Quantum periodicity and Kirillov-Reshetikhin modules

TL;DR

This paper proves the periodicity of quantum T-systems of type A with specific boundary conditions, using categorification and representation theory of quantum affine algebras, focusing on Kirillov-Reshetikhin modules.

Contribution

It provides a new proof of quantum T-system periodicity based on categorification and representation theory, linking modules and evaluation modules.

Findings

Established periodicity of quantum T-systems of type A

Connected T-system relations with Kirillov-Reshetikhin modules

Used categorification to relate algebraic structures

Abstract

We give a proof of the periodicity of quantum $T$-systems of type $A_n\times A_\ell$ with certain spiral boundary conditions. Our proof is based on categorification of the $T$-system in terms of the representation theory of quantum affine algebras, more precisely on relations between classes of Kirillov-Reshetikhin modules and of evaluation modules.

Equations

$$Q_{a,b}^2=Q_{a-1,b}{Q}_{a+1,b}+Q_{a,b+1}{Q}_{a,b-1},$$

$$T_{a,b}(u-1)T_{a,b}(u+1)=T_{a-1,b}(u)T_{a+1,b}(u)+T_{a,b+1}(u)T_{a,b-1}(u).$$

$$I=\{1,\cdots ,n\}\text{ and }J=\{1,\cdots ,\ell \}.$$

$$\Lambda =\{(a,b,u)\in I\times J\times \mathbb{Z}|a+b+u\in 2\mathbb{Z}\}.$$

$$T_{a,b}(u-1)T_{a,b}(u+1)=T_{a-1,b}(u)T_{a+1,b}(u)+T_{a,b+1}(u)T_{a,b-1}(u),$$

$$T_{0,b}(u)\text{ , }{T}_{a,\ell +1}(u)\text{ , }{T}_{n+1,b}(u)\text{ and }{T}_{a,0}(u).$$

$$T_{n+1,m}(u)=\{\begin{array}{ll}{\mathcal{F}}_{\frac{u+n+m+3}{2}}& \text{ for }-n-3\le u+m\le n-1\\ 1& \text{ for }0\le u+m-n+1\le 2\ell +2.\end{array}$$

$$T_{0,m}(u)=\{\begin{array}{ll}{\mathcal{F}}_{\frac{u-m+2}{2}}& \text{ for }-2\le u-m\le 2n,\\ 1& \text{ for }0\le u-m-2n\le 2(\ell +1).\end{array}$$

$$T_{k,\ell +1}(u)=\{\begin{array}{ll}{\mathcal{F}}_{\frac{u-k-\ell +1}{2}}& \text{ for }0\le u-\ell -k+1\le 2n+2,\\ 1& \text{ for }\ell +1\le u-2n-k\le 3(\ell +1).\end{array}$$

$$T_{k,0}(u)=\{\begin{array}{ll}{\mathcal{F}}_{\frac{u+k+2}{2}}& \text{ for }-2\le u+k\le 2n,\\ 1& \text{ for }0\le u-2n+k\le 2(\ell +1).\end{array}$$

$$X_{k,m}=T_{k,m}(k+m-2)\text{ for (k,m)∈I×J}$$

$$X_{k,0}=T_{k,0}(k-2)={\mathcal{F}}_k.$$

$$T_{a,b}(u)=T_{n+1-a,\ell +1-b}(u+n+\ell +2).$$

$$T_{0,1}(1)=T_{1,2}(3)=T_{2,1}(5)=T_{1,0}(7)={\mathcal{F}}_1.$$

$$X\text{ , }\frac{{\mathcal{F}}_1+1}{X}\text{ , }X\text{ , }\frac{{\mathcal{F}}_1+1}{X}\text{ , }X.$$

$$T_{0,1}(1)=T_{0,2}(2)=T_{1,3}(4)=T_{2,2}(6)=T_{2,1}(7)=T_{1,0}(9)={\mathcal{F}}_1.$$

$$X_1\text{ , }\frac{{\mathcal{F}}_1+X_2}{X_1}\text{ , }\frac{X_1+1}{X_2}\text{ , }{X}_2\text{ , }\frac{{\mathcal{F}}_1{X}_1+{\mathcal{F}}_1+X_2}{X_1{X}_2}\text{ , }{X}_1.$$

$$X_2\text{ , }\frac{{\mathcal{F}}_1{X}_1+{\mathcal{F}}_1+X_2}{X_1{X}_2}\text{ , }{X}_1\text{ , }\frac{{\mathcal{F}}_1+X_2}{X_1}\text{ , }\frac{X_1+1}{X_2}\text{ , }{X}_2.$$

