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: 30 January 2014

Schur functors

$R\text{-matrices}$ and quantum Casimir Elements

Let $U_h𝔤$ be the Drinfeld-Jimbo quantum group corresponding to a finite dimensional complex semisimple Lie algebra $𝔤\text{.}$ We shall use the notations and conventions for $U_h𝔤$ as in [LRa1977] and [ORa0401317]. There is an invertible element $ℛ=\sum a_i\otimes b_i$ in (a suitable completion of) $U_h𝔤\otimes U_h𝔤$ such that, for two $U_h𝔤$ modules $M$ and $N,$ the map $$\begin{array}{cccc}{Ř}_{MN}:& M\otimes N& ⟶& N\otimes M\\ & m\otimes n& ⟼& \sum b_{i}n\otimes a_{i}m\end{array}\begin{array}{c} M ⊗ N N ⊗ M \end{array}$$ is a $U_h𝔤$ module isomorphism. In order to be consistent with the graphical calculus these operators should be written on the right. The element $ℛ$ satisfies “quasitriangularity relations” (see [LRa1977, (2.1-2.3)]) which imply that, for $U_h𝔤$ modules $M,N,P$ and a $U_h𝔤$ module isomorphism ${\tau }_M:M\to M,$ $$\begin{array}{ccc}\begin{array}{c} M ⊗ N N ⊗ M τM \end{array}& =& \begin{array}{c} M ⊗ N N ⊗ M τM \end{array}\\ {Ř}_{MN}({\text{id}}_N\otimes {\tau }_M)& =& ({\tau }_M\otimes {\text{id}}_N){Ř}_{MN},\end{array}\begin{array}{ccccccc}\begin{array}{c} M⊗(N⊗P) (N⊗P)⊗M \end{array}& =& \begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}& & \begin{array}{c} (M⊗N)⊗P P⊗(M⊗N) \end{array}& =& \begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}\\ {Ř}_{M,N\otimes P}& =& ({Ř}_{MN}\otimes {\text{id}}_P)({\text{id}}_N\otimes {Ř}_{MP})& & {Ř}_{M\otimes N,P}& =& ({\text{id}}_M\otimes {Ř}_{NP})({Ř}_{MP}\otimes {\text{id}}_N),\end{array}$$ which, together, imply the braid relation $$\begin{array}{ccc}\begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}& =& \begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}\\ ({Ř}_{MN}\otimes {\text{id}}_P)({\text{id}}_N\otimes {Ř}_{MP})({Ř}_{NP}\otimes {\text{id}}_M)& =& ({\text{id}}_M\otimes {Ř}_{NP})({Ř}_{MP}\otimes {\text{id}}_N)({\text{id}}_P\otimes {Ř}_{MN}),\end{array}$$

Let $\rho$ be such that $⟨\rho ,{\alpha }_i⟩=1$ for all simple roots ${\alpha }_i\text{.}$ As explained in [LRa1977, (2.14)] and [Dri1990], there is a quantum Casimir element $e^{-h\rho }u$ in the center of $U_h𝔤$ and, for a $U_h𝔤$ module $M$ we define a $U_h𝔤$ module isomorphism $$\begin{array}{cccc}{C}_M:& M& ⟶& M\\ & m& ⟼& \left(e^{-h\rho }u\right)m\end{array}\begin{array}{c} M M CM \end{array}$$ and the elements $C_M$ satisfy $$\begin{array}{cc}{C}_{M\otimes N}={\left({Ř}_{MN}{Ř}_{NM}\right)}^{-1}(C_M\otimes C_N),\text{and}{C}_M=q^{-⟨\lambda ,\lambda +2\rho ⟩}{\text{id}}_M& \text{(3.1)}\end{array}$$ if $M$ is a $U_h𝔤$ module generated by a highest weight vector $v^{+}$ of weight $\lambda$ (see [LRa1977, Prop. 2.14] or [Dri1990, Prop. 3.2]). Note that $⟨\lambda ,\lambda +2\rho ⟩=⟨\lambda +\rho ,\lambda +\rho ⟩-⟨\rho ,\rho ⟩$ are the eigenvalues of the classical Casimir operator [Dix1996, 7.8.5]. From the relation (3.1) it follows that if $M=L\left(\mu \right),$ $N=L\left(\nu \right)$ are finite dimensional irreducible $U_h𝔤$ modules then ${Ř}_{MN}{Ř}_{NM}$ acts on the $\lambda$ isotypic component ${L\left(\lambda \right)}^{\oplus c_{\mu \nu }^{\lambda }}$ of the decomposition $$\begin{array}{cc}L\left(\mu \right)\otimes L\left(\nu \right)=\underset{\lambda }{⨁}{L\left(\lambda \right)}^{\oplus c_{\mu \nu }^{\lambda }}\text{by the constant}{q}^{⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩}\text{.}& \text{(3.2)}\end{array}$$

