Temperley–Lieb algebra

Let ${\displaystyle R}$ be a commutative ring and fix ${\displaystyle \delta \in R}$. The Temperley–Lieb algebra ${\displaystyle TL_{n}(\delta )}$ is the ${\displaystyle R}$-algebra generated by the elements ${\displaystyle e_{1},e_{2},\ldots ,e_{n-1}}$, subject to the Jones relations:

• ${\displaystyle e_{i}^{2}=\delta e_{i}}$ for all ${\displaystyle 1\leq i\leq n-1}$
• ${\displaystyle e_{i}e_{i+1}e_{i}=e_{i}}$ for all ${\displaystyle 1\leq i\leq n-2}$
• ${\displaystyle e_{i}e_{i-1}e_{i}=e_{i}}$ for all ${\displaystyle 2\leq i\leq n-1}$
• ${\displaystyle e_{i}e_{j}=e_{j}e_{i}}$ for all ${\displaystyle 1\leq i,j\leq n-1}$ such that ${\displaystyle |i-j|\neq 1}$

Using these relations, any product of generators ${\displaystyle e_{i}}$ can be brought to Jones' normal form:

${\displaystyle E={\big (}e_{i_{1}}e_{i_{1}-1}\cdots e_{j_{1}}{\big )}{\big (}e_{i_{2}}e_{i_{2}-1}\cdots e_{j_{2}}{\big )}\cdots {\big (}e_{i_{r}}e_{i_{r}-1}\cdots e_{j_{r}}{\big )}}$

where ${\displaystyle (i_{1},i_{2},\dots ,i_{r})}$ and ${\displaystyle (j_{1},j_{2},\dots ,j_{r})}$ are two strictly increasing sequences in ${\displaystyle \{1,2,\dots ,n-1\}}$. Elements of this type form a basis of the Temperley-Lieb algebra.[1]

The dimensions of Temperley-Lieb algebras are Catalan numbers:[2]

${\displaystyle \dim(TL_{n}(\delta ))={\frac {(2n)!}{n!(n+1)!}}}$

The Temperley–Lieb algebra ${\displaystyle TL_{n}(\delta )}$ is a subalgebra of the Brauer algebra ${\displaystyle {\mathfrak {B}}_{n}(\delta )}$,[3] and therefore also of the partition algebra ${\displaystyle P_{n}(\delta )}$. The Temperley–Lieb algebra ${\displaystyle TL_{n}(\delta )}$ is semisimple for ${\displaystyle \delta \in \mathbb {C} -F_{n}}$ where ${\displaystyle F_{n}}$ is a known, finite set.[4] For a given ${\displaystyle n}$, all semisimple Temperley-Lieb algebras are isomorphic.[3]

${\displaystyle TL_{n}(\delta )}$ may be represented diagrammatically as the vector space over noncrossing pairings of ${\displaystyle 2n}$ points on two opposite sides of a rectangle with n points on each of the two sides.

The identity element is the diagram in which each point is connected to the one directly across the rectangle from it. The generator ${\displaystyle e_{i}}$ is the diagram in which the ${\displaystyle i}$-th and ${\displaystyle (i+1)}$-th point on the left side are connected to each other, similarly the two points opposite to these on the right side, and all other points are connected to the point directly across the rectangle.

The generators of ${\displaystyle TL_{5}(\delta )}$ are:

From left to right, the unit 1 and the generators ${\displaystyle e_{1}}$, ${\displaystyle e_{2}}$, ${\displaystyle e_{3}}$, ${\displaystyle e_{4}}$.

Multiplication on basis elements can be performed by concatenation: placing two rectangles side by side, and replacing any closed loops by a factor ${\displaystyle \delta }$, for example ${\displaystyle e_{1}e_{4}e_{3}e_{2}\times e_{2}e_{4}e_{3}=\delta \,e_{1}e_{4}e_{3}e_{2}e_{4}e_{3}}$:

× = = ${\displaystyle \delta }$ .

The Jones relations can be seen graphically:

= ${\displaystyle \delta }$

=

=

The five basis elements of ${\displaystyle TL_{3}(\delta )}$ are the following:

.

From left to right, the unit 1, the generators ${\displaystyle e_{2}}$, ${\displaystyle e_{1}}$, and ${\displaystyle e_{1}e_{2}}$, ${\displaystyle e_{2}e_{1}}$.

For ${\displaystyle \delta }$ such that ${\displaystyle TL_{n}(\delta )}$ is semisimple, a complete set ${\displaystyle \{W_{\ell }\}}$ of simple modules is parametrized by integers ${\displaystyle 0\leq \ell \leq n}$ with ${\displaystyle \ell \equiv n{\bmod {2}}}$. The dimension of a simple module is written in terms of binomial coefficients as[4]

${\displaystyle \dim(W_{\ell })={\binom {n}{\frac {n-\ell }{2}}}-{\binom {n}{{\frac {n-\ell }{2}}-1}}}$

A basis of the simple module ${\displaystyle W_{\ell }}$ is the set ${\displaystyle M_{n,\ell }}$ of monic noncrossing pairings from ${\displaystyle n}$ points on the left to ${\displaystyle \ell }$ points on the right. (Monic means that each point on the right is connected to a point on the left.) There is a natural bijection between ${\displaystyle \cup _{\begin{array}{c}0\leq \ell \leq n\\\ell \equiv n{\bmod {2}}\end{array}}M_{n,\ell }\times M_{n,\ell }}$, and the set of diagrams that generate ${\displaystyle TL_{n}(\delta )}$: any such diagram can be cut into two elements of ${\displaystyle M_{n,\ell }}$ for some ${\displaystyle \ell }$.

