Calibrated representations of two boundary Temperley-Lieb algebras
aa r X i v : . [ m a t h . R T ] S e p CALIBRATED REPRESENTATIONS OF TWO BOUNDARYTEMPERLEY-LIEB ALGEBRAS
ZAJJ DAUGHERTY AND ARUN RAM
In memory of a friend and an inspiration, Vladimir Rittenberg 1934-2018
Abstract.
The two boundary Temperley-Lieb algebra
T L k arises in the transfer matrix for-mulation of lattice models in Statistical Mechanics, in particular in the introduction of inte-grable boundary terms to the six-vertex model. In this paper, we classify and study the cali-brated representations—those for which all the Murphy elements (integrals) are simultaneouslydiagonalizable—which, in turn, corresponds to diagonalizing the transfer matrix in the associatedmodel. Our approach is founded upon the realization of T L k as a quotient of the type C k affineHecke algebra H k . In previous work, we studied this Hecke algebra via its presentation by braiddiagrams, tensor space operators, and related combinatorial constructions. That work is directlyapplied herein to give a combinatorial classification and construction of all irreducible calibrated T L k -modules and explain how these modules also arise from a Schur-Weyl duality with the quantumgroup U q gl . Contents
1. Introduction 12. The two boundary Hecke algebra H k
33. The two boundary Temperley-Lieb algebra
T L k H ext k and T L ext k T L ext k and U q gl Introduction
The paper [DR] studied the calibrated representations of affine Hecke algebras of type C withunequal parameters and developed their combinatorics and their role in Schur-Weyl duality. Thispaper applies that information to the study of two boundary Temperley-Lieb algebras. The twoboundary Temperley-Lieb algebras appear in statistical mechanics for analysis of spin chains withgeneralized boundary conditions [GP, GNPR]. T he spectrum of the Hamiltonian for these spinchains with boundaries can be determined via the representation theory of the two boundaryTemperley-Lieb algebras. In fact, the need to understand the representation theory of the twoboundary Temperley-Lieb algebra better was a primary motivation for our preceding papers [Dau,DR] on two boundary Hecke algebras.
Department of Mathematics, The City College of New York, NAC 8/133, New York, NY 10031Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010 Australia
E-mail address : [email protected], [email protected] . Date : September 8, 2020.
In the first part of this paper, Section 2, we review the definition and structure of the twoboundary Hecke algebra H k (the affine Hecke algebra of type C with unequal parameters). Followingthis brief review we carefully analyze certain idempotents which, as we prove in Theorem 3.1,generate the ideal that one must quotient by to obtain the two boundary Temperley-Lieb algebrafrom the two boundary Hecke algebra. It is the expression of these idempotents in terms of theintertwiner presentation of H k (see Proposition 2.3) that will eventually provide understanding ofthe weights that can appear in two boundary Temperley-Lieb modules (the possible eigenvalues ofthe “Murphy elements” W i —see equation (2.9) and § symplectic blob algebra ) T L k following [GN, GMP07, GMP08, GMP12, Ree, KMP16, GMP17], and review the diagramalgebra calculus for these algebras. Part of our contribution is to extend this calculus to make itsconnection to the diagrammatic calculus of the Hecke algebra H k via braids. In Theorem 3.2 weuse these diagrammatics to give a proof of a result of [GN] that provides an expansion of a certaincentral element of H k inside T L k . Using the Hecke algebra point of view, this result enables us tounderstand that the center of T L k is a polynomial ring in one variable Z ( T L k ) = C [ Z ], and that T L k is of finite rank over this center. In retrospect, the algebra H k has a similar structure and soperhaps this should not be surprising but, nonetheless, it is pleasant to see it come out in such avivid and explicit form.We have used a different normalization of the parameters of the two boundary Hecke andTemperley-Lieb algebra from those used in [GN, GMP12]. Our normalization will be helpful,for example, for future applications of these algebras to the theory of Macdonald polynomials andto the study of the exotic nilpotent cone. In both of these cases the affine Hecke algebra of type C n plays an important role: the Koornwinder polynomials are the Macdonald polynomials for type( C ∨ n , C n ) [M03], and the K-theory of the Steinberg variety of the exotic nilpotent cone providesa geometric construction of the representations of the two boundary Hecke and Temperley-Liebalgebras at unequal parameters (see [Kat]).The calibrated representations are the irreducible representations of the two boundary Heckealgebra for which a large family of commuting operators (integrals, or Murphy elements) have asimple (joint) spectrum. This property makes these representations particularly attractive, andthe detailed combinatorics of these representations has been worked out in [DR]. In Section 4we use the detailed analysis of the idempotents done in Section 2 to determine exactly whichcalibrated irreducible representations of the two boundary Hecke algebra are representations of thetwo boundary Temperley-Lieb algebra (Theorem 4.3). In consequence, we obtain a full classificationof the calibrated irreducible representations of the two boundary Temperley-Lieb algebras.As explained in [DR], there is a Schur-Weyl type duality between the two boundary Heckealgebra and the quantum group U q gl n . The classical Schur-Weyl duality between U q gl n and thefinite Hecke algebra of type A becomes a Schur-Weyl duality for the finite Temperley-Lieb algebrawhen n = 2. In Theorem 5.1 we show that at n = 2 the Schur-Weyl duality of [DR] gives a Schur-Weyl duality for the two boundary Temperley-Lieb algebra. This method (coming from R-matricesfor the quantum group U q gl ) provides many many irreducible calibrated representations of the twoboundary Temperley-Lieb algebra T L k . Using our results from Section 4, we determine exactlywhich irreducible calibrated representations of T L k occur in the Schur-Weyl duality context.The seeds of this work were sown in a conversation between Pavel Pyatov, Arun Ram andVladimir Rittenberg at the Max Planck Institut in Bonn in 2006. Vladimir was the leader andprovided the inspiration by introducing us to spin chains with boundaries. The seed has now grownfrom a concept into fully formed and fruitful mathematics. We thank all the institutions which have ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 3 supported our work on this paper, particularly the University of Melbourne, the Australian Re-search Council (grants DP1201001942 and DP130100674), the National Science Foundation (grantDMS-1162010), the Simons Foundation (grant
The two boundary Hecke algebra H k The two boundary Hecke algebra is often called the affine Hecke algebra of type (
C, C ∨ ). In thissection we review the definitions of H ext k following our previous paper [DR]. In particular, we willneed the basic diagrammatics and the “Bernstein” presentation with a Laurent polynomial ring C [ W ± , . . . , W ± k ] and intertwiners τ , . . . , τ k . After this review we define the idempotent elements p (1 ) i , p ( ∅ , )0 , p (1 , ∅ )0 , p ( ∅ , )0 ∨ , p (1 , ∅ )0 ∨ , which we will need to quotient by in order to obtain the twoboundary Temperley-Lieb algebra. We derive expressions of these elements in terms of the differentchoices of generators: the braid generators T i , the cap/cup generators e i , and the intertwinergenerators τ i and W j .2.1. Graph notation for braid relations.
For generators g, h , encode relations graphically by g h means gh = hg , g h means ghg = hgh , and g h means ghgh = hghg . (2.1)For example, the group of signed permutations, W = (cid:26) bijections w : {− k, . . . , − , , . . . , k } → {− k, . . . , − , , . . . , k } such that w ( − i ) = − w ( i ) for i = 1 , . . . , k (cid:27) , (2.2)has a presentation by generators s , s , . . . , s k − , with relations s s k − s s k − s and s i = 1 for i = 0 , , , . . . , k −
1. (2.3)2.2.
The two boundary braid group.
The two boundary braid group is the group B k generatedby ¯ T , ¯ T , . . . , ¯ T k , with relations¯ T ¯ T ¯ T ¯ T k − ¯ T k − ¯ T k . (2.4)Pictorially, the generators of B k are identified with the braid diagrams¯ T k = , ¯ T = , and¯ T i = ii i +1 i +1 for i = 1 , . . . , k −
1, (2.5)
ZAJJ DAUGHERTY AND ARUN RAM and the multiplication of braid diagrams is given by placing one diagram on top of another (mul-tiplying generators left-to-right corresponds to stacking diagrams top-to-bottom).In some applications (notably to the Schur-Weyl duality of [DR, § σ = . (2.6)Define T i = σ ¯ T i σ − = ii i +1 i +1 , Y = σ ¯ T σ − = , (2.7)and X = T − T − · · · T − k − σ ¯ T k σ − T k − · · · T = . (2.8)Define Z = X Y and Z i = T i − T i − · · · T X Y T · · · T i − = ii , (2.9)for i = 2 , . . . , k . Let P = . The extended affine braid group is the group B ext k generated by B k and P with the additionalrelations P X P − = Z − X Z , P Y P − = Z − Y Z , (2.10) P Z P − = Z , and P T i P − = T i for i = 1 , . . . , k −
1. (2.11)The element Z = P Z · · · Z k is central in B ext k (c0)since the group B ext k is a subgroup of the braid group on k + 2 strands, and Z is the generator ofthe center of the braid group on k + 2 strands (see [GM, Theorem 4.2]). Soif D = { Z j | j ∈ Z } then B ext k = D × B k , with D ∼ = Z . (2.12)2.3. The extended affine Hecke algebra H ext k of type C k . Fix a , a , b , b , t ∈ C × and let t k = a ( − a ) − , t = b ( − b ) − . (2.13)The extended two boundary Hecke algebra H ext k with parameters t , t and t k is the quotient of B ext k by the relations( X − a )( X − a ) = 0 , ( Y − b )( Y − b ) = 0 , and ( T i − t )( T i + t − ) = 0 , (H) ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 5 for i = 1 , . . . , k −
1. Let T = b − ( − b ) − Y , T k = a − ( − a ) − T k − · · · T T X T − T − · · · T − k − . (2.14)Then T T T T k − T k − T k and( T − t )( T + t − ) = 0 , ( T i − t )( T i + t − ) = 0 , ( T k − t k )( T k + t − k ) = 0 , (2.15)for i ∈ { , . . . , k − } .Let a, a , a k ∈ C × and define a e = T − t , ae i = T i − t , a k e k = T k − t k , (2.16)for i ∈ { , . . . , k − } . The relations in (2.15) are equivalent to T e = − t − e , T i e i = − t − e i , T k e k = − t − k e k , (2.17)and to e = − ( t + t − ) a e , e i = − ( t + t − ) a e i , e k = − ( t k + t − k ) a k e k , (2.18)for i ∈ { , . . . , k − } . Remark 2.1.