The ${\stackrel{\sim }{ℬ}}_k$ module $M\otimes V^{\otimes k}$

Let $U_h𝔤$ be a quantum group and let $M$ and $V$ be $U\text{-modules}$ such that the operators ${Ř}_{MV},$ ${Ř}_{VM}$ and ${Ř}_{VV}$ are well defined. Define ${Ř}_i,$ $1\le i\le k-1,$ and ${Ř}_0^2$ in ${\text{End}}_U(M\otimes V^{\otimes k})$ by $${Ř}_i={\text{id}}_M\otimes {\text{id}}_V^{\otimes (i-1)}\otimes {Ř}_{VV}\otimes {\text{id}}_V^{\otimes (k-i-1)}\text{and}{Ř}_0^2=\left({Ř}_{MV}{Ř}_{VM}\right)\otimes {\text{id}}_V^{\otimes (k-1)}\text{.}$$ Then the braid relations $${Ř}_i{Ř}_{i+1}{Ř}_i=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}={Ř}_{i+1}{Ř}_i{Ř}_{i+1}$$ and $${Ř}_0^2{Ř}_1{Ř}_0^2{Ř}_1=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}={Ř}_1{Ř}_0^2{Ř}_1{Ř}_0^2\text{.}$$ imply that there is a well defined map $$\begin{array}{cc}\begin{array}{cccc}\Phi :& {\stackrel{\sim }{ℬ}}_k& ⟶& {\text{End}}_U(M\otimes V^{\otimes k})\\ & T_i& ⟼& {Ř}_i,\\ & X^{{\epsilon }_1}& ⟼& {Ř}_0^2,\end{array}1\le i\le k-1,& \text{(3.3)}\end{array}$$ which makes $M\otimes V^{\otimes k}$ into a right ${\stackrel{\sim }{ℬ}}_k$ module. By (3.1) and the fact that $$\begin{array}{cc}\Phi \left(X^{{\epsilon }_i}\right)={Ř}_{M\otimes V^{\otimes (i-1)},V}{Ř}_{V,M\otimes V^{\otimes (i-1)}}=\begin{array}{c} i \end{array}\text{.}& \text{(3.4)}\end{array}$$ the eigenvalues of $\Phi \left(X^{{\epsilon }_i}\right)$ are related to the eigenvalues of the Casimir. The Schur functors are the functors $$\begin{array}{cc}\begin{array}{cccc}{F}_V^{\lambda }:& \left\{U\text{-modules}\right\}& ⟶& \left\{{\stackrel{\sim }{ℬ}}_k\text{-modules}\right\}\\ & M& ⟼& {\text{Hom}}_U(M\left(\lambda \right),M\otimes V^{\otimes k})\end{array}& \text{(3.5)}\end{array}$$ where ${\text{Hom}}_U(M\left(\lambda \right),M\otimes V^{\otimes k})$ is the vector space of highest weight vectors of weight $\lambda$ in $M\otimes V^{\otimes k}\text{.}$