$$T_{0,1}(1)=T_{1,2}(3)=T_{2,2}(4)=T_{3,1}(6)=T_{2,0}(8)=T_{1,0}(9)={\mathcal{F}}_1$$

$$T_{1,0}(1)=T_{0,1}(3)=T_{1,2}(5)=T_{2,2}(6)=T_{3,1}(8)=T_{2,0}(11)={\mathcal{F}}_2$$

$$X_1\text{ , }\frac{{\mathcal{F}}_1{X}_2+{\mathcal{F}}_2}{X_1}\text{ , }\frac{X_1+{\mathcal{F}}_2}{X_2}\text{ , }{X}_2\text{ , }\frac{X_1+{\mathcal{F}}_2+{\mathcal{F}}_1{X}_2}{X_1{X}_2}\text{ , }{X}_1.$$

$$X_2\text{ , }\frac{X_1+{\mathcal{F}}_2+{\mathcal{F}}_1{X}_2}{X_1{X}_2}\text{ , }{X}_1\text{ , }\frac{{\mathcal{F}}_2+{\mathcal{F}}_1{X}_2}{X_1}\text{ , }\frac{X_1+{\mathcal{F}}_2}{X_2}\text{ , }{X}_2.$$

$$X_{a,b}\ast X_{c,d}=t^{\gamma (a,b;c,d)-\gamma (c,d;a,b)}{X}_{c,d}\ast X_{a,b}.$$

$$C(z)=((z+z^{-1}){\delta }_{i,j}-{\delta }_{i+1,j}-{\delta }_{i-1,j}{)}_{i,j\in I}.$$

$$\gamma (a,b;c,d)={\tilde{C}}_{a,c}(2\ell -2b+c-a+1)+{\tilde{C}}_{a,c}(2\ell -2b+c-a-1)+\cdots +{\tilde{C}}_{a,c}(2d-2b+c-a+1).$$

$${\mathcal{F}}_r\ast {\mathcal{F}}_{r^{\prime }}=t^{{\tilde{C}}_{r,r^{\prime }}(2\ell +r^{\prime }-r+1)+{\tilde{C}}_{r,r^{\prime }}(2\ell +r^{\prime }-r-1)+\cdots +{\tilde{C}}_{r,r^{\prime }}(2\ell +r-r^{\prime }+3)}{\mathcal{F}}_{r^{\prime }}\ast {\mathcal{F}}_r$$

$$X_1\ast {\mathcal{F}}_1=t^2{\mathcal{F}}_1\ast X_1.$$

$$X_1\ast X_2=t^{-2}{X}_2\ast X_1\text{ , }{X}_1\ast {\mathcal{F}}_1={\mathcal{F}}_1\ast X_1\text{ , }{X}_2\ast {\mathcal{F}}_1={\mathcal{F}}_1\ast X_2.$$

$$X_1\ast X_2=t{X}_2\ast X_1\text{ , }{X}_1\ast {\mathcal{F}}_1=t{\mathcal{F}}_1\ast X_1\text{ , }{X}_1\ast {\mathcal{F}}_2={\mathcal{F}}_2\ast X_1,$$

$$X_2\ast {\mathcal{F}}_2=t{\mathcal{F}}_2\ast X_2\text{ , }{X}_2\ast {\mathcal{F}}_1=t{\mathcal{F}}_1\ast X_2\text{ , }{\mathcal{F}}_1\ast {\mathcal{F}}_2=t^{-1}{\mathcal{F}}_2\ast {\mathcal{F}}_1.$$

Full text

To Nicolai Reshetikhin on his 60th birthday

1. 1 Introduction
2. 2 Periodicity and quantum periodicity
3. 3 Finite-dimensional representations of quantum affine algebras
4. 4 Relations in the Grothendieck ring
5. 5 Proof of periodicity

1. Introduction

The $Q$-system was introduced by Kirillov and Reshetikhin [KR] as a system of relations between characters of certain simple finite-dimensional representations of the quantum affine algebra $\mathcal{U}_{q}(\hat{sl}_{n+1})$, now called Kirillov-Reshetikhin modules

[TABLE]

where $1\leq a\leq n$ and $b$ is a non-negative integer. Inspired by this work, the $T$-system was written in [KNS1] as a refined version of the $Q$-system depending on a spectral parameter $u$ :

[TABLE]

It was conjectured that it is satisfied by the classes of the Kirillov-Reshetikhin modules.

In a fundamental work [FR], Frenkel and Reshetikhin introduced a character theory for finite-dimensional representations of quantum affine algebras, called the $q$-characters. Then the $T$-system is satisfied by $q$-characters of Kirillov-Reshetikhin modules in all types [N2, H2], and so by their classes as conjectured above.

