Moment matrices and multi-component KP, with applications to random matrix theory
Mark Adler1, Pierre van Moerbeke2 and Pol Vanhaecke3
Contents
1 Introduction 1
2 Tau functions and mixed multiple orthogonal polynomials 7
3 Cauchy transforms 12
4 Duality 18
5 The Riemann-Hilbert matrix and the bilinear identity 20
6 Consequences of the bilinear identities 23
7Examples 27 7.1 Biorthogonal polynomials (p = q =1)...... 27 7.2 Orthogonal polynomials ...... 29 7.3 Orthogonal polynomials on the circle ...... 30 7.4Non-intersectingBrownianmotions...... 31
1 Introduction
Random matrix theory has led to the discovery of novel matrix models and novel statistical distributions, which are defined by means of Fredholm determi- nants and which, in many cases, satisfy nonlinear ordinary or partial differential equations. A crucial observation is that these matrix integrals, upon appropri- ate deformation by means of exponentials containing one or several series of time parameters, satisfy (i) integrable equations and (ii) Virasoro constraints with respect to these time parameters. Most of the time, such matrix integrals can be written — by expressing the integrand in “polar coordinates” — as a multiple integral, which then can be expressed in terms of the determinant of
1Department of Mathematics, Brandeis University, Waltham, Mass 02454, USA, [email protected]. The support of a National Science Foundation grant # DMS-04-06287 is gratefully acknowledged 2D´epartement de Math´ematiques, Universit´e Catholique de Louvain, 1348 Louvain- la-Neuve, Belgium and Brandeis University, Waltham, Mass 02454, USA, vanmoer- [email protected]. The support of a National Science Foundation grant # DMS-04-06287, a European Science Foundation grant (MISGAM), a Marie Curie Grant (ENIGMA), Nato, FNRS and Francqui Foundation grants is gratefully acknowledged. 3The support of a European Science Foundation grant (MISGAM) and a Marie Curie Grant (ENIGMA) is gratefully acknowledged.
1 a moment matrix; this may be a moment matrix with regard to one or several weights. The extra time parameters are added in such a way that each weight has its own exponential time deformation. The main point is to show that this determinant satisfies (i) and (ii). These features turn out to be extremely robust! The purpose of the present paper is to show point (i) in great generality, which is the determinant of moment matrices associated with one or several weights and defined on various different domains, satisfies the multi-component KP hierarchy with regard to the time parameters. This is a very general class of integrable equations. This determinant will turn out to be the τ-function of this integrable hierar- chy; this τ-function with appropriate shifts of the deformation variables will be expressed in terms of the “orthogonal polynomials” defined by the weights and their Cauchy transform. We list below a number of examples having their origin in Hermitian random matrix theory, in random matrices coupled in a chain, in random permutations and in Dyson Brownian motions (non-intersecting Brow- nian motions) on R leaving from the origin, where some paths are forced to end upatonepointandothersatanotherpoint,etc...Theseexampleswillthenbe discussedindetailinSection7.
GUE: orthogonal polynomials. • n 1 ∞ k ∞ k 2 k=1 tk zℓ i+j k=1 tkz ∆n(z) e ρ(zℓ)dzℓ =det z e ρ(z)dz n! En P R P 0≤i,j≤n−1 ℓ=1 Coupled random matrices / Dyson Brownian motions: bi-orthogonal • polynomials. n ∞ k k 1 (tk x −sk y ) ∆n(x)∆n(y) e k=1 ℓ ℓ ρ(xℓ,yℓ)dxℓdyℓ n! En P ℓ=1 i j ∞ (t xk−s yk) =det x y e k=1 k k ρ(x, y)dxdy E P 0≤i,j≤n−1 Longest increasing subsequences in random permutations: orthogonal • polynomials on S1. n ∞ k −k 1 2 (tkz −skz ) dzℓ ∆n(z) e k=1 ℓ ℓ n! (S1)n | | P 2π√ 1zℓ ℓ=1 − dz i−j ∞ (t zk−s z−k) =det z e k=1 k k 1 2π√ 1z S − P 0≤i,j≤n−1 m + m non-intersecting Brownian motions on R leaving from 0 and • 1 2 m 1 paths forced to end up at a: multiple orthogonal polynomials on R. 2 ± 1 ∆m1+m2 (x, y) m !m ! m +m 1 2 E 1 2
2 m1 2 xℓ ∞ k − 2 +axℓ k=1(tk −sk)xℓ ∆m1 (x) e e dxℓ P ℓ=1 m2 2 yℓ ∞ k − 2 −ayℓ k=1(tk −uk)yℓ ∆m2 (y) e e dyℓ P ℓ=1 2 i+j − z +az ∞(t −s )zk z e 2 e 1 k k dz E P 0 ≤ i ≤ m1 − 1 =det 0 ≤ j ≤ m1 + m2 − 1 z2 ∞ k i+j − −az (tk −uk)z z e 2 e 1 dz E P 0 ≤ i ≤ m2 − 1 0 ≤ j ≤ m1 + m2 − 1 q m = p n non-intersecting Brownian motions on R,with • α=1 α β=1 β mα paths starting at aα R and nβ paths forced to end up at bβ: mixed multiple orthogonal∈ polynomials (mixed mops) on R.
