Chain models on Hecke algebra for corner type representations
aa r X i v : . [ m a t h . QA ] O c t Chain models on Hecke algebra for corner typerepresentations.
A.P. Isaev a , O.V. Ogievetsky b and A.F. Os’kin a a Bogoliubov Laboratory of Theoretical Physics, Joint Institute for NuclearResearch,Dubna, Moscow region 141980, RussiaE-mails: [email protected], [email protected] b Center of Theoretical Physics , Luminy, 13288 Marseille, Franceand P. N. Lebedev Physical Institute, Theoretical Department, Leninsky pr.53, 117924 Moscow, RussiaE-mail: [email protected] Abstract.
We consider the integrable open chain models formulated in termsof generators of the Hecke algebra. The spectrum of the Hamiltonians for theopen Hecke chains of finite size with free boundary conditions is deduced forspecial (corner type) irreducible representations of the Hecke algebra.
The A-type Hecke algebra H n +1 is generated by the elements σ i ( i = 1 , . . . , n )subject to the relations: σ i σ i +1 σ i = σ i +1 σ i σ i +1 , (1) σ i σ j = σ i σ j , | i − j | > , (2) σ i = ( q − q − ) σ i + 1 , (3)where q ∈ C \{ } is a parameter. In this paper we consider a restricted class ofrepresentations of H n +1 corresponding to so called corner Young diagrams (cor-ner representations) related to U q su (1 |
1) models [1]. In these representationswe calculate the spectrum of the Hamiltonian of the open Hecke chain models Unit´e Mixte de Recherche (UMR 6207) du CNRS et des Universit´es Aix–Marseille I,Aix–Marseille II et du Sud Toulon – Var; laboratoire affili´e `a la FRUMAM (FR 2291)
12, 3], i.e. the spectrum of the following element of the Hecke algebraˆ H = n X i =1 σ i , σ i ∈ H n +1 . (4)The paper is organized as follows. In Section 2 we describe the corner typerepresentations of the Hecke algebra. In Section 3 we calculate the spectrum ofthe Hamiltonian for the corner type representations for Young diagrams withtwo rows. In Section 4 we establish a relation between the corner type represen-tation with l rows and the l -th wedge power of the corner type representationwith two rows. Using this and the results of Section 3, we calculate the spectrumof the Hamiltonian (4) in the general corner type diagram. { k + 1 , l } The representation theory of the Hecke algebras is a well developed subject, see[4, 5, 6, 7] and references therein. For generic q , it is known that irreduciblerepresentations ρ λ of the Hecke algebra H n +1 are labeled by the Young diagrams λ with n + 1 boxes and basis elements in the representation space of ρ λ can beindexed by the standard Young tableaux of the shape λ . The standard Youngtableau for the corner diagram { k + 1 , l } is:1 j j . . . j k i i ... i l I (5)Here { i , . . . , i l , j , . . . , j k } is a ( l, k )-shuffle of { , , . . . , k + l, k + l + 1 } . Thestandard Young tableau is thus determined by the set I = { i , . . . , i l } .Denote the corresponding representation of the Hecke algebra by ρ ( k,l ) andits space by V ( k,l ) . The action of the generators σ p , 1 ≤ p ≤ k + l , is ρ ( k,l ) ( σ p ) v I = qv I if p, p + 1 / ∈ I, (6) ρ ( k,l ) ( σ p ) v I = − q − p ( p ) q v I + ( p − q ( p ) q v s p I if p / ∈ I, p + 1 ∈ I , (7) ρ ( k,l ) ( σ p ) v I = q p ( p ) q v I + ( p + 1) q ( p ) q v s p I if p ∈ I, p + 1 / ∈ I , (8) ρ ( k,l ) ( σ p ) v I = − q − v I if p, p + 1 ∈ I , (9)where ( p ) q = q p − q − p q − q − and v s p I is the basis vector corresponding to the Youngtableau with the numbers p and p + 1 interchanged. In the matrix form ( e I,J ∈ V k,l ) are the matrix units, e I,J e K,L = δ JK e I,L ): ρ ( k,l ) ( σ p ) = q P I : p,p +1 / ∈ I e I,I + P I : p/ ∈ I,p +1 ∈ I (cid:16) − q − p p q e I,I + ( p − q p q e s p I,I (cid:17) + P I : p ∈ I,p +1 / ∈ I (cid:16) q p p q e I,I + ( p +1) q p q e s p I,I (cid:17) − q − P I : p,p +1 ∈ I e I,I . (10) Proposition 2.1
For the Hamiltonian ˆ H ≡ k + l P i =1 σ i ∈ H k + l +1 we have tr V ( k,l ) ( ρ ( k,l ) ( ˆ H )) = ( qk − q − l ) dim V ( k,l ) . (11) Proof.