In another direction, Zamolodchikov initiated in [Z] a long series of work on the periodicity of solutions of $T$-systems with certain boundary conditions, which culminated in the work of Keller [Kel2] with a very general uniform proof of the periodicity of $T$-systems associated to a pair $(\Delta,\Delta^{\prime})$ of Dynkin diagrams (see [IIKNS, Kel1] for reviews and references).

In this note we propose a simple proof of the periodicity (and half-periodicity) of $T$-systems of type $A_{n}\times A_{\ell}$ and of its quantum version (in the sense of [N2, HL2]), with certain spiral boundary conditions (more general than the unit condition usually considered). We follow the approach in [HL1, Section 12.1] where the proof of the commutative periodicity in type $A_{n}\times A_{1}$ is obtained with formulas for solutions in terms of $q$-characters. Indeed we find solutions in terms of certain evaluation representations, containing Kirillov-Reshetikhin modules but not only, and more precisely in terms of their $q,t$-characters defined by Nakajima [N1].

The quantum periodicity (and half-periodicity) established in this note should also follow from the analog results in the commutative case (with unit boundary condition) mentioned above and from results in [BZ, CKLP] (indeed the approach in [Kel2] is based on the study of the periodicity of a sequence of mutations in a certain cluster algebra). Our direct method gives an explicit solution in terms of $q,t$-characters.

The paper is organized as follows. In Section 2 we state the main periodicity and quantum periodicity results and we give several examples. In Section 3 we give the necessary reminders on the representation theory of quantum affine algebras. In Section 4 we recall how the $T$-system appears in the Grothendieck ring of the category of representations and we prove it has also other incarnations. We conclude in Section 5 with the proof of the quantum periodicity.

**Acknowledgment : ** The author is very grateful to Bernard Leclerc for many discussions over the years and to Bernhard Keller for useful remarks and explanations about [Kel2] and its consequences. The author would you like to thank Laura Fedele for her careful reading and for pointing typos in a former version of this paper. The author is supported by the European Research Council under the European Union’s Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine.

2. Periodicity and quantum periodicity

In this section we state the periodicity and quantum periodicity of $T$-systems that we establish in this note.

2.1. Periodicity

Let us first state the commutative $A_{n}\times A_{\ell}$-periodicity. Let

[TABLE]

We work on the lattice

[TABLE]

Let us consider a family of commuting variables $(T_{a,b}(u))_{(a,b,u)\in\Lambda}$ satisfying the $T$-system (sometimes called octahedron relation) :

[TABLE]

for any $(a,b,u+1)\in\Lambda$.

So that the system is well-defined, we have to fix the boundary conditions, that is the values of

[TABLE]

The first choice is to set all values to $1$, this is called the unit boundary condition (see [IIKNS]). The system is already non trivial with such a choice. We will also consider the following boundary condition. Let $(\mathcal{F}_{r})_{r\in I}$ be formal variables that we call coefficients. We also set $\mathcal{F}_{0}=\mathcal{F}_{n+1}=1$. We set the following boundary conditions (for $u$ modulo $2(\ell+n+2)$) :

[TABLE]

[TABLE]

[TABLE]

[TABLE]

This is a particular case of the spiral boundary condition (see [IIKNS]).

Remark 2.1**.**

For $(a,b,u)\in\Lambda$, an induction on $u\geq a+b-2$ shows that $T_{a,b}(u)$ is a rational fraction in the

[TABLE]

and in the coefficients

[TABLE]

We have the following periodicity.

Theorem 2.2**.**

For any $(a,b,u)\in\Lambda$, we have the half-periodicity property :

[TABLE]

It implies that $T_{a,b}(u)$ is $2(n+\ell+2)$-periodic in $u$.

Example 2.3**.**

Let $n=\ell=1$. The non trivial boundary conditions are

[TABLE]

*Set $X=X_{1,1}=T_{1,1}(0)$.

The values of $T_{1,1}(u)$ for $u=0,2,4,6,8$ are respectively*

[TABLE]

Example 2.4**.**

Let $n=1$ and $\ell=2$. The non trivial boundary conditions are

[TABLE]

*Set $X_{1}=X_{1,1}=T_{1,1}(0)$ and $X_{2}=X_{1,2}=T_{1,2}(1)$.

The values of $T_{1,1}(t)$ for $t=0,2,4,6,8,10$ are*

[TABLE]

The values of $T_{1,2}(u)$ for $u=1,3,5,7,9,11$ are respectively

[TABLE]

Example 2.5**.**

Let $n=2$ and $\ell=1$. The non trivial boundary conditions are

[TABLE]

[TABLE]

*Set $X_{1}=X_{1,1}=T_{1,1}(0)$ and $X_{2}=X_{2,1}=T_{2,1}(1)$.