A moment matrix for several weights: Define two sets of weights ψ (x),...,ψ (x)andϕ (y),...,ϕ (y), with x, y R, 1 q 1 p ∈ and deformed weights depending on time parameters sα =(sα1,sα2,...)(1 α q)andt =(t ,t ,...)(1 β p), denoted by ≤ ≤ β β1 β2 ≤ ≤ ∞ k ∞ k −s − k=1 sαk x t k=1 tβky ψα (x):=ψα(x)e and ϕβ(y):=ϕβ (y)e . P P That is, each weight goes with its own set of times. For each set of positive integers4 m =(m ,...,m ),n=(n ,...,n )withm = n , 1 q 1 p | | | | consider the determinant of a moment matrix Tmn, composed of blocks and of size m = n , with regard to a (not necessarily symmetric) inner product | | | | ·|· τmn(s1,...,sq; t1,...,tp)
:= det Tmn
11 1p Tmn ... Tmn . . := det . . T q1 ... Tqp mn mn i −s j t i −s j t x ψ (x) y ϕ (y) ... x ψ (x) y ϕ (y) 0≤i 3 A typical inner product to keep in mind is f(x) g(y) = f(x)g(y) dµ(x, y), (2) | R2 where µ = µ(x, y) is a fixed measure on R2, perhaps having support on a line or curve. From moment matrices to polynomials and their Cauchy transforms: I. Then, for 1 β, β′ p, the following expressions are polynomials (with coefficients depending≤ on s≤and t)5 τ (t z−1 ) nβ mn β (β,β) n z − := Qmn (z)=z β + ... τmn −1 τm,n+e −e ′ (tβ′ z ) ′ ′ n ′ −1 β β (β,β ) of degree p i −s (β,β′) t 1 α q x ψα (x) Qmn (y)ϕβ′ (y) =0 for ≤ ≤ (4) 0 i mα 1. β′=1 ≤ ≤ − II. In the same way, the following expressions are polynomials (depending on s and t) τ (s +[z−1]) mα−1 m−eα,n−eβ α ∗(β,α) ǫβα(n, m) z = Pnm (z)ofdegree q ′ ∗(β,α) −s j t 1 β p, 0 j nβ′ 1 Pnm (x)ψα (x) y ϕβ′ (y) =0 for ≤ ≤′ ≤ ≤ − except β = β, j = nβ 1 α=1 − q ∗(β,α) −s nβ −1 t Pnm (x)ψα (x) y ϕβ(y) =1. α=1 (6) III. The following expressions are Cauchy transforms of the polynomials ob- tained in II: 5 α2 α3 C Introduce the notation [α]:=(α, 2 , 3 ,...)forα . Only shifted times will be −1 ∈ made explicit in the τ-functions; i.e., τmn(tℓ ˆz ˜)meansthatτmn still depends on all − time parameters, but the variable tℓ only gets shifted. Moreover, here and below all the expressions ε (n), ǫ (n,m), etc... all equal 1 and will be given later. Throughout the αβ αβ ± paper, we use the standard notation e1 =(1, 0, 0,...),e2 =(0, 1, 0,...). 4 τ (t + z−1 ) q ϕt (y) −nβ mn β ∗(β,α) −s β z = Pnm (x)ψα (x) τmn z y α=1 − q ′ −1 t τm,n+eβ′ −eβ (tβ + z ) ϕ ′ (y) −nβ′ −1 ∗(β,α) −s β εββ′ (n)z = Pnm (x)ψα (x) τmn z y α=1 − (7) IV. Similarly, the following expressions are Cauchy transforms of the polyno- mials obtained in I: −1 −s p τ (s [z ]) ′ −mα−1 m+eα,n+eβ α ψα (x) (β,β ) t ǫαβ(m, n) z − = Qmn (y)ϕβ′ (y) . τmn z x β′=1 − (8) The statements I, II, III and IV summarize sections 1, 2 and 3. As will appear in Section 2, the polynomials appearing in (I) are called Type II mixed |ϕt β multiple orthogonal polynomials, whereas those appearing in (II) Type I | t ψα mixed multiple orthogonal polynomials. These were introduced by E. Daems and A. Kuijlaars [9], in the context of non-intersecting Brownian motions; they are a generalization of multiple orthogonal polynomials, where instead of one set of weights, there are two sets (the classical orthogonal polynomials correspond to one set with one element). They were introduced and studied by Aptekarev, Bleher, Geronimo, Kuijlaars, Van Assche [6, 14, 15, 8]. Around the same time, they were introduced by Adler-van Moerbeke in the context of band matrices and vertex operator solutions to the KP hierarchy [2]. In [7, 8], they were used in the context of non-intersecting Brownian motions and random matrices with external source. ∗ The (p + q)-KP hierarchy: Define two matrices Wmn(z)andWmn(z)of size p + q,whoseentriesaregivenby ratios of determinants τmn of moment matrices as above, but with appropriately shifted t and s parameters. They turn out to be the wave and dual wave matrices for the (p + q)-KP hierarchy. It is remarkable that, upon setting all t and s parameters equal to zero, the matrix Wmn(z) below is precisely the Riemann-Hilbert matrix characterizing ∗ the mixed multiple orthogonal polynomials! Similarly Wmn(z)att = s = 0 satisfies the Riemann-Hilbert problem characterizing alternately the “dual” multiple orthogonal polynomials or the inverse transpose matrix of Wmn(z)at t = s = 0. The Riemann-Hilbert matrix for the multiple-orthogonal polynomials has been defined in Daems-Kuijlaars [9], which is a far generalization of the Riemann-Hilbert matrix of Fokas-Its-Kitaev [11] and Deift-Zhou [10]. Using identities as in I to IV, the two left blocks of Wmn and the two right blocks of ∗ Wmn are mixed multiple orthogonal polynomials, and the remaining blocks are Cauchy transforms of such polynomials; for explicit expressions, see Section 5. The matrix Wmn(z) is defined by 5 ∞ k ∞ k ∞ k ∞ k − t1k z − tpkz − s1k z − sqk z Wmn(z)diag e 1 ,...,e 1 ,e 1 ,...,e 1 := P P P P τ (t −[z−1]) τ s − z−1 m,n+e −e ′ β′ n δ − m+eα,n+e ( α [ ]) β β β′ + ββ′ 1 β −mα−1 ε ′ (n) z ǫαβ(m,n) z 0 ββ τmn 1 τmn 1≤β≤p ! 1≤β≤p B C ′ @ A 1≤β ≤p 1≤α≤q −1 −1 , τm−e ,n−e (tβ −[z ]) τm e −e ,n(sα−[z ]) α β n −1 + α α′ δ ′ −1−mα ǫ (m,n) z β ε ′ (m) z αα αβ τmn α α τmn 1≤α≤q 1≤α′≤q ! ! 1≤β≤p 1≤α≤q (9) with inverse transpose matrix given by ∞ k ∞ k ∞ k ∞ k ∗ 1 t1k z 1 tpkz 1 s1k z 1 sqk z Wmn(z)diag e ,...,e ,e ,...,e = P P P P τ (t +[z−1]) τ s z−1 m,n+e −e ′ β δ − −n m−eα,n−e ( α+[ ]) β β β′β 1 β β mα−1 ε ′ (n) z −ǫβα(n,m) z 0 β β τmn 1 τmn 1≤β′≤p ! 1≤β≤p B C @ A 1≤β≤p 1≤α≤q −1 −1 . τm+e ,n+e (tβ +[z ]) τ ′ (s ′ +[z ]) α β −n −1 m+eα−eα,n α δ ′ −1+m ′ −ǫ (n,m) z β ε ′ (m) z αα α βα τmn αα τmn ! 1≤α≤q ! 1≤α≤q ′ 1≤β≤p 1≤α ≤q (10) ∗ The matrices Wmn(z)andWm∗n∗ (z) satisfy the bilinear identities which char- acterize the τ-function of the (p + q)-KP hierarchy ∗ ∗ ∗ ⊤ Wmn(z; s, t)Wm∗n∗ (z; s ,t ) dz =0, (11) ∞ for all m, n, m∗,n∗ such that m = n , m∗ = n∗ and all s,t,s∗,t∗ C∞. The integral above is taken along| | a small| | | circle| about| | z = ; writing out∈ the identity above componentwise and using the expressions (9) and∞ (10) for W and W ∗, the bilinear identity (11) is equivalent to the single identity p ∞ ∗ k ∗ σ (n) −1 ∗ −1 (t −t )z n −n −2 (−1) β τ (t −[z ])τ ∗ ∗ (t +[z ])e 1 βk βk z β β dz= m,n−eβ β m ,n +eβ β ∞ P β=1 q ∞ ∗ k ∗ σα(m) −1 ∗ −1 1 (sαk −sαk)z mα−mα−2 (−1) τm+eα,n(sα−[z ])τm∗−eα,n∗ (sα+[z ])e z dz, P α=1 ∞ where m∗ = n∗ +1and m = n 1and | | | | | | | |− α β ∗ ∗ σα(m)= (mα′ mα′ )andσβ(n)= (nβ′ nβ′ ). ′ − ′ − α=1 β=1 It remains an open problem to have a clear understanding of why the Wmn(z; s, t)- matrix above, evaluated at t = s = 0, coincides with the Riemann-Hilbert matrix for the mixed multiple orthogonal polynomials. 6 PDE’s for the determinant of moment matrices: Upon actually comput- ing the residues in the contour integrals above, the functions τmn,with m = n , satisfy the following PDE’s expressed in terms of the Hirota symbol6:| | | | 2 2 ∂ ˜ τmn ln τmn = Sℓ+2δββ′ ∂tβ τm,n+eβ −eβ′ τm,n+eβ′ −eβ ∂tβ,ℓ+1∂tβ′,1 ◦ 2 2 ∂ ˜ τmn ln τmn = Sℓ+2δαα′ (∂sα )τm+eα′ −eα,n τm+eα−eα′ ,n ∂sα,ℓ+1∂sα′,1 ◦ 2 2 ∂ ˜ τmn ln τmn = Sℓ(∂tβ )τm+eα,n+eβ τm−eα,n−eβ ∂sα,1∂tβ,ℓ+1 − ◦ 2 2 ∂ ˜ τmn ln τmn = Sℓ(∂sα )τm−eα,n−eβ τm+eα,n+eβ . (12) ∂tβ,1∂sα,ℓ+1 − ◦ Whereas the formulae above have in their right hand side different τmn’s, one can