For i ∈ { , . . . , k − } , using T i = ae i + t to expand T i T i +1 T i and T i +1 T i T i +1 interms of the e i shows that in the presence of the relations (H), T i T i +1 T i = T i +1 T i T i +1 is equivalent to a e i e i +1 e i − ae i = a e i +1 e i e i +1 − ae i +1 . Similarly, T T T T = T T T T is equivalent to a a e e e e − a a ( t − t + t t − ) e e = a a e e e e − a a ( t − t + t t − ) e e . In the case that a a = a a ( t − t + t t − ) then T T T T = T T T T is equivalent to e e e e − e e = e e e e − e e . In the case that a = a then T i T i +1 T i = T i +1 T i T i +1 is equivalent to e i e i +1 e i − e i = e i +1 e i e i +1 − e i +1 . This is the explanation for why the favorite choices of a , a and a k satisfy a = ± , a a = t − t + t t − = [[ t t − ]] and a k a = t − k t + t k t − = [[ t k t − ]] , where we use the notation[[ t s ]] = ( t s + t − s ) = (cid:16) t s − t − s t − t − (cid:17)(cid:16) t − t − t s − t − s (cid:17) = [2 s ][ s ] . (2.19) ZAJJ DAUGHERTY AND ARUN RAM
The Bernstein presentation of H ext k . The
Murphy elements for H ext k are W = T − T − · · · T − k − T k T k − · · · T T T and W j = T j W j − T j , for j ∈ { , . . . , k } . Let W = P W · · · W k . Theorem 2.2. (See, for example, [DR, Theorem 2.2] .) Fix t , t k , t ∈ C × and use notations forrelations as defined in (2.1) . The extended affine Hecke algebra H ext k defined in (H) is presented bygenerators, T , T , . . . , T k − , W , W , . . . , W k and relations W ∈ Z ( H ext k ) , T T k − T T k − T ; (B1) W i W j = W j W i , for i, j = 0 , , . . . , k ; (B2) T W j = W j T , for j = 1 ; (B3) T i W j = W j T i for i = 1 , . . . , k − and j = 1 , . . . , k with j = i, i + 1 ; (B4)( T − t )( T + t − ) = 0 , and ( T i − t )( T i + t − ) = 0 for i = 1 , . . . , k − ; (H) for i = 1 , . . . , k − , T i W i = W i +1 T i + ( t − t − ) W i − W i +1 − W i W − i +1 , T i W i +1 = W i T i + ( t − t − ) W i +1 − W i − W i W − i +1 , (C1) and T W = W − T + (cid:18) ( t − t − ) + ( t k − t − k ) W − (cid:19) W − W − − W − . (C2)The two boundary Hecke algebra H k with parameters t , t and t k is the subalgebra of H ext k generated by T , T , . . . , T k . Then H ext k = H k ⊗ C [ W ± ] as algebras, (2.20)and, as proved for example in [DR, Theorem 2.3], the element Z = W + W − + W + W − + · · · + W k + W − k is central in H ext k . (2.21)2.5. The elements τ i . Define τ = T − ( t − t − ) + ( t k − t − k ) W − − W − , and τ i = T i − t − t − − W i W − i +1 , (2.22)for i ∈ { , . . . , k − } . Evoking the notation of [DR, § § f ε i = (1 − W − i )(1 + W − i ) = 1 − W − i ,f ε i − r = (1 − t t k W − i ) , f ε i − r = (1 + t t − k W − i ) ,f − ε i − r = (1 − t t k W i ) , f − ε i − r = (1 + t t − k W i ) ,f ε i − ε j = 1 − W i W − j , f ε i − ε j +1 = 1 − tW i W − j , (2.23)for i, j ∈ { , . . . , k } . Then a e = τ − t − f ε − r f ε − r f ε and a i e i = τ i − t − f ε i − ε i +1 +1 f ε i − ε i +1 , (2.24) ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 7 and, as proved in [DR, Proposition 2.4], τ τ k − τ τ k − τ and τ = W − t − f ε − r f − ε − r f ε − r f − ε − r f ε , W τ = τ W − , W r τ = τ W r ,τ i = t − f ε i − ε i +1 +1 f ε i +1 − ε i +1 f ε i − ε i +1 f ε i +1 − ε i , W i τ i = τ i W i +1 , W i +1 τ i = τ i W i , W j τ i = τ i W j , for r, j ∈ { , . . . , k } with r = 1 and j = i, i + 1.2.6. The elements p (1 ) i , p ( ∅ , )0 and p (1 , ∅ )0 . Fix i ∈ { , . . . , k − } . Let HS be the subalgebra of H ext k generated by T i and T i +1 , and let HB be the subalgebra of H ext k generated by T and T .The idempotent p (1 ) i in HS and the idempotents p ( ∅ , )0 and p (1 , ∅ )0 in HB are uniquely determinedby the equations ( p (1 ) i ) = p (1 ) i , ( p ( ∅ , )0 ) = p ( ∅ , )0 ( p (1 , ∅ )0 ) = p (1 , ∅ )0 , (2.25)and T i p (1 ) i = − t − p (1 ) i , T i +1 p (1 ) i = − t − p (1 ) i ,T p ( ∅ , )0 = − t − p ( ∅ , )0 T p ( ∅ , )0 = − t − p ( ∅ , )0 ,T p (1 , ∅ )0 = t p (1 , ∅ )0 , T p (1 , ∅ )0 = − t − p (1 , ∅ )0 . (2.26)The conditions in (2.26) are equivalent to ae i p (1 ) i = − ( t + t − ) p (1 ) i , ae i +1 p (1 ) i = − ( t + t − ) p (1 ) i ,a e p ( ∅ , )0 = − ( t + t − ) p ( ∅ , )0 , ae p ( ∅ , )0 = − ( t + t − ) p ( ∅ , )0 ,a e p (1 , ∅ )0 = 0 , ae p (1 , ∅ )0 = − ( t + t − ) p (1 , ∅ )0 . (2.27) Proposition 2.3.
Let p (1 ) i , p ( ∅ , )0 and p (1 , ∅ )0 be as defined in (2.25) and (2.26) and let N = t − (1 + t )(1 + t + t ) and N = N ′ = t − t − (1 + t )(1 + t )(1 + t t ) . Then the expansions of these idempotents in terms of the three favored generating sets is given by
N p (1 ) i = T i T i +1 T i − t T i T i +1 − t T i +1 T i + tT i + tT i +1 − t = a e i e i +1 e i − ae i = a e i +1 e i e i +1 − ae i +1 = τ i τ i +1 τ i − t − τ i +1 τ i f ε i +1 − ε i +2 +1 f ε i +1 − ε i +2 − t − τ i τ i +1 f ε i +1 − ε i +1 f ε i +1 − ε i + t − τ i f ε i +1 − ε i +2 +1 f ε i +2 − ε i +1 f ε i +1 − ε i +2 f ε i +2 − ε i + t − τ i +1 f ε i +2 − ε i +1 f ε i +1 − ε i +1 f ε i +2 − ε i f ε i +1 − ε i − t − f ε i +1 − ε i +2 +1 f ε i +2 − ε i +1 f ε i +1 − ε i +1 f ε i +1 − ε i +2 f ε i +2 − ε i f ε i +1 − ε i +1 , ZAJJ DAUGHERTY AND ARUN RAM N p ( ∅ , )0 = T T T T − t T T T − t T T T + t t T T + t t T T − t t T − t tT + t t = a a e e e e − a a ( t − t + t t − ) e e = a a e e e e − a a ( t − t + t t − ) e e = τ τ τ τ − t τ τ τ f ε − r f ε − r f ε − t − τ τ τ f ε − ε +1 f ε − ε + t t − τ τ f − ε − ε +1 f − ε − ε f ε − r f ε − r f ε − t t − τ τ f ε − r f ε − r f ε f ε − ε +1 f ε − ε − t t − τ f ε − r f ε − r f ε f − ε − ε +1 f − ε − ε f ε − r f ε − r f ε − t t − τ f − ε − ε +1 f − ε − ε f ε − r f ε − r f ε f ε − ε +1 f ε − ε + t t − f ε − r f ε − r f ε f − ε − ε +1 f − ε − ε f ε − r f ε − r f ε f ε − ε +1 f ε − ε , and N ′ p (1 , ∅ )0 = T T T T + t − T T T − t T T T − t − t T T − t − t T T − t t T + t − tT + t t = ( a a e e e e − a a ( t − t + t t − ) e e ) − ( a a e e e − a ( t − t + t t − ) e )= τ τ τ τ − t τ τ τ W − f − ε − r f − ε − r f ε − t − τ τ τ f ε − ε +1 f ε − ε + t t − τ τ W − f − ε − ε +1 f − ε − ε f − ε − r f − ε − r f ε + t t − τ τ W − f − ε − r f − ε − r f ε f ε − ε +1 f ε − ε − t t − τ W − W − f − ε − r f − ε − r f ε f − ε − ε +1 f − ε − ε f − ε − r f − ε − r f ε − t t − τ W − f − ε − ε +1 f − ε − ε f − ε − r f − ε − r f ε f ε − ε +1 f ε − ε + t t − W − W − f − ε − r f − ε − r f ε f − ε − ε +1 f − ε − ε f − ε − r f − ε − r f ε f ε − ε +1 f ε − ε . Proof.