The values of $T_{1,1}(t)$ for $t=0,2,4,6,8,10$ are*

[TABLE]

The values of $T_{2,1}(u)$ for $u=1,3,5,7,9,11$ are respectively

[TABLE]

2.2. Quantum periodicity

Let us now the state the quantum version of the $A_{n}\times A_{\ell}$-periodicity (see also [KN] and [DFK] for $n=1$).

We work now with quasi-commuting variables $(X_{a,b})_{(a,b)\in I\times J}$ :

[TABLE]

To define the power of $t$, we use the inverse $\tilde{C}(z)$ of the quantized Cartan matrix

[TABLE]

For $p\in\mathbb{Z}$ and $a,c\in I$, we denote by $\tilde{C}_{a,c}(p)$ the coefficient of $z^{p}$ in the expansion in $z$ of $\tilde{C}_{a,c}(p)$. We set

[TABLE]

The relation is also extended to $b=0$ or $d=0$ so that we get the quasi-commutation rule with the coefficients. In particular for $r<r^{\prime}$, we have

[TABLE]

and $\mathcal{F}_{r}*\mathcal{F}_{r+1}=t^{\tilde{C}_{r,r+1}(2\ell+2)}\mathcal{F}_{r+1}*\mathcal{F}_{r}$. This is derived from $\tilde{C}_{i,j}(k)=0$ if $k\leq|j-i|$ (which can be observed for example in the formula in [GTL, Appendix A.3]).

The quasi-commuting variables $(X_{a,b})_{(a,b)\in I\times J}$ with the $\mathcal{F}_{t}$ generate a quantum torus $\mathcal{T}_{t}$ over $\mathbb{Z}[t^{\pm 1/2}]$. We denote its fraction field by $K_{t}$. It has an antimultiplicative bar involution satisfying $\overline{t}=t^{-1}$ and so that the $X_{a,b}$, $\mathcal{F}_{t}$ are bar-invariant.

For each product $m$ of various $X_{a,b}^{\pm 1}$, $\mathcal{F}_{t}^{\pm 1}$, there is a unique $\alpha\in\mathbb{Z}$ so that $t^{\alpha/2}m$ is bar invariant. This is called a commutative monomial. The commutative monomials form a $\mathbb{Z}[t^{\pm 1/2}]$-basis of $\mathcal{T}_{t}$.

Example 2.6**.**

For $n=\ell=1$, we have :

[TABLE]

For $n=1$, $\ell=2$, we have :

[TABLE]

For $n=2$, $\ell=1$, we have :

[TABLE]

[TABLE]

We fix the same the boundary conditions as for the commutative setting above.

Theorem 2.7**.**

Consider a family of bar-invariant $T_{a,b}(u)\in K_{t}$ satisfying :

[TABLE]

for $(a,b,u+1)\in\Lambda$. We assume the same initial conditions as in Remark 2.1 and the same spiral boudary conditions as in the classical setting. Then for $(a,b,u)\in\Lambda$ :

[TABLE]

It implies that $T_{a,b}(u)$ is $2(n+\ell+2)$-periodic in $u$.

The classical periodicity in Theorem 2.2 follows directly from this Theorem. We propose a simple proof based on the representations theory of quantum affine algebras and on their Kirillov-Reshetikhin modules.

Example 2.8**.**

Let us study the examples 2.3, 2.4, 2.5 above. In these examples, let us just replace each Laurent monomial in the $X_{a,b}$ by the corresponding commutative monomial in the quantum torus. We get bar-invariant elements in $\mathcal{T}_{t}$ and we keep the notation $T_{a,b}(u)$. Let us verify they satisfy the quantum $T$-system.

Let $n=\ell=1$. We get :

[TABLE]

Let $n=1$, $\ell=2$.

[TABLE]

Let $n=2$, $\ell=1$.

[TABLE]

3. Finite-dimensional representations of quantum affine algebras

We recall the main definitions and properties of finite-dimensional representations of the quantum affine algebra associated to $sl_{n+1}$.

3.1. Quantum affine algebras

All vector spaces, algebras and tensor products are defined over $\mathbb{C}$.

Let $C=(C_{i,j})_{0\leq i,j\leq n}$ be the Cartan matrix of type $A_{n}^{(1)}$, that is

[TABLE]

where $n+1$ is identified with [math]. Fix $q\in\mathbb{C}^{*}$ which is not a root of unity.