combine these relations to yield PDE’s in a single τmn; so, these are PDE’s for the determinant of the moment matrix (1). In particular, one finds the following p+q 2 PDE’s, which play a fundamental role in chains of random matrices and in the transition probabilities for critical infinite-dimensional diffusions: ∂2 ∂2 ln τmn ln τmn ∂ ∂tβ,2∂tβ′,1 ∂ ∂tβ′,2∂tβ,1 ∂2 + ∂2 =0, ∂tβ′,1 ln τmn ∂tβ,1 ln τmn ∂tβ,1∂tβ′,1 ∂tβ′,1∂tβ,1 ∂2 ∂2 ln τmn ln τmn ∂ ∂sα,2∂sα′,1 ∂ ∂sα′,2∂sα,1 ∂2 + ∂2 =0, ∂sα′,1 ln τmn ∂sα,1 ln τmn ∂sα,1∂sα′,1 ∂sα′,1∂sα,1 ∂2 ∂2 ln τmn ln τmn ∂ ∂tβ,2∂sα,1 ∂ ∂sα,2∂tβ,1 ∂2 + ∂2 =0. ∂sα,1 ln τmn ∂tβ,1 ln τmn ∂tβ,1∂sα,1 ∂sα,1∂tβ,1 2 Tau functions and mixed multiple orthogonal polynomials Following [9] we introduce the notion of mixed multiple orthogonal polyno- mials (mixed mops), with regard to two sets of weights ϕ1,ϕ2,...,ϕp and ψ ,ψ ,...,ψ : { } { 1 2 q} 6 For a given polynomial p(t1,t2,...), the Hirota symbol between functions f = f(t1,t2,...) and g = g(t1,t2,...) is defined by: ∂ ∂ ∂ ∂ ˛ p( , ,...)f g := p( , ,...)f(t + y)g(t y)˛ . ∂t1 ∂t2 ◦ ∂y1 ∂y2 − ˛ ˛y=0 ∞ k tkz k We also need the elementary Schur polynomials Sℓ, defined by e 1 := Pk≥0 Sk(t)z for ℓ 0andS (t)=0forℓ<0; moreover, set P ≥ ℓ ∂ 1 ∂ 1 ∂ Sℓ(∂˜t):=Sℓ( , , ,...). ∂t1 2 ∂t2 3 ∂t3 7 Definition 2.1 Let A1,A2,...,Ap be p polynomials in y and set Q(y):=A (y)ϕ (y)+A (y)ϕ (y)+ + A (y)ϕ (y). 1 1 2 2 ··· p p Type I For α 1, 2,...,q the polynomials A1,A2,...,Ap are said to be Type I normalized∈{with respect} to ψ ,denotedType I ,ifdeg(A ) i ′ x ψ ′ (x) Q(y) = δ ′ δ ,i=0,...,m ′ 1, 1 α q. (13) α | αα i,mα−1 α − ≤ ≤ Type II For β 1,...,p the polynomials A ,A,...,A are said to be ∈{ } 1 2 p Type II normalized with respect to ϕβ,denotedType II ,ifAβ is monic of |ϕβ ′ ′ degree nβ and deg(Aβ′ ) xiψ (x) Q(y) =0,i=0,...,m 1, 1 α q. (14) α | α − ≤ ≤ In both cases, the polynomials A1,...,Ap are called multiple orthogonal poly- nomials of mixed type,ormixed mops for brevity. Proposition 2.2 For β =1,...,p,let Q(β) (y):=Q(β,1)(y)ϕt (y)+ + Q(β,p)(y)ϕt (y), (15) mn mn 1 ··· mn p ′ (β,β ) ′ where Qmn ,with1 β,β p are the polynomials, defined by ≤ ≤ τ (t z−1 ) (β,β) nβ mn β Qmn (z):=z − τmn −1 (16) ′ τm,n+e −e ′ (tβ′ z ) (β,β ) n ′ −1 β β ′ Qmn (z):=εββ′ (n)z β − ,β= β, τmn and ( 1)nβ′+1+nβ′+2+···+nβ +1 if β>β′, εββ′ (n)= − (17) ( 1)nβ+1+nβ+2+···+nβ′ if β<β′. − (β,1) (β,p) Then Qmn (y),...,Qmn (y) are Type II mixed mops. |ϕt β Proof β For j =0, 1, 2,... and β =1, 2,...,p we define a column vector Cj of size m by | | i1 −s j t x ψ1 (x) y ϕβ (y) 0≤i1 1 1 1 2 2 p τmn =det C0 ,C1 , ..., Cn1−1,C0 ,C1 , ..., Cnp−1 . (19) For future use, let us point out that the dependence of τmn on the t variables is as follows: 1 1 1 2 2 p τmn(t1)=det(C0 (t1),C1 (t1), ..., Cn1−1(t1),C0 ,C1 , ..., Cnp−1), 1 1 1 2 2 p τmn(t2)=det(C0 ,C1 , ..., Cn1−1,C0 (t2),C1 (t2), ..., Cnp−1), . . 1 1 1 2 2 p τmn(tnp )=det(C0 ,C1 , ..., Cn1−1,C0 ,C1 , ..., Cnp−1(tnp )). (20) Since p p nβ′ −1 ′ (β) (β,β ) t nβ j j t Q (y)= Q (y)ϕ ′ (y)= δ ′ y + A ′ y ϕ ′ (y), mn mn β ββ ββ β ′ ′ j=0 β=1 β=1 (21) (β) the orthogonality conditions (14) for Q = Qmn can be written as the linear system p nβ′−1 j β′ β A ′ C = C , ββ j − nβ ′ j=0 β=1 j ′ of m equations, in the n (= m ) unknowns A ′ ,where1 β p and 0 | | | | | | ββ ≤ ≤ ≤ 0 1 n1−1 0 j In order to connect these polynomials with the tau functions τmn we first −1 β −1 expand τmn(tβ z ) using (20). Thus, we need to compute Cj (tβ z ), which we claim− to be given by − Cβ(t z−1 )=Cβ(t ) z−1Cβ (t )=Cβ z−1Cβ , (23) j β − j β − j+1 β j − j+1 where the last equality is