The expressions in terms of T i are proved by using the relations T i = ( t − t − ) T i + 1 and T = ( t − t − ) T + 1 to show that the equations in (2.26) are satisfied. In view of the conditions(2.25), using the equations (2.26) to compute the product of the expansion in terms of the T i witheach element p (1 ) i , p ( ∅ , )0 and p (1 , ∅ )0 respectively, determines the normalizing constants N = − t − − t − − t − − t − t − t = t − (1 + t )(1 + t + t ) , and N = N ′ = t − t − + t − + t − + 1 + 1 + t + t + t t = t − t − (1 + t )(1 + t )(1 + t t ) . Checking the conditions (2.27) verifies that the expressions in terms of the e i for the elements N p (1 ) i , N p ( ∅ , )0 and N ′ p (1 , ∅ )0 are correct. Similarly, using the expressions for a e and ae i in termsof τ i given in (2.24) to check these same conditions verifies that the expressions for the elements N p (1 ) i , N p ( ∅ , )0 and N ′ p (1 , ∅ )0 in terms of the τ i are correct. (cid:3) ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 9
Setting up the relation a k a e k − e k e k − − a ( t − k t + t k t − ) e k − = 0 . As in [DR, Remark2.3], let w A be the longest element of W A k = h s , . . . , s k − i . Let T ∨ = T − w A T k T w A = a − ( − a ) − = a − ( − a ) − = W T − , and note that T − w A T k − T w A = T . Then( T ∨ − t k )( T ∨ + t − k ) = 0 and T ∨ T T ∨ T = T T ∨ T T ∨ . Let HB ∨ be the subalgebra of H ext k generated by T ∨ and T and define idempotents p ( ∅ , )0 ∨ and p (1 , ∅ )0 ∨ in HB ∨ by the equations( p ( ∅ , )0 ∨ ) = p ( ∅ , )0 ∨ , ( p (1 , ∅ )0 ∨ ) = p (1 , ∅ )0 ∨ ; (2.28)and T ∨ p ( ∅ , )0 ∨ = − t − k p ( ∅ , )0 ∨ , T p ( ∅ , )0 ∨ = − t − p ( ∅ , )0 ∨ ,T ∨ p (1 , ∅ )0 ∨ = t k p (1 , ∅ )0 ∨ , and T p (1 , ∅ )0 ∨ = − t − p (1 , ∅ )0 ∨ . (2.29)Let a k ∈ C × and define a k e ∨ = T ∨ − t k , so that e ∨ = T w A e k T − w A and e = T w A e k − T − w A . (2.30)The conditions in (2.29) are equivalent to a k e ∨ p ( ∅ , )0 ∨ = − ( t k + t − k ) p ( ∅ , )0 ∨ , ae p ( ∅ , )0 ∨ = − ( t + t − ) p ( ∅ , )0 ∨ ,a k e ∨ p (1 , ∅ )0 ∨ = 0 , and ae p (1 , ∅ )0 ∨ = − ( t + t − ) p (1 , ∅ )0 ∨ . (2.31)Using a k e ∨ = W T − − t k = W ( T − ( t − t − )) − t k = W ( τ + t − c α − ( t − t − )) − t k , ashort computation gives a k e ∨ = τ W − − t − W − f ε − r f − ε − r f ε . And, with N k = t − k t − (1 + t k )(1 + t )(1 + t k t ), we have N k p ( ∅ , )0 ∨ = a k a e ∨ e e ∨ e − a k a ( t − k t + t k t − ) e ∨ e = a k a e e ∨ e e ∨ − a k a ( t − k t + t k t − ) e e ∨ = τ τ τ τ ( W W ) − − t − τ τ τ ( W W ) − f ε − r f − ε − r f ε + t − τ τ τ ( W W ) − f ε − ε +1 f ε − ε − t − t − τ τ ( W W ) − f − ε − ε +1 f − ε − ε f ε − r f − ε − r f ε − t − t − τ τ ( W W ) − f ε − r f − ε − r f ε f ε − ε +1 f ε − ε + t − t − τ ( W W ) − f ε − r f − ε − r f ε f − ε − ε +1 f − ε − ε f ε − r f − ε − r f ε − t − t − τ ( W W ) − f − ε − ε +1 f − ε − ε f ε − r f − ε − r f ε f ε − ε +1 f ε − ε + t − t − ( W W ) − f ε − r f − ε − r f ε f − ε − ε +1 f − ε − ε f ε − r f − ε − r f ε f ε − ε +1 f ε − ε , and N k p (1 , ∅ )0 ∨ = ( a k a e ∨ e e ∨ e − a k a ( t − k t + t k t − ) e ∨ e ) − ( a k a e e ∨ e − a ( t − k t + t k t − ) e )= τ τ τ τ ( W W ) − − t − τ τ τ ( W W ) − f − ε − r f ε − r f ε + t − τ τ τ ( W W ) − f ε − ε +1 f ε − ε − t − t − τ τ ( W W ) − f − ε − ε +1 f − ε − ε f − ε − r f ε − r f ε − t − t − τ τ ( W W ) − f − ε − r f ε − r f ε f ε − ε +1 f ε − ε + t − t − τ ( W W ) − f − ε − r f ε − r f ε f − ε − ε +1 f − ε − ε f − ε − r f ε − r f ε − t − t − τ ( W − W ) − f − ε − ε +1 f − ε − ε f − ε − r f ε − r f ε f ε − ε +1 f ε − ε + t − t − ( W W ) − f − ε − r f ε − r f ε f − ε − ε +1 f − ε − ε f − ε − r f ε − r f ε f ε − ε +1 f ε − ε , in analogy with (and with the same proof as) Proposition 2.3.3. The two boundary Temperley-Lieb algebra
T L k In this section we define the two boundary Temperley-Lieb algebra
T L k (also called the symplec-tic blob algebra, see [GMP07, GMP08, GMP12, Ree, KMP16, GMP17]) and review its diagram-matic calculus. We extend the diagrammatic calculus to make clear the relationship to the twoboundary Hecke algebra and to set the stage for the proof of Theorem 3.2. Although Theorem 3.2takes the form of a computation, it is a computation that has amazing consequences as it determinesthe relationship between the center of H ext k and the center of T L k . The center of H ext k is a ring ofsymmetric functions (see [DR, Theorem 2.3]) and the center of T L k turns out to be a polynomialring C [ Z ] in a single variable Z . We shall see that, in the same way that H ext k is finite rank overits center, the algebra T L k is finite rank over C [ Z ]. However, whereas the former has the easily ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 11 classified rank of (2 k k !) over its center, the rank of T L k is as yet unclassified combinatorially. Forexample, dim( T L k ( b )) = 5, 19, 84, 335, and 1428, for k = 1, 2, 3, 4, and 5, respectively.3.1. The extended two boundary Temperley-Lieb algebra
T L ext k . Let H ext k be the extendedtwo boundary Hecke algebra as defined in (2.15). The extended two boundary Temperley-Liebalgebra T L ext k is the quotient of H ext k by the relations p ( ∅ , )0 ∨ = p (1 , ∅ )0 ∨ , p ( ∅ , )0 = p (1 , ∅ )0 and p (1 ) i = 0 for i ∈ { , . . . , k − } . Theorem 3.1.
The algebra
T L ext k is the quotient of H ext k by the relations a k a e k − e k e k − − a ( t − k t + t k t − ) e k − = 0 , a a e e e − a ( t − t + t t − ) e = 0 , and a e i e i +1 e i − ae i = a e i +1 e i e i +1 − ae i +1 = 0 for i ∈ { , . . . , k − } .Proof. Let F i = a e i e i +1 e i − ae i = a e i +1 e i e i +1 − ae i +1 for i ∈ { , . . . , k − } , F k = a k a e k − e k e k − − a ( t − k t + t k t − ) e k − , and F = a a e e e − a ( t − t + t t − ) e . By Proposition 2.3, N p (1 , ∅ )0 = e F , N p ( ∅ , )0 = ( e − F , F = N ( p (1 , ∅ )0 − p ( ∅ , )0 ) , and N p (1 ) i = F i ;and, by (2.30), T w A F k T − w A = N ∨ ( p (1 , ∅ )0 ∨ − p ( ∅ , )0 ∨ ) , T − w A p (1 , ∅ )0 ∨ T w A = e k F k , and T − w A p ( ∅ , )0 ∨ T w A = ( e k − F k . Thus, provided N , N and N k are invertible, the ideal H ext k F k H ext k is the same as the ideal generatedby ( p (1 , ∅ )0 ∨ and p ( ∅ , )0 ∨ ; the ideal H ext k F H ext k is the same as the ideal generated by p (1 , ∅ )0 and p ( ∅ , )0 ;and H ext k p (1 ) i H ext k = H ext k F i H ext k . (cid:3) The two boundary Temperley-Lieb algebra
T L k . The two boundary Temperley-Liebalgebra
T L k is the subalgebra of T L ext k generated by a e , ae , . . . , ae k − , a k e k (as defined in (2.16)).As in (2.12) and (2.20), where B ext k = B k × D and H ext k = H k ⊗ C [ W ± ], the extended two boundaryTemperley-Lieb algebra is T L ext k = T L k ⊗ C [ W ± ] , as algebras, where W = P W · · · W k . Diagrammatic calculus for
T L k . Pictorially, identify T k = , T = , T i = ii i +1 i +1 ,e = . . . , e k = . . . , and ae i = . . . . . . ii , for i ∈ { , . . . , k − } . Recall the notation[[ x ]] = x + x − from (2.19). With i ∈ { , . . . , k − } , the relations (2.16), (2.17) and (2.18) are T = a e + t , T i = ae i + t , T k = a k e k + t k , = a + t / = + t / = a k + t / k T e = e T = − t − e , T i ( ae i ) = ( ae i ) T i = − t − ( ae i ) , T k e k = e k T k = − t − k e k , = = − t − / = = − t − / = = − t − / k T − e = e T − = − t e , T − i ( ae i ) = ( ae i ) T − i = − t ( ae i ) , T − k e k = e k T − k = − t k e k , = = − t / = = − t / = = − t / k e = − [[ t ]] a e , ( ae i ) = − [[ t ]]( ae i ) , and e k = − [[ t k ]] a k e k . = − [[ t ]] a = − [[ t ]] = − [[ t k ]] a k In the quotient by ( ae i )( ae i +1 )( ae i ) = ae i , we have ae i T i +1 T i = aT i +1 T i e i +1 = t a e i e i +1 , ae i T − i +1 T − i = aT − i +1 T − i e i +1 = t − a e i e i +1 ,ae i +1 T i T i +1 = aT i T i +1 e i = t a e i +1 e i , ae i +1 T − i T − i +1 = aT − i T − i +1 e i = t − a e i +1 e i , (3.1)= = t / = = t − / = = t / = = t − / which are proved by using T ± i = ae i + t ± to expand both sides in terms of e i .When a ( ae ) e ( ae ) − [[ t t − ]]( ae ) = 0 and a k ( ae k − ) e k ( ae k − ) − [[ t k t − ]] ae k − = 0, then( ae ) T T = t ( ae ) T − , T T ( ae ) = t T − ( ae ) , ( ae k − ) T k T k − = t ( ae k − ) T − k , T k − T k ( ae k − ) = t T − k ( ae k − ) , (3.2)= t / = t / = t / = t / ( ae ) T T T T = t ( ae ) T − T − T − , (3.3)= t / ( ae ) T ( ae ) = − t ( t − t − )( ae ) , ( ae k − ) T k ( ae k − ) = − t ( t k − t − k )( ae k − ) , (3.4)= − t ( t − t − ) = − t ( t k − t − k )and e T − T − T − e = − t − [[ t ]] e ( ae ) e − t − t e . = − t − [[ t ]] − t − t T L k as a diagram algebra. Using the pictorial notation, the algebra