The quantum affine algebra $\mathcal{U}_{q}(\mathfrak{g})$ is defined by generators $k_{i}^{\pm 1}$, $x_{i}^{\pm}$ ($0\leq i\leq n$) and relations

[TABLE]

[TABLE]

where we denote $\left(x_{i}^{\pm}\right)^{(p)}=\left(x_{i}^{\pm}\right)^{p}/[p]_{q}$ for $0\leq p\leq 2$, where $[p]_{q}=(q^{p}-q^{-p})(q-q^{-1})^{-1}$.

It is a Hopf algebra with a coproduct $\Delta:\mathcal{U}_{q}(\mathfrak{g})\rightarrow\mathcal{U}_{q}(\mathfrak{g})\otimes\mathcal{U}_{q}(\mathfrak{g})$ defined for $0\leq i\leq n$ by

[TABLE]

Let $\overline{\mathfrak{g}}=sl_{n+1}$ be the finite-dimensional simple Lie algebra of Cartan matrix $(C_{i,j})_{i,j\in I}$. We denote respectively by $\omega_{i}$, $\alpha_{i}$, $\alpha_{i}^{\vee}$ ($i\in I$) the fundamental weights, the simple roots and the simple coroots of $\overline{\mathfrak{g}}$. We use the standard partial ordering $\leq$ on the weight lattice $P$ of $\overline{\mathfrak{g}}$.

The algebra $\mathcal{U}_{q}(\mathfrak{g})$ has another set of generators, the Drinfeld generators, denoted by

[TABLE]

We have $x_{i}^{\pm}=x_{i,0}^{\pm}$ for $i\in I$. A complete set of relations for Drinfeld generators was obtained in [B, D]. In particular the multiplication defines a surjective linear morphism

[TABLE]

where $\mathcal{U}_{q}^{\pm}(\mathfrak{g})$ is the subalgebra generated by the $x_{i,m}^{\pm}$ ($i\in I$, $m\in\mathbb{Z}$) and $\mathcal{U}_{q}(\mathfrak{h})$ is the subalgebra generated by the $k_{i}^{\pm 1}$, the $h_{i,r}$ and $c^{\pm 1/2}$ ($i\in I$, $r\in\mathbb{Z}\setminus\{0\}$).

3.2. Finite-dimensional representations

We refer to [CH] for generalities on the category $\mathcal{C}$ of finite-dimensional representations of $\mathcal{U}_{q}(\mathfrak{g})$. For $i\in I$, the action of $k_{i}$ on any object of $\mathcal{C}$ is diagonalizable with eigenvalues in $\pm q^{\mathbb{Z}}$. Without loss of generality, we can assume that $\mathcal{C}$ is the category of type 1 finite-dimensional representations (see [CP2]), i.e. we assume that for any object of $\mathcal{C}$, the eigenvalues of $k_{i}$ are in $q^{\mathbb{Z}}$ for $i\in I$. The simple objects of $\mathcal{C}$ are parametrized by $n$-tuples of polynomials $(P_{i}(u))_{i\in I}$ satisfying $P_{i}(0)=1$ (they are called Drinfeld polynomials) [CP1, CP2].

In type $A$, there is a family of evaluation morphisms $ev_{a}:\mathcal{U}_{q}(\mathfrak{g})\rightarrow\mathcal{U}_{q}(\overline{\mathfrak{g}})$ parametrized by $a\in\mathbb{C}^{*}$. Hence for $V$ a simple finite-dimensional representations of $\mathcal{U}_{q}(\overline{\mathfrak{g}})$, by pull-back we get an evaluation representation $(V)_{a}$. If the highest weight of $V$ is a multiple of a fundamental weight, then $V$ is a Kirillov-Reshetikhin module. In the particular case of a fundamental weight, we get the fundamental representations $V_{i}(a)=(V(\omega_{i}))_{a}$ of Drinfeld polynomials $(1,\cdots,1,1-za,1,\cdots,1)$ with a non-trivial polynomial in position $i$. Their classes generate the Grothendieck ring $K_{0}(\mathcal{C})$ of the category $\mathcal{C}$ which is a polynomial ring in the variables $[V_{i}(a)]$ as proved in [FR]. In general simple finite-dimensional representations are not evaluation modules.

For $\omega\in P$, the weight space $V_{\omega}$ of an object $V$ in $\mathcal{C}$ is the set of weight vectors of weight $\omega$, i.e. of vectors $v\in V$ satisfying $k_{i}v=q^{\left(\omega(\alpha_{i}^{\vee})\right)}v$ for any $i\in I$.