The dimension of the space V ( k,l ) is the number of l -element subsets of { , , . . . , k + l, k + l + 1 } , dim V ( k,l ) = (cid:18) k + ll (cid:19) . By eqs.(6)-(9), the action of the first generator σ is diagonal andtr( ρ ( k,l ) ( σ )) = qN − q − N , N = (cid:18) k + l − l (cid:19) , N = (cid:18) k + l − l − (cid:19) . (12)Here N (resp., N ) is the number of sets I with 2 / ∈ I (resp., 2 ∈ I ).Since Xσ i X − = σ i +1 ∀ i ∈ [1 , k + l − X = σ σ . . . σ k + l , theelements σ i , i >
1, are conjugate to σ . Thus tr( ρ ( k,l ) ( σ i )) = tr( ρ ( k,l ) ( σ )) = qN − q − N and (11) follows.It turns out that it is more convenient to work with the traceless Hamiltonian H ( k,l ) ( q ) := ρ ( k,l ) k + l X i =1 σ i ! − ( q k − q − l ) . (13) H ( k, ( q ) Consider the Hecke algebra H k +2 and its representation ρ ( k, for the cornerYoung diagram { k + 1 , } with only two rows (i.e. l = 1). The dimension of thisrepresentation is k +1. As we shall see in the sequel, the Hamiltonian H ( k, ( q ) inthis representation is a building block for the construction of the Hamiltonians H ( k,l ) ( q ), corresponding to all corner diagrams { k + 1 , l } . In the representation ρ ( k, the set I (5) consists of only one number, I = { i } , i ∈ { , . . . , k + 2 } , andwe use the notation v i = v I for basis vectors and e i,j for matrix units. Proposition 3.1
In the basis { v i } , the Hamiltonian (13) reads H ( k, ( q ) = k +1 X p =2 (cid:16) ( p +1) q e p,p +1 +( p − q e p +1 ,p ( p ) q − e p,p ( p ) q ( p − q (cid:17) + ( k ) q e k +2 ,k +2 ( k +1) q . (14)3 roof. According to the general formula (10), we have ρ ( k, ( σ p ) = q + ( p − q p q ( e p,p +1 − e p,p ) + ( p + 1) q p q ( e p +1 ,p − e p +1 ,p +1 ) (15)(here e , = 0 = e , ). Eq. (14) is a straightforward consequence of (15).Let D = diag(1 , q, q , . . . ). Then the operator D H ( k, ( q ) D − possesses afinite limit H ∞ ( k, = k +1 P p =2 ( e pp +1 + e p +1 p ) when q tends to infinity.For q ∈ C ∗ \ { q | ( k + 1) q ! = 0 } define an upper triangular matrix C ( q ), C ( q ) = k +2 X p =2 p − q e pp − k +1 X p =2 p ) q e pp +1 . (16) Proposition 3.2
We have H ( k, ( q ) C ( q ) = C ( q ) H ∞ ( k, . (17) Proof.
A direct calculation.Eq.(17) demonstrates the isospectrality (the q -independence of the spec-trum) of the family H ( k, ( q ). By (17), H ( k, ( q ) has the same spectrum as H ∞ ( k, . This spectrum is well known ( H ∞ ( k, is the incidence matrix of theDynkin diagram of type A ), Spec ( H ∞ ( k, ) = { πpk +2 ) } , 1 ≤ p ≤ k + 1. Wesummarize the results (obtained by a different method in [8]). Theorem 3.1
The spectrum of the Hamiltonian (13) for l = 1 is Spec( H ( k, ( q )) = { πpk + 2 } , p = 1 , , . . . , k + 1 . (18) Remark.
