3.11 to the case of problematic pairs.

Proof Proof (Corollary 3.11, problematic case)

The proof given for Proposition C.4 shows that $\mathsf {GP}_{N;0,q} \simeq \mathsf {L}_{N;2,-1} = \text {top } \mathsf {W}_{N;2,-1}$ which is a quotient of imi_N;0,q by Proposition 3.9. To repeat this argument for the other problematic pair (0,q^− 1), note that $\mathsf {W}_{N;0,q}\simeq \mathsf {W}_{N;0,q^{-1}}$ by definition of the cellular modules as the cokernel of some f_z (see Section 3.1). Note also that (0,q^− 1) ≼ (2,− 1) and (0,q^− 1) ≼ (2, 1) directly through condition A and B (respectively) by Lemma 3.10. Proposition C.4 thus gives $\mathsf {GP}_{N;0,q^{-1}} \simeq \mathsf {L}_{N;2,1} = \text {top } \mathsf {W}_{N;2,1}$ which is a quotient of $\text {im} i_{N;0,q^{-1}}$ by Proposition 3.9. □

Note that the generic part of W_N;0,q and $\mathsf {W}_{N;0,q^{-1}}$ are not isomorphic even though the modules W_N;0,q and $\mathsf {W}_{N;0,q^{-1}}$ are.

In the following proof (which follows almost exactly the steps of Section 5.3) we will often use implicitly the Loewy diagram of Proposition C.4 in order to identify the successors of the problematic pair (0,q) for the order ≼.

Proof Proof (Theorem 2.4, subcase (iii), problematic case)

Fix $N\in 2\mathbb {N}$ and $q\in \mathbb {C}$ with q² = − 1. Then, the reasoning of Section 5.3 shows that $\mathsf {X}_{N;0,q}^{+}$ admits a submodule M such that $\mathsf {M}\simeq \mathsf {X}_{N;2,-1}^{+}/\text {im} F^{+}_{(2,-1);(4,-q)}$ (namely $\mathsf {M}=\text {im} F^{+}_{(0,q);(2,-1)}$). However, (D,Z) = (2,− 1) is not a problematic pair and is such that q^D∉{Z²,Z^− 2}. The proof of Section 5.3 thus shows that $\mathsf {X}_{N;2,-1}^{+}$ has the Loewy diagram

where the image $\text {im} F^{+}_{(2,-1);(4,-q)}$ has been identified with red arrows. The submodule $\mathsf {M}\subseteq \mathsf {X}_{N;0,q}^{+}$ hence has the structure:

Using the above analysis for the problematic pair (0,q^− 1) and applying the ⋆-duality (with the help of Proposition 3.4), we can moreover deduce that $\mathsf {X}_{N;0,q}^{+}$ has a quotient with the Loewy diagram:

(C.1)

Also, the proof of Section 5.3 shows that $\mathsf {X}_{N;0,q}^{+}$ has another submodule, namely $\text {im} s F^{-}_{(0,-q);(2,-1)}$, which is isomorphic to $(\mathsf {X}_{N;2,-1}^{+}/\text {im} F^{+}_{(2,-1);(4,-q)})^{\circ } \simeq \mathsf {M}^{\circ }$. By the above analysis and Proposition 3.5, the structure of this submodule is

Repeating this argument for the pair (0,q^− 1) and applying once again the ⋆-duality (with the help of Proposition 3.4), we deduce that $\mathsf {X}_{N;0,q}^{+}$ has in addition a quotient with the following Loewy diagram:

We can now finish the proof precisely as in Section 5.3. Indeed, we have shown (see Proposition C.4) that all composition factors of W_N;0,q appear in some subquotient of $\mathsf {X}_{N;0,q}^{+}$. Since $\dim \mathsf {X}_{N;0,q}^{+} = \dim \mathsf {W}_{N;0,q}$, we have found all composition factors of the eigenspace $\mathsf {X}_{N;0,q}^{+}$ and its Loewy diagram must be the one announced in Theorem 2.4, that is

(C.2)

with perhaps some missing arrows and where the submodule $M=\text {im} F^{+}_{(0,q);(2,-1)}$ has been identified with red arrows. The fact that there is no missing arrows is shown exactly like in Section 5.3 if N > 2. When N = 2, the precedent diagram becomes the one of the module W_2;0,q (see Proposition C.4). It must therefore be missing an arrow since $\mathsf {W}_{2;0,q}\not \simeq \mathsf {X}_{2;0,q}^{+}$ (as an easy computation shows that the generator e₁ ∈a TL_N acts as zero on W_2;0,q but not of $\mathsf {X}_{2;0,q}^{+}$). However, the only arrow that can be added while still producing a diagram compatible with the submodules and quotients found above is $(2,1)\rightarrow (2,-1)$. The Loewy diagram of $\mathsf {X}_{2;0,q}^{+}$ must then be given by this arrow, that is

The structure of the other problematic eigenspace $\mathsf {X}_{N;0,q^{-1}}^{+} \simeq (\mathsf {X}_{N;0,q}^{+})^{\star }$ can also be deduced from Eq. C.2 with the help of Proposition 3.4. The resulting diagram for N = 2 is given by the arrow $(2,-1)\rightarrow (2,1)$. For N > 2, we rather obtain

(C.3)

where the image imF_{(0,−q);(2,1)} is again identified with red arrows (this image is the ⋆-dual of the quotient Eq. C.1 of $\mathsf {X}_{N;0,q}^{+}$). The diagrams obtained are compatible with Theorem 2.4 and we can thus conclude the proof. □

We conclude this appendix by noting that a comparison between the diagram Eq. C.2 (or the corresponding diagram for N = 2) and the one given in Proposition C.4 shows that, for the problematic pair (0,q), the image imi_N;0,q is exactly the generic part $\mathsf {GP}_{N;0,q} \simeq \mathsf {L}_{N;2,-1}$. This is also true for the other problematic pair (0,−q) as the generic part GP_N;0,−q is then isomorphic to the simple module L_N;2,1 (see the proof of Corollary 3.11 for the problematic case) which is a submodule of $\mathsf {X}_{N;0,-q}^{+}$ by diagram Eq. C.3 (or by the corresponding diagram for N = 2). This is a special case of Corollary 5.4.