Matrix biorthogonal polynomials on the unit circle and non-Abelian Ablowitz-Ladik hierarchy

Matrix biortho g onal p o lynomials on the unit ci rcle and no n-Ab elian Ablo witz-Ladik hie r arc h y . Ma ttia Caf asso Univ ersit ´ e Cat holique de Lo uv ain D ´ epartemen t de Ma th´ ematiques Chemin du Cyclotron, 2 1348 Louv ain-la-Neuv e Belgium Abstract Adler and v an Mo er be ke [13] describ ed a reductio n o f the 2D-T o da hiera rch y called T o eplitz lattice. This hierarch y t urns out to be equiv a lent to the o ne originally descr ibed by Ablowitz and Ladik [1] using semidisc r ete zero-c ur v ature equations. In this pap er we obtain the orig inal semidiscr ete zero- curv atur e eq uations starting directly from the T o eplitz la ttice and we gener alize these computations to the ma tr ix case. This gene- ralization lead us to the semidiscr ete zero-curv a ture equations for the non-abelia n (or m ulticomp onent) version of the Ablowitz-Ladik equations [3]. In this way we extend the link betw een biorthogo nal p o lynomials on the unit circle and the Ablowitz-Ladik hierar - ch y to the matr ix case. MSC classe s: 0 5E35, 37K10 . 1 In tro duction. The Ablo witz-Ladik hierarc hy has b een intro d uced in 1975 [1] as a spatial discretization of the AKNS hierarch y . As d escrib ed b y S uris in [2] Ablowitz and Ladik replaced the celebrated Zakharo v-Shabat sp ectral p r oblem      ∂ x Ψ = L Ψ ∂ τ Ψ = M Ψ with a discretized ve rsion of it; namely      Ψ k +1 = L k Ψ k ∂ τ Ψ k = M k Ψ k . Here Ψ and Ψ k are t wo-co mp onen t vecto rs while L , M , L k and M k are 2 × 2 matrices. In particular L :=  z x y − z  and L k :=  z x k y k z − 1  . 1 W e can consider, as usu al, the p erio dic case ( k ∈ Z n ), the infinite case ( k ∈ Z ) or the semi- infinite case ( k ∈ N ). The standard zero-curv ature equations for th e AKNS hierarch y are replaced with the semidiscrete zero-curv ature equations ∂ τ L k = M k +1 L k − L k M k . As an example, one of the most imp ortan t equation of this hierarc hy is the discrete complex- ified version of the nonlinear Schr¨ odinger equations      ∂ τ x k = x k +1 − 2 x k + x k − 1 − x k y k ( x k +1 + x k − 1 ) ∂ τ y k = − y k +1 + 2 y k − y k − 1 + x k y k ( y k +1 + y k − 1 ) . Quite r ecen tly different authors (see [13] and [15]) w ork ed on the lin k b et w een biorthogonal p olynomials on the un it circle and th e semi-infinite Ablo witz-Ladik h ierarc hy . This is an analogue of the celebrated link b et w een T o da hierarc hy and orthogonal p olynomials on the real line. Many articles h a v e b een wr itten ab ou t orthogonal p olynomials and the T o da hierarc hy and it is almost imp ossible to recall all of them. Let u s just men tion th e seminal pap er of Moser [8] and the monography [9]; there in terested readers ca n find man y references. The more r ecen t articles [11], [12] and [13] should b e also cited. In these articles the connection b etw een the theory of orthogonal p olynomials and the Sato’s theory of infinite Grassmannians app eared for the fi rst time. In particular in [13] the case of T o da and Ablowitz -Ladik hierarc h ies are treated in a sim ilar wa y as r ed uctions of 2D-T o d a. In this wa y it’s giv en a clear and unified explanation of the role p la yed b y orthogonal p olynomials in the theory of inte grable equations. As noted by th e authors their app roac h is quite different fr om the original one; act ually in their pap er [13] they alwa ys sp eak ab out T o eplitz lattice and th e coincidence with Ab lowitz - Ladik hierarc hy is just stated in the int ro du ction. Neve rtheless, as explained in the second section of this pap er, it is very easy to d ed uce semid iscr ete zero-curv ature equations starting from Adler-v an Mo erb ek e’s equations. Ha ving this app roac h in mind we addressed a question arising fr om the follo wing facts: • Time ev olution for orthogonal p olynomials on the real line leads to th e T o da hierarc hy . • Time evol ution for biorthogonal p olynomials on the unit circle leads to the Ablowitz - Ladik h ierarc h y . • Time ev olution for matrix orthogonal p olynomials on the r eal line leads to th e non- ab elian T o da h ierarch y (see f or instance [17] and references therein; our approac h is close to [20]). What ab out time evolution for matrix b iortho gonal p olynomials on the unit cir cle? In other wo rds our goal wa s to replace the qu estion mark in the table b elo w w ith the corresp ondin g hierarch y . Orthogonal p olynomials on R Biorthogonal p olynomials on S 1 scalar case T o da Ablo witz-Ladik matrix case non-Ab elian T o d a ? 