T L k has a basis (see[GMP12, Theorem 3.4]) of non-crossing diagrams with k dots in the top row, k dots in the bottomrow, edges connecting pairs of dots, an even number of left boundary to right boundary edges, and( − { left boundary edges } = 1 and ( − { right boundary edges } = 1 . For example, d = and d =are both basis elements of T L k . Multiplication of basis elements can be computed pictorially byvertical concatenation, with self-connected loops and strands with both ends on the left or on theright replaced by constant coefficients according to the following local rules:= − [[ t ]] , if even = [[ t t − ]] a , if even = [[ t k t − ]] a k , if odd = − [[ t ]] a , and if odd = − [[ t k ]] a k . For example with d and d as above, d d = = = ( − [[ t ]]) (cid:16) − [[ t k ]] a k (cid:17)(cid:16) [[ t k t − ]] a k (cid:17) (where the dashed strand is removed with a coefficient of [[ t k t − ]] a k , and the thick strand is removedwith a coefficient of − [[ t k ]] a k ). The through-strand filtration of
T L k . A through-strand is an edge that connects a topvertex to a bottom vertex. Define the ideals T L ( ≤ j ) k = C -span { diagrams with ≤ j through-strands } . Then the algebra TL k is filtered by ideals as T L k = T L ( ≤ k ) k ⊇ T L ( ≤ k − k ⊇ · · · ⊇ T L ( ≤ k ⊇ T L ( ≤ k ⊇ . (3.5)If T L ( j ) k = T L ( ≤ j ) k T L ( ≤ j − k , then dim( T L ( j ) k ) < ∞ , for j ≥
1, and dim(
T L ( ≤ k ) = ∞ , as there can be an arbitrarily large number of edges which connect the left and right sides indiagrams with no through strands: ... . The elements I and I . As in [GN, § I = ( ( ae )( ae ) · · · ( ae k − ) , if k is even,( ae )( ae ) · · · ( ae k − ) e k , if k is odd, = · · · if k is even, · · · if k is odd, (3.6)and I = ( e ( ae ) · · · ( ae k − ) e k , if k is even, e ( ae ) · · · ( ae k − ) , if k is odd. = · · · if k is even, · · · if k is odd. (3.7)Up to a constant multiple the elements I and I are idempotents and I I I = · · ·· · · if k is even, · · ·· · · if k is odd, I I I = · · ·· · · if k is even, · · ·· · · if k is odd.Proposition 3.2 gives another striking formula for the elements I I I and I I I .3.7. The element ZI in T L k . Conceptually, the diagram F = · · · would be a central element of H k if it represented a true element of the algebra H k . Though the diagram F does not naturallyrepresent an element of H k , the diagrams D even = I (cid:0) T − ( ae )( ae ) · · · ( ae k − ) T k (cid:1) I = · · · , and D odd = I (cid:0) T − T − T − ( ae )( ae ) · · · ( ae k − ) T k (cid:1) I = · · · ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 15 do appear in the algebra
T L k and play an important role in the proof of the following theorem.See also [GN, Thm. 4.1], using Remark 3.4 below as a guide. Theorem 3.2.
Let Z = W + W − + · · · + W k + W − k which, as noted in (2.21) is a central elementof H k . As elements of T L k ,if k is even, then D even = a a k I I I + [[ t t k t − ]] I and ZI = [ k ] D even , andif k is odd, then D odd = t − (cid:16) − [[ t ]] a (cid:17) (cid:0) a a k I I I − [[ t t − k ]] I (cid:1) and t − (cid:16) − [[ t ]] a (cid:17) ZI = [ k ] D odd . Proof.
Case: k even. Let L even = I (cid:0) ( ae )( ae ) · · · ( ae k − ) e k (cid:1) I = · · ·· · · = (cid:16) [[ t k t − ]] a k (cid:17) I ,M even = I (cid:0) e ( ae )( ae ) · · · ( ae k − ) (cid:1) I = · · ·· · · = (cid:16) [[ t t − ]] a (cid:17) I , and P even = I (cid:0) ( ae )( ae ) · · · ( ae k − ) (cid:1) I = · · ·· · · = − [[ t ]] I . Using T − = a e + t − for the left pole and T k = a k e k + t k for the right pole, D even = a a k I I I + a t k M even + a k t − L even + t t k P even = a a k I I I + ( t k [[ t t − ]] + t − [[ t k t − ]] − t − t k [[ t ]]) I = a a k I I I + [[ t t k t − ]] I , which completes the proof of the first statement.Using ( ae ) T − = ( − t )( ae ) and ( ae ) T T ( ae ) = t ( ae ) T − ( ae ) gives R even = I (cid:0) T − ( ae )( ae ) · · · ( ae k − ) T k T T (cid:1) I = = ( − t ) t D even , and using T k − ( ae k − ) = ( − t − )( ae k − ) and T − k − T − k ( ae k − ) = t − T k ( ae k − ) gives S even = I (cid:0) T ( ae )( ae ) · · · ( ae k − ) T − k − T − k T k − (cid:1) I = = ( − t − ) t − D even . Pictorially, I W i I = , I W i I = ,I W − i I = , and I W − i I = . Working left to right removing loops, I W i I = (cid:0) t t ( − [[ t ]]) (cid:1) i ( t − t ( − [[ t ]])) k − − i R even = ( − [[ t ]]) k − t i + ( − t ) D even ,I W − i I = (cid:0) t − t − ( − [[ t ]]) (cid:1) i (cid:0) t − t ( − [[ t ]]) (cid:1) k − − i S = ( − [[ t ]]) k − t − ( i + ) ( − t − ) D even , for i ∈ { , . . . , k − } . Since I W i I and I W i I only differ by two twists (similarly I W − i I and I W − i I only differ by two twists) the relations T ± i ( ae i ) = ( ae i ) T ± i = ( − t ∓ )( ae i ) give I W i I = ( − t − )( − t − ) t − I W i I = ( − [[ t ]]) k − t i + ( − t − ) D even and I W − i I = ( − t )( − t − ) I W − i I = ( − [[ t ]]) k − t − ( i + ) ( − t ) D even , for i ∈ { , . . . , k − } .Thus ( − [[ t ]]) k ZI = ZI = I ZI = k − X i =0 I ( W i + W i + W − i + W − i ) I = − ( − [[ t ]]) k − D even k − X i =0 (cid:0) t i + ( t + t − ) + t − ( i + ) ( t + t − ) (cid:1) = ( − [[ t ]]) k D even (cid:16) t k − t − k t − t − (cid:17) = ( − [[ t ]]) k [ k ] D even . Case: k odd. Let L odd = I (cid:0) ( ae )( ae ) · · · ( ae k − ) e k (cid:1) I = · · ·· · · = (cid:16) − [[ t ]] a (cid:17)(cid:16) [[ t k t − ]] a k (cid:17) I ,M odd = I (cid:0) ( ae )( ae ) · · · ( ae k − ) (cid:1) I = · · ·· · · = (cid:16) − [[ t ]] a (cid:17) I , and P odd = I (cid:0) ( ae )( ae ) · · · ( ae k − ) (cid:1) I = · · ·· · · = (cid:16) − [[ t ]] a (cid:17) ( − [[ t ]]) I . Using e T − T − T − e = − t − [[ t ]] e ( ae ) e − t − t e and T k = a k e k + t k gives D odd = − t − [[ t ]] a k I I I − t − t a k L odd − t − [[ t ]] t k M odd − t − t t k P odd = t − (cid:16) − [[ t ]] a (cid:17) (cid:18) a a k I I I + (cid:0) − t − t [[ t k t − ]] − t k [[ t ]] + t − t t k [[ t ]] (cid:1) I (cid:19) = t − (cid:16) − [[ t ]] a (cid:17) (cid:0) a a k I I I − [[ t t − k ]] I (cid:1) , which completes the proof of the first statement.Using ( ae ) T − = − t ( ae ) and T T T T ( ae ) = t T − T − T − ( ae ), R odd = I ( T − ( ae )( ae ) · · · ( ae k − ) T k T T T T ) I = · · · = ( − t ) t D odd . ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 17
Using T k − ( ae k − ) = − t − ( ae k − ) and T − k − T − k ( ae k − ) = t − T k ( ae k − ) gives S odd = I ( T − T − T − ( ae )( ae ) · · · ( ae k − ) T − k − T − k T k − ) I = · · · = ( − t − ) t − D odd . Pictorially, I W i I = , I W i I = ,I W − i I = , and I W − i I = . Working left to right removing loops, I W i I = (cid:0) t t ( − [[ t ]]) (cid:1) i (cid:0) t − t ( − [[ t ]]) (cid:1) k − − − i R odd = ( − [[ t ]]) k − t i +1 ( − t ) D odd ,I W − i I = (cid:0) t − t − ( − [[ t ]]) (cid:1) i (cid:0) t − t ( − [[ t ]]) (cid:1) k − − − i S odd = ( − [[ t ]]) k − t − ( i +1) ( − D odd , for i ∈ { , . . . , k − − } . Since I W i I and I W i I only differ by two twists (similarly I W − i I and I W − i I only differ by two twists) the relations T ± i e i = e i T ± i = ( − t ∓ ) e i give I W i I = ( − t − )( − t − ) I W i I = ( − [[ t ]]) k − t i +1 ( − D odd , and I W − i I = ( − t )( − t ) I W − i I = ( − [[ t ]]) k − t − ( i +1) ( − t ) D odd , for i ∈ { , . . . , k − − } .Next, I W I = · · · = ( − t − )( − [[ t ]]) k − I (( ae )( ae ) · · · ( ae k − ) T k ) I , and I W − I = · · · = ( − t )( − [[ t ]]) k − I (( ae )( ae ) · · · ( ae k − ) T − k ) I . Using − t − T k − t T − k = − t − ( a k e k + t k ) − t ( a k e k + t − k ) = − [[ t ]] a k e k − [[ t o t − k ]], I ( W + W − ) I = ( − [[ t ]]) k − (cid:16) − [[ t ]] a k I I I − [[ t t − k ]] M odd (cid:17) = ( − [[ t ]]) k − (cid:16) − [[ t ]] a k I I I − [[ t t − k ]] (cid:16) − [[ t ]] a (cid:17) I (cid:17) = ( − [[ t ]]) k − (cid:16) − [[ t ]] a (cid:17)(cid:16) a a k I I I − [[ t t − k ]] I (cid:17) = − ( t + 1)( − [[ t ]]) k − D odd . Thus (cid:16) − [[ t ]] a (cid:17) ( − [[ t ]]) k − ZI = ZI = I ZI = I ( W + W − ) I + k − − X i =0 I ( W i + W i + W − i + W − i ) I = − ( t + 1)( − [[ t ]]) k − t D odd + ( − [[ t ]]) k − (cid:16) k − X i =0 ( t i +1 − t − ( i +1) )( − t − D odd (cid:17) = − ( − [[ t ]]) k − ( t + 1) D odd (cid:16) k − X i =0 ( t i +1 − t − ( i +1) ) (cid:17) = ( − [[ t ]]) k − t D odd [ k ] . (cid:3) Corollary 3.3.