The elements $c^{\pm 1/2}$ act by identity on any object $V$ of $\mathcal{C}$, and so the action of the $h_{i,r}$ commute. Since the $h_{i,r}$, $i\in I$, $r\in\mathbb{Z}\setminus\{0\}$, also commute with the $k_{i}$, $i\in I$, every object in $\mathcal{C}$ can be decomposed as a direct sum of generalized eigenspaces of the $h_{i,r}$ and $k_{i}$. More precisely, by Frenkel-Reshetikhin theory of $q$-characters [FR], the eigenvalues of the $h_{i,r}$ and $k_{i}$ can be encoded by monomials $m$ in formal variables $Y_{i,a}^{\pm 1}$ ($i\in I,a\in\mathbb{C}^{*}$). Let $\mathcal{M}$ be the set of such monomials (also called $l$-weights). Given $m\in\mathcal{M}$ and an object $V$ in $\mathcal{C}$, let $V_{m}$ be the subspace of $V$ of common pseudo-eigenvectors of the $h_{i,r}$, $k_{i}$ with pseudo-eigenvalues associated to $m$ (also called $l$-weight space). Thus,

[TABLE]

If $v\in V_{m}$, then $v$ is a weight vector of weight

[TABLE]

where we denote $m=\prod_{i\in I,a\in\mathbb{C}^{*}}Y_{i,a}^{u_{i,a}(m)}$. For $v\in V_{m}$, we set $\omega(v)=\omega(m)$.

The $q$-character morphism is an injective ring morphism

[TABLE]

[TABLE]

If $V_{m}\neq\{0\}$ we say that $m$ is an $l$-weight of $V$.

A monomial $m\in\mathcal{M}$ is said to be dominant if $u_{i,a}(m)\geq 0$ for any $i\in I,a\in\mathbb{C}^{*}$. For $V$ a simple object in $\mathcal{C}$, let $M(V)$ be the highest weight monomial of $\chi_{q}(V)$, that is so that $\omega(M(V))$ is maximal for the partial ordering on $P$. $M(V)$ is dominant and characterizes the isomorphism class of $V$ (it is equivalent to the data of the Drinfeld polynomials). Hence to a dominant monomial $M$ is associated a simple representation $L(M)$. For $i\in I$ and $a\in\mathbb{C}^{*}$, we have for example the fundamental representation $V_{i}(a)=L\left(Y_{i,a}\right)$. The simple modules of highest weight monomial

[TABLE]

for some $i\in I,\alpha\in\mathbb{Z},\beta\geq 1$ are Kirillov-Reshetikhin modules. We will also use the notation $X_{i,\alpha}^{\beta}=1$ for $\beta\leq 0$.

Example 3.1**.**

The $q$-character of the fundamental representation $L(Y_{a})$ of $\mathcal{U}_{q}(\hat{sl_{2}})$ is

[TABLE]

The $q$-characters of evaluation modules, including Kirillov-Reshetikhin modules and fundamental modules, are known explicitly (see references in the introduction of [H3]). The formulas involve the monomials $A_{i,a}$ defined in [FR] for $i\in I,a\in\mathbb{C}^{*}$ by

[TABLE]

3.3. Quantum Grothendieck ring

The Grothendieck ring $K_{0}(\mathcal{C})$ has a $t$-deformation called quantum Grothendieck ring $K_{t}(\mathcal{C})$ as constructed in [VV, N1, H1] (we use the version of [H1, HL2]). It is a $\mathbb{Z}[t^{\pm 1/2}]$-subalgebra of a quantum torus $\mathcal{Y}_{t}$ and simple objects $L(m)$ have corresponding classes $[L(m)]_{t}\in K_{t}(\mathcal{C})$. A quantum version of a result in [FM] gives the following [N1] :

[TABLE]

In other words, $m$ is maximal for the Nakajima partial ordering on monomials, that is $M\preceq M^{\prime}$ if $M^{\prime}M^{-1}$ is a product of variables $A_{i,c}$.

If a simple module $V$ is thin, that is if its $\ell$-weight spaces are of dimension $1$, then $[V]_{t}$ is a sum of commutative monomials (defined as in section 2.2) and can be identified with its $q$-character (see [HL2, Corollary 5.3]). In type $A$, all simple evaluation modules are thin.

4. Relations in the Grothendieck ring

We recall how the $T$-system originally occurs in the representation theory of quantum affine algebras and we also establish another incarnation of the $T$-system in the Grothendieck ring $K_{0}(\mathcal{C})$ (that we call horizontal $T$-system).

We denote $I=\{1,\cdots,n\}$ and $J=\{1,\cdots,\ell\}$ as above.

4.1. Original $T$-systems

For $1\leq i\leq n$ and $0\leq m\leq p\leq\ell$, consider the Kirillov-Reshetikhin module

[TABLE]

We extend the notation to $i=0$ and $i=n+1$ by setting $\beta(m,p)^{i}=1$ in these cases.

