## Commuting families in Hecke and Temperley-Lieb Algebras

Last update: 30 January 2014

## Schur functors

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

Let ${U}_{h}𝔤$ 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 ${U}_{h}𝔤$ as in [LRa1977] and [ORa0401317]. There is an invertible element $ℛ=\sum {a}_{i}\otimes {b}_{i}$ in (a suitable completion of) ${U}_{h}𝔤\otimes {U}_{h}𝔤$ such that, for two ${U}_{h}𝔤$ modules $M$ and $N,$ the map $ŘMN: M⊗N ⟶ N⊗M m⊗n ⟼ ∑bin⊗aim M ⊗ N N ⊗ M$ is a ${U}_{h}𝔤$ 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 ${U}_{h}𝔤$ modules $M,N,P$ and a ${U}_{h}𝔤$ module isomorphism ${\tau }_{M}:M\to M,$ $M ⊗ N N ⊗ M τM = M ⊗ N N ⊗ M τM ŘMN (idN⊗τM) = (τM⊗idN) ŘMN, M⊗(N⊗P) (N⊗P)⊗M = M ⊗ N ⊗ P P ⊗ N ⊗ M (M⊗N)⊗P P⊗(M⊗N) = M ⊗ N ⊗ P P ⊗ N ⊗ M ŘM,N⊗P = (ŘMN⊗idP) (idN⊗ŘMP) ŘM⊗N,P = (idM⊗ŘNP) (ŘMP⊗idN),$ which, together, imply the braid relation $M ⊗ N ⊗ P P ⊗ N ⊗ M = M ⊗ N ⊗ P P ⊗ N ⊗ M (ŘMN⊗idP) (idN⊗ŘMP) (ŘNP⊗idM) = (idM⊗ŘNP) (ŘMP⊗idN) (idP⊗ŘMN),$

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 ${U}_{h}𝔤$ and, for a ${U}_{h}𝔤$ module $M$ we define a ${U}_{h}𝔤$ module isomorphism $CM: M ⟶ M m ⟼ (e-hρu)m M M CM$ and the elements ${C}_{M}$ satisfy $CM⊗N= (ŘMNŘNM)-1 (CM⊗CN),and CM= q-⟨λ,λ+2ρ⟩ idM (3.1)$ if $M$ is a ${U}_{h}𝔤$ 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 ${U}_{h}𝔤$ modules then ${Ř}_{MN}{Ř}_{NM}$ acts on the $\lambda$ isotypic component ${L\left(\lambda \right)}^{\oplus {c}_{\mu \nu }^{\lambda }}$ of the decomposition $L(μ)⊗L(ν)= ⨁λ L(λ)⊕cμνλ by the constant q ⟨λ,λ+2ρ⟩- ⟨μ,μ+2ρ⟩- ⟨ν,ν+2ρ⟩ . (3.2)$

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

Let ${U}_{h}𝔤$ 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 ${\text{End}}_{U}\left(M\otimes {V}^{\otimes k}\right)$ by $Ři=idM⊗ idV⊗(i-1)⊗ ŘVV⊗ idV⊗(k-i-1) andŘ02= (ŘMVŘVM) ⊗idV⊗(k-1).$ Then the braid relations and imply that there is a well defined map $Φ: ℬ∼k ⟶ EndU(M⊗V⊗k) Ti ⟼ Ři, Xε1 ⟼ Ř02, 1≤i≤k-1, (3.3)$ which makes $M\otimes {V}^{\otimes k}$ into a right ${\stackrel{\sim }{ℬ}}_{k}$ module. By (3.1) and the fact that $Φ(Xεi)= ŘM⊗V⊗(i-1),V ŘV,M⊗V⊗(i-1)= i . (3.4)$ the eigenvalues of $\Phi \left({X}^{{\epsilon }_{i}}\right)$ are related to the eigenvalues of the Casimir. The Schur functors are the functors $FVλ: {U-modules} ⟶ {ℬ∼k-modules} M ⟼ HomU(M(λ),M⊗V⊗k) (3.5)$ where ${\text{Hom}}_{U}\left(M\left(\lambda \right),M\otimes {V}^{\otimes k}\right)$ 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 $ρ=(n-1)ε1+ (n-2)ε2+⋯+ εn-1. (3.6)$ The dominant integral weights of ${𝔤𝔩}_{n}$ are $λ=λ1ε1+⋯+ λnεn,where λ1≥λ2≥⋯≥ λn,andλ1 ,…,λn∈ℤ$ 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]) $L(λ)≅Δλn ⊗L(λ+(-λn,…,-λn)) with λ+(-λn,…,-λn) a partition. (3.7)$

Identify each partition $\lambda$ with the configuration of boxes which has ${\lambda }_{i}$ boxes in row $i\text{.}$ For example, $λ= =5ε1+5ε2 +3ε3+3ε4 +ε5+ε6. (3.8)$ 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 $μ= thenλ/μ= .$ If $b$ is a box in position $\left(i,j\right)$ of $\lambda$ the content of $b$ is $c(b)=j-i= the diagonal number of b,so that 0 1 2 3 4 -1 0 1 2 3 -2 -1 0 -3 -2 -1 -4 -5 (3.9)$ are the contents of the boxes for the partition in (3.8).