Let Z = W + W − + · · · + W k + W − k and let I and I be as defined in (3.6) and (3.7) . If k is even, then a a k I I I = (cid:16) k ] Z − [[ t t k t − ]] (cid:17) I and a a k I I I = (cid:16) k ] Z − [[ t t k t − ]] (cid:17) I . If k is odd, then a a k I I I = (cid:16) k ] Z + [[ t t − k ]] (cid:17) I and a a k I I I = (cid:16) k ] Z + [[ t t − k ]] (cid:17) I . Proof.
As observed in the proof of Theorem 3.2, the products I ZI and I Z reduce to computationof the diagram with a single string going around all the poles ( D even or D odd ). These diagrammaticsgive that there are constants C, C , C and D, D , D such that I = CI , I I I = ( C Z + C ) I , I = DI , I I I = ( D Z + D ) I . Then, computing ( I I I ) in two different ways, we have I I I I I I = CI I I I I = C ( D Z + D ) I I I , and I I I I I I = ( C Z + C ) I I I I = C ( C Z + C ) I I I , which indicates that C Z + C = D Z + D .Theorem 3.2 gives that, if k is even, then a a k I I I = D even − [[ t t k t − ]] I = 1[ k ] ZI − [[ t t k t − ]] I , and if k is odd, then a a k I I I = t (cid:16) a − [[ t ]] (cid:17) D odd + [[ t t − k ]] I = 1[ k ] ZI + [[ t t − k ]] I . (cid:3) Remark 3.4. Comparison to de Gier-Nichols.
Let us explain how to relate the constants inCorollary 3.3 and Proposition 4.4 to the values which appear in [GN]. Let t = − iq ω , t = q − , t k = − iq ω ,T = − i g , T i = − g i , T k = − i g k ,e = e , e i = e i , e k = e k . ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 19
Then ( g − q ω )( g − q − ω ) = 0 , ( g i + q − )( g i − q ) = 0 , ( g k − q ω )( g − q − ω ) = 0 , as in [GN, Definitions 2.4, 2.6, and 2.8], and g = q ω − ( q ω − q − (1+ ω ) ) e , g i = e i − q − , g k = q ω − ( q ω − q − (1+ ω ) ) e k , as in [GN, (5)]. Following [GN, Definitions 2.8 and (9)], J ( C )0 = g − · · · g − k − g k g k − · · · g g g = ( − k − ( − i )( − i )( − k − T − · · · T − k − T k T k − · · · T T = − W , J ( C ) i = g i J ( C ) i − g i = ( − T i ( − W i ) T i = − W i +1 for i ∈ { , . . . , k − } , and Z k = k − X i =0 ( J ( C ) i + ( J ( C ) i ) − ) = − ( W + w − + · · · + W k + W − k ) = − Z. Use the notation [ x ] = t x − t − x t − t − = q x − q − x q − q − and let a , a and a k take the favorite values from Remark2.1 so that a = − , a = − [[ t t − ]] , and a k = − [[ t k t − ]] , and set θ = c + k −
12 and z = [[ t θ ]][ k ] , as in Proposition 4.4. Following [GN, Theorem 4.1] and remembering that Z k = − Z , letΘ = θ + 1log q iπ so that − [ k ][[ t θ ]] = − [ k ]( t θ + t − θ ) = [ k ]( − q − θ − q θ )= [ k ]( q − ( θ + q iπ ) + q θ + q iπ ) = [ k ]( q − Θ + q Θ ) = [ k ] [2Θ][Θ] . Note that a a k = [[ t t − ]][[ t k t − ]] = ( t t − + t − t )( t k t − + t − k t )= ( − iq − ω − + iq ω +1 )( − iq − ω − + iq ω +1 ) = − [ ω + 1][ ω + 1]( q − q − ) . Then the constant b that appears in [GN, Definition 3.6 and Theorem 4.1] to make I I I = bI and I I I = bI as operators on a simple T L k -module is computed from Corollary 3.3 as follows: b = k ] z − [[ t t k t − ]] a a k = k ] [ k ][[ t θ ]] − [[ t t k t − ]][[ t t − ]][[ t k t − ]] = [[ t θ ]] − [[ t t k t − ]][[ t t − ]][[ t k t − ]]= − ( q Θ + q − Θ ) + ( − iq ω )( − iq ω ) q + ( iq − ω )( iq − ω ) q − − [ ω + 1][ ω + 1]( q − q − ) = q Θ + q − Θ − q ω + ω +1 − q − ( ω + ω +1) [ ω + 1][ ω + 1]( q − q − ) = (( q ω + ω +1+Θ ) − ( q ω + ω +1+Θ ) − )(( q ω + ω +1 − Θ ) − ( q ω + ω +1 − Θ ) − ))[ ω + 1][ ω + 1]( q − q − ) = [ ( ω + ω + 1 + Θ)][ ( ω + ω + 1 − Θ)][ ω + 1][ ω + 1] when k is even, and b = k ] z + [[ t t − k ]] a a k = k ] [ k ][[ t θ ]] + [[ t t − k ]][[ t t − ]][[ t k t − ]] = [[ t θ ]] + [[ t t − k ]][[ t t − ]][[ t k t − ]]= − ( q Θ + q − Θ ) + ( − iq ω )( iq − ω ) + ( iq − ω )( − iq ω ) − [ ω + 1][ ω + 1]( q − q − ) = − q Θ − q − Θ + q ω − ω + q − ( ω − ω ) − [ ω + 1][ ω + 1]( q − q − ) = − (( q ω − ω − Θ ) − ( q ω − ω − Θ ) − )(( q ω − ω +Θ ) − ( q ω − ω +Θ ) − )[ ω + 1][ ω + 1]( q − q − ) = − [ ( ω − ω − Θ)][ ( ω − ω + Θ)][ ω + 1][ ω + 1] when k is odd.4. Calibrated representations of H ext k and T L ext k In this section we classify and construct all irreducible calibrated representations of the extendedtwo boundary Temperley-Lieb algebras
T L ext k . This is done by using the classification of irreduciblecalibrated H ext k -modules from [DR], which we begin by reviewing in Sections 4.1 and 4.2. Usingthe formulas for the elements p (1 ) i , p ( ∅ , )0 , p (1 , ∅ )0 , p ( ∅ , )0 ∨ , and p (1 , ∅ )0 ∨ that one quotients H ext k byto obtain T L ext k , we determine exactly which irreducible calibrated representations of H ext k factorthrough the quotient, thus providing a full classification of irreducible calibrated representations of T L ext k .4.1. Calibrated representations of H ext k . A calibrated H ext k -module is an H ext k -module M suchthat W , W , . . . , W k are simultaneously diagonalizable as operators on M . Let r , r ∈ C such that − t r = − t k t − and − t r = t k t . (4.1)For c = ( c , . . . , c k ) ∈ C k let c − i = − c i and define Z ( c ) = { ε i | c i = 0 } ⊔ { ε j − ε i | < i < j and c j − c i = 0 } , ⊔ { ε j + ε i | < i < j and c j + c i = 0 } , (4.2) P ( c ) = { ε i | c i ∈ {± r , ± r }} ⊔ { ε j − ε i | < i < j and c j − c i = ± }⊔ { ε j + ε i | < i < j and c j + c i = ± } , (4.3)where { ε , . . . , ε n } is an orthonormal basis for the weights corresponding to gl n (see [DR, § local region is a pair ( c , J ) with c ∈ C k and J ⊆ P ( c ). The set of standard tableaux of shape ( c , J )is F ( c ,J ) = { w ∈ W | R ( w ) ∩ Z ( c ) = ∅ , R ( w ) ∩ P ( c ) = J } (4.4)(see the following section for a visualization of this set as fillings of box arrangements). A skewlocal region is a local region ( c , J ), c = ( c , . . . , c k ), such thatif w ∈ F ( c ,J ) then w c = (( w c ) , . . . , ( w c ) n ) satisfies( w c ) = 0 , ( w c ) = 0 , ( w c ) = − ( w c ) , ( w c ) i = ( w c ) i +1 for i = 1 , . . . , k −
1, and ( w c ) i = ( w c ) i +2 for i = 1 , . . . , k −
2. (4.5)The following theorem constructs and classifies the calibrated irreducible representations of H ext k . ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 21
Theorem 4.1. [DR, Theorem 3.3]
Assume t , t , and t k are invertible, t is not a root of unity,and t t k , − t − t k
6∈ { , − , t ± , − t ± , t ± , − t ± } and t t k = ( − t − t k ) ± . Let r , r be as in (4.1) .(a) Let ( c , J ) be a skew local region and let z ∈ C × . Define H ( z, c ,J ) k = span C { v w | w ∈ F ( c ,J ) } , (4.6) so that the symbols v w are a labeled basis of the vector space H ( z, c ,J ) k . Let γ i = − t c i for i = 1 , , . . . , k , and γ = zγ − w − (1) · · · γ − w − ( k ) . Then the following formulas make H ( z, c ,J ) k into an irreducible H ext k -module: P W · · · W k v w = zv w , P v w = γ v w , W i v w = γ w − ( i ) v w , (4.7) T i v w = [ T i ] ww v w + q − ([ T i ] ww − t )([ T i ] ww + t − ) v s i w , for i = 1 , . . . , k − , (4.8) T v w = [ T ] ww v w + q − ([ T ] ww − t )([ T ] ww + t − ) v s w , (4.9) where v s i w = 0 if s i w
6∈ F ( c ,J ) , and [ T i ] ww = t − t − − γ w − ( i ) γ − w − ( i +1) and [ T ] ww = ( t − t − ) + ( t k − t − k ) γ − w − (1) − γ − w − (1) . (4.10) (b) The map C × × { skew local regions ( c , J ) } ←→ { irreducible calibrated H ext k -modules } ( z, c , J ) H ( z, c ,J ) k is a bijection. Configurations of boxes.