the notational simplification agreed upon. To prove the first equality in (23), which is an equality of formal series in z−1,letuswrite 9 β a typical entry of the column vector Cj (tβ) with its explicit time-dependence on tβ, ∞ k i −s j t i −s j k=1 tβky x ψα (x) y ϕβ(y) = x ψα (x) y ϕβ(y)e , P ! " where 1 α q and 0 i i − ∞ x e 1 i =1 x. (24) P − In view of the latter, the same entry of Cβ(t z−1 ) (as above) is given by j β − y 1 xiψ−s(x) yjϕt (y) 1 = xiψ−s(x) yjϕt (y) xiψ−s(x) yj+1ϕt (y) α β − z α β −z α β ! " which proves (23). Using the fact that the determinant is a skew-symmetric multilinear function of its columns, which vanishes when two columns are equal, it follows from (20), (23) and (22) that znβ τ (t z−1 ) mn β − 1 β−1 β β β β β+1 p =det(C0,...,C ,zC C ,...,zC Cn ,C ,...,C ) nβ−1−1 0 − 1 nβ −1 − β 0 np−1 nβ (∗) = zj det(C1,C1,...,Cβ , Cβ ,..., Cβ ,Cβ+1,...,Cp ) 0 1 j−1 − j+1 − nβ 0 np−1 j=0 nβ = zj det(C1,C1,...,Cβ , Cβ ,Cβ ,...,Cβ ,Cβ+1,...,Cp ) 0 1 j−1 − nβ j+1 nβ −1 0 np−1 j=0 nβ j j = z Aββτmn j=0 (β,β) = τmn Qmn (z). β β In ( ) it is understood that all the columns between Cj+1 and Cnβ come with negative∗ signs and no others; this notation shall be− used freely− in the sequel, without further mention. ′ (β,β ) ′ For Qmn with β = β we also need to keep track of signs and of shifts in the first index of the tau function, as is seen in the following computation, wherewesupposethatβ<β′: n ′ −1 z β −1 τ (t ′ z ) m,n+eβ −eβ′ β − ′ ′ ′ ′ ′ =det(C1,...,Cβ ,Cβ,...,zCβ Cβ ,...,zCβ Cβ ,Cβ +1,...,Cp ) 0 nβ −1 nβ 0 − 1 nβ′ −2 − nβ′ −1 0 np−1 nβ′ −1 ′ ′ ′ ′ = zj det(C1,...,Cβ ,...,Cβ , Cβ ,..., Cβ ,Cβ +1,...,Cp ) 0 nβ j−1 − j+1 − nβ′−1 0 np−1 j=0 10 nβ′ −1 j 1 β β+1 β′ β β′ p = εββ′ (n) z det(C ,..., C ,C ,...,C , C ,C ,...,C ) 0 nβ−1 0 j−1 − nβ j+1 np−1 j=0 nβ′ −1 j j = εββ′ (n) z Aββ′ τmn j=0 (β,β′) = εββ′ (n) τmn Qmn (z). ′ β The sign εββ (n) which we introduced when moving the column Cnβ to the right is given by ( 1)nβ+1+···+nβ′ , in agreement with (17). When β>β′ the column − n ′ +···+n β ′ β +1 β Cnβ is moved to the left, which yields a sign εββ (n)= ( 1) ,asis easily checked. − − ✷ The tau functions τmn also lead to Type I normalized mixed mops, as given in the following proposition. Proposition 2.3 For α =1,...,q, let P (α)(y):=P (α,1)(y)ϕt (y)+ + P (α,p)(y)ϕt (y), (25) mn mn 1 ··· mn p (α,β) where Pmn are the polynomials, defined by τ (t z−1 ) (α,β) nβ −1 m−eα,n−eβ β Pmn (z):=ǫαβ(m, n)z − , (26) τmn with leading sign ǫ (m, n)=( 1)m1+···+mα ( 1)n1+···+nβ . (27) αβ − − (α,1) (α,p) Then Pmn (y),...,Pmn (y) are Type I | −s mixed mops. ψα Proof Letting p p nβ −1 (α) (α,β) t j j t Pmn (y)= Pmn (y)ϕβ(y)= Bαβ y ϕβ(y), (28) j=0 β=1 β=1 (α) the orthogonality conditions (13) for Q = Pmn can be written as the linear system p mβ −1 j β α BαβCj = Emα , j=0 β=1 where Eα denotes the column vector of size m with a 1 at position m of the mα | | α α-th block (so at position m1 + m2 + + mα), and zeros elsewhere. Cramer’s rule now yields ··· det(C1,C1,...,Cβ ,Eα ,Cβ ,...,Cp ) j 0 1 j−1 mα j+1 np−1 Bαβ = , τmn 1 1 β β β p det(D0,D1,...,Dj−1, Dj ,Dj+1,...,Dn −1) = ǫ (m, n)( 1)j+1−nβ p , αβ − τ mn# 11 α where the last line was obtained by expanding the determinants along the Em γ γ α column, ǫαβ(m, n) is given by (27) and Dk is the column vector Dk with its (m + + m )-th entry removed, i.e., Dγ := Cγ (m e ). This yields ex- 1 ··· α k k − α plicit expressions for the Type I | −s mixed mops. To connect them with tau ψα γ functions, we notice on the one hand that the columns Dk appear in the ma- trices which define the