Open Hecke chains for corner type representations

A.P. Isaev", O.V. Ogievetsky" and A.F. Os'kin11

" Bogoliubov Lnboratmy of Thcm-etica.l Physir:s, .Joint Institutr~ for Nuclear Hesmn:h, Dubna, Moscow n~gion 141980, Russia

isaevap@theor.jinr.ru, aoskin@theor.jinr.ru

° Crn.lff of Theoretical Physir:s 1, Lmniny, 1:J288 Marsr~illc , Frnn1:1'

and P . N. Lebcrle-11 Physical Institute, Theoretical Dcpa1'lnwnt, Lcninsky pr. 5.'I, 1J7924

Moscow, Russia oleg@cpt .univ-mrs.fr

Abstract

This report is based on the paper [1]. We consider the integrable open chain models formulated in terms of generators of the Hecke algebra. The spectrum of the Hamiltonians for the open Hecke chains of finite size with free boundary conditions is deduced for special (corner type) irreducible representations of the Hecke algebra.

1 Introduction

The A-type Hecke algebra Hn+t is generated by the elements a; (i = 1, .. . , n) subject to the relations:

a;a;+iai a;a~

U;

U;+JUiUi+t 1

aiaj, (q - q- 1)a; + 1,

li-jj>l,

where q E C\ { 0} is a parameter. In this paper we consider a restricted class of representa­tions of H,.+ 1 corresponding to so called corner Young diagrams (corner representations) related to Uq(su(ljl)) models [2). In these representations we calculate the spectrum of the Hamiltonian of the open Hecke chain models [3, 4], i.e. the spectrum of the following element of the Hecke algebra

n

il = LCT;, CT; E Hn+l· (4) i=l

The paper is organized as follows. In Section 2 we describe the corner type represen­tations of the Hecke algebra. In Section 3 we calculate the spectrum of the Hamiltonian for the corner type representations for Young diagrams with two rows. In Section 4 we establish a relation between the corner type representation with l rows and the l-th wedge power of the corner type representation with two rows. Using this and the results of Section 3, we calculate the spectrum of the Hamiltonian ( 4) in the general corner type diagram.

1 Unite Mixte de Recherche (UMR 6207) du CNRS et des Universites Aix- Marseille I, Aix- Maiseille II et du Sud Toulon - Var; laboratoire affilie a la FRUMAM (FR 2291)

217

2 Representation for the corner type diagrams {k + 1, 11}

The 1 cprcs< ~ 111 atio11 1 lwory of the Hecke algehrns is a well dPvelopcd subject , sec [5 ·8] and references t.ll('rl'iu. For generic q, it is known that irreducible representations P>. of the Hecke alg<'bra ll,,+ 1 ate labeled by the Young diagrams A with 11 + 1. boxes and basis c~lemcnts in th<~ rcp1csent.at.ion space of P>. can be indexed by the staudard Young tableaux of the shape.>-. ThP sl.anclard Young tableau for the corner diagram {A:+ 1, 11

} is:

I (5)

Hen~ {·it,. . ., ·idt,. . ., jk} is a (l, k)-shuffie of {2, 3, . .. , k + l, k + l + 1 }. The standard Young t.ableau is thus determined by the set I= {i1, ... , ii}.

Denote the corresponding representation of the Hecke algebra by P(k,I) and its space by \'(k,I) · The action of the generators aµ, 1 :::; p:::; k + l, is

P(k,IJ(ap)vr = qvr if p,p + 1 f:. I, (6)

- q·-p (p - 1), fi(k,l)(ap)vr = -(-)-v1 + - (- )- 1

·11. ,,1 p q p q

if p f:. I, p + 1 E I , (7)

qP (p+ J),, P(k,l)(ap)v1 = (p)qv1 + ~ v,1, 1 if p E l,p + 1 f:. I , (8)

P(k,t)(ap)Vt = -q-1v1 ifp,p+l EI, (9)

where (p)q = 9:~qq_-." and Vs,.J is the basis vector corresponding to the Young tableau with

the numbers p and p + 1 interchanged. In the matrix form (er,J E End(Vi,1) are the matrix units, e1,Jer<,L = bJ 1<<'-1,L):

P(k,l)(ap) = q L e1,1 + L (-~ e1.r + (p;.1l• es.1,1) l:p,p+l</cl l:prtl,p+lEI

' I:: ('I". (p+l) ) -1 I:: • e1 1 + _fl es Ir - q e1 1 . Pq J Pq p I I

J:pEl,p+l</cl l:p,p+lEl

• k+I Proposition 2.1 For the Hamiltonian 1-l = I:: ai E Hk+t+i we have

i=l

(10)

(11)

Proof. The dimension of the space V(k,1) is the number of I-element subsets of {2, 3, ... , k+

l,k+l+l},dim'~k,l)= e71) .