Let N = P k +2 p =2 e pp − P k +1 p =2 e pp +1 . Then N − C ( q ) C ( r ) − N =diag(1 , r q , r q , . . . ). In particular, the operators C ( q ) C ( r ) − commute for differ-ent values of q and r . H ( k,l ) ( q ) For arbitrary k and l we realize the Hamiltonian H ( k,l ) ( q ) in terms of the Hamil-tonian H ( k + l − , ( q ). The best way to do this is to relate the representations ρ ( k,l ) and ρ ( k + l − , of the Hecke algebra H k + l +1 .For a vector space V let A n be the antisymmetrizer in V ⊗ n defined by A n ( v ⊗ v ⊗ . . . ⊗ v n ) = n ! P s ∈ S n ( − l ( s ) v s (1) ⊗ v s (2) ⊗ . . . ⊗ v s ( n ) ( S n is thepermutation group and l ( s ) is the length of a permutation s ). Denote σ ( m ) p = ⊗ ( m − ⊗ ρ k + l − , ( σ p ) ⊗ ⊗ ( l − m ) ∈ End( V ⊗ l ), where is the identity matrixin V . Denote by V ∧ l the wedge power of V , V ∧ l = A l V ⊗ l .4 roposition 4.1 The following identity holds q − l A l ( σ (1) p σ (2) p . . . σ ( l ) p ) A l = A l l X m =1 σ ( m ) p − ( l − q ⊗ l ! A l . (19) Proof.
The formula (19) is proved by induction using the l = 2 case, q − A (cid:16) σ (1) p σ (2) p (cid:17) A = A (cid:16) σ (1) p + σ (2) p − q ⊗ (cid:17) A , (20)which can be written in the form A (cid:16) σ (1) p − q (cid:17) (cid:16) σ (2) p − q (cid:17) A = 0 (21)and directly deduced from (15). Proposition 4.2
The set of matrices ˜ ρ ( k,l ) ( σ p ) = q − l A l ( ρ k + l − , ( σ p ) ⊗ . . . ⊗ ρ k + l − , ( σ p )) A l (22) defines a representation of the Hecke algebra H k + l +1 in V ∧ lk + l − , . Proof.
The braid relations (1) and the locality (2) follows from the multiplica-tive structure of ˜ ρ ( k,l ) ( σ p ) and the fact that ρ k + l − , ( σ p ) is a representation.The Hecke condition (3) can be proved by induction using (19).Proposition can be generalized as follows. Proposition 4.3
Let ρ and ρ be representations of the Hecke algebra H n in spaces V and V , respectively. Assume that an idempotent A ∈ End( V ⊗ V ) , A = A , commutes with ρ ( σ p ) ⊗ ρ ( σ p ) and ρ ( σ p ) ⊗ ⊗ ρ ( σ p ) for any p = 1 , , . . . , n − and satisfies (cid:16) ( q − q − − α ) ρ ( σ p ) ⊗ ρ ( σ p )+ ρ ( σ p ) ⊗ ⊗ ρ ( σ p )+ − α q − q − ⊗ (cid:17) A = 0 (23) for some α = 0 . Then ρ ( σ p ) := α − A ρ ( σ p ) ⊗ ρ ( σ p ) is a representation of theHecke algebra H n in the image A ( V ⊗ V ) of A . Proof.
A direct calculation, as in the previous Proposition.The condition in (23) factorizes as in (21) only if α = q, − q − .The map ι : v i ∧ v i ∧ . . . ∧ v i l v I , I = { i , . . . , i l } , i < i < . . . < i l , isan isomorphism of the vector spaces V ∧ lk + l − , and V ( k,l ) (and we use the samenotation v I for basis vectors of both spaces). Now we identify the representation(22) with the irreducible representation ρ ( k,l ) . Proposition 4.4
The map ι intertwines the representations ˜ ρ ( k,l ) and ρ ( k,l ) . Proof.
