T-system

The \(T\)-system or octahedron relation is the following recursion relation for variables \(T_{i,j,k}>0\), \(i,j,k\in {\mathbb Z}\) \[\label{Tsys} T_{i,j,k+1}\, T_{i,j,k-1}=T_{i+1,j,k}\, T_{i-1,j,k}+T_{i,j+1,k}\, T_{i,j-1,k} .\] It may be interpreted as a discrete time \(k\) evolution for the variable \(T\), expressing its value at the time \(k+1\) vertex of an octahedron in terms of the values at the 4 vertices at time \(k\) and at a single vertex at time \(k-1\). Note that this is a coefficient free \(T\)-system. The solution of \(T_{i,j,k}\), i.e. algebraic expression of initial data, is unique once we fix admissible initial data along any given “stepped surface" \({\mathbf k}\) made of the vertices \((i_0,j_0,k_{i_0,j_0})\), \(i_0,j_0\in {\mathbb Z}\), where the height function \(k_{i,j}:{\mathbb Z}^2\to{\mathbb Z}\) obeys \(|k_{i+1,j}-k_{i,j}|=|k_{i,j+1}-k_{i,j}|=1\) for all \(i,j\in {\mathbb Z}\). The initial data assignments read \[\label{init} T_{i_0,j_0,k_{i_0,j_0}}=t_{i_0,j_0}\, , \qquad (i_0,j_0\in {\mathbb Z})\] for some fixed initial variables \(t_{i_0,j_0}>0\), \(i_0,j_0\in {\mathbb Z}\).

In , Di Francesco defined the general inhomogenous coefficient \(T\)-system to be: \[\label{inhomogenous Tsys} T_{i,j,k+1}\, T_{i,j,k-1}=\lambda_iT_{i+1,j,k}\, T_{i-1,j,k}+ \mu_jT_{i,j+1,k}\, T_{i,j-1,k} .\] where the initial data can be defined similarly. The solution of this recurrence relation is found explicitly for the case of uniform flat initial data \(T_{i,j,0}=T_{i,j,1}=1\) in .

Q-system

We consider the family of variables \[\{Q_{{\alpha},k}| ~ {\alpha}\in I_r, k\in {\mathbb Z}\}.\] Let \({\mathfrak{g}}\) be a simple Lie algebra with Cartan matrix \(C\). We denote the simple roots \({\alpha}\) by the corresponding integers in \(I_r=\{1,...,r\}\). The \(Q\)-system associated with \({\mathfrak{g}}\) is a recursion relation of the form \[\label{Qsystem} Q_{{\alpha},k+1} Q_{{\alpha},k-1} = Q_{{\alpha},k}^2 - \prod_{\beta\sim{\alpha}} \mathcal T^{({\alpha},\beta)}_k, ~{\alpha}\in I_r, ~k\in {\mathbb Z},\]

where \(\alpha\sim\beta\) means \(\alpha\) is connected to \(\beta\) in the Dynkin diagram of \(\mathfrak g\), and

\[ \mathcal{T}^{(\alpha,\beta)}_k = \prod_{i=0}^{|C_{\alpha,\beta}|-1} Q_{\beta,\lfloor\frac{t_\beta k+i}{t_\alpha} \rfloor}, \]

where \(\lfloor a \rfloor\) is the integer part of \(a\). Here \(t_\alpha\) are the integers symmetrizing the Cartan matrix. Namely, \(t_r=2\) for \(B_r\), \(t_\alpha=2\) (\(\alpha<r\)) for \(C_r\), \(t_3=t_4=2\) for \(F_4\) and \(t_2=3\) for \(G_2\), and \(t_\alpha=1\) otherwise. The recursion relation explores all other lattice points \((\alpha,k)\) associated to \(Q_{\alpha,k}\) once \(2r\) initial data points are fixed. One common initial data is \(\{Q_{\alpha,0}, Q_{\alpha,1}~|~\alpha\in I_r\}\).

Then,

\[ Q_{\alpha,k+1} = \frac{Q_{\alpha,k}^2 - \prod_{\beta\sim \alpha} \mathcal {T}_k^{\alpha,\beta}}{Q_{\alpha,k-1}}, \ k\geq 1, \]

and

\[ Q_{\alpha,k-1}=\frac{Q_{\alpha,k}^2 - \prod_{\beta\sim \alpha} \mathcal{T}_k^{\alpha,\beta}}{Q_{\alpha,k+1}}, \ k\leq0. \]

References

1. R. J. Baxter, Exactly Solved Models in Statistical Mechanics , 1982

2. Philippe Di Francesco, Q-system cluster algebras, paths and total positivity, Symmetry, Integrability and Geometry: Methods and Applications, Feb 2010

3. Philippe Di Francesco, T-systems, networks and dimers, Communications in Mathematical Physics, 331(3):1237{1270, May 2014.

4. Philippe Di Francesco and Rinat Kedem, Q-systems, Heaps, Paths and Cluster Positivity, Communications in Mathematical Physics, 293(3):727{802, Nov 2009

5. Atsuo Kuniba, Tomoki Nakanishi, and Junji Suzuki, T-systems and Y-systems in Integrable Systems. , Journal of Physics A: Mathematical and Theoretical, 44(10):103001, Feb 2011.