Temperley–Lieb algebra#Representations

Let $R$ be a commutative ring and fix $\backslash delta\; \backslash in\; R$. The Temperley–Lieb algebra $TL\_n(\backslash delta)$ is the $R$-algebra generated by the elements $e\_1,\; e\_2,\; \backslash ldots,\; e\_\{n-1\}$, subject to the Jones relations:

• $e\_i^2\; =\; \backslash delta\; e\_i$ for all $1\; \backslash leq\; i\; \backslash leq\; n-1$
• $e\_i\; e\_\{i+1\}\; e\_i\; =\; e\_i$ for all $1\; \backslash leq\; i\; \backslash leq\; n-2$
• $e\_i\; e\_\{i-1\}\; e\_i\; =\; e\_i$ for all $2\; \backslash leq\; i\; \backslash leq\; n-1$
• $e\_i\; e\_j\; =\; e\_j\; e\_i$ for all $1\; \backslash leq\; i,j\; \backslash leq\; n-1$ such that $|i-j|\; \backslash neq\; 1$

Using these relations, any product of generators $e\_i$ can be brought to Jones' normal form:

:$$

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 $(i\_1,i\_2,\backslash dots,i\_r)$ and $(j\_1,j\_2,\backslash dots,j\_r)$ are two strictly increasing sequences in $\backslash \{1,2,\backslash dots,n-1\backslash \}$. Elements of this type form a basis of the Temperley-Lieb algebra.

The dimensions of Temperley-Lieb algebras are Catalan numbers:

:$$

\dim(TL_n(\delta)) = \frac{(2n)!}{n!(n+1)!}

The Temperley–Lieb algebra $TL\_n(\backslash delta)$ is a subalgebra of the Brauer algebra $\backslash mathfrak\{B\}\_n(\backslash delta)$, and therefore also of the partition algebra $P\_n(\backslash delta)$. The Temperley–Lieb algebra $TL\_n(\backslash delta)$ is semisimple for $\backslash delta\backslash in\backslash mathbb\{C\}-F\_n$ where $F\_n$ is a known, finite set. For a given $n$, all semisimple Temperley-Lieb algebras are isomorphic.

$TL\_n(\backslash delta)$ may be represented diagrammatically as the vector space over noncrossing pairings of $2n$ 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 $e\_i$ is the diagram in which the $i$-th and $(i+1)$-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 $TL\_5(\backslash delta)$ are:

File:Temperley-Lieb (generateurs).svg

From left to right, the unit 1 and the generators $e\_1$, $e\_2$, $e\_3$, $e\_4$.

Multiplication on basis elements can be performed by concatenation: placing two rectangles side by side, and replacing any closed loops by a factor $\backslash delta$, for example $e\_1\; e\_4\; e\_3\; e\_2\backslash times\; e\_2\; e\_4\; e\_3=\backslash delta\backslash ,e\_1\; e\_4\; e\_3\; e\_2\; e\_4\; e\_3$:

50px × 50px = 50px50px = $\backslash delta$ 50px.

The Jones relations can be seen graphically:

50px 50px = $\backslash delta$ 50px

50px 50px 50px = 50px

50px 50px = 50px 50px

The five basis elements of $TL\_3(\backslash delta)$ are the following:

File:Temperley-lieb (horizontal).svg.

From left to right, the unit 1, the generators $e\_2$, $e\_1$, and $e\_1\; e\_2$, $e\_2\; e\_1$.

For $\backslash delta$ such that $TL\_n(\backslash delta)$ is semisimple, a complete set $\backslash \{W\_\backslash ell\backslash \}$ of simple modules is parametrized by integers $0\backslash leq\; \backslash ell\backslash leq\; n$ with $\backslash ell\backslash equiv\; n\backslash bmod\; 2$. The dimension of a simple module is written in terms of binomial coefficients as

:$$

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

A basis of the simple module $W\_\backslash ell$ is the set $M\_\{n,\backslash ell\}$ of monic noncrossing pairings from $n$ points on the left to $\backslash ell$ 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 $\backslash cup\_\{\backslash begin\{array\}\{c\}\; 0\backslash leq\; \backslash ell\backslash leq\; n\; \backslash \backslash \; \backslash ell\backslash equiv\; n\backslash bmod\; 2\backslash end\{array\}\}M\_\{n,\backslash ell\}\backslash times\; M\_\{n,\backslash ell\}$, and the set of diagrams that generate $TL\_n(\backslash delta)$: any such diagram can be cut into two elements of $M\_\{n,\backslash ell\}$ for some $\backslash ell$.