For $i\in I$ and $0\leq m\leq p<\ell$, we have the $T$-system in $K_{0}(\mathcal{C})$:

[TABLE]

See the list of references in the introduction of [H3]. It can be deformed into the quantum $T$-system in $K_{t}(\mathcal{C})$ (see [N2, HL2]) :

[TABLE]

for some $\lambda,\mu\in\mathbb{Z}/2$ which depend in $m,p,i$ (they can be explicitly computed but this is not relevant for the following).

4.2. Horizontal $T$-systems

The $T$-system has another incarnation in $K_{0}(\mathcal{C})$.

For $0\leq i\leq j\leq n+1$ and $0\leq m\leq\ell+1$, consider the evaluation module

[TABLE]

Some of these representations are Kirillov-Reshetikhin modules :

[TABLE]

For $0\leq i\leq n+1$ and $j\geq 0$, we will denote

[TABLE]

Note that $F_{0}=F_{n+1}=1$.

Theorem 4.1**.**

For $0\leq i<j\leq n$ and $m\in J$, there are $\lambda,\lambda^{\prime}\in\mathbb{Z}/2$, so that :

[TABLE]

Remark 4.2**.**

(i) This relation is ”orthogonal” to the original quantum $T$-system in the sense that the spectral parameter is replaced by the vertex of the Dynkin diagram.

(ii) At $t=1$, it can be shown that the relation comes from a non-split exact sequence obtained by a normalized $R$-matrix, as for the original $T$-system (see [N2, H2]). In fact, it can be checked that the two tensor products associated to the right hand terms correspond to simple modules. Using [C, Theorem 4], the proof is analogous to the one for the $T$-system.

(iii) At $t=1$, this relation can be seen as an extended $T$-systems in [MY].

(iv) For the limit values of $i,j$, the relation involves both the $\alpha$ and the $\beta$-modules and so connect the two families. The specialization at $t=1$ reads :

[TABLE]

[TABLE]

Proof.

By [FR, FM], a $q$-character is determined uniquely by the multiplicity of its dominant monomials. We will use the notation $A_{i,\lambda}$ instead of $A_{i,q^{\lambda}}$ for $i\in I$, $\lambda\in\mathbb{Z}$. First we prove that $\alpha(i,j)^{m}*\alpha(i+1,j+1)^{m}$ has $2m+1$ dominant monomials :

[TABLE]

[TABLE]

where $1\leq r\leq m$. It is clear that these monomials occur. Indeed $M_{1}$ is the product of the highest monomials. For $2\leq r\leq 2m+1$, we decompose

[TABLE]

Now consider $M\neq M_{1}$ a dominant monomial which occurs. We factorize

[TABLE]

where $M^{\prime}$ (resp. $M^{\prime\prime}$) is a monomial of $(M_{[i,j]}^{m})^{-1}\alpha(i,j)^{m}$ (resp. of $(M_{[i+1,j+1]}^{m})^{-1}\alpha(i+1,j+1)^{m}$). As $M$ is dominant, we have $M^{\prime}M^{\prime\prime}\in\mathbb{Z}[A_{k,r}^{-1}]_{k\in I,r\leq j+2\ell}$. Then

[TABLE]

and that there is $R\geq 0$ such that

[TABLE]

The monomial $\tilde{M}=M(X_{j+1,j+1+2m}^{\ell+1-m}X_{j,j+2m}^{\ell+1-m})^{-1}$ is a monomial of $\chi_{q}(L(X_{i,i}^{m}X_{i+1,i+1}^{m}))$ which has a unique dominant monomial. Hence $\tilde{M}Y_{j,j+2m}$ is dominant. So

[TABLE]

is a monomial of $\chi_{q}(L(X_{i,i}^{m}X_{i+1,i+1}^{m}))$. If $\tilde{M}^{\prime}=X_{i,i}^{m}X_{i+1,i+1}^{m}$, then $M=M_{2}$. Otherwise, $\tilde{M}A_{i,i+2m-1}$ is a monomial of $\chi_{q}(L(X_{i,i}^{m}X_{i+1,i+1}^{m}))$. We continue by induction, and so $M$ is one of the $M_{r}$.

This also implies that each $M_{r}$ occurs with multiplicity which is a power of $t$.

Similarly, we get that $\alpha(i,j+1)^{m}*\alpha(i+1,j)^{m}$ has $m+1$ dominant monomials which are the $M_{2r+1}$ for $0\leq r\leq m$. We also get that $\alpha(i,j)^{m+1}*\alpha(i+1,j+1)^{m-1}$ has $m$ dominant monomials which are the $M_{2r}$ for $1\leq r\leq m$.