If $\nu$ is a partition and $V=L(▫)then L(ν)⊗V= ⨁λ∈ν+ L(λ), (3.10)$ 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 $L(μ)⊗V⊗k= ⨁λL(λ)⊗ H∼kλ/μ, k∈ℤ≥0, (3.11)$ are encoded by the graph ${\stackrel{ˆ}{H}}^{/\mu }$ with $vertices on level k: {skew shapes λ/μ with k boxes} edges: λ/μ⟶γ/μ, if γ is obtained from λ by adding a box labels on edges: content of the added box. (3.12)$ For example if $\mu =\left(3,3,3,2\right)=\begin{array}{c} \end{array},$ then the first $4$ rows of ${\stackrel{ˆ}{H}}^{/\mu }$ are $3 -1 -4 4 2 -1 3 -4 3 -4 -1 -3 -5 (3.13)$

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 ${\stackrel{ˆ}{H}}^{/\mu }$ with standard tableaux of shape $\lambda /\mu$ by filling the boxes, successively, with $1,2,\dots ,k$ as they appear. In the example graph ${\stackrel{ˆ}{H}}^{/\mu }$ above $1 2 3 corresponds to the path ⟶ ⟶ ⟶$

### 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 ${\stackrel{ˆ}{H}}^{/\mu }$ all have $\le 2$ rows. For example if $\mu =\left(42\right)=\begin{array}{c} \end{array}$ then the first few rows of ${\stackrel{ˆ}{H}}^{/\mu }$ are $k=0: k=1: k=2: k=3: k=4: (3.14)$ 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 $\left(q+{q}^{-1}\right)·\text{pr}$ where pr is the unique ${U}_{h}𝔤\text{-invariant}$ 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 $= =-q-1·$ it follows that ${\Phi }_{2}\left({e}_{1}{X}^{{\epsilon }_{1}}{T}_{1}{X}^{{\epsilon }_{1}}\right)$ acts as $-\left(q+{q}^{-1}\right){q}^{-1}·{Ř}_{L\left(\begin{array}{c} \end{array}\right),L\left(\mu \right)}{Ř}_{L\left(\mu \right),L\left(\begin{array}{c} \end{array}\right)}\left({\text{id}}_{L\left(\mu \right)}\otimes \text{pr}\right)\text{.}$ By (3.1), this is equal to $-q-1 ( CL(μ)⊗ C L ( ) ) C L(μ)⊗L ( ) -1 Φ2 (idL(μ)⊗e1) =-q-1 q-⟨μ,μ+2ρ⟩ q-⟨ε1+ε2,ε1+ε2+2ρ⟩ CL(μ+ε1+ε2)-1 Φ2(idL(μ)⊗e1).$ and the coefficient $-{q}^{-1}{q}^{-⟨\mu ,\mu +2\rho ⟩}{q}^{-⟨{\epsilon }_{1}+{\epsilon }_{2},{\epsilon }_{1}+{\epsilon }_{2}+2\rho ⟩}{C}_{L\left(\mu +{\epsilon }_{1}+{\epsilon }_{2}\right)}^{-1}$ simplifies to $-q-1 q-⟨μ,μ+2ρ⟩ q-⟨ε1+ε2,ε1+ε2+2ρ⟩ q⟨μ+ε1+ε2,μ+ε1+ε2+2ρ⟩ =-q-1q2(μ1+μ2) =-q2m-1,$ where $m={\mu }_{1}+{\mu }_{2}=|\mu |\text{.}$ $\square$

### 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 $\lambda =\left({\lambda }_{1},\dots ,{\lambda }_{n}\right)$ with ${\lambda }_{n}=0\text{.}$ Hence, the graph which describes the ${U}_{h}{𝔰𝔩}_{2}\text{-module}$ decompositions of $L(μ)⊗V⊗k= ⨁λL(λ)⊗ T∼kλ/μ, k∈ℤ≥0 (3.15)$ 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 ${\stackrel{ˆ}{T}}^{/\mu }$ with $vertices on level k: { μ1-μ2+k, μ1-μ2+k-2, …,μ1-μ2-k } ∩ℤ≥0 edges: ℓ⟶ℓ±1. (3.16)$ For example if $m=7$ and ${\mu }_{1}-{\mu }_{2}=3$ then the first few rows of ${\stackrel{ˆ}{T}}^{/\mu }$ are $k=0: k=1: ∅ k=2: k=3: ∅ k=4: (3.17)$ 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.