## Commuting families in Hecke and Temperley-Lieb Algebras

Last update: 30 January 2014

## Eigenvalues

### Eigenvalues of the ${X}^{{\epsilon }_{i}}$ in the affine Hecke algebra

Recall, from (2.8), that 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 $Ti2=(q-q-1) Ti+1. (4.1)$ As observed in Proposition 3.2 the map $\Phi$ in (3.3) makes the module $L\left(\mu \right)\otimes {V}^{\otimes k}$ in (3.11) into an ${\stackrel{\sim }{H}}_{k}$ module. Thus the vector spaces ${\stackrel{\sim }{H}}_{k}^{\lambda /\mu }$ in (3.11) are the ${\stackrel{\sim }{H}}_{k}\text{-modules}$ given by $H∼kλ/μ= FVλ(L(μ)) ,where FVλ are the Schur functors of (3.5).$ The following theorem is well known (see, for example, [Che1987]).

 (a) The ${X}^{{\epsilon }_{i}},$ $1\le i\le k,$ mutually commute in the affine Hecke algebra ${\stackrel{\sim }{H}}_{k}\text{.}$ (b) The eigenvalues of ${X}^{{\epsilon }_{i}}$ are given by the graph ${\stackrel{ˆ}{H}}^{/\mu }$ of (3.13) in the sense that if $Hˆk/μ = {skew shapes λ/μ with k boxes} and Hˆkλ/μ = {standard tableaux T of shape λ/μ}$ for $\lambda /\mu \in {\stackrel{ˆ}{H}}_{k}^{/\mu },$ then $Hˆk/μ is an index set for the simple H∼k modules H∼kλ/μ appearing in L(μ)⊗ V⊗k,$ and $H∼kλ/μ has a basis {vT | T∈Hˆkλ/μ} withXεi vT=q2c(T(i)) vT,$ where $c\left(T\left(i\right)\right)$ is the content of box $i$ of $T\text{.}$ (c) $\kappa ={X}^{{\epsilon }_{1}}\cdots {X}^{{\epsilon }_{k}}$ is a central element of ${\stackrel{\sim }{H}}_{k}$ and $κ acts on H∼kλ/μ by the constant q2∑b∈λ/μc(b).$

 Proof. (a) is a restatement of (2.4). (b) Since the ${\stackrel{\sim }{H}}_{k}$ action and the ${U}_{h}{𝔤𝔩}_{n}$ action commute on $L\left(\mu \right)\otimes {V}^{\otimes k}$ it follows that the decomposition in (3.11) is a decomposition as $\left({U}_{h}{𝔤𝔩}_{n},{\stackrel{\sim }{H}}_{k}\right)$ bimodules, where the ${\stackrel{\sim }{H}}_{k}^{\lambda /\mu }$ are some ${\stackrel{\sim }{H}}_{k}\text{-modules.}$ Comparing the $L\left(\lambda \right)$ components on each side of $⨁λL(λ)⊗ H∼ℓλ/μ ≅ L(μ)⊗V⊗ℓ= L(μ)⊗V⊗(ℓ-1) ⊗V≅ ( ⨁νL(ν)⊗ H∼ℓ-1ν/μ ) ⊗V ≅ ⨁λ⨁λ/ν=▫ L(ν)⊗H∼ℓ-1ν/μ ≅⨁λ ( L(λ)⊗ ( ⨁λ/ν=▫ H∼ℓ-1ν/μ ) )$ gives $H∼ℓλ/μ≅ ⨁λ/ν=▫ H∼ℓ-1ν/μ, (4.2)$ for any $\ell \in {ℤ}_{\ge 0}$ and skew shape $\lambda /\mu$ with $\ell$ boxes. Iterate (4.2) (with $\ell =k,k-1,\dots \text{)}$ to produce a decomposition $H∼kλ/μ= ⨁T∈Hˆkλ/μ H∼1T,$ where the summands ${\stackrel{\sim }{H}}_{1}^{T}$ are 1-dimensional vector spaces. This determines a basis (unique up to multiplication of the basis vectors by constants) $\left\{{v}_{T} | T\in {\stackrel{ˆ}{H}}_{k}^{\lambda /\mu }\right\}$ of ${\stackrel{\sim }{H}}_{k}^{\lambda /\mu }$ which respects the decompositions in (4.2) for $1\le \ell \le k\text{.}$ Combining (3.1), (3.2) and (3.4) gives that ${X}^{{\epsilon }_{i}}$ acts on the $L\left(\lambda \right)$ component of the decomposition (3.10) by the constant $q ⟨λ,λ+2ρ⟩- ⟨ν,ν+2ρ⟩- ⟨ε1,ε1+2ρ⟩ =q2c(λ/ν)$ since if $\lambda =\nu +{\epsilon }_{j},$ so that $\lambda$ is the same as $\nu$ except with an additional box in row $j,$ then $\nu \subseteq \lambda ,$ $\lambda /\nu =▫$ and $⟨λ,λ+2ρ⟩- ⟨ν,ν+2ρ⟩- ⟨ε1,ε1+2ρ⟩ = ⟨ν+εj,ν+εj+2ρ⟩ -⟨ν,ν+2ρ⟩ -(1+2(n-1)) = 2νj+⟨εj,εj+2ρ⟩ -2n+1=2νj+ (1+2(n-j))-2n+1 = 2(νj+1)-2j=2c (λ/ν).$ Hence, $XεivT= q2c(T(i)) vT,for 1≤i≤k,$ where $T\left(i\right)$ is the box containing $i$ in $T\text{.}$ The remainder of the proof, including the simplicity of the ${\stackrel{\sim }{H}}_{k}\text{-modules}$ ${\stackrel{\sim }{H}}_{k}^{\lambda /\mu },$ is accomplished as in [Ram0401326, Thm. 4.1]. (c) The element ${X}^{{\epsilon }_{1}}\cdots {X}^{{\epsilon }_{k}}$ is central in ${\stackrel{\sim }{ℬ}}_{k}$ (it is a full twist) and hence its image is central in ${\stackrel{\sim }{H}}_{k}\text{.}$ The constant describing its action on ${\stackrel{\sim }{H}}_{k}^{\lambda /\mu }$ follows from the formula ${X}^{{\epsilon }_{i}}{v}_{T}={q}^{2c\left(T\left(i\right)\right)}{v}_{T}\text{.}$ $\square$