2 In this article we pro v e that the relev an t hierarch y is the non-Ab elian version of the Ablo witz-Ladik hierarc hy . T h is hierarch y has b een already stu died by different authors since 1983 (see [3], [4] and [5]) but, at our b est kno wledge, the connection with matrix biorthogonal p olynomials on the unit circle w as never established b efore. W e also remark that this h ierarch y is usually called matrix, vec tor or multicomp on ent Ablo witz-Ladik. Here we p refer to call it non-Ab elian to stress the analogy with T o da. In our setting, instead of ha ving one Lax op erator L k of size 2 × 2, we hav e t w o Lax op erators L l k and L r k of size 2 n × 2 n . These op erators dep end on matrices x l k , x r k , y l k and y r k of size n × n . S emidiscrete zero cur v ature equations are giv en by ∂ τ L l k = M l k +1 L l k − L l n M l k ∂ τ L r k = L r k M r k +1 − M r k L r k . where also M l k and M r k are blo c k m atrices. F or in stance w e hav e, in this hierarc hy , tw o v ersions of the n on-Ab elian complexified discrete n on lin ear Sc hr¨ odinger:      ∂ τ x l k = x l k +1 − 2 x l k + x l k − 1 − x l k +1 y r k x l k − x l k y r k x l k − 1 ∂ τ y r k = − y r k +1 + 2 y r k − y r k − 1 + y r k +1 x l k y r k + y r k x l k y r k − 1      ∂ τ x r k = x r k +1 − 2 x r k + x r k − 1 − x r k − 1 y l k x r k − x r k y l k x r k +1 ∂ τ y l k = − y l k +1 + 2 y l k − y l k − 1 + y l k − 1 x r k y l k + y l k x r k y l k +1 (see [6] for a review ab out th ese equations). Sections are organized as follo ws: • In the second section we recall some basic facts ab out 2D-T o da and the connection with biorthogonal p olynomials. W e use the approac h dev elop ed in [13]. • Thir d section starts with the d escrip tion of the T o eplitz lattice (see [13]) and sh o ws how to dedu ce semidiscrete zero-curv ature equations for the Ablo witz-Ladik hierarc hy . • In the fourth section we extend the T o eplitz lattic e to the case of blo c k T o eplitz matrices. • Fifth section gives r ecursion relations for matrix biorthogonal p olynomials on the unit circle; these formulas sligh tly generalize formulas con tained in [19] and [22] for matrix orthogonal p olynomials on th e unit circle. • In the sixth section we deriv e blo ck semidiscrete zero-curv ature equations d efining the non-Ab elian Ablo witz-Ladik hierarc h y . As an example we w rite the non-ab elian ana- logue of the discrete nonlinear Schr¨ odinger. 2 2D-T o da; linearization and biorthogonal p olynomials. In this section we recall some b asic facts ab out the 2D- T o da hierarch y as presente d in [7]. Moreo ver we describ e the connection with b iorth ogonal p olynomials as originally p resen ted in [13]. W e are interested in th e semi-infinite case; w e start denoting with Λ the shif t matrix Λ := ( δ i +1 ,j ) i,j ≥ 0 . 3 F or the transp ose we use the notation Λ T = Λ − 1 . Then we define tw o Lax matrices      L 1 := Λ + P i ≤ 0 a (1) i Λ i L 2 := a (2) − 1 Λ − 1 + P i ≥ 0 a (2) i Λ i where { a ( s ) i , s = 1 , 2 } are some diagonal matrices. 2D-T o da equations, expressed in Lax form , arise as compatibilit y cond itions for the follo wing Zak aro v-Shabat sp ectral pr oblem:                                              L 1 Ψ 1 = z Ψ 1 L T 2 Ψ ∗ 2 = z − 1 Ψ ∗ 2 ∂ t n Ψ 1 = ( L n 1 ) + Ψ 1 ∂ t n Ψ ∗ 2 = − ( L n 1 ) T + Ψ ∗ 2 ∂ s n Ψ 1 = ( L n 2 ) − Ψ 1 ∂ s n Ψ ∗ 2 = − ( L n 2 ) T − Ψ ∗ 2 . Here w e in tro du ced t w o in finite sets of times { t i , i ≥ 0 } and { s i , i ≥ 0 } . W e denoted with N + the u pp er triangular part of a matrix N (including the main d iagonal) and with N − the low er triangular part (excludin g the m ain diagonal). Ψ 1 and Ψ ∗ 2 are semi-infinte column vecto rs of t yp e Ψ 1 ( z ) = (Ψ 1 , 0 ( z ) , Ψ 1 , 1 ( z ) , . . . ) T Ψ ∗ 2 ( z ) = (Ψ ∗ 2 , 0 ( z ) , Ψ ∗ 2 , 1 ( z ) , . . . ) T . F or ev ery k the tw o expressions e − ξ ( t,z ) Ψ 1 ,k ( z ) and e − ξ ( s,z ) Ψ ∗ 2 ,k ( z − 1 ) are p olynomials in z of order k . Lax equations are written as ∂ t n L i =  ( L n 1 ) + , L i  ∂ s n L i =  ( L n 2 ) − , L i  , i = 1 , 2 . 2D-T o da equations can b e linearized as explained in [7]. W e start w ith an initial v alue matrix M (0 , 0) = { M ij (0 , 0) } i,j ≥ 0 and w e define its time ev olution through the equation M ( t ; s ) := exp  ξ ( t, Λ)  M (0 , 0) exp  − ξ ( s , Λ − 1 )  . W e assume that there exist a factorization M (0 , 0) = S 1 (0 , 0) − 1 S 2 (0 , 0) . Here S 1 is lo wer triangular while S 2 is upp er triangular. W e assume that b oth S 1 and S 2 ha v e non zero elemen ts on the main diagonal and w e norm alize them in such a wa y that ev ery 4 elemen t on the m ain diagonal of S 1 is equ al to 1. Moreo v er we consid er v alues of t and s for whic h we can write M ( t, s ) = S 1 ( t, s ) − 1 S 2 ( t, s ) (1) with S 1 and S 2 ha ving the same prop erties as ab o ve . It can b e pr o ven that suc h factoriza tion exists if and only if all the p rincipal minors of M ( t, s ) do not v anish. In p articular this condition is satisfied when M ( t, s ) is the m atrix of the m oments of a p ositiv e-defin ite measure. No w w e denote with χ ( z ) the infinite vec tor χ ( z ) := (1 , z , z 2 , . . . ) T . W a ve v ectors f or 2D-T o da and Lax matrices are constru cted in the follo wing wa y . Theorem 2.1 ([7]) . The wave ve ctors Ψ 1 ( z ) := exp ( ξ ( t, z )) S 1 χ ( z ) Ψ ∗ 2 ( z ) := exp ( − ξ ( s, z − 1 ))( S − 1 2 ) T χ ( z − 1 ) . and the two L ax op er ators L 1 := S 1 Λ S − 1 1 and L 2 := S 2 Λ − 1 S − 1 2 satisfy the 2D -T o da Zakhar ov- Shab at sp e ctr al pr oblem. Pr oof W e ju s t sket c h the pro of and m ake r eference to the article [7]. It is clear that the matrix M ( t, s ) satisfies differentia l equations ∂ t i M = Λ i M ∂ s i M = − M Λ − i . Then it is easy to deduce S ato’s equations ∂ t n S 1 = − ( L n 1 ) − S 1 ∂ t n S 2 = ( L n 1 ) + S 2 ∂ s n S 1 = ( L n 2 ) − S 1 ∂ s n S 2 = − ( L n 2 ) + S 2 . and Zakharo v-Sh abat’s equations can b e deduced from th e expression of wa v e ve ctors in terms of S 1 and S 2 .  