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

Schur functors

R-matrices and quantum Casimir Elements

Let Uh𝔤 be the Drinfeld-Jimbo quantum group corresponding to a finite dimensional complex semisimple Lie algebra 𝔤. We shall use the notations and conventions for Uh𝔤 as in [LRa1977] and [ORa0401317]. There is an invertible element =aibi in (a suitable completion of) Uh𝔤Uh𝔤 such that, for two Uh𝔤 modules M and N, the map ŘMN: MN NM mn binaim M N N M is a Uh𝔤 module isomorphism. In order to be consistent with the graphical calculus these operators should be written on the right. The element satisfies “quasitriangularity relations” (see [LRa1977, (2.1-2.3)]) which imply that, for Uh𝔤 modules M,N,P and a Uh𝔤 module isomorphism τM:MM, M N N M τM = M N N M τM ŘMN (idNτM) = (τMidN) ŘMN, M(NP) (NP)M = M N P P N M (MN)P P(MN) = M N P P N M ŘM,NP = (ŘMNidP) (idNŘMP) ŘMN,P = (idMŘNP) (ŘMPidN), which, together, imply the braid relation M N P P N M = M N P P N M (ŘMNidP) (idNŘMP) (ŘNPidM) = (idMŘNP) (ŘMPidN) (idPŘMN),

Let ρ be such that ρ,αi=1 for all simple roots αi. As explained in [LRa1977, (2.14)] and [Dri1990], there is a quantum Casimir element e-hρu in the center of Uh𝔤 and, for a Uh𝔤 module M we define a Uh𝔤 module isomorphism CM: M M m (e-hρu)m M M CM and the elements CM satisfy CMN= (ŘMNŘNM)-1 (CMCN),and CM= q-λ,λ+2ρ idM (3.1) if M is a Uh𝔤 module generated by a highest weight vector v+ of weight λ (see [LRa1977, Prop. 2.14] or [Dri1990, Prop. 3.2]). Note that λ,λ+2ρ=λ+ρ,λ+ρ-ρ,ρ are the eigenvalues of the classical Casimir operator [Dix1996, 7.8.5]. From the relation (3.1) it follows that if M=L(μ), N=L(ν) are finite dimensional irreducible Uh𝔤 modules then ŘMNŘNM acts on the λ isotypic component L(λ)cμνλ 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 k module MVk

Let Uh𝔤 be a quantum group and let M and V be U-modules such that the operators ŘMV, ŘVM and ŘVV are well defined. Define Ři, 1ik-1, and Ř02 in EndU(MVk) by Ři=idM idV(i-1) ŘVV idV(k-i-1) andŘ02= (ŘMVŘVM) idV(k-1). Then the braid relations ŘiŘi+1Ři = = =Ři+1Ři Ři+1 and Ř02Ř1Ř02Ř1= = = = =Ř1Ř02Ř1Ř02. imply that there is a well defined map Φ: k EndU(MVk) Ti Ři, Xε1 Ř02, 1ik-1, (3.3) which makes MVk into a right k module. By (3.1) and the fact that Φ(Xεi)= ŘMV(i-1),V ŘV,MV(i-1)= i . (3.4) the eigenvalues of Φ(Xεi) are related to the eigenvalues of the Casimir. The Schur functors are the functors FVλ: {U-modules} {k-modules} M HomU(M(λ),MVk) (3.5) where HomU(M(λ),MVk) is the vector space of highest weight vectors of weight λ in MVk.

The quantum group Uh𝔤𝔩n

Although the Lie algebra 𝔤𝔩n 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 αi=εi-εi+1, 1in-1, and ρ=(n-1)ε1+ (n-2)ε2++ εn-1. (3.6) The dominant integral weights of 𝔤𝔩n are λ=λ1ε1++ λnεn,where λ1λ2 λn,andλ1 ,,λn and these index the simple finite dimensional Uh𝔤𝔩n-modules L(λ). A partition with n rows is a dominant integral weight with λn0. If λn<0 and Δ denotes the 1-dimensional “determinant” representation of Uh𝔤𝔩n then (see [FHa1991, §15.5]) L(λ)Δλn L(λ+(-λn,,-λn)) withλ+(-λn,,-λn) a partition. (3.7)