The quantum group $U_h{𝔤𝔩}_n$

Although the Lie algebra ${𝔤𝔩}_n$ is reductive, not semisimple, all of the general setup of Sections 3.1 and 3.2 can be applied without change. The simple roots are ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1},$ $1\le i\le n-1,$ and $$\begin{array}{cc}\rho =(n-1){\epsilon }_1+(n-2){\epsilon }_2+\cdots +{\epsilon }_{n-1}\text{.}& \text{(3.6)}\end{array}$$ The dominant integral weights of ${𝔤𝔩}_n$ are $$\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n,\text{where}{\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n,\text{and}{\lambda }_1,\dots ,{\lambda }_n\in \mathbb{Z}$$ and these index the simple finite dimensional $U_h{𝔤𝔩}_n\text{-modules}$ $L\left(\lambda \right)\text{.}$ A partition with $\le n$ rows is a dominant integral weight with ${\lambda }_n\ge 0\text{.}$ If ${\lambda }_n<0$ and $\Delta$ denotes the $1\text{-dimensional}$ “determinant” representation of $U_h{𝔤𝔩}_n$ then (see [FHa1991, §15.5]) $$\begin{array}{cc}L\left(\lambda \right)\cong {\Delta }^{{\lambda }_n}\otimes L(\lambda +(-{\lambda }_n,\dots ,-{\lambda }_n))\text{with}\hspace{0.17em}\lambda +(-{\lambda }_n,\dots ,-{\lambda }_n)\hspace{0.17em}\text{a partition.}& \text{(3.7)}\end{array}$$

Identify each partition $\lambda$ with the configuration of boxes which has ${\lambda }_i$ boxes in row $i\text{.}$ For example, $$\begin{array}{cc}\lambda =\begin{array}{c} \end{array}=5{\epsilon }_1+5{\epsilon }_2+3{\epsilon }_3+3{\epsilon }_4+{\epsilon }_5+{\epsilon }_6\text{.}& \text{(3.8)}\end{array}$$ If $\mu$ and $\lambda$ are partitions with $\mu \subseteq \lambda$ (as collections of boxes) then the skew shape $\lambda /\mu$ is the collection of boxes of $\lambda$ that are not in $\mu \text{.}$ For example, if $\lambda$ is as in (3.8) and $$\mu =\begin{array}{c} \end{array}\text{then}\lambda /\mu =\begin{array}{c} \end{array}\text{.}$$ If $b$ is a box in position $(i,j)$ of $\lambda$ the content of $b$ is $$\begin{array}{cc}c\left(b\right)=j-i=\text{the diagonal number of}\hspace{0.17em}b,\text{so that}\begin{array}{c} 0 1 2 3 4 -1 0 1 2 3 -2 -1 0 -3 -2 -1 -4 -5 \end{array}& \text{(3.9)}\end{array}$$ are the contents of the boxes for the partition in (3.8).

If $\nu$ is a partition and $$\begin{array}{cc}V=L\left(▫\right)\text{then}L\left(\nu \right)\otimes V=\underset{\lambda \in {\nu }^{+}}{⨁}L\left(\lambda \right),& \text{(3.10)}\end{array}$$ where the sum is over all partitions $\lambda$ with $\le n$ rows that are obtained by adding a box to $\nu$ [Mac1995, I App. A (8.4) and I (5.16)], Hence, the $U_h{𝔤𝔩}_n\text{-module}$ decompositions of $$\begin{array}{cc}L\left(\mu \right)\otimes V^{\otimes k}=\underset{\lambda }{⨁}L\left(\lambda \right)\otimes {\stackrel{\sim }{H}}_k^{\lambda /\mu },k\in {\mathbb{Z}}_{\ge 0},& \text{(3.11)}\end{array}$$ are encoded by the graph ${\hat{H}}^{/\mu }$ with $$\begin{array}{cc}\begin{array}{cc}\text{vertices on level}\hspace{0.17em}k\text{:}& \{\text{skew shapes}\hspace{0.17em}\lambda /\mu \hspace{0.17em}\text{with}\hspace{0.17em}k\hspace{0.17em}\text{boxes}\}\\ \text{edges:}& \lambda /\mu ⟶\gamma /\mu \text{, if}\hspace{0.17em}\gamma \hspace{0.17em}\text{is obtained from}\hspace{0.17em}\lambda \hspace{0.17em}\text{by adding a box}\\ \text{labels on edges:}& \text{content of the added box.}\end{array}& \text{(3.12)}\end{array}$$ For example if $\mu =(3,3,3,2)=\begin{array}{c} \end{array},$ then the first $4$ rows of ${\hat{H}}^{/\mu }$ are $$\begin{array}{cc}\begin{array}{c} 3 -1 -4 4 2 -1 3 -4 3 -4 -1 -3 -5 \end{array}& \text{(3.13)}\end{array}$$