By eqs.(6)-(9), the action of the first generator a 1 is diagonal and

- j (k + l - 1) (k + l - 1) tr(P(k,l) (ai)) = qN 1 - q N2, N 1 = l , N 2 = l - l · (12)

218

Hctc N 1 (t'csp., N2 ) is the utnllher of sets 1with2¢1 (resp., 2 E !). Siucc Xo;x - 1 = o,+ 1 Vi E [l,J.: -1- l - l], when~ X = o 1(f2 ... okf-1, t.lu! c}(!l11Cnts

o;, ·i > l, an! conj11gaU· Lo a 1 Tlius tr(f!(k,l)(o;)) = tr(f!(k,1)(01)) = 11N1 -- q·- 1JV2 and (Ll) follows. •

It. t.11rns out t.hat it is 111<>rP convenient Lo work with tlw traccl<!SS Hamiltonian

(

k-1-1 )

1t(A,l)(IJ) := P(k,l) t;a, -- (qi.: - 1( 1 l)L (13)

3 Spectrum of 1-l(l.:,l)(q)

Consider tlir. lfodw algebra fh-1-2 and its repn~sentation /l(k,I) for the corner Young diagram {!.: + 1, l} with only two rnws (i.e. l = 1). The dimension of this rcpreseutatiou is k + 1. As we shall see later, the Hamiltonian 1i(k,l)(q) in this representation is a building block for t.he construction of the Hamiltonians 1i(k,l) ( q), corresponding to all corner diagrams{!.:+ 1, 11}. In the representation P(k,t) the sl!t 1 (5) rnnsists of only one number, 1 = {i}, ·i. E {2, ... , !.: + 2}, and we use the notation v; = v1 for basis vectors and e.;,1 for matrix units.

Proposition 3.1 Jn the Im.sis {11;}, tlw Hamiltonian {13) reads

(14)

Proof. According to the general formula (10), we have

(15)

(here e1,2 = 0 = e.2, 1). Eq. (14) is a straightforward consequence of (15). •

Let D = diag(l, q, q2, ... ). Then the operator D1i(k,1)(1J)D--1 possesses a finite limit

k+l

1i(k',i) = L (epp-H + ep+1p) wheu q tends to infinity. p= 2

For q E c· \ {q I (k + l)q! = O} define an upper triangular matrix C(q),

k+2 1 k+l 1 C(q) = L (p _ l) ePP - 2:-(p-), epp+t .

7>=2 q p=2 I

Proposition 3.2 We have

1i(k,1J(1J)C(q) = C(q)1i'tf,1i

Proof. A direct calculation.

(16)

(17)

• Eq.(17) demonstrates the isospectrality (the q-inclependence of the spectrum) of the

family 1i(A-,l)(!J). I3y (17), 1i(k,I)(q) has the same spectrum as 1i{f,ii· This spectrum is well known (11{!'.,t) is the incicl<!Tlce matrix of the Dynkin diagram of type A), Spec (1i(k',l)) =

219

{2 cos({/':;)}, 1'.SJi'.Sk+1. W<~ surnrri;Hi,,e the results (obtairn~d by a different meLhod in [9]).

Theorem 3.1 The sper:irum of the Ho.rni:ltonian {iii) fo·r· l = l is

' I 1f]) ( Spcr:(7-l(k i)(q)) = {2 cos -- }, J! = 1, 2, ... , k + 1. , /,: + 2

(18)

R k I 't ,, - "'k·l 2 - "'k+I , 'I'! ' ,,_IC( )C'(· ·i-· l 'I - I" ·(1 :I.,. :1,. ) emar. A. 1' -- L., =2c,,,, L., =2Cf'p+I· 1Ll1JV .. q .. I h - (J<lg ,-2 ,c-l , .... p p 'I , '/

In particular, the operators C(q)C(1·)- 1 commute for diffon~ut valut•s of q awl ·r.

4 Spectrum of Hu,,n(q)

For arbitrary k and I we realize the Hamiltonian 1-l(k,l)(IJ) in terms of the Hamiltonian 1-l(k+t--1,L)(IJ) . The best way to do this is to rnlate the rcpres<~ntations P(k,t) and P(k-H--l,l)

of the Hecke algebra Hk-il-1 1.

