Commuting families in Hecke and Temperley-Lieb Algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 29 January 2014

Affine braid groups, Hecke and Temperley-Lieb algebras

The Affine Braid Group ${\stackrel{\sim }{ℬ}}_k$

The affine braid group is the group ${\stackrel{\sim }{ℬ}}_k$ of affine braids with $k$ strands (braids with a flagpole). The group ${\stackrel{\sim }{ℬ}}_k$ is presented by generators $T_1,T_2,\dots ,T_{k-1}$ and $X^{{\epsilon }_1},$ $$\begin{array}{cc}{T}_i=\begin{array}{c} i i+1 \end{array}\text{and}{X}^{{\epsilon }_1}=\begin{array}{c} \end{array}\text{.}& \text{(2.1)}\end{array}$$ with relations $$\begin{array}{cc}\begin{array}{ccc}{X}^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1}{T}_1& =& T_1{X}^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1}\\ X^{{\epsilon }_1}{T}_i& =& T_i{X}^{{\epsilon }_1},& \text{for}\hspace{0.17em}i>1,\\ T_i{T}_j& =& T_j{T}_i,& \text{if}\hspace{0.17em}|i-j|>1,\\ T_i{T}_{i+1}{T}_i& =& T_{i+1}{T}_i{T}_{i+1},& \text{if}\hspace{0.17em}1\le i\le k-2\text{.}\end{array}& \text{(2.2)}\end{array}$$

For $1\le i\le k$ define $$\begin{array}{cc}{X}^{{\epsilon }_i}=T_{i-1}{T}_{i-2}\cdots T_2{T}_1{X}^{{\epsilon }_1}{T}_1{T}_2\cdots T_{i-2}{T}_{i-1}=\begin{array}{c} i \end{array}\text{.}& \text{(2.3)}\end{array}$$ By drawing pictures of the corresponding affine braids it is easy to check that $$\begin{array}{cc}{X}^{{\epsilon }_i}{X}^{{\epsilon }_j}=X^{{\epsilon }_j}{X}^{{\epsilon }_i},\text{for}\hspace{0.17em}1\le i,j\le k,& \text{(2.4)}\end{array}$$ so that the elements $X^{{\epsilon }_1},\dots ,X^{{\epsilon }_k}$ are a commuting family for ${\stackrel{\sim }{ℬ}}_k\text{.}$ Thus $X=⟨X^{{\epsilon }_i}\hspace{0.17em}|\hspace{0.17em}1\le i\le k⟩$ is an abelian subgroup of ${\stackrel{\sim }{ℬ}}_k\text{.}$ The free abelian group generated by ${\epsilon }_1,\dots ,{\epsilon }_k$ is ${\mathbb{Z}}^k$ and $$\begin{array}{cc}X=\left\{X^{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in {\mathbb{Z}}^k\}\text{where}{X}^{\lambda }={\left(X^{{\epsilon }_1}\right)}^{{\lambda }_1}{\left(X^{{\epsilon }_2}\right)}^{{\lambda }_2}\cdots {\left(X^{{\epsilon }_k}\right)}^{{\lambda }_k},& \text{(2.5)}\end{array}$$ for $\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_k{\epsilon }_k$ in ${\mathbb{Z}}^k\text{.}$

An alternate presentation of ${\stackrel{\sim }{ℬ}}_k$ can be given using the generators $T_0,T_1,\dots ,T_{k-1}$ and $\tau$ where $$\tau =X^{-{\epsilon }_1}{T}_1^{-1}\cdots T_{k-1}^{-1}=\begin{array}{c} \end{array}\text{and}{T}_0={\tau }^{-1}{T}_1\tau =\begin{array}{c} \end{array}\text{.}$$

The affine braid group ${\stackrel{\sim }{ℬ}}_k$ is the affine braid group of type ${GL}_k\text{.}$ The affine braid groups of type ${SL}_k$ and ${PGL}_k$ are the subgroup $${\stackrel{\sim }{ℬ}}_Q=⟨T_0,T_1,\dots ,T_{k-1}⟩\text{and the quotient}{\stackrel{\sim }{ℬ}}_P=\frac{{\stackrel{\sim }{ℬ}}_k}{⟨{\tau }^k⟩},\text{respectively.}$$ Then ${\tau }^k=X^{-{\epsilon }_1}{X}^{-{\epsilon }_2}\cdots X^{-{\epsilon }_k}$ is a central element of ${\stackrel{\sim }{ℬ}}_k,$ $\tau T_i{\tau }^{-1}=T_{i+1}$ (where the indices are taken mod $n\text{),}$ and $\tau X^{{\epsilon }_i}{\tau }^{-1}=X^{e_{i+1}}$ and $$Z\left({\stackrel{\sim }{ℬ}}_k\right)=⟨{\tau }^k⟩,{\stackrel{\sim }{ℬ}}_k=⟨\tau ⟩⋉{\stackrel{\sim }{ℬ}}_Q,{\stackrel{\sim }{ℬ}}_P=⟨\bar{\tau }⟩⋉{\stackrel{\sim }{ℬ}}_Q\text{.}$$ In ${\stackrel{\sim }{ℬ}}_k$ we have $⟨\tau ⟩\cong \mathbb{Z},$ and $\bar{\tau }\in {\stackrel{\sim }{ℬ}}_P$ is defined to be the image of $\tau$ under the homomorphism $\mathbb{Z}\to \mathbb{Z}/k\mathbb{Z}$ so that $⟨\bar{\tau }⟩\cong \mathbb{Z}/k\mathbb{Z}\text{.}$