The following result is well known (see [Jim1986] or [LRa1977, (4.4)]).

If $U=U_h{𝔤𝔩}_n$ and $V=L\left({\epsilon }_1\right)=L\left(▫\right)$ is the $n\text{-dimensional}$ “standard” representation of ${𝔤𝔩}_n$ then the map $\Phi$ of (3.3) factors through the surjective homomorphism (2.8) to give a representation of the affine Hecke algebra.

For a skew shape $\lambda /\mu$ with $k$ boxes identify paths from $\mu$ to $\lambda /\mu$ in ${\hat{H}}^{/\mu }$ with standard tableaux of shape $\lambda /\mu$ by filling the boxes, successively, with $1,2,\dots ,k$ as they appear. In the example graph ${\hat{H}}^{/\mu }$ above $$\begin{array}{c} 1 2 3 \end{array}\text{corresponds to the path}\begin{array}{c} \end{array}⟶\begin{array}{c} \end{array}⟶\begin{array}{c} \end{array}⟶\begin{array}{c} \end{array}$$

The quantum group $U_h{𝔤𝔩}_2$

In the case when $n=2,$ $U=U_h{𝔤𝔩}_2$ and the partitions which appear in (3.11) and in the graph ${\hat{H}}^{/\mu }$ all have $\le 2$ rows. For example if $\mu =\left(42\right)=\begin{array}{c} \end{array}$ then the first few rows of ${\hat{H}}^{/\mu }$ are $$\begin{array}{cc}\begin{array}{c} k=0: k=1: k=2: k=3: k=4: \end{array}& \text{(3.14)}\end{array}$$ and this is the graph which describes the decompositions in (3.11).

If $U=U_h{𝔤𝔩}_2,$ $M=L\left(\mu \right)$ where $\mu$ is a partition of $m$ with $\le 2$ rows, and $V=L\left({\epsilon }_1\right)=L\left(▫\right)$ is the 2-dimensional "standard" representation of ${𝔤𝔩}_2$ then the map $\Phi$ of (3.3) factors through the surjective homomorphism of (2.13) with ${\alpha }^2=-q^{2m-1}$ to give a representation of the affine Temperley-Lieb algebra $T_k^a\text{.}$