For a vector space V let An be the antisymrnetrizer in V0 " defined by An (v 1 0v2 ® · · · ® Vn) = ;h L (-l)l(s)Vs(l) 0 1!8 (2) 0 · · · ® Vs(n) (Sn is the permutation group and l(s) is the

.c;ESn

length of a permutati011 s). Denotr. a~"')= I®(m-IJ ® fh+t-i, 1(a,,) ® 1°<t-m) E End(V®1),

where 1 is the identity matrix in V . Denote by V/\1 the wedge power of V, V/\1 = A1Vt91 •

Proposition 4.1 The following identity holds

Proof. The formula (19) is proved by induction using the l = 2 case,

q- 1 A2 ( a~1 lab2l) A2 = A2 (a~') + a~2 ) - q 1 ®2) A2,

which can be written in the form

A2 (a~') - q 1) ( a~2l - q 1) A2 = 0

and directly deduced from (15).

Proposition 4.2 The set of matrices

defines a representation of the Hecke algebra Hk+l+I in vkr.:.t-1,1 ·

(19)

(20)

(21)

•

(22)

Proof. The braid relations (1) and the locality (2) follows from the multiplicative struc­ture of P(k,l)(ap) and the fact that Pk+L-i,1(ap) is a representation. The Hecke condition (3) can be proved by induction using (19). •

Proposition 4.2 can be generalized as follows.

Proposition 4.3 Let p1 and p2 be representations of the Hecke algebra H 11 in space8 Vi and V2 , respectively. Assume that an idempotent A E End(Vi ® V2), A2 = A, commutes with p 1 (ap) ® p2(0'p) and p1 (O'p) ® 1+1 ® P2(ap) for any p = 1, 2, ... , n - 1 and satisfies

((q- q- 1 - a)p1(0'p)®p2(ap) +p1(0'p)®l+l©p2(ap)+ q1~,~.;.101)A=O (23)

220

for some a=/= 0. Then p(o,,) := a- 1Ap1(op) 0 P2(op) i8 a 1·epresentation of thr~ Hecke algcbm H,, in the -image A(Vi@ \~) of A.

Proof. A direct calculation, as in the previous Prnposition . • The condition in (23) factorir,c~:; as in (21) only if n = q, -q- 1

.

The map 1 : v;, /\ v;, /\ · · · /\ v;, >--+ v 1, I = { i 1 , ... , i1}, i 1 < i2 < · · · < i1, is an isomorphism of the vector spacc~s \ 1cr.;1_ 1,1 and \;(k,1) (and W<' lliil' the same notation v1 fo1·

basis vectors of both spaces). Now we identify the rcprrsrntation (22) with the irn~duciblC' representation P(k,t).

Proposition 4.4 The rna.p l inte1·twines the repre8entati:ons fJ(k,t) a.nd P(k,l).

Proof. We directly verify eqs.(6)-(9) for P(k,t)· Note that thl' matrices P(k,l)(op) can llC' written in the form

/ik,1(op) = q1- 1A1(Pk+l-1,1(ap) 0 · · · 0 Pk+1-1,1(ap)) = q1

-1(Pk+1-1,1(ap) 0 · · · 0 Pk+1-1,1(op))A1,

where the antisymmetrizers act from the left or from the right only. If p, p + 1 ¢ I then (omitting the sign of the tensor product)

- ( )( ) _ I-IA ( (I) (2) (I)) Pk,I aP v;, /\ ···/\vi, - q 1 ap ap ... Op v;, vi, .. . V; 1

= qA1(vi, ... v;,) = qv1,

which proves (6). If p E I,p + 1 ¢I then

- ( )( ) I-IA ( (!) (I)) Pk,1 ap vi, /\ ... /\ Vp /\ ... /\ V;, = q I Op . .. Op v;, ... Vp .. . v;,

( qP (p+l)q ) qP (p+l)q

= A1 Vi 1 ••• ((p)q Vp + ~Vp+1) ... vi, = (p),1V1 +~Vsµ!,

(24)

(25)

(26)

which coincides with (8). We used that v;, /\ · · · /\ Vp+! /\ · · · /\ v;1 = v8• 1 since p + 1 ¢ I . Eq. (7) can be proved in the same way. Finally, if p, p + 1 E I then

Pk,1(ap)(v; 1 /\ • • • /\ Vp /\ Vp+I /\···/\vi,)=

_ qP (p+l)q q-P (p-l)q -A1(v;1 ••• (-() Vp + -(-)-Vp+1)(--vp+1 +---vp) ... v;,) =

P q P q Pq Pq q-1

= (p)~A,(-vi, ... VpVp+I ... v;, + qP(p - l)qvi, ... 'VpVp ... v;,-

- q-P(p + l)qvi, ... Vp+JVp+I ... v;, + (p + l)q(p - l)qvi, ... Vp+1Vp ... v;,) = -1

= - ~)~ (v1 + (p + l)q(p- l)qv1) = -q-1v1, (27)

which coincides with (9). •

Now one can find eigenvalues for 1i(k,l)(q) using the results of Section.

