Commuting families in Hecke and Temperley-Lieb Algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 30 January 2014


Eigenvalues of the Xεi in the affine Hecke algebra

Recall, from (2.8), that the affine Hecke algebra Hk is the quotient of the group algebra of the affine braid group k by the relations Ti2=(q-q-1) Ti+1. (4.1) As observed in Proposition 3.2 the map Φ in (3.3) makes the module L(μ)Vk in (3.11) into an Hk module. Thus the vector spaces Hkλ/μ in (3.11) are the Hk-modules given by Hkλ/μ= FVλ(L(μ)) ,whereFVλ are the Schur functors of (3.5). The following theorem is well known (see, for example, [Che1987]).

(a) The Xεi, 1ik, mutually commute in the affine Hecke algebra Hk.
(b) The eigenvalues of Xεi are given by the graph Hˆ/μ of (3.13) in the sense that if Hˆk/μ = {skew shapesλ/μwithkboxes} and Hˆkλ/μ = {standard tableauxTof shapeλ/μ} for λ/μHˆk/μ, then Hˆk/μ is an index set for the simpleHk modules Hkλ/μ appearing inL(μ) Vk, and Hkλ/μ has a basis {vT|THˆkλ/μ} withXεi vT=q2c(T(i)) vT, where c(T(i)) is the content of box i of T.
(c) κ=Xε1Xεk is a central element of Hk and κacts on Hkλ/μ by the constant q2bλ/μc(b).


(a) is a restatement of (2.4). (b) Since the Hk action and the Uh𝔤𝔩n action commute on L(μ)Vk it follows that the decomposition in (3.11) is a decomposition as (Uh𝔤𝔩n,Hk) bimodules, where the Hkλ/μ are some Hk-modules. Comparing the L(λ) components on each side of λL(λ) Hλ/μ L(μ)V= L(μ)V(-1) V ( νL(ν) H-1ν/μ ) V λλ/ν= L(ν)H-1ν/μ λ ( L(λ) ( λ/ν= H-1ν/μ ) ) gives Hλ/μ λ/ν= H-1ν/μ, (4.2) for any 0 and skew shape λ/μ with boxes. Iterate (4.2) (with =k,k-1,) to produce a decomposition Hkλ/μ= THˆkλ/μ H1T, where the summands H1T are 1-dimensional vector spaces. This determines a basis (unique up to multiplication of the basis vectors by constants) {vT|THˆkλ/μ} of Hkλ/μ which respects the decompositions in (4.2) for 1k.

Combining (3.1), (3.2) and (3.4) gives that Xεi acts on the L(λ) component of the decomposition (3.10) by the constant q λ,λ+2ρ- ν,ν+2ρ- ε1,ε1+2ρ =q2c(λ/ν) since if λ=ν+εj, so that λ is the same as ν except with an additional box in row j, then νλ, λ/ν= 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,for1ik, where T(i) is the box containing i in T.

The remainder of the proof, including the simplicity of the Hk-modules Hkλ/μ, is accomplished as in [Ram0401326, Thm. 4.1].

(c) The element Xε1Xεk is central in k (it is a full twist) and hence its image is central in Hk. The constant describing its action on Hkλ/μ follows from the formula XεivT=q2c(T(i))vT.

Eigenvalues of the mi in Tka

Let m1,m2,,mk 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 mi in the (generically) irreducible representations.

(a) The elements mi, 1ik, mutually commute in Tka.
(b) The eigenvalues of the elements mi are given by the graph Tˆ/μ 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 } 0,and Tˆkλ/μ = { pathsp= ( μ=p(0) p(1) p(k)=λ/μ ) toλ/μ inTˆ/μ } , for λ/μTˆk/μ then Tˆk/μ is an index set for the simpleTka modulesTkλ/μ appearing inL(μ) Vk, and Tkλ/μhas a basis {vp|pTˆkλ/μ} with mivp= { ±[p(i-1)+1] vp, ifp(i-1) ±1=p(i-2)= p(i), 0, otherwise. where p(i) is the partition (a single part in this case) on level i of the path p.
(c) κ=mk+[2]mk-1++[k]m1 is a central element of Tka and κ acts on Tkλ/μ 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] ) .


(a) The elements Xεi commute with one another in the affine Hecke algebra (see (2.4) and the mj are by definition linear combinations of the Xεi (see 2.14), so they commute.

(b) Let p be a path to λ/μ in Tˆμ and let T be the corresponding standard tableau on 2 rows. If p(i)=p(i-1)-1=p(i-2)-2 or if p(i)=p(i-1)+1=p(i-2)+2 then c(T(i-1))=c(T(i))-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(i)=p(i-2)=p(i-1)-1 with T(i-1)=(a,b) then c(T(i))=a and c(T(i-1))=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=|μ|=a+b-i+1. If p(i)=p(i-2)=p(i-1)+1 with T(i-1)=(a,b) then c(T(i-1))=a-1 and c(T(i))=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=|μ|=a+b-i+1.

(c) Let k=|λ/μ|. The identity qλ1+λ2-2 bλ/μ q-2c(b)= ( i=μ2λ2-1 [λ1+λ2-2i] (q-q-1) ) +[k] qμ2-μ1+1, is best visible in an example: With λ=(10,6) and μ=(4,2), 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 mk+[2]mk-1++[k]m1 acts on Tkλ/μ by the constant (q-q-1)-1 qk-2 bλ/μ q-2c(b) = (q-q-1)-1 q-(μ1+μ2) qλ1+λ2-2 bλ/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 μ1+μ2=m, μ1-μ2=p(0), λ1+λ2=m+k and λ1-λ2=p(k).

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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