The Temperley-Lieb algebra $T{L}_k\left(n\right)$

A Temperley-Lieb diagram on $k$ dots is a graph with $k$ dots in the top row, $k$ dots in the bottom row, and $k$ edges pairing the dots such that the graph is planar (without edge crossings). For example, $$d_1=\begin{array}{c} \end{array}\text{and}{d}_2=\begin{array}{c} \end{array}\text{.}$$ are Temperley-Lieb diagrams on 7 dots. The composition $d_1\circ d_2$ of two diagrams $d_1,d_2\in T_k$ is the diagram obtained by placing $d_1$ above $d_2$ and identifying the bottom vertices of $d_1$ with the top dots of $d_2$ removing any connected components that live entirely in the middle row. If $T_k$ is the set of Temperley-Lieb diagrams on $k$ dots then the Temperley-Lieb algebra ${TL}_k\left(n\right)$ is the associative algebra with basis $T_k,$ $${TL}_k\left(n\right)=\text{span}\{d\in T_k\}\text{with multiplication defined by}{d}_1{d}_2=n^{\ell }(d_1\circ d_2),$$ where $\ell$ is the number of blocks removed from the middle row when constructing the composition $d_1\circ d_2$ and $n$ is a fixed element of the base ring. For example, using the diagrams $d_1$ and $d_2$ above, we have $$d_1{d}_2=\begin{array}{c} \end{array}=n\begin{array}{c} \end{array}\text{.}$$ The algebra ${TL}_k\left(n\right)$ is presented by generators $$\begin{array}{cc}{e}_i=\begin{array}{c} i i+1 ⋯ ⋯ ⋯ ⋯ \end{array},1\le i\le k-1,& \text{(2.6)}\\ \text{and relations}{e}_i^2=n{e}_i,e_i{e}_{i\pm 1}{e}_i=e_i,\text{and}{e}_i{e}_j=e_j{e}_i,\text{if}\hspace{0.17em}|i-j|>1& \text{(2.7)}\end{array}$$ (see [GHJ1989, Lemma 2.8.4]).

In the definition of the Temperley-Lieb algebra, and for other algebras defined in this paper, the base ring could be any one of several useful rings (e.g. $ℂ,$ $ℂ\left(q\right),$ $ℂ\left[\left[h\right]\right],$ $\mathbb{Z}[q,q^{-1}],$ $\mathbb{Z}\left[n\right]$ or localizations of these at special primes). The most useful approach is to view the results of computations as valid over any ring $R$ with $n,q,h\in R$ such that the formulas make sense.

The Surjection ${\stackrel{\sim }{H}}_k\left(q\right)\mapsto {TL}_k\left(n\right)$

The affine Hecke algebra ${\stackrel{\sim }{H}}_k$ is the quotient of the group algebra of the affine braid group $ℂ{\stackrel{\sim }{ℬ}}_k$ by the relations $$\begin{array}{cc}{T}_i^2=(q-q^{-1})T_i+1,\text{so that}ℂ{\stackrel{\sim }{ℬ}}_k⟶{\stackrel{\sim }{H}}_k& \text{(2.8)}\end{array}$$ is a surjective homomorphism $\text{(}q$ is a fixed element of the base ring). The affine Hecke algebra ${\stackrel{\sim }{H}}_k$ is the affine Hecke algebra of type ${GL}_k\text{.}$ The affine Hecke algebras of types ${SL}_k$ and ${PGL}_k$ are, respectively, the quotients ${\stackrel{\sim }{H}}_Q$ and ${\stackrel{\sim }{H}}_P$ of the group algebras of ${\stackrel{\sim }{ℬ}}_Q$ and ${\stackrel{\sim }{ℬ}}_P$ (see Remark 2.2) by the relations (2.8).