Theorem 4.1 The family 1i(k,l)(q) is isospectral with the spectrum

Spec(1i(k,l)(q)) = { t 2 cos k ;7~ 1

, 1 S: m 1 < m 2 · · · < m1 S: k + l}. (28)

221

Proof. Due to propositions 4.1 and 4.3, the Hamiltonian 1-l(k,l)(q) equals

'H. (A·.1) (11 ) = (

11

/iu( rr1,) - (r1k - lr1 1 )1) A1 1•=•

= (t t(,,.~•l - (I - 1)111) - (kq - lq 1)1) .I,= (t ·7-1. (<l ) _11.

11=. I 1.::..., ' I

(2!J)

where 1-£(i) = l~'(H) ® 1-lu +i-i,L)(rJ) ® 1°(l-i). The isospectrality of 1-l(k,l)(q) follows from proposition 3. 2.

L(~t {1/1m}, 1 ::; m ::; /; + l, be the eigenbasis of 1-l(k+l-t,tJ(q). By (29), {4Jt}, where 1/J 1 = 'lj;m, /\ ·1/im, /\ · · · /\ ·~'"'', 1 ::; m 1 < · · · < m1 ::; k + l, is the eigenbasis of the Hamiltonian 1-l(i ,I) ( q) an cl (28) follows. •

Remark. Similiar l'<'Hults were obtained in the paper [10].

5 Connection with weak Bruhat order.

Tlw same rnsults can be obtained from combinatorial approach, connected with partial ordcting of the symnwtrical group Sn+t · Define word w E Sn+! as a multiplication of a 1111rnhcr elementary transpositions, i.e. w = si, si, ... si,. Then we can introduce specific partial order on Sn+i namecl weo.k Bruho.t order, which is defined as follows. For any two words w1, w2 E Sn+i we have w 1 > w2 if there is elementary transpositions E Sn+t such that 'll!J = S'W2.

Consider arbitrary corner-type representation of the Hecke algebra. From eqs.(6)-(9) it can be seen, that all of the basis vectors of such a representation can be ordered with weak Bruhat order if we consider the following mapping µ : V°(k,1) --t Sk+l+t

µ(v{2,3,4,. . .,1-1})

µ(vs,I)

where 1 is the unit element in Sk+L+t· The partial ordering can be graphically represented with Hasse diagram. It is a graph, which nodes are basis vectors, and only nodes of the type v1, vs,i are connected. For example, in case of P(2,2) representation we have the following Hasse diagram.

222

As it can be seen incidence matrix of the Hasse diagram of the representation P(k,l) coincide with the matrix 1-l{;.',i) and so it's spectrum is the same as the spectrum of the 1-l(k,l)(<J).

Proposition 5.1 Incidence matrix G(k,l) of the Hasse diagram. of the irredur.ibfo n~p­

rcsentation P(k,L) can be presentc~d in the form

Gck.1) = flU-1lG(k+1,1),

where D.(l-I) is a (l - 1)-th degn~e of the cornv.ltiplication D., and

D.G(k+l,1J = G(k+t,1) ® 1+1 ® Gck+t,1),

where 1 is the un-it (k + l) x (k + l) ·· matrix. Proof. Follows from the explicit form of the corresponding incidence matrices. •

Using Proposition 5.1 and some simple but long calculations one can reproduce Tlw­orem 4.1. Indeed, we needed to prove the fact, that the matrix 1-l(k,l)(<J) has the same spectrum as the incidence matrix G(k,l). It can be proved by noticing that

1-l(k,i)(q) C'(q) = C'(q) G(k,1),

where C'(q) is the similarity transformation matrix, which is defined as

C'(q) = C(q) ® C(q) ® · · · ® C(q)

e - 1 times

Acknowledgement. The first author (A.P. Isaev) was supported by the RFBR grant No. 05-01-01086-a; the second author (0. Ogievetsky) was supported by the ANR project GIMP No. ANR-05-BLAN-0029-01.

References

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