### Eigenvalues of the ${m}_{i}$ in ${T}_{k}^{a}$

Let ${m}_{1},{m}_{2},\dots ,{m}_{k}$ be the commuting family in the affine Temperley-Lieb algebra as defined in (2.14). We will use the results of Theorem 4.1 to determine the eigenvalues of the ${m}_{i}$ in the (generically) irreducible representations.

 (a) The elements ${m}_{i},$ $1\le i\le k,$ mutually commute in ${T}_{k}^{a}\text{.}$ (b) The eigenvalues of the elements ${m}_{i}$ are given by the graph ${\stackrel{ˆ}{T}}^{/\mu }$ of (3.17) in the sense that if the set of vertices on level $k$ is $Tˆk/μ = { μ1-μ2+k, μ1-μ2+k-2, …,μ1-μ2-k } ∩ℤ≥0, and Tˆkλ/μ = { paths p= ( μ=p(0)→ p(1)→⋯→ p(k)=λ/μ ) to λ/μ in Tˆ/μ } ,$ for $\lambda /\mu \in {\stackrel{ˆ}{T}}_{k}^{/\mu }$ then $Tˆk/μ is an index set for the simple Tka modules Tkλ/μ appearing in L(μ) ⊗V⊗k,$ and $Tkλ/μhas a basis {vp | p∈Tˆkλ/μ}$ with $mivp= { ±[p(i-1)+1] vp, if p(i-1) ±1=p(i-2)= p(i), 0, otherwise.$ where ${p}^{\left(i\right)}$ is the partition (a single part in this case) on level $i$ of the path $p\text{.}$ (c) $\kappa ={m}_{k}+\left[2\right]{m}_{k-1}+\cdots +\left[k\right]{m}_{1}$ is a central element of ${T}_{k}^{a}$ and $\kappa$ acts on ${T}_{k}^{\lambda /\mu }$ by the constant $[k] q-(μ1+μ2)-(μ1-μ2)+1 q-q-1 +q-(μ1+μ2) ( [λ1-λ2+2]+ [λ1-λ2+4]+⋯+ [μ1-μ2+k] ) .$