The last thin g we need is the link b et wee n the f actoriza tion of M and biorthogonal p oly- nomials. W e introd uce a b ilinear pairing on the space of p olynomials in z definin g < z i , z j > M := M ij . The follo wing prop osition is a direct consequence of (1 ). Prop osition 2.2 ([1 3]) . q (1) = ( q (1) i ) i ≥ 0 := S 1 χ ( z ) q (2) = ( q (2) i ) i ≥ 0 := ( S − 1 2 ) T χ ( z ) ar e biorthonorma l p olynomials with r esp e ct to the p airing <, > M ; i.e. < q (1) i , q (2) j > M = δ ij ∀ i, j ∈ N . 5 3 F rom the T o eplitz lattice hierarc h y to the semidiscrete zero- curv ature equations for the Ablo witz-Ladik hierarc h y . In this section we briefly recall the r ed uction fr om 2D-T o da to the T o eplitz lat tice as describ ed in [13]. Then we sho w ho w the Ablo witz-Ladik equ ations are easily obtained fr om the T o eplitz lattice . S upp ose that ou r initial v alue M (0 , 0) is a T o eplitz matrix; i.e. we h a v e M (0 , 0) = T ( γ ) =            γ (0) γ ( − 1) γ ( − 2) . . . γ (1) γ (0) γ ( − 1) . . . γ (2) γ (1) γ (0) . . . . . . . . . . . . . . .            for some formal p o wer series γ ( z ) = P n ∈ Z γ ( n ) z n . Since Λ = T ( z − 1 ) is an up p er triangular T o eplitz matrix it follo ws easily (see for instance [23]) that M ( t, s ) = exp  ξ  t, Λ   M (0 , 0) exp  − ξ  s, Λ − 1   = T  exp  ξ ( t, z − 1 )  γ ( z ) exp  − ξ ( s , z )   . This means th at T o eplitz form in conserv ed along 2D-T o d a fl ow, hence we are dealing with a reduction of it. Th is redu ction is called T o eplitz lattice in [13]. In that article th e authors noticed, in the in tro du ction, that this is nothing but the Ablo witz-Ladik hierarch y . No w we will d escrib e how to obtain the original formulatio n of the Ablo witz-Ladik equations starting from Adler-v an Moerb ek e’s formulation. The ke y obser v ation is that, in th is case, the bilinear pairing < p, q > M b et we en t w o arb itrary p olynomials is giv en by < p, q > M = I p ( z ) γ ( z ) q ∗ ( z ) dz 2 π iz . Here the sym b ol of inte gration means th at we are taking the residue of the form al series p ( z ) γ ( z ) q ∗ ( z ) and q ∗ ( z ) = q ( z − 1 ). In other w ords q (1) i and q (2) j are nothing but orthonormal p olynomials on the unit circle. W e also define monic b iorthogonal p olynomials p (1) = ( p (1) i ) i ≥ 0 := S 1 χ ( z ) p (2) = ( p (2) i ) i ≥ 0 := h ( S − 1 2 ) T χ ( z ) with h = d iag( h 0 , h 1 , h 2 , . . . ) some diagonal matrix. Giv en an arb itrary p olynomial q ( z ) of degree n w e d efine its reversed p olynomial ˜ q ( z ) := z n q ∗ ( z ) and reflection coefficients x n := p (1) n (0) y n := p (2) n (0) . W e can state th e standard recursion r elation asso ciated to biorthogonal p olynomials on the unit circle. The equation b elow w as already known to Szeg¨ o [18]. It h as b een u s ed in [14 ] in relation w ith the theory of inte grable equations. 6 Prop osition 3.1. The fol lowing r e cursion r elation holds: p (1) n +1 ( z ) ˜ p (2) n +1 ( z ) ! = L n p (1) n ( z ) ˜ p (2) n ( z ) ! =  z x n +1 z y n +1 1  p (1) n ( z ) ˜ p (2) n ( z ) ! (2) Using th is recursion relation Adler and v an Mo erb ek e in [13] wrote the p eculiar form of Lax op erators f or the T o eplitz reduction. Prop osition 3.2 ([1 3]) . L ax op er ators of T o eplitz lattic e ar e of the fol lowing form: h − 1 L 1 h =         − x 1 y 0 1 − x 1 y 1 0 . . . . . . − x 2 y 0 − x 2 y 1 1 − x 2 y 2 0 . . . − x 3 y 0 − x 3 y 1 − x 3 y 2 1 − x 3 y 3 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         L 2 =         − x 0 y 1 − x 0 y 2 − x 0 y 3 . . . . . . 1 − x 1 y 1 − x 1 y 2 − x 1 y 3 . . . . . . 0 1 − x 2 y 2 − x 2 y 3 . . . . . . . . . 0 1 − x 3 y 3 . . . . . . . . . . . . 0 . . . . . .         . F r om this p rop osition it follo ws easily the follo wing corollary . Corollary 3.3. R efle ction c o efficie nts satisfy the fol lowing e quation h n +1 h n = 1 − x n +1 y n +1 . No w w e can state th e theorem relating the T o eplitz lattice to the original form of the Ablo witz-Ladik hierarc hy . Theorem 3.4. The T o eplitz lattic e flow c an b e written in the form ∂ t i p (1) n ( z ) ˜ p (2) n ( z ) ! = M t i ,n p (1) n ( z ) ˜ p (2) n ( z ) ! (3) ∂ s i p (1) n ( z ) ˜ p (2) n ( z ) ! = M s i ,n p (1) n ( z ) ˜ p (2) n ( z ) ! (4) for some matric es M t i ,n , M s i ,n dep ending on { x j , y j , z } . Pr oof W e pro v e it for the set of times den oted with t . W e denote d ([ z ]) = diag(1 , z , z 2 , z 3 , . . . ). W e h a v e the iden tities Ψ 1 ( z ) = exp( ξ ( t, z )) p (1) ( z ) Ψ ∗ 2 ( z ) = h − 1 d ([ z − 1 ]) exp( − ξ ( s, z − 1 )) ˜ p (2) ( z ) 7 giving the follo win g time ev olution for orthogonal p olynomials ∂ t i p (1) ( z ) = − z i p (1) ( z ) + ( L i 1 ) + p (1) ( z ) ∂ t i ˜ p (2) ( z ) = − hd ([ z ])( L i 1 ) T ++ h − 1 d ([ z − 1 ]) ˜ p (2) ( z ) . Here ( L i 1 ) ++ denotes the strictly upp er diagonal part of L n 1 . The form ulas ab o ve are obtained from a straigh tforw ard computation and using the fact, prov en in [13], that ∂ t i log( h n ) = ( L i 1 ) nn . Hence w e ha ve that, for every k , ∂ t i p (1) k is a linear com bination of { p (1) k , p (1) k +1 , p (1) k +2 , . . . } with co efficien ts in C [ x j , y j ]. In the same w a y , for ev ery k , ∂ t i ˜ p (2) k is a linear combination of { ˜ p (2) k , ˜ p (2) k − 1 , ˜ p (2) k − 2 , . . . } with co efficien ts in C [ x j , y j ]. Using the r ecursion relation (2) and its in v erse p (1) n ( z ) ˜ p (2) n ( z ) ! = L − 1 n p (1) n +1 ( z ) ˜ p (2) n +1 ( z ) ! = h n h n +1  z − 1 − z − 1 x n +1 − y n +1 1  p (1) n +1 ( z ) ˜ p (2) n +1 ( z ) ! w e can obtain the desired matrices M t i ,n .  