To conclude, we have to check that the powers of $t$ match : this can be done using positivity in the quantum Grothendieck ring as in [HL2, Section 5.10] or directly as in [HO, section 9]. ∎

5. Proof of periodicity

In this section we finish the proof of the quantum periodicity.

It suffices to identify the $T_{a,b}(t)$ with variables satisfying the $T$-system, the half-periodicity and such that the variables corresponding to the $X_{a,b}$ are algebraically independent. We will identify the $T_{a,b}(t)$ with certain $q,t$-characters of minimal affinizations, that is elements of the quantum torus $\mathcal{Y}_{t}$.

For $0\leq k\leq n+1$, $0\leq m\leq\ell+1$ and $u\in\mathbb{Z}$ so that $k+m+u\in 2\mathbb{Z}$, we set :

[TABLE]

This defines $T_{k,m}(u)$ for $0\leq u-m-k+2\leq 2n+2\ell+4$, and we extend the definition for any $u$ by $2(n+\ell+2)$-periodicity.

Remark 5.1**.**

(i) The formulas in all cases are compatible thanks to relations (5).

(ii) Identifying the class $F_{r}$ defined in (6) with $\mathcal{F}_{r}$, we recover boundary conditions of Section 2.

The $X_{k,m}$ quasi-commute, with the same rules as in Section 2. The relations (4) and Theorem 4.1 imply that the $T_{a,b}(u)$ satisfy the quantum $T$-system for a distinguished choice of the powers of $t$ (let us call it the distinguished powers).

The $(X_{k,m})_{(k,m)\in I\times(J\cup\{0\})}$ form a family of algebraically independent variables. We may argue as in [HL4]. Let us explain this point for completeness : all the representations we consider belong to the monoidal category $\mathcal{C}_{\ell}^{o}$ of representations whose classes belong to the subring of the Grothendieck ring $K_{0}(\mathcal{C})$ generated by the classes of fundamental representations $[L(Y_{k,k+2m})]$ for $(k,m)\in I\times(J\cup\{0\})$. Then there is an injective ring morphism

[TABLE]

called truncated $q$-character morphism [HL3] : it is defined so that for $L(m)$ a simple module in $\mathcal{C}_{\ell}^{o}$, $\chi_{q}^{T}(L(m))$ is obtained from $\chi_{q}(L(m))$ by removing the monomials $m^{\prime}$ so that in $m^{\prime}m^{-1}$ contains a factor of the form $A_{k,k+2\ell+1}^{-1}$, $k\in I$. Now by [H2], the $\chi_{q}^{T}(X_{k,m})=X_{k,k+2m}^{\ell+1-m}$ are just monomials which are clearly algebraically independent.

As by construction we have $T_{a,b}(u)=T_{n+1-a,\ell+1-b}(u+n+\ell+2)$, we get the result for the quantum $T$-system with the distinguished powers of $t$.

To conclude it suffices to check that the powers of $t$ correspond automatically to the distinguished choice. We consider a solution and we prove by induction on $u\geq a+b-2$ that the $T_{a,b}(u)$ correspond to the $q,t$-characters and that the powers of $t$ are given by the distinguished choice. As discussed above, the $X_{a,b}$ are algebraically independent so we can identify the $T_{a,b}(u)$ for $u=a+b-2$, with the corresponding $q,t$-characters. In general, we have a relation

[TABLE]

for some $\alpha,\beta\in\mathbb{Z}/2$. For $u\leq U$, we have $T_{a,b}(u)=M_{a,b}(u)\chi_{a,b}(u)$ where $M_{a,b}(u)$ is a monomial in the quantum torus and $\chi_{a,b}(u)$ is a polynomial in the $A_{i,c}^{-1}$ with coefficients in $\mathbb{Z}[t^{\pm 1}]$ and with constant term $1$ (see (3)). Then $(\chi_{a,b}(u))^{-1}$ is a formal power series in the $A_{i,c}^{-1}$. Each term of the sum

[TABLE]

is a monomial multiplied by such a formal power series. The highest monomial is

[TABLE]

which only appears in the first term. As $T_{a,b}(U+1)$ is bar invariant, it imposes that $\alpha$ is the power of the distinguished choice. Then one may consider

[TABLE]

The same arguments identifies $\beta$ with the distinguished choice. Hence $T_{a,b}(U+1)$ satisfies the equation as the corresponding $q,t$-character and so is equal to it.

Example 5.2**.**

Let us study the examples 2.3, 2.4, 2.5 above. Let $n=\ell=1$. We get :

[TABLE]

Let $n=1$, $\ell=2$.

[TABLE]

Let $n=2$, $\ell=1$.

[TABLE]