Then $TL\_n(\backslash delta)$ acts on $W\_\backslash ell$ by diagram concatenation from the left. (Concatenation can produce non-monic pairings, which have to be modded out.) The module $W\_\backslash ell$ may be called a standard module or link module.

If $\backslash delta\; =\; q+q^\{-1\}$ with $q$ a root of unity, $TL\_n(\backslash delta)$ may not be semisimple, and $W\_\backslash ell$ may not be irreducible:

:$$

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

If $W\_\backslash ell$ is reducible, then its quotient by its maximal proper submodule is irreducible.

Simple modules of the Brauer algebra $\backslash mathfrak\{B\}\_n(\backslash delta)$ 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:

:$$

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 $c\_\backslash ell^\backslash lambda$ do not depend on $n,\backslash delta$, and are given by

:$$

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

The affine Temperley-Lieb algebra $aTL\_n(\backslash delta)$ is an infinite-dimensional algebra such that $TL\_n(\backslash delta)\backslash subset\; aTL\_n(\backslash delta)$. It is obtained by adding generators $e\_n,\backslash tau,\backslash tau^\{-1\}$ such that

• $\backslash tau\; e\_i\; =\; e\_\{i+1\}\backslash tau$ for all $1\; \backslash leq\; i\; \backslash leq\; n$,
• $e\_1\backslash tau^2\; =\; e\_1e\_2\; \backslash cdots\; e\_\{n-1\}$,
• $\backslash tau\; \backslash tau^\{-1\}=\backslash tau^\{-1\}\backslash tau\; =\; \backslash text\{id\}$.

The indices are supposed to be periodic i.e. $e\_\{n+1\}=e\_1,e\_n=e\_0$, and the Temperley-Lieb relations are supposed to hold for all $1\; \backslash leq\; i\; \backslash leq\; n$. Then $\backslash tau^n$ is central. A finite-dimensional quotient of the algebra $aTL\_n(\backslash delta)$, sometimes called the unoriented Jones-Temperley-Lieb algebra, is obtained by

assuming $\backslash tau^n=\backslash text\{id\}$, and replacing non-contractible lines with the same factor $\backslash delta$ as contractible lines (for example, in the case $n=4$, this implies $e\_1e\_3e\_2e\_4e\_1e\_3\; =\; \backslash delta^2\; e\_1e\_3$).

The diagram algebra for $aTL\_n(\backslash delta)$ is deduced from the diagram algebra for $TL\_n(\backslash delta)$ by turning rectangles into cylinders. The algebra $aTL\_n(\backslash delta)$ is infinite-dimensional because lines can wind around the cylinder. If $n$ is even, there can even exist closed winding lines, which are non-contractible.

The Temperley-Lieb algebra is a quotient of the corresponding affine Temperley-Lieb algebra.

The cell module $W\_\{\backslash ell,z\}$ of $aTL\_n(\backslash delta)$ is generated by the set of monic pairings from $n$ points to $\backslash ell$ points, just like the module $W\_\{\backslash ell\}$ of $TL\_n(\backslash delta)$. However, the pairings are now on a cylinder, and the right-multiplication with $\backslash tau$ is identified with $z\backslash cdot\backslash text\{id\}$ for some $z\backslash in\backslash mathbb\{C\}^*$. If $\backslash ell=0$, there is no right-multiplication by $\backslash tau$, and it is the addition of a non-contractible loop on the right which is identified with $z+z^\{-1\}$. Cell modules are finite-dimensional, with

:$$

\dim(W_{\ell,z}) = \binom{n}{\frac{n-\ell}{2}}

The cell module $W\_\{\backslash ell,z\}$ is irreducible for all $z\backslash in\backslash mathbb\{C\}^*-R(\backslash delta)$, where the set $R(\backslash delta)$ is countable. For $z\backslash in\; R(\backslash delta)$, $W\_\{\backslash ell,z\}$ has an irreducible quotient. The irreducible cell modules and quotients thereof form a complete set of irreducible modules of $aTL\_n(\backslash delta)$. Cell modules of the unoriented Jones-Temperley-Lieb algebra must obey $z^\backslash ell=1$ if $\backslash ell\backslash neq\; 0$, and $z+z^\{-1\}\; =\; \backslash delta$ if $\backslash ell=0$.