Let ( c , J ) be a local region with c = ( c , . . . , c k ), c ∈ Z k or c ∈ ( Z + ) k , and 0 ≤ c ≤ · · · ≤ c k . (4.11)Start with an infinite arrangement of NW to SE diagonals, numbered consecutively from Z or Z + ,increasing southwest to northeast (see Example 4.2). The configuration κ of boxes correspondingto the local region ( c , J ) has 2 k boxes (labeled box − k , . . . , box − , box , . . . , box k ) with the followingconditions.( κ
1) Location: box i is on diagonal c i , where c − i = − c i for i ∈ {− k, . . . , − } .( κ
2) Same diagonals: box i is NW of box j if i < j and box i and box j are on the same diagonal.( κ
3) Adjacent diagonals:If ε j − ε i ∈ J , then box j is NW (strictly north and weakly west) of box i : ji If ε j − ε i ∈ P ( c ) − J , then box j is SE (weakly south and strictly east) of box i : i j ( κ
4) Markings: There is a marking on each of the diagonals r , − r , r and − r .If ε i ∈ J , box i is NW of the marking on diagonal c i : i If ε i ∈ P ( c ) − J , then box i is SE of the marking in diagonal c i : i A standard filling of the boxes of κ is a bijective function S : κ → {− k, . . . , − , , . . . k } such that(S1) Symmetry: S (box − i ) = − S (box i ).(S2) Same diagonals:If 0 < i < j and box i and box j are on the same diagonal then S (box i ) < S (box j ).(S3) Adjacent diagonals:If 0 < i < j , box i and box j are on adjacent diagonals, and box j is NW of box i , then S (box j ) < S (box i ).If 0 < i < j , box i and box j are on adjacent diagonals, and box j is SE of box i , then S (box j ) > S (box i ).(S4) Markings:If box i is on a marked diagonal and is SE of the marking, then S (box i ) > i is on a marked diagonal and is NW of the marking, then S (box i ) < identity filling of a configuration κ is the filling F of the boxes of κ given by F (box i ) = i , for i = − k, . . . , − , , . . . , k . The identity filling of κ is usually not a standard filling of κ (see Example4.2). Example.
Let k = 4 , r = 1 , and r = 3 . Consider c = (2 , , . Then Z ( c ) = { ε − ε } and P ( c ) = (cid:8) ε , ε − ε , ε − ε (cid:9) . The box configurations corresponding to J = { ε − ε } and J = { ε , ε − ε , ε − ε } (filled withtheir identity fillings) are - - - - - - - - - - - - - - - - J = { ε − ε } J = { ε , ε − ε , ε − ε } For both configurations, the identity filling is not a standard filling. Examples of standard fillingsof the configuration corresponding to J = { ε − ε } include - - - , - - - , and - - - , but not - - - . Proposition 4.2. [DR, Proposition 3.1]
Let κ be a configuration of boxes corresponding to a localregion ( c , J ) with c ∈ Z k or c ∈ ( Z + ) k . For w ∈ W let S w be the filling of the boxes of κ givenby S w (box i ) = w ( i ) , for i = − k, . . . , − , , . . . , k .The map F ( c ,J ) −→ { standard fillings S of the boxes of κ } w S w is a bijection . ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 23
Calibrated representations of
T L ext k . The following theorem determines which calibratedirreducible representations of H ext k are T L ext k -modules. In Theorem 4.3 the answer is stated in termsof the configuration of boxes κ . By ( κ κ
4) of Section 4.2 the local region ( c , J ) is determinedby κ . See Theorem 5.1 for the explicit conversion from κ to ( c , J ) for the irreducible calibrated T L k -modules. Theorem 4.3.
Assume that if r , r ∈ Z or r , r ∈ Z + , then r > r + 1 . Let κ be theconfiguration of boxes corresponding to a skew local region ( c , J ) with c ∈ Z k or c ∈ ( Z + ) k . Theirreducible calibrated H ext k -module H ( z, c ,J ) k is a T L ext k -module if and only if κ is a ◦ rotationallysymmetric skew shape with two rows of k boxes each (with or without markings),or . (4.12) Proof.
Let P = { p ( ∅ , )0 , p (1 , ∅ )0 , p ( ∅ , )0 ∨ , p (1 , ∅ )0 ∨ , p (1 )1 , p (1 )2 , . . . , p (1 ) k − } so that T L k is the quotient of H k by the ideal generated by the set P . For w ∈ F ( c ,J ) let S w be the standard tableau of shape κ corresponding to w as given in Proposition 4.2. For j ∈ {− k, . . . , − , , . . . , k } ,( w c ) j is the diagonal number of S w ( j ),where S w ( j ) is the box containing j in S w . Step 1: Rewriting of the conditions for pv w = 0 . By the construction of H ( z, c ,J ) k in Theorem 4.1,the module H ( z, c ,J ) k has basis { v w | w ∈ F ( c ,J ) } and, if w ∈ F ( c ,J ) then τ i v w = 0 if and only if ( w c ) i +1 = ( w c ) i ± ,f ε i − r v w = 0 if and only if ( w c ) i = r , and f ε i − ε j +1 v w = 0 if and only if ( w c ) i = ( w c ) j − . Let i ∈ { , . . . , k − } . Using the expansion of p (1 ) i in terms of the τ i from Proposition 2.3, p (1 ) i v w = τ i τ i +1 τ i v w − t − τ i +1 τ i f ε i +1 − ε i +2 +1 f ε i +1 − ε i +2 v w − t − τ i τ i +1 f ε i +1 − ε i +1 f ε i +1 − ε i v w + t − τ i f ε i +1 − ε i +2 +1 f ε i +2 − ε i +1 f ε i +1 − ε i +2 f ε i +2 − ε i v w + t − τ i +1 f ε i +2 − ε i +1 f ε i +1 − ε i +1 f ε i +2 − ε i f ε i +1 − ε i v w − t − f ε i +1 − ε i +2 +1 f ε i +2 − ε i +1 f ε i +1 − ε i +1 f ε i +1 − ε i +2 f ε i +2 − ε i f ε i +1 − ε i +1 v w , we consider the condition p (1 ) i v w = 0 term-by-term. First, τ i τ i +1 τ i v w = 0 exactly when ( wc ) i +1 =( wc ) i ± s i wc ) i +2 = ( s i wc ) i +1 ± s i +1 s i w ) i +1 = ( s i +1 s i w ) i = ±
1, i.e. when( w c ) i +1 = ( w c ) i ± w c ) i +2 = ( w c ) i ± w c ) i +2 = ( w c ) i +1 ± . Next, − t − f εi +1 − εi +2+1 f εi +2 − εi +1 f εi +1 − εi +1 f εi +1 − εi +2 f εi +2 − εi f εi +1 − εi +1 v w = 0 exactly when( w c ) i +1 = ( w c ) i + 1 or ( w c ) i +2 = ( w c ) i + 1 or ( w c ) i +1 = ( w c ) i +1 + 1 . Thus − t − f εi +1 − εi +2+1 f εi +2 − εi +1 f εi +1 − εi +1 f εi +1 − εi +2 f εi +2 − εi f εi +1 − εi +1 v w = 0 already implies τ i τ i +1 τ i v w = 0, and similarly forthe other terms in the expansion of p (1 ) i v w = 0. Thus p (1 ) i v w = 0 if and only if( w c ) i = ( w c ) i +1 − w c ) i = ( w c ) i +2 − w c ) i +1 = ( w c ) i +2 − . (4.13) Similarly, p ( ∅ , )0 v w = 0 if and only if( w c ) ∈ { r , r } or ( w c ) ∈ { r , r } or ( w c ) = ( w c ) + 1 or ( w c ) = ( w c ) − + 1; (4.14) p (1 , ∅ )0 v w = 0 if and only if( w c ) ∈ {− r , − r } or ( w c ) ∈ {− r , − r } or ( w c ) = ( w c ) + 1 or ( w c ) = ( w c ) − + 1; (4.15) p ( ∅ , )0 ∨ v w = 0 if and only if( w c ) ∈ {− r , r } or ( w c ) ∈ {− r , r } or ( w c ) = ( w c ) + 1 or ( w c ) = ( w c ) − + 1; (4.16)and p (1 , ∅ )0 ∨ v w = 0 if and only if( w c ) ∈ { r , − r } or ( w c ) ∈ { r , − r } or ( w c ) = ( w c ) + 1 or ( w c ) = ( w c ) − + 1 . (4.17) Step 2: If κ is as in (4.12) and w ∈ F ( c ,J ) and p ∈ P then pv w = 0 . Assume κ has the form givenin (4.12) and let w ∈ F ( c ,J ) . Since κ has only two rows the positions of ( − , − , ,