The Iwahori-Hecke algebra is the subalgebra $H_k$ of ${\stackrel{\sim }{H}}_k$ generated by $T_1,\dots ,T_{k-1}\text{.}$ In the Iwahori-Hecke algebra $H_k,$ define $$\begin{array}{cc}{e}_i=q-T_i,\text{for}\hspace{0.17em}i=1,2,\dots ,k-1\text{.}& \text{(2.9)}\end{array}$$ Direct calculations show that $e_i^2=(q+q^{-1})e_i$ and that $e_1{e}_2{e}_1=e_1$ and $e_2{e}_1{e}_2=e_2$ if and only if $$\begin{array}{cc}$ q^3$-q^2{T}_1-q^2{T}_2+q{T}_1{T}_2+q{T}_2{T}_1-T_1{T}_2{T}_1=0\text{.}& \text{(2.10)}\end{array}$$ Thus, setting $n=\left[2\right]=q+q^{-1},$ there are surjective algebra homomorphisms given by $$\begin{array}{cc}\begin{array}{cccccc}\psi :& {\stackrel{\sim }{H}}_k\left(q\right)& ⟶& H_k\left(q\right)& ⟶& {TL}_k\left(n\right)\\ & X^{{\epsilon }_1}& ⟼& 1& ⟼& 1\\ & T_i& ⟼& T_i& ⟼& q-e_i\text{.}\end{array}& \text{(2.11)}\end{array}$$ The kernel of $\psi$ is generated by the element on the left hand side of equation (2.10). In the notation of Theorem 4.1, the representations of $H_k$ correspond to the case when $\mu =\varnothing \text{.}$ Writing ${\stackrel{\sim }{H}}_k^{\lambda /\varnothing }$ as ${\stackrel{\sim }{H}}_k^{\lambda },$ the element from (2.10) acts as $0$ on the irreducible Iwahori-Hecke algebra modules ${\stackrel{\sim }{H}}_3^{\begin{array}{c} \end{array}}$ and ${\stackrel{\sim }{H}}_3^{\begin{array}{c} \end{array}},$ and (up to a scalar multiple) it is a projection onto ${\stackrel{\sim }{H}}_3^{\begin{array}{c} \end{array}}\text{.}$

There is an alternative surjective homomorphism that instead sends $T_i\mapsto e_i-q^{-1}\text{.}$ This alternative surjection has kernel generated by $$q^{-3}+q^{-2}{T}_1+q^{-2}{T}_2+q^{-1}{T}_1{T}_2+q^{-1}{T}_2{T}_1+T_2{T}_1{T}_2\text{.}$$ This element is $0$ on ${\stackrel{\sim }{H}}_3^{\begin{array}{c} \end{array}}$ and ${\stackrel{\sim }{H}}_3^{\begin{array}{c} \end{array}},$ and (up to a scalar multiple) it is a projection onto ${\stackrel{\sim }{H}}_3^{\begin{array}{c} \end{array}}\text{.}$

A priori, there are two different kinds of integrality for the Temperley-Lieb algebra: coefficients in $\mathbb{Z}\left[n\right]$ or coefficients in $\mathbb{Z}[q,q^{-1}]$ (in terms of the basis of Temperley-Lieb diagrams). The relation between these is as follows. If $$\left[2\right]=q+q^{-1}=n\text{then}q=\frac{1}{2}(n+\sqrt{n^2-4}),q^{-1}=\frac{1}{2}(n-\sqrt{(n^2-4)}),$$ since $q^2-nq+1=0\text{.}$ Then $$\left[k\right]=\frac{q^k-q^{-k}}{q-q^{-1}}=\frac{1}{2^{k-1}}\sum _{m=1}^{(k+1)/2}\left(\genfrac{}{}{0ex}{}{k}{2m-1}\right)n^{k-2m+1}{(n^2-4)}^{m-1}$$ so that $\left[k\right]$ is a polynomial in $n\text{.}$ The polynomials $$n^k={(q+q^{-1})}^k\text{and}\left\{k\right\}=q^k+q^{-k}\text{and}\left[k\right]=\frac{q^k-q^{-k}}{q-q^{-1}},$$ all form bases of the ring $ℂ\left[(q+q^{-1})\right]\text{.}$ The transition matrix $B$ between the $\left[k\right]$ and the $\left\{k\right\}$ is triangular (with 1s on the diagonal) and the transition matrix $C$ between the $n^k$ and the $\left\{k\right\}$ is also triangular (the non zero entries are binomial coefficients). Hence, the transition matrix ${BC}^{-1}$ between $\left[k\right]$ and $n^k$ has integer entries and so $\left[k\right]$ is, in fact, a polynomial in $n$ with integer coefficients.