Corollary 3.5 (Ablo witz-Ladik semidiscrete zero-curv ature equati ons) . The matric es L n satisfy the fol lowing time evolution ∂ t i L n = M t i ,n +1 L n − L n M t i ,n (5) ∂ s i L n = M s i ,n +1 L n − L n M s i ,n . (6) Pr oof These equations are nothing but the compatibilit y conditions of recursion relation (2) with time ev olution (3) and (4).  R emark 3.6 . Actually our Lax op erator L n is sligh tly differen t from the Lax op erator L n used in [1] b y Ablo witz and Ladik and written in th e in tro duction ab ov e. Neve rtheless, as sho wn in [16], these t wo Lax op erators are linked through a simple change of sp ectral parameter. Example 3.7 (The first flo ws; discrete nonlinear Schr¨ odinger.) . The fir st matrices M t i ,n and M s i ,n are easily computed. W e h av e ∂ t 1 p (1) k = − z p (1) k − x k +1 y k p (1) k + p (1) k +1 = − x k +1 y k p (1) k + x k +1 ˜ p (2) k ∂ t 1 ˜ p (2) k = − z h k +1 h k ˜ p (2) k − 1 = z y k p (1) k − z ˜ p (2) k giving immediately M t 1 ,k =  − x k +1 y k x k +1 z y k − z  An analogue computation for s 1 can b e easily done obtaining M s 1 ,k =  z − 1 − z − 1 x k − y k +1 x k y k +1  . 8 W e can already write, with t 1 and s 1 , the well kno wn inte grable discretiza tion of nonlinear Sc hr¨ odinger. W e just n eed t w o more trivial rescaling times in tro duced with sub s titutions p (1) 7− → exp( t 0 ) p (1) ˜ p (2) 7− → exp( − s 0 ) ˜ p (2) and corresp onding to matrices M t 0 ,k :=  1 0 0 0  M s 0 ,k :=  0 0 0 − 1  . No w we can constru ct the matrix M τ ,k = M t 1 ,k + M s 1 ,k − M t 0 ,k − M s 0 ,k =  z − 1 − 1 − x k +1 y k x k +1 − z − 1 x k z y k − y k +1 x k y k +1 + 1 − z  asso ciated to the time τ = t 1 + s 1 − t 0 − s 0 so that semidiscrete zero-curv ature equation ∂ τ L k = M τ ,k +1 L k − L k M τ ,k is equiv alent to the system      ∂ τ x k = x k +1 − 2 x k + x k − 1 − x k y k ( x k +1 + x k − 1 ) ∂ τ y k = − y k +1 + 2 y k − y k − 1 + x k y k ( y k +1 + y k − 1 ) . (7) This is exactly the complexified version of the discrete nonlinear Schr¨ odinger. Rescaling τ 7→ iτ and imp osing y k = ± x ∗ k w e ob tain − i∂ τ x k = x k +1 − 2 x k + x k − 1 ∓ k x k k 2 ( x k +1 + x k − 1 ) . (8) 4 T o da flo w for b lo c k T o eplitz matrices and the related Lax op erators. No w w e generalize the T o eplitz lattice’s equations to th e blo c k case. W e start with a matrix- v alued form al series γ ( z ) = X k ∈ Z γ ( k ) z k . Here eve ry elemen t γ ( k ) is a n × n matrix. Th en we define its time evo lution as γ ( t, s ; z ) := exp  − ξ ( s, z − 1 I)  γ ( z ) exp  ξ ( t, z I)  . 9 where I is the n × n iden tit y matrix. Differen tly from the scalar case we don’t consider just one T o ep litz matrix b ut the tw o blo ck T o eplitz matrices, righ t and left, give n by T r ( γ ) :=            γ (0) γ ( − 1) γ ( − 2) . . . γ (1) γ (0) γ ( − 1) . . . γ (2) γ (1) γ (0) . . . . . . . . . . . . . . .            T l ( γ ) :=            γ (0) γ (1) γ (2) . . . γ ( − 1) γ (0) γ (1) . . . γ ( − 2) γ ( − 1) γ (0) . . . . . . . . . . . . . . .            In this wa y w e obtain the follo wing linear time ev olution for our b lo c k T o eplitz matrices (in the follo wing w e will omit th e symb ol γ ): ∂ t i T l = Λ i T l ∂ t i T r = T r Λ − i (9) ∂ s i T l = − T l Λ − i ∂ s i T r = − Λ i T r (10) where, in this case, we hav e Λ = T r ( z − 1 I). Then w e assume that there exist tw o factorizations T l = S − 1 1 S 2 T r = Z 2 Z − 1 1 . Here S 1 , Z 2 are block-l o we r triangular while S 2 , Z 1 are block-upp er triangular. W e assume that all these matrices hav e non d egenerate b lo c ks on the main diagonal (i.e . these blocks m ust ha v e non zero d eterminan ts). No rmalizations are c hosen in suc h a wa y that ev ery elemen t on the main blo c k-diagonal of S 1 and Z 2 is equal to the iden tit y matrix I. As w e did b efore w e assu me that these conditions hold when ev ery time is equal to 0 and the we consider ju s t the v alues of t and s for whic h these conditions still hold. In the matrix case w e can d efi ne tw o bilinear pairings giv en b y the follo wing definition. Definition 4.1. < P , Q > r := I P ∗ ( z ) γ ( z ) Q ( z ) dz 2 π iz < P , Q > l := I P ( z ) γ ( z ) Q ∗ ( z ) dz 2 π iz wher e P and Q ar e two arbitr ary matrix p olynomials and P ∗ ( z ) := ( P ( z − 1 )) T Our t wo fact orizations give exactly biorthonorm al p olynomials for <, > r and <, > l . In the follo wing w e denote χ ( z ) := (I , z I , z 2 I , z 3 I , . . . ) T . 10 Prop osition 4.2. Q (1) l :=    Q (1) l 0 Q (1) l 1 . . .    = S 1 χ ( z ) (11) Q (2) l :=    Q (2) l 0 Q (2) l 1 . . .    = ( S − 1 2 ) T χ ( z ) (12) Q (1) r :=  Q (1) r 0 Q (1) r 1 . . .  = χ ( z ) T Z 1 (13) Q (2) r :=  Q (2) r 0 Q (2) r 1 . . .  = χ ( z ) T ( Z − 1 2 ) T (14) ar e the biorthonormal p olynomials asso ciate d to the p airing <, > l and < , > r . Henc e for eve ry i, j we have < Q (1) l i , Q (2) l j > l = δ ij < Q (2) r i , Q (1) r j > r = δ ij Pr oof W e just prov e, as an example, the pr op osition for the righ t p olynomials. On the other hand the one for left p olynomials is identic al to the usual pro of for 2D-T o da. W e ha ve < Q (2) r i , Q (1) r j > r ! i,j ≥ 0 = X k ,l ≥ 0 ( Z − 1 2 ) k i < z k I , z l I > r ( Z 1 ) lj ! = Z − 1 2 T l Z 1 = I ⇐ ⇒ T l = Z 2 Z − 1 . (it should b e noted th at, in this case, subscripts of t yp e ( Z 1 ) ij denote the blo ck in p osition ( i, j ) and not th e elemen t ( i, j ).)  No w w e ca n write the r elated Sato’s equations for S i and Z i . It is conv enient to in tro d uce the follo wing Lax op erators. Definition 4.3. L 1 := S 1 Λ S − 1 1 L 2 := S 2 Λ − 1 S − 1 2 (15) R 1 := Z − 1 1 Λ − 1 Z 1 R 2 := Z − 1 2 Λ Z 2 . (16) Prop osition 4.4. The fol lowing Sato’s e quations ar e satisfie d. ∂ t n S 1 = − ( L n 1 ) − S 1 ∂ t n Z 1 = − Z 1 ( R n 1 ) + (17) ∂ t n S 2 = ( L n 1 ) + S 2 ∂ t n Z 2 = Z 2 ( R n 1 ) − (18) ∂ s n S 1 = ( L n 2 ) − S 1 ∂ s n Z 1 = Z 1 ( R n 2 ) + (19) ∂ s n S 2 = − ( L n 2 ) + S 2 ∂ s n Z 2 = − Z 2 ( R n 2 ) − (20) Pr oof W e will just prov e, as an example, the equations in vo lving t -deriv ativ e of Z 1 and Z 2 . W e assume, as an ansatz, that we h a v e ∂ t n Z 1 = Z 1 A ∂ t n Z 2 = Z 2 B . 11 for some matrices A and B . Then exploiting time evol ution of T r w e can write T r Λ − n = ∂ t n T r = ∂ t n ( Z 2 Z − 1 1 ) = Z 2 B Z − 1 1 − Z 2 Z − 1 1 Z 1 AZ − 1 1 = Z 2 ( B − A ) Z − 1 1 hence we must hav e ( B − A ) Z − 1 1 = Z − 1 1 Λ − n . W e rewrite it as B − A = Z − 1 1 Λ − n Z 1 = R n 1 . Moreo ver we ha ve that B must b e strictly blo c k-lo w er triangular and A blo ck- upp er triangular. This is b ecause Z 1 is b lo c k-upp er triangular and Z 2 is b lo c k-lo wer triangular with constan t en tries on the main diagonal. Hence we conclude that A = − ( R n 1 ) + and B = ( R n 1 ) − .  No w it’s just a matter of trivial computations to write d o w n the corresp onding Lax equations for L i and R i . Prop osition 4.5. The fol lowing L ax e quations ar e satisfie d: ∂ t n L i = h ( L n 1 ) + , L i i ∂ t n R i = h R i , ( R n 1 ) − i (21) ∂ s n L i = h ( L n 2 ) − , L i i ∂ s n R i = h R i , ( R n 2 ) + i . (22) The definition of our Lax op erators will giv e us eigen v alue equations for su itably defin ed w a v e v ectors. Definition 4.6. Ψ 1 ( z ) := exp( ξ ( t, z I)) S 1 χ ( z ) (23) Φ 1 ( z ) := exp( ξ ( t, z I)) h χ ( z ) i T Z 1 (24) Ψ ∗ 2 ( z ) := exp ( − ξ ( s, z − 1 I))( S − 1 2 ) T χ ( z − 1 ) (25) Φ ∗ 2 ( z ) := exp( − ξ ( s, z − 1 I)) χ ( z − 1 ) T ( Z − 1 2 ) T . (26) Prop osition 4.7. The fol lowing e quations hold true: L 1 Ψ 1 ( z ) = z Ψ 1 ( z ) Φ 1 ( z ) R 1 = z Φ 1 ( z ) (27) L T 2 Ψ ∗ 2 ( z ) = z − 1 Ψ ∗ 2 ( z ) Φ ∗ 2 ( z ) R T 2 = z − 1 Φ ∗ 2 ( z ) . (28) Pr oof W e will just pr o v e the last equation, all th e other ones are p ro v ed in a sim ilar wa y . F r om the very definition w e hav e Φ ∗ 2 ( z ) R T 2 = z − 1 IΦ ∗ 2 ( z ) ⇐ ⇒ [ χ ( z − 1 )] T ( Z − 1 2 ) T R T 2 = z − 1 [ χ ( z − 1 )] T ( Z − 1 2 ) T = [ χ ( z − 1 )] T Λ − 1 ( Z − 1 2 ) T ⇐ ⇒ R T 2 = Z T 2 Λ − 1 ( Z − 1 2 ) T ⇐ ⇒ R 2 = Z − 1 2 Λ Z 2  The pr o of of the follo wing prop osition is straigh tforwa rd. 12 Prop osition 4.8. The L ax e qu ations (21) and (22) ar e the c omp atibility c onditions of the eigenvalue e quations (27) and (28) with the fol lowing e quations: ∂ t n Ψ 1 = ( L n 1 ) + Ψ 1 ∂ t n Φ 1 = Φ 1 ( R n 1 ) − (29) ∂ s n Ψ 1 = ( L n 2 ) − Ψ 1 ∂ s n Φ 1 = Φ 1 ( R n 2 ) + (30) ∂ t n Ψ ∗ 2 = − ( L n 1+ ) T Ψ ∗ 2 ∂ t n Φ ∗ 2 = − Φ ∗ 2 ( R n 1 − ) T (31) ∂ s n Ψ ∗ 2 = − ( L n 2 − ) T Ψ ∗ 2 ∂ s n Φ ∗ 2 = − Φ ∗ 2 ( R n 2+ ) T . (32) R emark 4.9 . Actually our Lax equations as w ell as equations (27)-(32) can b e d educed from the equations of multic omp onent 2D-T o da [10]. 5 Recursion relations for matrix biorthogonal p olynomials on the unit circle. In order to generalize the scalar theory w e h a ve to construct an analogue of the recursion relation giv en b y Prop osition 3.1. Recursion r elations for m atrix orthogonal p olynomial on the u nit circle are already k n o wn, see [19] and [22]. Here we sligh tly generalize to the case of matrix b iorthogonal p olynomials on the unit circle. W e defin e the follo w ing imp ortan t n × n matrices: Definition 5.1. h r N := SC( T r N +1 ) h l N := SC( T l N +1 ) where SC denote the n × n Sc hur complemen t of a blo ck m atrix with r esp ect to the upp er left blo ck (see for instance [21]). F or example we ha ve SC( T r N +1 ) = γ (0) −  γ ( N ) . . . . . . γ (1)  T − r N     γ ( − N ) . . . . . . γ ( − 1)     (here and b elo w T − r N := ( T r N ) − 1 and similarly for h r N , h l N and T l N ). Prop osition 5.2. Monic biortho gonal p olynomials such that < P (2) r k , P (1) r j > r = δ k j h r k < P (1) l k , P (2) l j > l = δ k j h l k . ar e given by the fol lowing formulas: P (1) r N = SC       γ (0) . . . . . . γ ( − N +1) γ ( − N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . γ ( N − 1) . . . . . . γ (0) γ ( − 1) I z I . . . z N − 1 I z N I       ( P (2) r N ) T = SC       γ (0) . . . . . . γ ( − N +1) I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . γ ( N − 1) . . . . . . γ (0) z N − 1 I γ ( N ) . . . . . . γ (1) z N I       13 P (1) l N = SC       γ (0) . . . . . . γ ( N − 1) I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . γ ( − N +1) . . . . . . γ (0) z N − 1 I γ ( − N ) . . . . . . γ ( − 1) z N I       ( P (2) l N ) T = SC       γ (0) . . . . . . γ ( N − 1) γ (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . γ ( − N +1) . . . . . . γ (0) γ ( N − 1) I z I . . . z N − 1 I z N I       . Pr oof W e will ju st prov e the fir st formula, the second one is pro v ed similarly . First of all we hav e ∀ 0 ≤ m ≤ N − 1 < z m I , P (1) r N ( z ) > r = I z − m γ ( z ) z N I −  I . . . . . . z N − 1 I  T − r N     γ ( − N ) . . . . . . γ ( − 1)     ! dz 2 π iz = γ ( m − N ) − γ ( m − N ) = 0 . In the same wa y ∀ 0 ≤ m ≤ N − 1 < P (2) r N ( z ) , z m I > r = I z − N I −  γ ( N ) . . . . . . γ (1)  T − r N     I . . . . . . z − N + 1 I     ! γ ( z ) z m dz 2 π iz = γ ( N − m ) − γ ( N − m ) = 0 . Finally < P (2) r N ( z ) , P (1) r N ( z ) > r = < z N I , P (1) r N > r = γ (0) −  γ ( N ) . . . . . . γ (1)  T − r N     γ ( − N ) . . . . . . γ ( − 1)     = h r N . This completes the pr o of of the first form ula, the second one is p ro v ed similarly .  R emark 5.3 . Note that, imp osing Q (1) l k := P (1) l k Q (2) l k := ( h − l k ) T P (2) l k Q (1) r k := P (1) r k ( h − r k ) Q (2) r k := P (2) r k . w e ob tain biorthonormal p olynomials. 14 No w we will write a long list of relatio ns among this p olynomials an d reflection coefficien ts. Giv en an y matrix p olynomial Q ( z ) of d egree n w e d efine the asso ciated r ev ersed p olynomial as ˜ Q ( z ) = z n Q ∗ ( z ) . The reflection coefficients are defined as follo ws: x l N := P (1) l N (0) x r N := P (1) r N (0) y l N := ( P (2) l N (0)) T y r N := ( P (2) r N (0)) T . Prop osition 5.4. The fol lowing formulas hold true: P (1) l N +1 − z P (1) l N = x l N +1 ˜ P (2) r N (33) ˜ P (2) r N +1 − ˜ P (2) r N = z y r N +1 P (1) l N (34) P (1) r N +1 − z P (1) r N = ˜ P (2) l N x r N +1 (35) ˜ P (2) l N +1 − ˜ P (2) l N = z P (1) r N y l N +1 (36) P (1) r N +1 = z P (1) r N (I − y l N +1 x r N +1 ) + ˜ P (2) l N +1 x r N +1 (37) P (1) l N +1 = z (I − x l N +1 y r N +1 ) P (1) l N + x l N +1 ˜ P (2) l N +1 (38) ˜ P (2) r N +1 = (I − y r N +1 x l N +1 ) ˜ P (2) r N + y r N +1 P (1) l N +1 (39) ˜ P (2) l N +1 = ˜ P (2) l N (I − x r N +1 y l N +1 ) + P (1) r N +1 y l N +1 (40) x l N h r N = h l N x r N (41) y r N h l N = h r N y l N (42) h − r N h r N +1 = I − y l N +1 x r N +1 (43) h l N +1 h − l N = I − x l N +1 y r N +1 . (44) Pr oof The fi rst four f orm ulas are pro ve d observing, for instance for the first case, that ∀ 1 ≤ i ≤ N we ha ve 0 = < P (1) l N +1 − z P (1) l N , z i I > l = < ˜ P (2) r N , z i I > l so that P (1) l N +1 − z P (1) l N and ˜ P (2) r N m ust b e prop ortional. Setting z = 0 yo u also find the constan t of prop ortionalit y . In particular, wh en p ro ving (34) and (36), w e fin d a formula and then we ha v e to take the rev ersed one. (37) is pr ov ed su b stituting (36) int o (35) an d similarly for (38),(39),(40). (41) and (42) are p ro v en resp ectiv ely observing that we ha ve < ˜ P (1) l N , P (1) r N > r = < P (1) l N , ˜ P (1) r N > l and < P (2) r N , ˜ P (2) l N > r = < ˜ P (2) r N , P (2) l N > l and then doing explicit computations. Finally (43) is obtained rewriting (37 ) as P (1) r N +1 z N +1 = P (1) r N z N (I − y l N +1 x r N +1 ) + ( P (2) l N +1 ) ∗ x r N +1 , 15 m ultiplying from the left for P (1) l N γ and then taking the residue. (44) is pro ve d similarly .  No w w e define tw o sets of block matrice s {L r N } N ≥ 0 and {L l N } N ≥ 0 . Th ey will ha v e, in the matrix case, the same role pla ye d by {L n } n ≥ 0 in the scalar case. Definition 5.5. L l N :=  z I x l N +1 z y r N +1 I  (45) L r N :=  z I z y l N +1 x r N +1 I  . (46) Corollary 5.6. The fol lowing blo ck matric es r e cursion r elations ar e satisfie d P (1) l N +1 ˜ P (2) r N +1 ! = L l N P (1) l N ˜ P (2) r N ! (47)  P (1) r N +1 ˜ P (2) l N +1  =  P (1) r N ˜ P (2) l N  L r N . (48) Pr oof These are nothing but (33),(34),(35) and (36).  6 Explicit expressions for Lax op erators and related semidis- crete zero-curv ature equations. Using our recursion relations we w an t to fin d explicit expressions for L i and R i in terms of our reflection co efficien ts x l k , x r k , y l k , y r k . First of all w e underlin e a r emark able symmetry that will allo w u s to redu ce the amount of our compu tations. Doing the follo wing thr ee substitutions z 7→ z − 1 t 7→ − s s 7→ − t w e ob tain immediately the follo w ing prop osition. Prop osition 6.1. Under the symmetry ab ove the dr essings, the ortho gonal p olynomials, the L ax op er ators and the r efle ction c o efficients change as f ol lows: T r 7→ T l T l 7→ T r S 1 7→ Z − 1 2 S 2 7→ Z − 1 1 L 1 7→ R 2 L 2 7→ R 1 Q (1) l 7→ ( Q (2) r ) ∗ Q (2) l 7→ ( Q (1) r ) ∗ P (1) l 7→ ( P (2) r ) ∗ P (2) l 7→ ( P (2) r ) ∗ x l 7→ y r y l 7→ x r h l k 7→ h r k . 16 Hence we can write ju s t the left theory and w e w ill ha v e the right one as we ll. Actually ev ery compu tation made ab o ve for the righ t theory can b e deduced from the left theory and this symmetry whic h will b e called in the sequel t − s symmetry . In the theorem b elo w the sym b ol Q M − j = N +2 means that the terms in the pro du ct must b e written from the smallest ind ex to the b iggest, going from left to r igh t. The sym b ol Q M + j = N +2 means that the p ro duct must b e take n in the opp osite dir ection. Theorem 6.2 (Lax op erators for blo c k T o eplitz lattice) . L ax op er ators L i and R i ar e ex- pr esse d i n terms of r efle ction c o efficients ac c or ding to the f ol lowing formulas: ∀ N > M ≥ − 1 ( L 1 ) N ,M +1 = − x l N +1  M − Y j = N +2 (I − y r j x l j )  y r M +1 (49) ( R 2 ) N ,M +1 = − y r N +1  M − Y j = N +2 (I − x l j y r j )  x l M +1 (50) ( L 2 ) M +1 ,N = − h − l M +1 x r M +1  M + Y j = N +2 (I − y l j x r j )  y l N +1 h l N (51) ( R 1 ) M +1 ,N = − h − r M +1 y l M +1  M + Y j = N +2 (I − x r j y l j )  x r N +1 h r N . (52) Mor e over ( L 1 ) N ,N +1 = ( R 2 ) N ,N +1 = I (53) ( L 2 ) N +1 , N = h l N +1 h − l N ( R 1 ) N +1 , N = h r N +1 h − r N . (54) Pr oof (53 ) and (54 ) follo w trivially from the expr essions of d ressings S i , Z i . In fact, b ecause of our n ormalization, we hav e that S 1 and Z 2 are equal to th e identit y matrix plu s a strictly lo wer triangular matrix whic h is what stated in (53). (54) is obtained observing that the b lo c k on the main diago nal of S 2 and Z 1 are giv en resp ectiv ely by matrices h l k and h r k as follo ws from Prop osition 4.2 and Remark 5.3 . No w let’s b egin with (49) and (50); the imp ortan t p oint is that we h a v e Ψ 1 = exp( ξ ( t, z I )) P (1) l . Hence as done in [13] we can find that ∀ N > M ≥ − 1 w e ha ve ( L 1 ) N ,M +1 = − x l N +1 h r N h − r M +1 y r M +1 . (55) Infact ∀ N > M ≥ − 1 < P (1) l N +1 − z P (1) l N , P (2) l M +1 − z P (2) l M > l = − < z P (1) l N , P (2) l M +1 > l = − < P (1) l N +1 + . . . + ( L 1 ) N ,M +1 P (1) l M +1 + . . . , P (2) l M +1 > l = − ( L 1 ) N ,M +1 h l M +1 . On the other hand , u sing recursion relations, I also ha ve ∀ N ≥ M ≥ − 1 < P (1) l N +1 − z P (1) l N , P (2) l M +1 − z P (2) l M > l = < x l N +1 ˜ P (2) r N , ( y l M +1 ) T ˜ P (1) r M > l = x l N +1 I z N − M ( P (2) r N ) ∗ γ ( z ) P (1) r M dz 2 π iz ! y l M +1 = x l N +1 < P (2) r N , z N − M P (1) r M > r y l M +1 = x l N +1 h r N y l M +1 17 and comparing them we find (55). No w w e use t − s s y m metry to s im p lify this expression. W e obtain ( R 2 ) N ,M +1 = − y r N +1 h l N h − l M +1 x l M +1 . No w w e apply sev eral times the recursion (44) to the piece h l N h − l M +1 and w e ge t (5 0). (49) is ob- tained using t − s symmetry . F or (51) and (52) we start defining ˜ R 1 suc h that z P (1) r = P (1) r ˜ R 1 ; then we will hav e R 1 = h r ˜ R 1 h − r and computations for ˜ R 1 is carried on similarly as for L 1 .  R emark 6.3 . Our equations (49),(50),(51) and (52) extend to the m atrix biorthogonal setting the equ ations written in [22] for matrix orthogonal p olynomials (see equ ation (4.2),(4.3)). In that article prop erties of M are applied to stu d y some p roblems in computational mathema- tics (m ultiv ariate time series analysis and multic hann el signal pro cessing) an d no relation is established w ith the Lax theory and in tegrable s ystems. The theorem ab o ve describ es the blo ck-a nalogue of T o eplitz lattice; we are no w in the p osition to prov e th e analogue of Theorem 3.4. Theorem 6.4. Blo ck T o eplitz lattic e flow c an b e written in the form ∂ t i /s i P (1) l N ( z ) ˜ P (2) r N ( z ) ! = M l t i /s i ,N P (1) l N ( z ) ˜ P (2) r N ( z ) ! (56) ∂ t i /s i  P (1) r N ( z ) ˜ P (2) l N ( z )  =  P (1) r N ( z ) ˜ P (2) l N ( z )  M r t i /s i ,N (57) for some blo c k matric es M r t i ,N , M r s i ,N , M l t i ,N , M l s i ,N dep ending on the matric es { x l j , y l j , x r j , y r j } and the sp e ctr al p ar ameter z . Pr oof As w e did for the scalar case we pro ve it ju s t for t times. Th e relev an t equations linking b iorthogonal p olynomials with wa v e v ectors are Ψ 1 ( z ) = exp( ξ ( t, z I)) P (1) l ( z ) Φ ∗ 2 ( z ) = exp( − ξ ( s, z − 1 I))( ˜ P (2) r ) T d ([ z − 1 ]) Φ 1 ( z ) = exp ( ξ ( t, z I)) P (1) r h − r Ψ ∗ 2 ( z ) = exp( − ξ ( s, z − 1 I))( h − l ) T d ([ z − 1 ])( ˜ P (2) l ) T . T rivial compu tations giv e the follo w ing time ev olution: ∂ t n P (1) l = ( L n 1 ) + P (1) l − z n P (1) l ∂ t n ˜ P (2) r = − d ([ z ])( R n 1 ) − d ([ z − 1 ]) ˜ P (2) r ∂ t n P (1) r = P (1) r ( h − r ( R n 1 ) − h r + h − r ( ∂ t n h r ) − z n I) ∂ t n ˜ P (2) l = ˜ P (2) l ( − h − l ( L n 1 ) + h l + h − l ( ∂ t n h l )) . the last t w o can b e s implified giving ∂ t n P (1) l = ( L n 1 ) + P (1) l − z n I P (1) l (58) ∂ t n ˜ P (2) r = − d ([ z ])( R n 1 ) − d ([ z − 1 ]) ˜ P (2) r (59) ∂ t n P (1) r = P (1) r ( h − r ( R n 1 ) −− h r − z n I) (60) ∂ t n ˜ P (2) l = − ˜ P (2) l  d ([ z − 1 ]) h − l ( L n 1 ) ++ h l d ([ z ])  . (61) 18 where ( R n 1 ) −− means th e lo wer triangular part including the main diagonal and ( L n 1 ) ++ means the strictly upp er diagonal part. This simplification can b e obtained ev aluating the terms h − r ( ∂ t n h r ) and h − l ( ∂ t n h l )) through Sato’s equations. Al so they can b e obtained ob s erving that P (1) r N and P (2) l N are monic so that the deriv ative of the leading term is equal to 0. Then the pro of is obtained as we did in the scalar case u sing f orw ard and bac kward r ecursion relations (33),(35 ),(39) and (40).  