tau functions τm−eα,⋆, and on the other hand that these γ columns behave in the same way (23) as Ck under shifts. Therefore we can compute, as before znβ−1τ (t z−1 ) m−eα,n−eβ β − 1 β−1 β β β β β+1 p =det(D0,...,D ,zD D ,...,zD D ,D ,...,D ) nβ−1−1 0 − 1 nβ−2 − nβ −1 0 np−1 nβ −1 = zj det(D1,...,Dβ , Dβ ,..., Dβ ,Dβ+1,...,Dp ) 0 j−1 − j+1 − nβ−1 0 np−1 j=0 nβ −1 = ( 1)nβ −j−1zj det(D1,...,Dβ , Dβ,Dβ ,...,Dp ) − 0 j−1 j j+1 np−1 j=0 nβ −1 # j j = ǫαβ(m, n) z Bαβ τmn j=0 (α,β) = ǫαβ(m, n) Pmn (z) τmn. ✷ 3 Cauchy transforms We now show that certain shifts of the tau function, appearing in the Riemann- Hilbert matrix of [9], are (formal) Cauchy transforms. For a function F and a weight ψ, define its Cauchy transform as ψ(x) ∞ 1 G(z):= G(y) = xiψ(x) G(y) , (29) Cψ z x | zi+1 | $ − % i=0 i.e., our Cauchy transforms will be formal in the sense that we always think of z as being large, and this is precisely how it will be used. The first type of Cauchy transforms which we are interested in are given in the following proposition. Proposition 3.1 For α =1,...,q and β =1,...,p, the Cauchy transforms (β) (β,1) t (β,p) t −s of Qmn(y)=Qmn (y)ϕ1(y)+ + Qmn (y)ϕp(y), with respect to ψα can be expressed in terms of tau functions··· as follows. τ (s z−1 ) (β) −mα−1 m+eα,n+eβ α −s Q (z)=ǫ (m, n) z − . (30) ψα mn αβ C τmn 12 Proof The proof is based on an investigation of the moment matrix by row. i Therefore we define, for α =1,...,q and for i =0, 1, 2,...the row Rα of size n by | | i i −s j1 t i −s jp t R := x ψα (x) y ϕ1(y) x ψα (x) y ϕp(y) . α 0≤j1 0 R1 1 R1 . . τ =det Rm1−1 . (31) mn 1 0 R2 . . m −1 R q q −1 The tau function which we need to compute is τm+eα,n+eβ (sα z ); so i i ′ − throughout the proof, R ′ stands for R ′ (n + e ) for all 1 α q and i = α α β ≤ ≤ 0, 1, 2,.... Notice that the only rows which depend on the time variables sα = i −s (sα1,sα2,...)aretherowsRα. Recall the dependence of ψα on sα as follows ∞ k −s − k=1 sαkx ψα (x)=ψα(x)e , P −1 so that, according to the identity (24), when sα gets replaced by sα z , 2 − then ψ−s(x) gets replaced by ψ−s(x) 1+ x + x + . It follows that α α z z2 ··· x x2 i −1 i −s jβ′ t R (sα z )= x ψ (x) 1+ + + y ϕ ′ (y) , α α 2 β 1≤β′ z−mα−1τ (s z−1 ) (32) m+eα,n+eβ α − 13 0 0 R1 R1 . . . . mα−1 mα−1 Rα Rα = z−mα−1 det Rmα (s z−1 ) =det R˜ (z) . α α − α 0 0 Rα+1 Rα+1 . . . . m −1 m −1 R q R q q q For the second equality, in which we have put ψ−s(x) ψ−s(x) ˜ α yj1 ϕt (y) α yjp ϕt (y) Rα(z):= z−x 1 ′ z−x p ′ , 0≤j1 mα−1 Rmα (s z−1 )+ zmα−iRi (s )=zmα+1R˜ (z), α α − α α α i=0 and hence to the proof of the second equality in (32). In order to make the connection with mixed mops, we introduce for β′ = ′ 1,...,p the row Sβ (z)ofsize n + e = n + 1 which has zeroes everywhere, β | β| | | except in its β′-th block, namely7 β′ j S (y)= 0 ...0 y ′ 0 ...0 . β 0≤j 14 Notice that with this notation, definition (33) of R˜α(z) can be rewritten as p β′ t R˜ (z)= −s S (z)ϕ ′ (z) . (35) α ψα β β C ′ β=1 It suggests the introduction of the following polynomials (in y) 0 R1 . . mα−1 Rα β′ ′ S (y):=det Sβ (y) . (36) αβ β 0 Rα+1 . . Rmq −1 q Expanding this determinant along its (m1 + + mα +1)-th row, which is the ′ ···′ β ′ (unique) row that contains y, it is clear that if β = β,thendegSαβ(y) β nβ nβ −1 Sαβ(y)=ǫαβ (m, n) τmny + O(y ). ′ Moreover, for any α =1,...,q and i =0,...,mα′ 1, we have by linearity of the determinant − p i −s β′ t x ψ ′ (x) S (y)ϕ ′ (y) α αβ β β′=1 0 R1 . . . mα−1 Rα p ′ i −s β t S (y)ϕ ′ (y) = x ψα′ (x) det β β (37) ′ β=1 0 Rα+1 . . . mq −1 Rq 8 See (27) for the definition of ǫαβ(m,n). 