Affine Temperley-Lieb algebras

The affine Temperley-Lieb algebra $T_k^a$ is the diagram algebra generated by $$e_0=\begin{array}{c} \end{array},e_i=\begin{array}{c} i \end{array}\hspace{0.17em}(1\le i\le k-1),\text{and}\tau =\begin{array}{c} \end{array}\text{.}$$ The generators of $T_k^a$ satisfy $e_i^2=n{e}_i,$ $e_i{e}_{i\pm 1}{e}_i=e_i,$ $\tau e_i{\tau }^{-1}=e_{i+1}$ (where the indices are taken mod $k\text{)}$ and $$\begin{array}{cc}{\tau }^2{e}_{k-1}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=e_1{e}_2\cdots e_{k-1}& \text{(2.12)}\end{array}$$ (see [GLe2004, 4.15(iv)]). In $T_k^a,$ we let $X^{{\epsilon }_1}=T_1^{-1}{T}_2^{-1}\cdots T_{k-1}^{-1}{\tau }^{-1}$ (see Remark 2.1).

Graham and Lehrer [GLe2004, §4.3] define four slightly different affine Temperley-Lieb algebras, the diagram algebra $T_k^a$ and the algebras defined as follows: $$\begin{array}{cc}\text{Type}\hspace{0.17em}{GL}_k\text{:}& {\widehat{TL}}_k^a\hspace{0.17em}\text{is}\hspace{0.17em}{\stackrel{\sim }{H}}_k\hspace{0.17em}\text{with the relation (2.10),}\\ \text{Type}\hspace{0.17em}{SL}_k\text{:}& {TL}_k^a\hspace{0.17em}\text{is}\hspace{0.17em}{\stackrel{\sim }{H}}_Q\hspace{0.17em}\text{with the relation (2.10),}\\ \text{Type}\hspace{0.17em}{PGL}_k\text{:}& {\widetilde{TL}}_k^a\hspace{0.17em}\text{is}\hspace{0.17em}{\tilde{H}}_P\hspace{0.17em}\text{with the relation (2.10).}\end{array}$$ For each invertible element $\alpha$ in the base ring there is a surjective homomorphism $$\begin{array}{cc}\begin{array}{ccccc}{\stackrel{\sim }{H}}_k& ⟶& {\widehat{TL}}_k^a& ⟶& T_k^a\\ \tau & ⟼& \tau & ⟼& \alpha \tau \\ T_i& ⟼& q-e_i& ⟼& q-e_i\end{array}& \text{(2.13)}\end{array}$$ and every irreducible representation of ${\widehat{TL}}_k^a$ factors through one of these homomorphisms (see [GLe2004, Prop. 4.14(v)]). In Proposition 3.2 we shall see that these homomorphisms arise naturally in the Schur-Weyl duality setting.

A commuting family in the affine Temperley-Lieb algebra

View the elements $X^{-{\epsilon }_i}$ in the affine Temperley-Lieb algebra ${\widehat{TL}}_k^a$ via the surjective algebra homomorphism of (2.13). Define $$\begin{array}{cc}(q-q^{-1})m_1=q^{-1}{X}^{-{\epsilon }_1}\text{and}(q-q^{-1})m_i=q^{i-2}(X^{-{\epsilon }_i}-q^{-2}{X}^{-{\epsilon }_{i-1}}),& \text{(2.14)}\end{array}$$ for $i=2,3,\dots ,k\text{.}$ Since $X^{-{\epsilon }_i}{X}^{-{\epsilon }_j}=X^{-{\epsilon }_j}{X}^{-{\epsilon }_i}$ for all $1\le i,j\le k,$ and the $m_i$ are linear combinations of the $X^{-{\epsilon }_i},$ $$m_i{m}_j=m_j{m}_i\text{in}\hspace{0.17em}{\widehat{TL}}_k^a,\text{for all}\hspace{0.17em}1\le i,j\le k\text{.}$$

For $1\le i\le k,$

(a)	$X^{-{\epsilon }_i}=q^{-(i-2)}(q-q^{-1})(m_i+q^{-1}{m}_{i-1}+q^{-2}{m}_{i-2}+\cdots +q^{-(i-1)}{m}_1),$
(b)	$X^{-{\epsilon }_1}+\cdots +X^{-{\epsilon }_i}=q^{-(i-2)}(q-q^{-1})(m_i+\left[2\right]m_{i-1}+\cdots +\left[i\right]m_1)\text{.}$