2) in S w takeone of the following forms: ( w c ) − ( w c ) -1 2-2 1 ( w c ) ( w c ) −
12 12 -2 -11 2 0 1-1 -1 2-2 1 ( wc ) < − , ( wc ) > , ( wc ) = − , ( wc ) = 0 . In each of these cases, the conditions in (4.14)–(4.17) give that p ( ∅ , )0 v w = 0, p (1 , ∅ )0 v w = 0, p ( ∅ , )0 ∨ v w = 0 and p (1 , ∅ )0 ∨ v w = 0. Next, let i ∈ { , . . . , k − } . Since κ has only two rows, theneither i or i + 1 are in the same row ( w c ) i ( w c ) i +1 i i +1 ( w c ) i ( w c ) i +2 i i +2 or i and i + 2 are in the same row. Thus, by (4.13), p i v w = 0. This completes the proof that if κ is of the form (4.12) then H ( z, c ,J ) k is a T L ext k -module. Step 3: If κ is not as in (4.12) then there exists w ∈ F ( c ,J ) and p ∈ P such that pv w = 0 . Let 2 k be the number of boxes in κ . The proof is by induction on k .First, if k = 2, then the condition (4.13) does not apply. If c = ( r , r ) then there are 8 possi-bilities for w c : ( r , r ), ( − r , r ), ( r , − r ), ( − r , − r ), ( r , r ), ( − r , r ), ( r , − r ) and ( − r , − r ).None of these satisfy all of the conditions (4.14)–(4.17). If c = ( c , c + 1), then s c = ( c + 1 , c )does not satisfy (4.14) and s s s s c = ( − c, − c −
1) does not satisfy (4.17). Thus that only theshaded local regions in Figure 1 can have pv w = 0 for all p ∈ P and all w ∈ F ( c ,J ) . For all of these, κ is as in (4.12).Next, assume k > H ( z, c ,J ) k is a calibrated T L ext k -module thenRes T L ext k T L ext k − ( H ( z, c ,J ) k ) is calibrated T L ext k − -module. This means that if S w is a standard tableau ofshape κ and S ′ w is S w except with the boxes S w ( k ) and S w ( − k ) removed and κ ′ is the shape of S ′ w , ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 25 then κ ′ must be as in (4.12) and have only two rows. The box S w ( k ) is in a SE corner of κ and thebox S w ( − k ) is in a NW corner of κ . c c + k − − c − c − k + 2 or c c + k − − c − c − k + 2 . Given that κ ′ has only two rows and κ is obtained from κ ′ by adding boxes that could contain k and − k in a standard tableau, the following are possibilities that we discard for κ : k - k k -2 k -1 , k - k k -2 k -1 , and k - k k -2 k -1 . Namely, in each case there is a standard tableaux that has k − k − k in positions that donot satisfy the conditions in (4.13). Thus, in these cases, there exists an S w of shape κ for which p (1 ) k − v w = 0. Further, in the case k - k k -2 k -1 , the shape κ does not satisfy the ( w c ) k − = ( w c ) k from (4.5) and the module H ( z, c ,J ) k is notcalibrated. In summary, unless κ is of the form given in (4.12) k - k or k - k , then either H ( z, c ,J ) k is not calibrated or there exists an S w of shape κ for which p (1 ) k − v w = 0. (cid:3) Figure 1.
Calibrated representations of H have regular central character. Foreach ( c , J ) the corresponding configuration of boxes κ is displayed in the local regionof chambers corresponding to the elements of F ( c ,J ) ; only the boxes on positivediagonals are shown, since they determine κ when c is regular. The local regionsmarked in blue are those that factor through the Temperley-Lieb quotient. c = 0 c = c c = 0 c = c + 1 c = − c + 1 c = r c = r c = r c = r ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 27
The following proposition determines the action of the central element Z on each of the irreduciblecalibrated T L ext k -modules. Proposition 4.4.
Let Z = W + W − + · · · + W k + W − k be the central element of T L ext k studiedin Theorem 3.2. Assume that c = ( c, c + 1 , . . . , c + k − and H ( z, c ,J ) k is an irreducible calibrated T L ext k as in Theorem 4.3. If v ∈ H ( z, c ,J ) k , then Zv = [[ t θ ]][ k ] v, where θ = c + k − , [[ t θ ]] = t θ + t − θ and [ k ] = t k − t − k t − t − . Proof.
Let v ∈ H ( z, c ,J ) k be such that W i v = q c + i − for i ∈ { , . . . , k } . Then Zv w = zv w where z = t − ( c + k − + · · · + t − ( c +1) + t − c + t c + t c +1 + · · · + t c + k − = ( t c + k − + t − ( c + k − ) )( t − k − + · · · + t k − ) = ( t θ + t − θ ) t k − t − k t − t − = [[ t θ ]][ k ] . Since Z is a central element of H ext k and H ( z, c ,J ) k is a simple H ext k -module, Schur’s lemma impliesthat if v ∈ H ( z, c ,J ) k then Zv = zv . (cid:3) Schur-Weyl duality between
T L ext k and U q gl In this section we show that the Schur-Weyl duality studied in [DR] provides calibrated irre-ducible representations of the two boundary Temperley-Lieb algebra. We classify these represen-tations using the combinatorial classification of irreducible calibrated
T L ext k modules obtained inTheorem 4.3. We follow the combinatorics of [Dau, §
4] and [DR, § gl with sl —see, for example, [Dau, § L ( λ ) of U q ( gl ) are indexed by λ = ( λ , λ ) ∈ Z with λ ≥ λ . By the Clebsch-Gordan formula or the Littlewood-Richardson rule (see [Mac,(5.16)]) L ( a, ⊗ L ( b,
0) = L ( a + b, ⊕ L ( a + b − , ⊕ · · · ⊕ L ( a + 1 , b − ⊕ L ( a, b ) , and L ( λ , λ ) ⊗ L (1 ,
0) = ( L ( λ + 1 , λ ) ⊕ L ( λ , λ + 1) , if λ > λ ,L ( λ + 1 , λ ) , if λ = λ . Let a, b ∈ Z ≥ with a ≥ b and fix the simple U q gl -modules M = L ( a, , N = L ( b, V = L (1 , . (5.1)We identify ( λ , λ ) ∈ Z with a left-justified arrangement of boxes with λ i boxes in the i th row.As in [DR, (5.28)] the shifted content of a box in row i and column j as˜ c (box) = j − i − ( a + b −
2) (5.2)i.e. the shifted content is its diagonal number, where the box in the upper left corner has shiftedcontent − ( a + b − j ∈ Z ≥− let P ( j ) be an index set for the irreducible U q gl -modules that appear in M ⊗ N ⊗ V ⊗ j . In particular, P ( − = ( a, P (0) = { ( a + b − j, j ) | j = 0 , , . . . , b } and P ( j ) = { ( a + b + j − ℓ, ℓ ) | ≤ ℓ ≤ ( j + a + b ) } , for j ≥ § vertices on level j labeled by the partitions in P ( j ) ,an edge ( a, −→ µ for each µ ∈ P (0) , andfor each j ≥ µ ∈ P ( j ) and λ ∈ P ( j +1) , there isan edge µ → λ if λ is obtained from µ by adding a box.The case when a = 6 and b = 3 is illustrated in Figure 2.Assume q ∈ C × and a > b + 2 so that the generality condition ( a + 1) − ( b + 1)
6∈ { , ± , ± } of[DR, Theorem 5.5] is satisfied. Define r = ( a − b ) and r = ( a + b + 2) , (5.3)and let H ext k be the extended two boundary Hecke algebra with parameters t , t k , and t given by t = q, t = − t r − r = − q ( b +1) , and t k = − t r + r = − q a +1) , (5.4)so that − t k t − = − t r and t k t = − t r as in [DR, (3.5), (5.21)]. By [DR, Theorem 5.4 and (5.21)]there are commuting actions of U q gl and H ext k on M ⊗ N ⊗ V ⊗ k ,where the H ext k action is given via R-matrices for the quantum group U q gl . Theorem 5.1.