Consider an interaction-round-a-face model e.g. a square lattice model and let $n$ be the number of sites on the lattice. Following Temperley and Lieb{{cite journal|first1=Neville|last1=Temperley|author1-link=Harold Neville Vazeille Temperley|author2-link=Elliott H. Lieb|first2=Elliott|last2= Lieb|title=Relations between the 'percolation' and 'colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the 'percolation' problem| journal= Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=322|issue=1549 |year=1971|pages= 251–280|mr=0498284 |doi=10.1098/rspa.1971.0067|jstor=77727|bibcode=1971RSPSA.322..251T|s2cid=122770421}} we define the Temperley–Lieb Hamiltonian (the TL Hamiltonian) as

$\backslash mathcal\{H\}\; =\; \backslash sum\_\{j=1\}^\{n-1\}\; (\backslash delta\; -\; e\_j)$

In what follows we consider the special case $\backslash delta=1$.

We will firstly consider the case $n\; =\; 3$. The TL Hamiltonian is $\backslash mathcal\{H\}\; =\; 2\; -\; e\_1\; -\; e\_2$, namely

$\backslash mathcal\{H\}$ = 2 50px - 50px - 50px.

We have two possible states,

40px and 40px.

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

$\backslash mathcal\{H\}$ 40px = 2 50px40px - 50px40px - 50px40px = 40px - 40px,

and

$\backslash mathcal\{H\}$ 40px = 2 50px40px - 50px40px - 50px40px = - 40px + 40px.

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

$\backslash mathcal\{H\}\; =\; \backslash left(\backslash begin\{array\}\{rr\}$

1 & -1\\

-1 & 1

\end{array}\right)

The eigenvector of $\backslash mathcal\{H\}$ with the lowest eigenvalue is known as the ground state. In this case, the lowest eigenvalue $\backslash lambda\_0$ for $\backslash mathcal\{H\}$ is $\backslash lambda\_0\; =\; 0$. The corresponding eigenvector is $\backslash psi\_0\; =\; (1,\; 1)$. As we vary the number of sites $n$ we find the following table{{cite journal|last1=Batchelor|first1= Murray|author1-link=Murray Batchelor|last2= de Gier|first2= Jan |last3= Nienhuis|first3= Bernard|year=2001|title= The quantum symmetric $XXZ$ chain at $\backslash Delta\; =\; -1/2$, alternating-sign matrices and plane partitions|journal= Journal of Physics A |volume=34| issue=19| pages= L265–L270|doi=10.1088/0305-4470/34/19/101| mr=1836155|arxiv= cond-mat/0101385|s2cid= 118048447}}

class="wikitable"
$n$ ! $\backslash psi\_0$ ! $n$ ! $\backslash psi\_0$
2 | (1) |3 |(1, 1)
4 |(2, 1) |5 |$(3\_3,\; 1\_2)$
6 | $(11,\; 5\_2,4,\; 1)$ |7 |$(26\_4,\; 10\_2,\; 9\_2,\; 8\_2,\; 5\_2,\; 1\_2)$
8 |$(170,\; 75\_2,\; 71,\; 56\_2,\; 50,\; 30,\; 14\_4,\; 6,\; 1)$ |9 |$(646,\; \backslash ldots)$
$\backslash vdots$ |$\backslash vdots$ |$\backslash vdots$ |$\backslash vdots$

where we have used the notation $m\_j\; =\; (m,\; \backslash ldots,\; m)$ $j$-times e.g., $5\_2\; =\; (5,\; 5)$.

An interesting observation is that the largest components of the ground state of $\backslash mathcal\{H\}$ have a combinatorial enumeration as we vary the number of sites,{{cite journal|last= de Gier|first= Jan|year= 2005|title= Loops, matchings and alternating-sign matrices|journal= Discrete Mathematics | volume =298|issue= 1–3|pages =365–388|doi=10.1016/j.disc.2003.11.060|mr=2163456|arxiv= math/0211285|s2cid= 2129159}} as was first observed by Murray Batchelor, Jan de Gier and Bernard Nienhuis. Using the resources of the on-line encyclopedia of integer sequences, Batchelor et al. found, for an even numbers of sites

$$

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

$$

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 $n$ even, this {{OEIS|id=A051255}} corresponds to cyclically symmetric transpose complement plane partitions and for $n$ odd, {{OEIS|id=A005156}}, these correspond to alternating sign matrices symmetric about the vertical axis.

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