		Proof.
	Rewrite (2.14) as $$X^{-{\epsilon }_i}=q^{-(i-2)}(q-q^{-1})m_i+q^{-2}{X}^{-{\epsilon }_{i-1}}$$ and use induction, $$X^{-{\epsilon }_i}=q^{-(i-2)}(q-q^{-1})m_i+q^{-2}\left(q^{-(i-1-2)}(q-q^{-1})(m_{i-1}+q^{-1}{m}_{i-2}+\cdots +q^{-(i-2)}{m}_1)\right),$$ to obtain the formula for $X^{-{\epsilon }_i}$ in (a). Summing the formula in (a) over $i$ gives $$\sum _{j=1}^i{X}^{-{\epsilon }_j}=\sum _{j=1}^i\left(q^{-(j-2)}(q-q^{-1})\sum _{\ell =0}^{j-2}{q}^{-\ell }{m}_{j-\ell }\right)=q^{-(i-2)}(q-q^{-1})\sum _{j=1}^i\sum _{\ell =0}^{j-1}{q}^{i-j-\ell }{m}_{j-\ell }$$ and, thus, formula (c) follows from $$\sum _{j=1}^i\sum _{\ell =0}^{j-1}{q}^{i-j-\ell }{m}_{j-\ell }=\sum _{j=1}^i\sum _{r=1}^j{q}^{i-j-(j-r)}{m}_r=\sum _{r=1}^i\sum _{j=r}^i{q}^{i+r-2j}{m}_r=\sum _{r=1}^i[i-r+1]m_r\text{.}$$ $\square$

The following Lemma is a transfer of the recursion $X^{{\epsilon }_i}=T_{i-1}{X}^{{\epsilon }_{i-1}}{T}_{i-1}$ to the $m_i\text{.}$ The following are the base cases of Lemma 2.7. $$m_1=\frac{q^{-1}}{q-q^{-1}}{X}^{-{\epsilon }_1}\text{and}{m}_2=\frac{x}{q-q^{-1}}{e}_1-(e_1{m}_1+m_1{e}_1)$$

Let $x$ be the constant defined by the equation $e_1{X}^{-{\epsilon }_1}{e}_1=x{e}_1\text{.}$ For $2\le i\le k,$ $$m_i=\frac{q^{i-2}x}{q-q^{-1}}{e}_{i-1}-(e_{i-1}{m}_{i-1}+m_{i-1}{e}_{i-1})-\sum _{\ell =1}^{i-2}([i-\ell ]-[i-\ell -2])m_{\ell }{e}_{i-1}\text{.}$$

		Proof.
	From (2.3) and (2.9) we have $X^{-{\epsilon }_i}=(q^{-1}-e_{i-1})X^{-{\epsilon }_{i-1}}(q^{-1}-e_{i-1})\text{.}$ Substituting this into the definition of $m_i$ gives $$\begin{array}{ccc}(q-q^{-1})m_i& =& q^{i-2}(X^{-{\epsilon }_i}-q^{-2}{X}^{-{\epsilon }_{i-1}})=q^{i-2}(q^{-1}-e_{i-1})X^{-{\epsilon }_{i-1}}(q^{-1}-e_{i-1})-q^{i-4}{X}^{-{\epsilon }_{i-1}}\\ & =& q^{i-2}{e}_{i-1}{X}^{-{\epsilon }_{i-1}}{e}_{i-1}-q^{i-3}(e_{i-1}{X}^{-{\epsilon }_{i-1}}+X^{-{\epsilon }_{i-1}}{e}_{i-1})\text{.}\end{array}$$ Use Proposition 2.6 (a) to substitute for $X^{-{\epsilon }_{i-1}},$ $$\begin{array}{ccc}(q-q^{-1})m_i& =& (q-q^{-1})q^{-(m+i-3)}(q^{i-2}{e}_{i-1}{m}_{i-1}{e}_{i-1}-q^{i-3}(e_{i-1}{m}_{i-1}+m_{i-1}{e}_{i-1}))\\ & & +(q-q^{-1})q^{-(i-3)}(q^{-1}{m}_{i-2}+\cdots +q^{-(i-2)}{m}_1)(q^{i-2}{e}_{i-1}^2-2q^{i-3}{e}_{i-1})\\ & =& (q-q^{-1})(q{e}_{i-1}{m}_{i-1}{e}_{i-1}-(e_{i-1}{m}_{i-1}+m_{i-1}{e}_{i-1}))\\ & & +(q-q^{-1})(m_{i-2}+\cdots +q^{-(i-3)}{m}_1)(q+q^{-1}-2q^{-1})e_{i-1}\\ & =& (q-q^{-1})\left(\begin{array}{c}q{e}_{i-1}{m}_{i-1}{e}_{i-1}-(e_{i-1}{m}_{i-1}+m_{i-1}{e}_{i-1})\\ +(q-q^{-1})(m_{i-2}+\cdots +q^{-(i-3)}{m}_1)e_{i-1}\end{array}\right),\end{array}$$ which gives $$\begin{array}{cc}\begin{array}{ccc}{m}_i& =& q{e}_{i-1}{m}_{i-1}{e}_{i-1}-(e_{i-1}{m}_{i-1}+m_{i-1}{e}_{i-1})\\ & & +(q-q^{-1})(m_{i-2}+q^{-1}{m}_{i-3}+q^{-2}{m}_{i-4}+\cdots +q^{-(i-3)}{m}_1)e_{i-1}\text{.}\end{array}& \text{(2.15)}\end{array}$$ Using induction, substitute for the first $m_{i-1}$ in this equation to get $$\begin{array}{ccc}{m}_i& =& -(e_{i-1}{m}_{i-1}+m_{i-1}{e}_{i-1})+(q-q^{-1})\sum _{\ell =1}^{i-2}{q}^{-(i-2-\ell )}{m}_{\ell }{e}_{i-1}\\ & & +q(\frac{q^{i-3}x}{q-q^{-1}}{e}_{i-1}-2m_{i-2}{e}_{i-1}-\sum _{\ell =1}^{i-3}([i-\ell -1]-[i-\ell -3])m_{\ell }{e}_{i-1})\\ & =& \frac{q^{i-2}x}{q-q^{-1}}{e}_{i-1}-(e_{i-1}{m}_{i-1}+m_{i-1}{e}_{i-1})-\sum _{\ell =2}^{i-2}([i-\ell ]-[i-\ell -2])m_{\ell }{e}_{i-1}\text{.}\end{array}$$ $\square$