		Proof.
	The proof that the kernel of $\Phi$ contains the element (2.12) is exactly as in the proof of [ORa0401317, Thm. 6.1(c)]: The element $e_1$ in $T_2^a$ acts on $V^{\otimes 2}$ as $(q+q^{-1})·\text{pr}$ where pr is the unique $U_h𝔤\text{-invariant}$ projection onto $L\left(\begin{array}{c} \end{array}\right)$ in $V^{\otimes 2}\text{.}$ Using $e_1{T}_1=-q^{-1}{e}_1$ and the pictorial equalities $$\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=-q^{-1}·\begin{array}{c} \end{array}$$ it follows that ${\Phi }_2\left(e_1{X}^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1}\right)$ acts as $-(q+q^{-1})q^{-1}·{Ř}_{L\left(\begin{array}{c} \end{array}\right),L\left(\mu \right)}{Ř}_{L\left(\mu \right),L\left(\begin{array}{c} \end{array}\right)}({\text{id}}_{L\left(\mu \right)}\otimes \text{pr})\text{.}$ By (3.1), this is equal to $$\begin{array}{c}-q^{-1}(C_{L\left(\mu \right)}\otimes C_{L\left(\begin{array}{c} \end{array}\right)})C_{L\left(\mu \right)\otimes L\left(\begin{array}{c} \end{array}\right)}^{-1}{\Phi }_2({\text{id}}_{L\left(\mu \right)}\otimes e_1)\\ =-q^{-1}{q}^{-⟨\mu ,\mu +2\rho ⟩}{q}^{-⟨{\epsilon }_1+{\epsilon }_2,{\epsilon }_1+{\epsilon }_2+2\rho ⟩}{C}_{L(\mu +{\epsilon }_1+{\epsilon }_2)}^{-1}{\Phi }_2({\text{id}}_{L\left(\mu \right)}\otimes e_1)\text{.}\end{array}$$ and the coefficient $-q^{-1}{q}^{-⟨\mu ,\mu +2\rho ⟩}{q}^{-⟨{\epsilon }_1+{\epsilon }_2,{\epsilon }_1+{\epsilon }_2+2\rho ⟩}{C}_{L(\mu +{\epsilon }_1+{\epsilon }_2)}^{-1}$ simplifies to $$-q^{-1}{q}^{-⟨\mu ,\mu +2\rho ⟩}{q}^{-⟨{\epsilon }_1+{\epsilon }_2,{\epsilon }_1+{\epsilon }_2+2\rho ⟩}{q}^{⟨\mu +{\epsilon }_1+{\epsilon }_2,\mu +{\epsilon }_1+{\epsilon }_2+2\rho ⟩}=-q^{-1}{q}^{2({\mu }_1+{\mu }_2)}=-q^{2m-1},$$ where $m={\mu }_1+{\mu }_2=\left|\mu \right|\text{.}$ $\square$

The quantum group $U_h{𝔰𝔩}_2$

The restriction of an irreducible representation $L\left(\lambda \right)$ of $U_h{𝔤𝔩}_n$ to $U_h{𝔰𝔩}_n$ is irreducible and all irreducible representations of $U_h{𝔰𝔩}_n$ are obtained in this fashion. Since the “determinant” representation is trivial as an $U_h{𝔰𝔩}_n$ module it follows from (3.7) that the irreducible representations $L_{{𝔰𝔩}_n}\left(\lambda \right)$ of $U_h{𝔰𝔩}_n$ are indexed by partitions $\lambda =({\lambda }_1,\dots ,{\lambda }_n)$ with ${\lambda }_n=0\text{.}$ Hence, the graph which describes the $U_h{𝔰𝔩}_2\text{-module}$ decompositions of $$\begin{array}{cc}L\left(\mu \right)\otimes V^{\otimes k}=\underset{\lambda }{⨁}L\left(\lambda \right)\otimes {\stackrel{\sim }{T}}_k^{\lambda /\mu },k\in {\mathbb{Z}}_{\ge 0}& \text{(3.15)}\end{array}$$ is exactly the same as the graph for $U_h{𝔤𝔩}_2$ except with all columns of length 2 removed from the partitions. More precisely, the decompositions are encoded by the graph ${\hat{T}}^{/\mu }$ with $$\begin{array}{cc}\begin{array}{cc}\text{vertices on level}\hspace{0.17em}k\text{:}& \{{\mu }_1-{\mu }_2+k,{\mu }_1-{\mu }_2+k-2,\dots ,{\mu }_1-{\mu }_2-k\}\cap {\mathbb{Z}}_{\ge 0}\\ \text{edges:}& \ell ⟶\ell \pm 1\text{.}\end{array}& \text{(3.16)}\end{array}$$ For example if $m=7$ and ${\mu }_1-{\mu }_2=3$ then the first few rows of ${\hat{T}}^{/\mu }$ are $$\begin{array}{cc}\begin{array}{c} k=0: k=1: ∅ k=2: k=3: ∅ k=4: \end{array}& \text{(3.17)}\end{array}$$ Paths in (3.17) correspond to paths in (3.14) which correspond to standard tableaux $T$ of shape $\lambda /\mu \text{.}$

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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