 Proof. (a) The elements ${X}^{{\epsilon }_{i}}$ commute with one another in the affine Hecke algebra (see (2.4) and the ${m}_{j}$ are by definition linear combinations of the ${X}^{{\epsilon }_{i}}$ (see 2.14), so they commute. (b) Let $p$ be a path to $\lambda /\mu$ in ${\stackrel{ˆ}{T}}^{\mu }$ and let $T$ be the corresponding standard tableau on 2 rows. If ${p}^{\left(i\right)}={p}^{\left(i-1\right)}-1={p}^{\left(i-2\right)}-2$ or if ${p}^{\left(i\right)}={p}^{\left(i-1\right)}+1={p}^{\left(i-2\right)}+2$ then $c\left(T\left(i-1\right)\right)=c\left(T\left(i\right)\right)-1$ and, from (2.14) and Theorem 4.1(b), $mivT=qi-2 q-2c(T(i)) -q-2 q-2c(T(i-1)) q-q-1 vT=qi-2 q-2c(T(i)) -q-2 q-2c(T(i))+2 q-q-1 vT=0.$ If ${p}^{\left(i\right)}={p}^{\left(i-2\right)}={p}^{\left(i-1\right)}-1$ with ${T}^{\left(i-1\right)}=\left(a,b\right)$ then $c\left(T\left(i\right)\right)=a$ and $c\left(T\left(i-1\right)\right)=b-2$ and $mivT=qi-2 q-2a-q-2 q-2b+4 q-q-1 vT=qi q-(a+b+1) ( q-(a-b+1)- q(a-b+1) ) q-q-1 =-q-m [a-b+1]vT,$ where $m=|\mu |=a+b-i+1\text{.}$ If ${p}^{\left(i\right)}={p}^{\left(i-2\right)}={p}^{\left(i-1\right)}+1$ with ${T}^{\left(i-1\right)}=\left(a,b\right)$ then $c\left(T\left(i-1\right)\right)=a-1$ and $c\left(T\left(i\right)\right)=b-1$ and $mivT=qi-2 q-2b+2- q-2q-2a+2 q-q-1 vT=qi q-(a+b+1) ( q(a-b+1)- q-(a-b+1) ) q-q-1 =q-m [a-b+1]vT,$ where $m=|\mu |=a+b-i+1\text{.}$ (c) Let $k=|\lambda /\mu |\text{.}$ The identity $qλ1+λ2-2 ∑b∈λ/μ q-2c(b)= ( ∑i=μ2λ2-1 [λ1+λ2-2i] (q-q-1) ) +[k] qμ2-μ1+1,$ is best visible in an example: With $\lambda =\left(10,6\right)$ and $\mu =\left(4,2\right),$ $q16-2 ( +0 +0 +0 +0 +q-8 +q-10 +q-12 +q-14 +q-16 +q-18 +0 +0 +q-2 +q-4 +q-6 +q-8 ) = +0 +0 +0 +0 +q6 +q4 +q2 +q- +q-2 +q-4 +0 +0 +q12 +q10 +q8 +q6 ( +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +(q12-q-12) +(q10-q-10) +(q8-q-8) +(q6-q-6) ) + ( +0 +0 +0 +0 +q-4 +q-2 +q0 +q2 +q4 +q6 +0 +0 +q-12 +q-10 +q-8 +q-6 ) = ( ∑i=26-1 q16-2i- q-(16-2i) ) +[10] q4-2+1.$ Then Proposition 2.6 says $X-ε1+⋯+ X-εk= q-(k-2) (q-q-1) ( mk+[2] mk-1+⋯+ [k]m1 ) ,$ and so ${m}_{k}+\left[2\right]{m}_{k-1}+\cdots +\left[k\right]{m}_{1}$ acts on ${T}_{k}^{\lambda /\mu }$ by the constant $(q-q-1)-1 qk-2 ∑b∈λ/μ q-2c(b) = (q-q-1)-1 q-(μ1+μ2) qλ1+λ2-2 ∑b∈λ/μ q-2c(b) = (q-q-1)-1 q-(μ1+μ2) ( [k] qμ2-μ1+1+ ∑i=μ2λ2-1 [λ1+λ2-2i] (q-q-1) ) = [k] q-m-p(0)+1 q-q-1 +∑i=μ2λ2-1 q-m[m+k-2i] = [k] q-m-p(0)+1 q-q-1 +q-m ( [p(k)+2]+ [p(k)+4]+⋯+ [p(0)+k-2]+ [p(0)+k] ) ,$ since ${\mu }_{1}+{\mu }_{2}=m,$ ${\mu }_{1}-{\mu }_{2}={p}^{\left(0\right)},$ ${\lambda }_{1}+{\lambda }_{2}=m+k$ and ${\lambda }_{1}-{\lambda }_{2}={p}^{\left(k\right)}\text{.}$ $\square$

## 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.