Let a, b ∈ Z ≥ with a > b + 2 . Let q ∈ C × not a root of unity and let H ext k be the twoboundary Hecke algebra with parameters t , t k and t as in (5.4) . Let U q gl be the Drinfeld-Jimboquantum group corresponding to gl and let M , N and V be the simple U q gl -modules given in (5.1) . Then the H ext k action factors through T L ext k and, as ( U q gl , T L ext k ) -bimodules, M ⊗ N ⊗ V ⊗ k ∼ = M λ ∈P ( k ) L ( λ ) ⊗ B λk with B ( a + b + k − ℓ,ℓ ) k ∼ = H ( z, c ,J ) , where z = ( − k q ( a + b − ℓ )( a + b − ℓ − ℓ ( ℓ − − a ( a − − b ( b − − k ( a + b − and ( c , J ) is the local region corre-sponding to the configuration κ of k boxes r − ℓℓ + 1 − r − k r − k − ℓℓ − r (5.5) that has k boxes in each row, the shifted content of the leftmost box in the first row is r − ℓ , theshifted content of the leftmost box in the second row is ℓ + 1 − r − k . Between the rows there are bluemarkers in diagonals with shifted content ± r and there are red markers in diagonals with shiftedcontent ± r , as pictured. Explicitly, c = ( c , c , . . . , c k ) is the sequence ofabsolute values of c, c + 1 , · · · , c + k − , where c = ( a + b ) − ℓ + 1 , arranged in increasing order; and J is the union of J = ∅ , if a ≥ b ≥ ℓ , { ε ℓ − b } , if a ≥ ℓ > b , { ε a − b } , if ℓ > a > b, ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 29 and J = ∅ , if ( a + b + 2) > ℓ , { ε − ε , ε − ε , . . . , ε ℓ − a − b − ε ℓ − a − b − } , if ℓ ≥ ( a + b + 2) and a + b even, { ε − ε , ε − ε , . . . , ε ℓ − a − b − ε ℓ − a − b − } , if ℓ ≥ ( a + b + 2) and a + b odd.Proof. Fix λ = ( a + b + k − ℓ, ℓ ) ∈ P ( k ) . The sum of the contents of the boxes in λ is X box ∈ λ c (box) = (0 + 1 + . . . + ( a + b + k − ℓ − − · · · + ℓ − ( a + b + k − ℓ − a + b + k − ℓ ) + ℓ ( ℓ − . By [DR, Theorem 5.5 and (5.35)], B λk ∼ = H ( z, c ,J ) k where z = ( − k q c , where c = − ( k ( a + b −
2) + a ( a −
1) + b ( b − X box ∈ λ c (box) , and c and J and the corresponding configuration κ of 2 k boxes are determined as follows.Place markers at the NW corner of the boxes at positions (1 , a + b + 1), (2 , a + 1), (2 , b + 1), and(3 ,
1) so that these markers are in the diagonals with shifted contents ± r and ± r . λ = ( a + b + k − ℓ, ℓ ) = b ℓ − ba b k − ℓℓ − r r − k − ℓ Following [DR, (5.27)], let S (0)max = ( ( a + b − ℓ, ℓ ) , if a ≥ b ≥ ℓ ,( a, b ) , if a ≥ ℓ ≥ b (since a ≥ b we are in the left case of [DR, (5.15)] with c = d = 1 so that µ c = min( ℓ, b ) and S (0)max = ˚ µ = ( a + b − µ c , µ c )): S (0)max ℓ a + b − ℓ k if a ≥ b ≥ ℓ S (0)max b ℓ − ba k − ( ℓ − b ) if a ≥ ℓ > b or ℓ > a ≥ b . By [DR, (5.35)], the corresponding configuration of boxes is κ = rot( λ/S (0)max ) ∪ λ/S (0)max , as picturedabove in (5.5).To determine ( c , J ), use the conditions ( κ κ
4) of Section 4.2 which specify the relation between κ and ( c , J ). First index the boxes of κ with − k, . . . , − , , . . . , k by diagonals, left to right, andNW to SE along diagonals. The sequence c = ( c , . . . , c k ) with 0 ≤ c ≤ c ≤ · · · ≤ c k is thesequence of the absolute values of the shifted contents of boxes in the first row of κ . Next, the set J is determined as follows.1. By ( κ J contains ε i if i > i is NW of the marker in the diagonal with shiftedcontent r or r in κ . This occurs on diagonal r whenever ℓ > b (marked in blue), ε ℓ − b ∈ J if a ≥ ℓ > b and ε a − b ∈ J if ℓ > a ≥ b ; and J contains no roots of the form ε j when a ≥ b ≥ ℓ .2. By ( κ J contains ε j − ε i if j > i > i and box j are in the same column of κ (so that box i and box j are in adjacent diagonals and box j is NW of box i ). This occurs exactlywhen 0 ≥ r − ℓ = ( a + b + 2) − ℓ . If ℓ ≥ ( a + b + 2) and a + b is even then the boxes indexed1 , , . . . , ℓ − ( a + b + 2)) = 2 ℓ − ( a + b + 1) are in the second row directly below boxes ofindex 2 , , . . . , ℓ − a − b . If ℓ ≥ ( a + b + 2) and a + b is odd then boxes 2 , , . . . , ℓ − ( a + b + 1)),directly below boxes of index 3 , , . . . , ℓ − a − b : r − ℓ ℓ − r
12 34-1-2-3-4 · · ·· · · · · · ∗ +1 k - ∗ -1 - ∗ - k · · ·· · ·· · · ℓ − a − b − ∗ if a + b is even, r − ℓ ℓ − r − · · ·· · · · · · ∗ +1 k -1-2-3 - ∗ -1 - ∗ - k · · ·· · ·· · · ℓ − a − b − ∗ if a + b is odd.So J contains ε − ε , ε − ε , . . . , ε ℓ − a − b − ε ℓ − a − b − if ℓ ≥
12 ( a + b + 2) and a + b is even, or ε − ε , ε − ε , . . . , ε ℓ − a − b − ε ℓ − a − b − if ℓ ≥
12 ( a + b + 2) and a + b is odd.3. Also by ( κ J contains ε j + ε i if j > i >
0, and box j is directly above box − i , whichdoes not occur.In this way c and J are determined from κ . Since all of these H ( z, c ,J ) k satisfy the conditions ofTheorem 4.3, it follows that the H ext k -action on M ⊗ N ⊗ V ⊗ k factors through T L ext k . (cid:3) Remark 5.2.
The dimension of B ( a + b + k − ℓ,ℓ ) k is the number of paths in the Bratteli diagram froma shape on level 0 to the shape λ = ( a + b + k − ℓ, ℓ ) on level k . Summing over the shapes on level0 for which there is a path to λ givesdim( B ( a + b + k − ℓ,ℓ ) ) = min( b,ℓ ) X c =max(0 ,ℓ − k ) f λ/ ( a + b − c,c ) , where f λ/µ is the number of standard tableaux of skew shape λ/µ . If ℓ ≤ a + b − c then the secondrow of λ/ ( a + b − ℓ, ℓ ) does not overlap the first row and thus f λ/ ( a + b − c,c ) = (cid:18) kℓ − c (cid:19) if ℓ ≤ a + b − c .Since c ≤ min( b, ℓ ), the case ℓ > a + b − c can occur only when ℓ > a ≥ b , in which case( a + b + k − ℓ, ℓ ) / ( a + b − c, c ) = c ℓ ℓ − ( a + b − c ) a + b − c k − ℓ + c , ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 31 so that f ( a + b + k − ℓ,ℓ ) / ( a + b − c,c ) = k + ℓ − c X j = ℓ − ( a + b − c ) f ( k − j,j ) = min( k − ( ℓ − c ) ,ℓ − c ) X j = ℓ − ( a + b − c ) (cid:18) kj (cid:19) − (cid:18) kj − (cid:19) = (cid:18) kℓ − c (cid:19) − (cid:18) kℓ − ( a + b − c ) − (cid:19) . Namely, the first equality comes from the Pieri formula and the expansion of a skew Schur functionby Littlewood-Richardson coefficients (see [Mac, (5.16)] for the Pieri formula and [Mac, (5.2) and(5.3)] for Littlewood-Richardson coefficients) and the second equality comes from the number ofstandard tableaux of a two row shape as given, for example, in [GHJ, Theorem 2.8.5 and Lemma2.8.4].The following examples reference the node label styles in Figure 2.
Example.
Let a = 7 and b = 3 . The markers are in the diagonals with shifted contents ± r and ± r , where r = 2 and r = 6 . An example where ℓ > a ≥ b : Let ℓ = 11 and k = 14 , then λ = (13 ,
11) = with S (0)max = (7 , .The boxes of λ/S (0)max haveshifted contents: - - Then c is the rearrangement of the absolute values of ( − , − , , , , , , , , , , , , into in-creasing order and J = { ε , ε − ε , ε − ε , ε − ε , ε − ε , ε − ε , ε − ε } . The configurationof boxes κ corresponding to ( c , J ) has indexing of boxes - - - - - - - - - - - -
11 13 14 - - Example.
Let a = 6 and b = 3 to take advantage of the setting and notation of Figure 2. Themarkers are in the diagonals with shifted contents ± r and ± r , where r = and r = .(1) An example where ℓ > a ≥ b : Let ℓ = 8 and k = 9 , then λ = (10 ,
8) = with S (0)max = (6 , .The boxes of λ/S (0)max haveshifted contents: - -
12 12 3252 5272 92 112
Then c is the rearrangement of the absolute values of ( − , − , − , , , , , , ) into increas-ing order and J = { ε , ε − ε , ε − ε , ε − ε } . The configuration of boxes κ corresponding to ( c , J ) has indexing of boxes − − − − − − − − − with P ( c ) = ε , ε , ε + ε , ε − ε , ε − ε ε − ε , ε − ε , ε − ε , ε − ε ,ε − ε , ε − ε , ε − ε , ε − ε . (2) An example with a ≥ ℓ > b : Let k = 3 and ℓ = 5 , so that a + b + k − ℓ = 7 . λ = (7 ,
5) = with S (0)max = (6 , .The boxes of λ/S (0)max have shifted contents: - - Then c is the rearrangement of the absolute values of ( , , ) in increasing order and J = { ε } .The configuration of boxes κ corresponding to ( c , J ) is − − −
11 2 3 with P ( c ) = { ε , ε − ε , ε − ε } (3) An example with a ≥ b ≥ ℓ : Let k = 3 and ℓ = 2 , so that a + b + k − ℓ = 10 . Then λ = (10 ,
2) = with S (0)max = (7 , .The boxes of λ/S (0)max have shifted contents:
72 92 112 . Then c is the rearrangement of the absolute values of ( , , ) in increasing order and J = ∅ .The configuration of boxes κ corresponding to ( c , J ) is − − − with P ( c ) = { ε − ε , ε − ε } . ALIBRATED REPRESENTATIONS OF TWO BOUNDARY TEMPERLEY-LIEB ALGEBRAS 33
Figure 2.
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