Corollary 6.5 (Non-ab elian AL semid iscr ete zero-curv ature equations) . The matric e s L r n and L l n satisfy the fol lowing time evolution ∂ t i /s i L l n = M l t i /s i ,n +1 L l n − L l n M l t i /s i ,n (62) ∂ t i /s i L r n = L r n M r t i /s i ,n +1 − M r t i /s i ,n L r n . (63) Pr oof T hese equations are nothing but compatibilit y conditions of recursion r elations (47 ) and (48) with time ev olution (56) and (57).  R emark 6.6 . It sh ould b e noticed that, with resp ect to the equations originally written in [3], here we h a v e t wo coupled non-ab elian Ablowit z-Ladik equations. Example 6.7 (The fir s t flo ws; n on-ab elian analogue of discrete n onlinear Schr¨ odinger) . As we did f or th e scalar case we will compute the fir st matrices M r /l t 1 /s 1 ,k and u s e them to constru ct the n on-ab elian ve rsion of discrete nonlinear S c h r¨ odinger. W e start with M l t 1 ,k ; (58) giv es us immediately ∂ t 1 P (1) l k = P (1) l k +1 − x l k +1 y r k P (1) l k − z P (1) l k = z P (1) l k + x l k +1 ˜ P (2) r k − x l k +1 y r k P (1) l k − z P (1) l k = − x l k +1 y r k P (1) l k + x l k +1 ˜ P (2) r k while we obtain immediately from (59) that ∂ t 1 ˜ P (2) r k = − z h r k h − r k − 1 ˜ P (2) r k − 1 . Then we u se recursion relation (39) com b ined with h r k h − r k − 1 = (I − y r k x l k ) (this one comes from r ecursion relation (41) combined w ith t − s s ymmetry) to arriv e to ∂ t 1 ˜ P (2) r k = z y r k P (1) l k − z P (2) r k . These computations giv e us M l t 1 ,k =   − x l k +1 y r k x l k +1 z y r k − z I   . (64) Exploiting t − s symmetry we can wr ite immediately M l s 1 ,k =   z − 1 I − z − 1 x l k − y r k +1 y r k +1 x l k   . (65) 19 Analogue compu tations for M r t 1 giv es us ∂ t 1 P (1) r k = P (1) r k +1 − P (1) r k y l k x r k +1 − z P 1( r ) k = z P 1( r ) k + ˜ P (2) l k x r k +1 − P (1) r k y l k x r k +1 − z P 1( r ) k = ˜ P (2) l k x r k +1 − P (1) r k y l k x r k +1 and ∂ t 1 ˜ P (2) l k = − z ˜ P (2) l k − 1 ( h − l k − 1 h l k ) = − ˜ P (2) l k z + P (1) r k z y l k (here we started from (60) and (61) and we used recursion relations (37),(40 ),(42 ), the last one combined w ith t − s s ymmetry). Then we arrive to M r t 1 ,k =   − y l k x r k +1 z y l k x r k +1 − z I   (66) and us ing again t − s symmetry we also get M r s 1 ,k =   z − 1 I − y l k +1 − z − 1 x r k x r k y l k +1   . (67) As we d id for the scalar case w e introd uce times t 0 and s 0 that giv e matrices M r /l t 0 ,k =  I 0 0 0  (68) M r /l s 0 ,k =  0 0 0 − I  . (69) Then we constru ct the matrices M l τ ,k = M l t 1 ,k + M l s 1 ,k − M l t 0 ,k − M l s 0 ,k =   z − 1 I − I − x l k +1 y r k x l k +1 − z − 1 x l k z y r k − y r k +1 y r k +1 x l k + I − z I   M r τ ,k = M r t 1 ,k + M r s 1 ,k − M r t 0 ,k − M r s 0 ,k =   z − 1 I − I − y l k x r k +1 z y l k − y l k +1 x r k +1 − z − 1 x r k x r k y l k +1 − z I + I   asso ciate to the time τ = t 1 + s 1 − t 0 − s 0 . Semidiscrete zero-curv ature equations ∂ τ L l k = M l τ ,k +1 L l k − L l k M l τ ,k ∂ τ L r k = L r k M r τ ,k +1 − M r τ ,k L r k 20 are equiv alen t to the systems      ∂ τ x l k = x l k +1 − 2 x l k + x l k − 1 − x l k +1 y r k x l k − x l k y r k x l k − 1 ∂ τ y r k = − y r k +1 + 2 y r k − y r k − 1 + y r k +1 x l k y r k + y r k x l k y r k − 1 (70)      ∂ τ x r k = x r k +1 − 2 x r k + x r k − 1 − x r k − 1 y l k x r k − x r k y l k x r k +1 ∂ τ y l k = − y l k +1 + 2 y l k − y l k − 1 + y l k − 1 x r k y l k + y l k x r k y l k +1 . (71) Note that b oth of th em are equiv alen t to the d iscrete matrix NLS as w ritten, for instance, in [6]. Using (70) and (71) together w e p erform the reduction to the hermitian case in a differen t w a y from [6]. First of all we rescale τ 7→ iτ and th en w e imp ose y r k = ± ( x r k ) ∗ y l k = ± ( x l k ) ∗ Note that this reduction (with the s ign plus) corresp onds to stud ying the theory of matrix orthogonal p olynomials on the u nit circle as describ ed in [19] and [22], hence it is v ery n atural. This r eduction giv es us the t wo coupled equations      − i∂ τ x l k = x l k +1 − 2 x l k + x l k − 1 ∓ x l k +1 ( x r k ) ∗ x l k ∓ x l k ( x r k ) ∗ x l k − 1 − i∂ τ x r k = x r k +1 − 2 x r k + x r k − 1 ∓ x r k − 1 ( x l k ) ∗ x r k ∓ x r k ( x l k ) ∗ x r k +1 (72) already stud ied in [4] and generalized in [5]. R emark 6.8 . In [16] the authors studied fin ite gap solutions of the Ab lowitz- Ladik hierarc h y . It could b e inte resting to generalize their results to the non-abelian case. W e will consider this pr ob lem in a subsequent pub lication. Ac kno wledgmen ts I am ve ry grateful to Professor B. D ub r o vin for his constan t supp ort and many su gges- tions giv en du ring th e preparation of this p ap er. Also I wish to thank Professor T ak a yuki Tsuc hida that ga ve me useful referen ces to the already existing pap ers ab out the non-Ab elian Ablo witz-Ladik equations and Professor Man uel Manas for some clarifying discussions ab out m ulticomp onent 2D-T o da hierarch y and its relationship with th is work. I am also grateful to Professors Pierre v an Mo erb eke , Mark Adler, Arno Kuijlaars and W alter V an Ass c h e for s ome general discuss ions on the theory of orthogonal p olynomials and its relations with in tegrable equations. 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