Then ${\displaystyle TL_{n}(\delta )}$ acts on ${\displaystyle W_{\ell }}$ by diagram concatenation from the left.[3] (Concatenation can produce non-monic pairings, which have to be modded out.) The module ${\displaystyle W_{\ell }}$ may be called a standard module or link module.[1]

If ${\displaystyle \delta =q+q^{-1}}$ with ${\displaystyle q}$ a root of unity, ${\displaystyle TL_{n}(\delta )}$ may not be semisimple, and ${\displaystyle W_{\ell }}$ may not be irreducible:

${\displaystyle W_{\ell }{\text{ reducible }}\iff \exists j\in \{1,2,\dots ,\ell \},\ q^{2n-4\ell +2+2j}=1}$

If ${\displaystyle W_{\ell }}$ is reducible, then its quotient by its maximal proper submodule is irreducible.[1]

Simple modules of the Brauer algebra ${\displaystyle {\mathfrak {B}}_{n}(\delta )}$ can be decomposed into simple modules of the Temperley-Lieb algebra. The decomposition is called a branching rule, and it is a direct sum with positive integer coefficients:

${\displaystyle W_{\lambda }\left({\mathfrak {B}}_{n}(\delta )\right)=\bigoplus _{\begin{array}{c}|\lambda |\leq \ell \leq n\\\ell \equiv |\lambda |{\bmod {2}}\end{array}}c_{\ell }^{\lambda }W_{\ell }\left(TL_{n}(\delta )\right)}$

The coefficients ${\displaystyle c_{\ell }^{\lambda }}$ do not depend on ${\displaystyle n,\delta }$, and are given by[4]

${\displaystyle c_{\ell }^{\lambda }=f^{\lambda }\sum _{r=0}^{\frac {\ell -|\lambda |}{2}}(-1)^{r}{\binom {\ell -r}{r}}{\binom {\ell -2r}{\ell -|\lambda |-2r}}(\ell -|\lambda |-2r)!!}$

where ${\displaystyle f^{\lambda }}$ is the number of standard Young tableaux of shape ${\displaystyle \lambda }$, given by the hook length formula.

Consider an interaction-round-a-face model e.g. a square lattice model and let ${\displaystyle n}$ be the number of sites on the lattice. Following Temperley and Lieb[7] we define the Temperley–Lieb Hamiltonian (the TL Hamiltonian) as

${\displaystyle {\mathcal {H}}=\sum _{j=1}^{n-1}(\delta -e_{j})}$

In what follows we consider the special case ${\displaystyle \delta =1}$.

We will firstly consider the case ${\displaystyle n=3}$. The TL Hamiltonian is ${\displaystyle {\mathcal {H}}=2-e_{1}-e_{2}}$, namely

${\displaystyle {\mathcal {H}}}$ = 2 - - .

We have two possible states,

and .

In acting by ${\displaystyle {\mathcal {H}}}$ on these states, we find

${\displaystyle {\mathcal {H}}}$ = 2 - - = - ,

and

${\displaystyle {\mathcal {H}}}$ = 2 - - = - + .

Writing ${\displaystyle {\mathcal {H}}}$ as a matrix in the basis of possible states we have,

${\displaystyle {\mathcal {H}}=\left({\begin{array}{rr}1&-1\\-1&1\end{array}}\right)}$

The eigenvector of ${\displaystyle {\mathcal {H}}}$ with the lowest eigenvalue is known as the ground state. In this case, the lowest eigenvalue ${\displaystyle \lambda _{0}}$ for ${\displaystyle {\mathcal {H}}}$ is ${\displaystyle \lambda _{0}=0}$. The corresponding eigenvector is ${\displaystyle \psi _{0}=(1,1)}$. As we vary the number of sites ${\displaystyle n}$ we find the following table[8]

${\displaystyle n}$	${\displaystyle \psi _{0}}$	${\displaystyle n}$	${\displaystyle \psi _{0}}$
2	(1)	3	(1, 1)
4	(2, 1)	5	${\displaystyle (3_{3},1_{2})}$
6	${\displaystyle (11,5_{2},4,1)}$	7	${\displaystyle (26_{4},10_{2},9_{2},8_{2},5_{2},1_{2})}$
8	${\displaystyle (170,75_{2},71,56_{2},50,30,14_{4},6,1)}$	9	${\displaystyle (646,\ldots )}$
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$

where we have used the notation ${\displaystyle m_{j}=(m,\ldots ,m)}$ ${\displaystyle j}$-times e.g., ${\displaystyle 5_{2}=(5,5)}$.

An interesting observation is that the largest components of the ground state of ${\displaystyle {\mathcal {H}}}$ have a combinatorial enumeration as we vary the number of sites,[9] as was first observed by Murray Batchelor, Jan de Gier and Bernard Nienhuis.[8] Using the resources of the on-line encyclopedia of integer sequences, Batchelor et al. found, for an even numbers of sites

${\displaystyle 1,2,11,170,\ldots =\prod _{j=0}^{\frac {n-2}{2}}\left(3j+1\right){\frac {(2j)!(6j)!}{(4j)!(4j+1)!}}\qquad (n=2,4,6,\dots )}$

and for an odd numbers of sites

${\displaystyle 1,3,26,646,\ldots =\prod _{j=0}^{\frac {n-3}{2}}(3j+2){\frac {(2j+1)!(6j+3)!}{(4j+2)!(4j+3)!}}\qquad (n=3,5,7,\dots )}$

Surprisingly, these sequences corresponded to well known combinatorial objects. For ${\displaystyle n}$ even, this (sequence A051255 in the OEIS) corresponds to cyclically symmetric transpose complement plane partitions and for ${\displaystyle n}$ odd, (sequence A005156 in the OEIS), these correspond to alternating sign matrices symmetric about the vertical axis.