Identify each partition λ with the configuration of boxes which has λi boxes in row i. For example, λ= =5ε1+5ε2 +3ε3+3ε4 +ε5+ε6. (3.8) If μ and λ are partitions with μλ (as collections of boxes) then the skew shape λ/μ is the collection of boxes of λ that are not in μ. For example, if λ is as in (3.8) and μ= thenλ/μ= . If b is a box in position (i,j) of λ the content of b is c(b)=j-i= the diagonal number ofb,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 ν is a partition and V=L()then L(ν)V= λν+ L(λ), (3.10) where the sum is over all partitions λ with n rows that are obtained by adding a box to ν [Mac1995, I App. A (8.4) and I (5.16)], Hence, the Uh𝔤𝔩n-module decompositions of L(μ)Vk= λL(λ) Hkλ/μ, k0, (3.11) are encoded by the graph Hˆ/μ with vertices on levelk: {skew shapesλ/μwithkboxes} edges: λ/μγ/μ, if γis obtained fromλ by adding a box labels on edges: content of the added box. (3.12) For example if μ=(3,3,3,2)= , then the first 4 rows of Hˆ/μ 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=Uh𝔤𝔩n and V=L(ε1)=L() is the n-dimensional “standard” representation of 𝔤𝔩n then the map Φ of (3.3) factors through the surjective homomorphism (2.8) to give a representation of the affine Hecke algebra.

For a skew shape λ/μ with k boxes identify paths from μ to λ/μ in Hˆ/μ with standard tableaux of shape λ/μ by filling the boxes, successively, with 1,2,,k as they appear. In the example graph Hˆ/μ above 1 2 3 corresponds to the path

The quantum group Uh𝔤𝔩2

In the case when n=2, U=Uh𝔤𝔩2 and the partitions which appear in (3.11) and in the graph Hˆ/μ all have 2 rows. For example if μ=(42)= then the first few rows of Hˆ/μ 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=Uh𝔤𝔩2, M=L(μ) where μ is a partition of m with 2 rows, and V=L(ε1)=L() is the 2-dimensional "standard" representation of 𝔤𝔩2 then the map Φ of (3.3) factors through the surjective homomorphism of (2.13) with α2=-q2m-1 to give a representation of the affine Temperley-Lieb algebra Tka.


The proof that the kernel of Φ contains the element (2.12) is exactly as in the proof of [ORa0401317, Thm. 6.1(c)]: The element e1 in T2a acts on V2 as (q+q-1)·pr where pr is the unique Uh𝔤-invariant projection onto L ( ) in V2. Using e1T1=-q-1e1 and the pictorial equalities = =-q-1· it follows that Φ2(e1Xε1T1Xε1) acts as -(q+q-1)q-1· Ř L ( ) ,L(μ) Ř L(μ),L ( ) (idL(μ)pr) . 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-μ,μ+2ρ q-ε1+ε2,ε1+ε2+2ρ CL(μ+ε1+ε2)-1 simplifies to -q-1 q-μ,μ+2ρ q-ε1+ε2,ε1+ε2+2ρ qμ+ε1+ε2,μ+ε1+ε2+2ρ =-q-1q2(μ1+μ2) =-q2m-1, where m=μ1+μ2=|μ|.

The quantum group Uh𝔰𝔩2

The restriction of an irreducible representation L(λ) of Uh𝔤𝔩n to Uh𝔰𝔩n is irreducible and all irreducible representations of Uh𝔰𝔩n are obtained in this fashion. Since the “determinant” representation is trivial as an Uh𝔰𝔩n module it follows from (3.7) that the irreducible representations L𝔰𝔩n(λ) of Uh𝔰𝔩n are indexed by partitions λ=(λ1,,λn) with λn=0. Hence, the graph which describes the Uh𝔰𝔩2-module decompositions of L(μ)Vk= λL(λ) Tkλ/μ, k0 (3.15) is exactly the same as the graph for Uh𝔤𝔩2 except with all columns of length 2 removed from the partitions. More precisely, the decompositions are encoded by the graph Tˆ/μ with vertices on levelk: { μ1-μ2+k, μ1-μ2+k-2, ,μ1-μ2-k } 0 edges: ±1. (3.16) For example if m=7 and μ1-μ2=3 then the first few rows of Tˆ/μ 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 λ/μ.

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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