Diagram Representation of Murphy Elements

Label the vertices from left to right in the top row of a diagram $d\in T_k$ with $1,2,\dots ,k,$ and label the corresponding vertices in the bottom row with $1\prime ,2\prime ,\dots ,k\prime \text{.}$ The cycle type of a diagram $d\in T_k$ is the set partition $\tau \left(d\right)$ of $\{1,2,\dots ,k\}$ obtained from $d$ by setting $1=1\prime ,$ $2=2\prime ,$ $\dots ,$ $k=k\prime \text{.}$ If $\tau \left(d\right)$ is a set partition of the form $\{\{1,2,\dots ,{\gamma }_1\},\{{\gamma }_1+1,{\mu }_1+2,\dots ,{\gamma }_1+\gamma +2\},\dots ,\{{\gamma }_1+\cdots +{\gamma }_{\ell -1}+1,\dots ,k\}\}$ where $({\gamma }_1,\dots ,{\gamma }^{\ell })$ is a composition of $k,$ then we simplify notation by writing $\tau \left(d\right)=({\gamma }_1,\dots ,{\gamma }^{\ell })\text{.}$ For example $$d=\begin{array}{c} \end{array}\text{has}\tau \left(d\right)=(5,3,4)\text{.}$$ There are diagrams whose cycle type cannot be written as a composition (for example $d=\begin{array}{c} \end{array}$ has cycle type $\{\{1,4\},\{2,3\}\}\text{)}$ but all of the diagrams needed here have cycle types that are compositions.