We directly verify eqs.(6)-(9) for ˜ ρ ( k,l ) . Note that the matrices ˜ ρ ( k,l ) ( σ p )can be written in the form˜ ρ k,l ( σ p ) = q − l A l ( ρ k + l − , ( σ p ) ⊗ . . . ⊗ ρ k + l − , ( σ p ))= q − l ( ρ k + l − , ( σ p ) ⊗ . . . ⊗ ρ k + l − , ( σ p )) A l , (24)5here the antisymmetrizers act from the left or from the right only.If p, p + 1 / ∈ I then (omitting the sign of the tensor product)˜ ρ k,l ( σ p )( v i ∧ . . . ∧ v i l ) = q − l A l ( σ (1) p σ (2) p . . . σ ( l ) p ) v i v i . . . v i l = qA l ( v i . . . v i l ) = qv I , (25)which proves (6). If p ∈ I, p + 1 / ∈ I then˜ ρ k,l ( σ p )( v i ∧ . . . ∧ v p ∧ . . . ∧ v i l ) = q − l A l ( σ (1) p . . . σ ( l ) p ) v i . . . v p . . . v i l = A l (cid:18) v i . . . ( q p ( p ) q v p + ( p + 1) q ( p ) q v p +1 ) . . . v i l (cid:19) = q p ( p ) q v I + ( p + 1) q ( p ) q v s p I , (26)which coincides with (8). We used that v i ∧ . . . ∧ v p +1 ∧ . . . ∧ v i l = v s p I since p + 1 / ∈ I . Eq.(7) can be proved in the same way. Finally, if p, p + 1 ∈ I then˜ ρ k,l ( σ p )( v i ∧ . . . ∧ v p ∧ v p +1 ∧ . . . ∧ v i l )= A l ( v i . . . ( q p ( p ) q v p + ( p + 1) q ( p ) q v p +1 )( − q − p p q v p +1 + ( p − q p q v p ) . . . v i l )= q − ( p ) q A l ( − v i . . . v p v p +1 . . . v i l + q p ( p − q v i . . . v p v p . . . v i l − q − p ( p + 1) q v i . . . v p +1 v p +1 . . . v i l + ( p + 1) q ( p − q v i . . . v p +1 v p . . . v i l )= − q − ( p ) q ( v I + ( p + 1) q ( p − q v I ) = − q − v I , (27)which coincides with (9).Now one can find eigenvalues for H ( k,l ) ( q ) using the results of Section . Theorem 4.1
The family H ( k,l ) ( q ) is isospectral with the spectrum Spec( H ( k,l ) ( q )) = ( l X i =1 πm i k + l + 1 , ≤ m < m . . . < m l ≤ k + l ) . (28) Proof.
Due to propositions and , the Hamiltonian H ( k,l ) ( q ) equals H ( k,l ) ( q ) = n X p =1 ˜ ρ k,l ( σ p ) − ( qk − lq − ) ! A l = n X p =1 l X i =1 ( σ ( i ) p − ( l − q ) − ( kq − lq − ) ! A l = l X i =1 H ( i ) ! A l , (29)where H ( i ) = ⊗ ( i − ⊗ H ( k + l − , ( q ) ⊗ ⊗ ( l − i ) . The isospectrality of H ( k,l ) ( q )follows from proposition .Let { ψ m } , 1 ≤ m ≤ k + l , be the eigenbasis of H ( k + l − , ( q ). By (29), { ψ I } ,where ψ I = ψ m ∧ ψ m ∧ . . . ∧ ψ m l , 1 ≤ m < . . . < m l ≤ k + l , is the eigenbasisof the Hamiltonian H ( k,l ) ( q ) and (28) follows.6 cknowledgement. The first author (A.P. Isaev) was supported by theRFBR grant No. 05-01-01086-a; the second author (O. Ogievetsky) was sup-ported by the ANR project GIMP No. ANR-05-BLAN-0029-01.
Note added in proof.
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