If $\gamma =({\gamma }_1,\dots ,{\gamma }^{\ell })$ is a composition of $k$ define $$\begin{array}{cc}{d}_{\gamma }=\sum _{\tau \left(d\right)=\gamma }d& \text{(2.16)}\end{array}$$ as the sum of the Temperley-Lieb diagrams on $k$ dots with cycle type $\gamma \text{.}$ Define $d_{\gamma }^{*}$ be the sum of diagrams obtained from the summands of $d_{\gamma }$ by wrapping the first edge in each row around the pole, with the orientation coming from $X^{-{\epsilon }_1}$ as shown in the examples below. When the first edge in the top row connects to the first vertex in the bottom row only one new diagram is produced, otherwise there are two. For example, in ${\widehat{TL}}_4^a,$ $$\begin{array}{cc}\begin{array}{ccc}{d}_{31}& =& \begin{array}{c} \end{array}+\begin{array}{c} \end{array}\\ d_{13}& =& \begin{array}{c} \end{array}+\begin{array}{c} \end{array}\\ d_{22}& =& \begin{array}{c} \end{array}\end{array}& \begin{array}{ccc}{d}_{31}^{*}& =& \begin{array}{c} \end{array}+\begin{array}{c} \end{array}+\begin{array}{c} \end{array}+\begin{array}{c} \end{array}\\ d_{13}^{*}& =& \begin{array}{c} \end{array}+\begin{array}{c} \end{array}\\ d_{22}^{*}& =& \begin{array}{c} \end{array}+\begin{array}{c} \end{array}\end{array}\end{array}$$ View $d_{\gamma }$ and $d_{\gamma }^{*}$ as elements of ${\widehat{TL}}_k^a$ by setting $$d_{\gamma }=d_{\gamma 1^{k-i}},\text{if}\hspace{0.17em}\gamma \hspace{0.17em}\text{is a composition of}\hspace{0.17em}i\hspace{0.17em}\text{with}\hspace{0.17em}i<k\text{.}$$ With this notation, expanding the first few $m_i$ in terms of diagrams gives $$\begin{array}{ccc}(q-q^{-1})m_1& =& q^{-1}{d}_1^{*},(q-q^{-1})m_2=x{d}_2-q^{-1}{d}_2^{*},\\ (q-q^{-1})m_3& =& qx{d}_{1,2}-q^{-1}\left[2\right]d_{1,2}^{*}-x{d}_3+q^{-1}{d}_3^{*},\\ (q-q^{-1})m_4& =& q^{2}x{d}_{1^2,2}-q^{-1}(\left[3\right]-\left[1\right])d_{1^2,2}^{*}-x\left[2\right]d_{2,2}+q^{-1}\left[2\right]d_{2,2}^{*}\\ & & -qx{d}_{1,3}+q^{-1}\left[2\right]d_{1,3}^{*}+x{d}_4-q^{-1}{d}_4^{*},\\ (q-q^{-1})m_5& =& q^{3}x{d}_{1^3,2}-q^{-1}(\left[4\right]-\left[2\right])d_{1^3,2}^{*}-q^{2}x{d}_{1^2,3}+q^{-1}(\left[3\right]-\left[1\right])d_{1^2,3}^{*}\\ & & +qx{d}_{1,4}-q^{-1}\left[2\right]d_{1,4}^{*}-qx\left[2\right]d_{1,2,2}+q^{-1}{\left[2\right]}^2{d}_{1,2,2}^{*}+qx\left[2\right]d_{2,3}-q^{-1}\left[2\right]d_{2,3}^{*}\\ & & -x(\left[3\right]-\left[1\right])d_{2,1,2}+q^{-1}(\left[3\right]-\left[1\right])d_{2,1,2}^{*}+x\left[2\right]d_{3,2}-q^{-1}\left[2\right]d_{3,2}^{*}-x{d}_5+q^{-1}{d}_5^{*},\end{array}$$ where, as in Lemma 2.7, $x$ is the constant defined by the equation $e_1{X}^{-{\epsilon }_1}{e}_1=x{e}_1\text{.}$

Let $x$ be the constant defined by the equation $e_1{X}^{-{\epsilon }_1}{e}_1=x{e}_1\text{.}$ Then $(q-q^{-1})m_1=q^{-1}{d}_1,$ $(q-q^{-1})m_2=x{d}_2-q^{-1}{d}_2^{*}$ and, for $i\ge 2,$ $$m_i=\sum _{\text{compositions}\hspace{0.17em}\gamma }{\left(m_i\right)}_{\gamma }{d}_{\gamma }+{\left(m_i\right)}_{\gamma }^{*}{d}_{\gamma }^{*},$$ where the sum is over all compositions $\gamma =1^{b_1}{r}_1{1}^{b_2}{r}_2\cdots 1^{b_{\ell }}{r}_{\ell }$ of $i$ with $r_{\ell }>1,$ and $$\begin{array}{ccc}{\left(m_i\right)}_{\gamma }& =& {(-1)}^{\left|\gamma \right|-\ell \left(\gamma \right)-1}\frac{q^{b_1}x}{q-q^{-1}}\prod _{b_j\ge 0,j>1}([b_j+2]-\left[b_j\right]),\text{and}\\ {\left(m_i\right)}_{\gamma }^{*}& =& {(-1)}^{\left|\gamma \right|-\ell \left(\gamma \right)}\frac{q^{-1}}{q-q^{-1}}([b_1+1]-[b_1-1])\prod _{b_j\ge 0,j>1}([b_j+2]-\left[b_j\right]),\end{array}$$ with $\ell \left(\gamma \right)=\ell +b_1+\cdots +b_{\ell }\text{.}$

		Proof.
	From our computations above, $m_1=A{d}_1^{*}$ and $m_2=B{d}_2-A{d}_2^{*},$ where $$A=\frac{q^{-1}}{q-q^{-1}}\text{and}B=\frac{x}{q-q^{-1}}\text{.}$$ Let $m_1=A{d}_1^{*}\text{.}$ For $i>2$ the recursion in Lemma 2.7 gives $$\begin{array}{ccc}{m}_i& =& q^{i-2}B{e}_{i-1}-(e_{i-1}{m}_{i-1}+m_{i-1}{e}_{i-1})-\sum _{\ell =1}^{i-2}([i-\ell ]-[i-\ell -2])m_{\ell }{e}_{i-1}\\ & =& q^{i-2}B{d}_{1^{i-2},2}-([i-1]-[i-3])A{d}_{1^{i-2},2}^{*}-({\left(m_{i-1}\right)}_{\gamma \prime r}{d}_{\gamma \prime ,r+1}+{\left(m_{i-1}\right)}_{\gamma \prime r}^{*}{d}_{\gamma \prime ,r+1}^{*})\\ & & +\sum _{\ell =2}^{i-2}-([i-\ell ]-[i-\ell -2])({\left(m_{\ell }\right)}_{\gamma \prime }{d}_{\gamma \prime 1^{i-2-\ell }2}+{\left(m_{\ell }\right)}_{\gamma \prime }^{*}{d}_{\gamma \prime 1^{i-2-\ell }2}^{*})\text{.}\end{array}$$ So if $d$ has cycle type $\gamma =1^{b_1}{r}_1{1}^{b_2}{r}_2\cdots 1^{b_{\ell }}{r}_{\ell }$ with $r_{\ell }>0,$ then (a) Each part of size $r$ $(r>1)$ contributes ${(-1)}^{r-1}$ to the coefficient. Thus, there is a total contribution of ${(-1)}^{\left|\gamma \right|-\ell \left(\gamma \right)}$ from these parts. (b) Each inner $1^b$ $(b\ge 0)$ contributes a factor of $[b+2]-\left[b\right]$ to the coefficient. (c) The first $1^b$ $(b>0)$ contributes a $-q^{b}B$ in a nonstarred class, (c') The first $1^b$ $(b=0)$ contributes a $-B$ in a nonstarred class, which is the same as case (c) with $b=0\text{.}$ (d) The first $1^b$ $(b>0)$ contributes a $([b+1]-[b-1])A$ in a starred class. (d') The first $1^b$ $(b=0)$ contributes an $A$ in a starred class, which is the same as case (d) with $b=0$ assuming $[-1]=0\text{.}$ $\square$

To view $m_1,\dots ,m_k$ in the (nonaffine) Temperley-Lieb algebra ${TL}_k\left(n\right)$ (via (2.11)) let $X^{-{\epsilon }_1}=1$ so that $x=q+q^{-1}\text{.}$ If $b_1>1$ then $d_{\gamma }^{*}=d_{\gamma }$ and if $b_1=0$ then $d_{\gamma }^{*}=2d_{\gamma }\text{.}$ In both cases the coefficients in Theorem 2.8 specialize to $${\left(m_i\right)}_{\gamma }+{\left(m_i\right)}_{\gamma }^{*}={(-1)}^{\left|\gamma \right|-\ell \left(\gamma \right)-1}[b_1+1]\prod _{b_j\ge 0,j>1}([b_j+2]-\left[b_j\right])$$ and $$m_i=\sum _{\gamma }({\left(m_i\right)}_{\gamma }+{\left(m_i\right)}_{\gamma }^{*})d_{\gamma },$$ where the sum is over compositions $\gamma =1^{b_1}{r}_1{1}^{b_2}{r}_2\cdots 1^{b_{\ell }}{r}_{\ell }$ of $i$ with $r_{\ell }>1\text{.}$ The first few examples are $$\begin{array}{ccc}{m}_1& =& \frac{q^{-1}}{q-q^{-1}}=\frac{q^{-1}}{q-q^{-1}}{d}_1,m_2=e_2=d_2,m_3=\left[2\right]d_{12}-d_3,\\ m_4& =& \left[3\right]d_{1^2,2}-\left[2\right]d_{2,2}-\left[2\right]d_{1,3}+d_4,\\ m_5& =& \left[4\right]d_{1^3,2}-\left[3\right]d_{1^2,2}+\left[2\right]d_{1,4}-{\left[2\right]}^2{d}_{1,2,2}+\left[2\right]d_{2,3}-(\left[3\right]-\left[1\right])d_{2,1,2}+\left[2\right]d_{3,2}-d_5\text{.}\end{array}$$

Notes and references

This is a typed version of Commuting families in Hecke and Temperley-Lieb Algebras by Tom Halverson, Manuela Mazzocco and Arun Ram.

AMS Subject Classifications: Primary 20G05; Secondary 16G99, 81R50, 82B20.

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