15 R0 1 R0 . 1 . . . mα−1 Rα mα−1 Rα i −s j t ′ x ψα′ (x) y ϕβ′ (y) 1≤β 9 (β) (β,1) t (β,p) t for any α =1,...,q.SinceQmn(y)=Qmn (y)ϕ1(y)+ + Qmn (y)ϕp(y), it follows from (32), (35), (38) and (20) that ··· −1 p τ (s z ) ′ −mα−1 m+eα,n+eβ α 1 β t z − = −s S (z)ϕ ′ (z) ψα αβ β τmn τmn C β′=1 p (β,β′) t = ǫ (m, n) −s Q (z)ϕ ′ (z) αβ ψα mn β C ′ β=1 (β) = ǫαβ(m, n) −s Q (z) (39) Cψα mn This finishes the proof. ✷ (β) Observe our proof shows, as a byproduct, that each Qmn(y) is expressible nat- urally as a determinant, like in the classical case, namely 0 R1 . . mα−1 Rα p ′ ǫαβ(m, n) β t (β) S (y)ϕ ′ (y) Qmn(y)= det β β . (40) τmn β′=1 R0 α+1 . . mq−1 Rq 9Theformulasforthedifferentvaluesofα are all the same, up to a sign, as they amount to changing the location of a row in the evaluation of a determinant. 16 We now get to the second type of Cauchy transforms which correspond to the (α) Type I mops Pmn (y). ′ ′ (α ) Proposition 3.2 For 1 α, α q the Cauchy transforms of Pmn (y)= ′ ′ (α ,1) t ≤(α ,p) ≤t −s Pmn (y) ϕ1(y)+ + Pmn (y)ϕp(y) with respect to ψα can be expressed in terms of tau functions··· as follows: τ (s z−1 ) (α) −mα mn α −s P (z)=z − , (41) ψα mn C τmn −1 ′ τ (s z ) (α ) −1−mα m+eα−eα′ ,n α ′ −s P (z)=ε ′ (m) z − ,α= α.(42) ψα mn α α C τmn Proof Up to a relabeling of the indices, the shifted tau functions in question were already expressed as polynomials in the previous proof. Let us show how this leads to a quick proof of (41). Shifting the mα and nβ indices down by 1, it follows from (32) that 0 R1 . . mα−2 Rα −mα −1 ˜ z τmn(sα z )=det Rα(z) , − 0 R α+1 . . mq −1 Rq while the orthogonality relations (37) become 0 R1 . . mα−2 p Rα ′ i −s β t i x ψ ′ (x) S (y)ϕ ′ (y) =det R ′ = δ ′ δ τ . (43) α αβ β α αα i,mα−1 mn β′=1 0 Rα+1 . . m −1 R q q ′ β ′ Since deg Sαβ 1 1 1 p Sαβ(y) , ... , Sαβ(y) τmn τmn −s are type I mixed mops, normalized with respect to ψα , so they coincide ac- (α,1) (α,p) cording to Proposition 2.3 with the polynomials Pmn (y),...,Pmn (y). We 17 conclude, as in (39), that −1 p τ (s z ) ′ −mα mn α 1 β t (α) z − = −s S (z)ϕ ′ (z) = −s P (z). ψα αβ β ψα mn τmn τmn C C β′=1 ′ Similarly, one obtains (42) from (30) by shifting mα and nβ down by 1; the sign in this case is determined (for α′ <α) from the right hand side of (43) now taking the form 0 0 R1 R1 . . . . mα′ −2 mα′ −2 Rα′ Rα′ 0 i R ′ R ′′ α +1 α . 0 . Rα′+1 . ′ ′ ′ ′′ det = εα α(m)det . = εα α(n) δα α δi,mα′ −1τmn. mα−1 . Rα . i mα−1 R ′′ R α α 0 0 Rα+1 Rα+1 . . . . m −1 m −1 R q R q q q ✷ 4 Duality t −s By interchanging the rˆoles of the weights ψα with the weights ϕβ we obtain Type I mixed mops and Type II mixed mops, expressed in terms of tau |ϕt | −s β ψα functions, leading to a duality. As a general rule, in order to dualize a formula one does the following exchanges q p, m n, ψ ϕ, s t, x y. (44) ↔ ↔ ↔ ↔− ↔ At the level of the indices, duality amounts to α β, i j. (45) ↔ ↔ As for the mixed mops which we have constructed, they will correspond to new mixed mops for which we will use the same letter, but adding a star. Thus, (α) ∗(β) (α,β) ∗(β,α) (β) ∗(α) (β,β′) ∗(α,α′) Pmn Pnm ,Pmn Pnm ,Qmn Qnm ,Qmn Qnm . ↔ ↔ ↔ ↔ (46) What happens to the tau functions τmn? To see this, pick a typical shifted tau function τ (s z−1 ) and make its dependence on the weights and m+eα−eα′ ,n α − 18 on all times explicit, writing τ (s z−1 e ,t; ψ, ϕ). According to m+eα−eα′ ,n − α the above rule it becomes τ ( t z−1 e , s; ϕ, ψ) which is equal to n+eβ −eβ′ ,m − − β − τ (s, t + z−1 e ; ψ, ϕ), since transposing the moment matrix has no m,n+eβ −eβ′ β effect on the determinant, while it permutes the indices in the tau function, it permutes the time-dependence (with signs) and it permutes the weights. Thus, τ τ mn ↔ mn τ (t z−1 ) τ (s + z−1 ) mn β − ↔ mn α τ (t z−1 ) τ (s + z−1 ) m−eα,n−eβ β − ↔ m−eα,n−eβ α τ (s z−1 ) τ (t + z−1 ), m+eα−eα′ ,n α − ↔ m,n+eβ −eβ′ β and so on. Dualizing Propositions 2.2 and 2.3, we get the following proposition. Proposition 4.1 For α =1,...,q and β =1,...,p,let ∗(β) ∗(β,1) −s ∗(β,q) −s Pnm (x):=Pnm (x)ψ (x)+ + Pnm (x)ψ (x), 1 ··· q (47) ∗(α) ∗(α,1) −s ∗(α,q) −s Qnm (x):=Qnm (x)ψ (x)+ + Qnm (x)ψ (x), 1 ··· q ∗(β,α) ∗(α,α′) where Pnm and Qnm are the polynomials, defined by τ (s + z−1 ) ∗(β,α) mα−1 m−eα,n−eβ α Pnm (z):=ǫβα(n, m)z , (48) τmn and τ (s + z−1 ) ∗(α,α) mα mn α Qnm (z):=z τmn ′ ′ −1 ∗(α,α ) m ′ −1 τm+eα−eα′ ,n(sα + z ) ′ Qnm (z):=εαα′ (m)z α ,α= α. τmn (49) ∗(β,1) ∗(β,q) ∗(α1) ∗(α,q) Then Pnm (x),...,Pnm (x) are Type I mixed mops, while Qnm (x),...,Qnm (x) |ϕt β are Type II | −s mixed mops. ψα ✷ Dualizing Definition (29) we get the following definition for the dual Cauchy transform: for any function F and a weight ϕ we put ∗ ϕ(y) ϕ(y) ϕF (z):= F (x)dµ(x, y)= F (x) . (50) C R2 z y z y $ % − − If we dualize now Propositions 3.1 and 3.2, then we get the following proposition. Proposition 4.2 For α =1,...,q and β,β′ =1,...,p, the Cauchy transforms ′ ∗(β ) t ∗(α) t of Pnm (x) with respect to ϕβ,andofQnm (x) with respect to ϕβ can be ex- pressed in terms of tau functions as follows: τ (t + z−1 ) ∗ ∗(β) −nβ mn β t P (z)=z , ϕβ nm C τmn 19 −1 ′ τm,n+eβ −e ′ (tβ + z ) ∗ ∗(β ) −1−nβ β ′ ϕt Pnm (z)=εβ′β(n) z ,β= β, C β τmn and τ (t + z−1 ) ∗ ∗(α) −nβ −1 m+eα,n+eβ β t Q (z)=ǫβα(n, m) z . ϕβ nm C τmn ✷ 5 The Riemann-Hilbert matrix and the bilinear identity Orthogonal polynomials were shown to be characterized by a Riemann-Hilbert problem in [11] and [10]. This was generalized by Daems and Kuijlaars to the case of mixed mops. According to [9]10 the corresponding Riemann-Hilbert matrix is given by the (p + q) (p + q)matrix × (β,β′) (β) Q −s Q mn 1≤β≤p ψα mn 1≤β≤p 1≤β′≤p C 1≤α≤q Ymn(z):= = (α,β) (α′) Pmn −s Pmn ′ 1≤α≤q ψα 1≤α ≤q 1≤β≤p C 1≤α≤q −1 −1 τm,n+e −e (t ′ − z ) τm+e ,n+e (sα− z ) β β′ β [ ] n ′ +δ ′ −1 α β [ ] −mα−1 0ε (n) z β ββ 1 ǫ (m,n) z ββ′ τ αβ τmn mn 1≤β≤p B C 1≤β≤p ! B C ′ 1≤α≤q @ A 1≤β ≤p −1 −1 τm−e ,n−e (tβ − z ) τm+e −e ,n(sα− z ) α β [ ] nβ −1 α α′ [ ] δ ′ −1−mα ǫαβ (m,n) z ε ′ (m) z αα τmn α α τmn ′ ! 1≤α≤q ! 1≤α ≤q 1≤β≤p 1≤α≤q whose inverse transpose matrix is given by ′ ∗ ∗(β ) ∗(β,α) ϕt Pnm ′ Pnm 1≤β≤p C β 1≤β ≤p − ∗ 1≤β≤p 1≤α≤q Ymn(z)= = ′ ∗ ∗(α) ∗(α,α ) t Qnm Qnm ϕβ 1≤α≤q 1≤α≤q −C 1≤β≤p 1≤α′≤q −1 −1 τm,n+e −e (tβ + z ) τm−e ,n−e (sα+ z ) β β′ [ ] δ ′ −1−nβ α β [ ] mα−1 0ε (n) z β β 1 −ǫ (n,m) z β′β τ βα τmn mn ′ 1≤β≤p B C 1≤β ≤p ! B C 1≤α≤q @ A 1≤β≤p −1 −1 τm+e ,n+e (tβ + z ) τm e −e ,n(s ′ + z ) α β [ ] −n −1 + α α′ α [ ] δ ′ −1+m ′ −ǫ (n,m) z β ε ′ (m) z αα α βα τmn αα τmn ! 1≤α≤q ! 1≤α≤q ′ 1≤β≤p 1≤α ≤q We will obtain bilinear identities for these tau functions from an identity which is satisfied by the Riemann-Hilbert matrix and its adjoint. We define the wave 11 matrix Wmn(z)byYmn(z)∆(z), where ∆(z) is the diagonal matrix ∆(z):=diag(eξ(t1,z),...,eξ(tp,z),eξ(s1,z),eξ(sq ,z)), 10 Up to a factor diag(Ip, 2π√ 1Iq) which we suppress. 11 − − ∞ k Throughout this section, we set ξ(t, z):=P1 tkz . 20 ∗ −1 with adjoint wave matrix Ymn(z)∆ (z). In order to make the dependence onthetimevariables(s, t) explicit, we will write Wmn(z; s, t)forW (z)and ∗ ∗ Wmn(z; s, t)forWmn(z). Theorem 5.1 The tau functions τmn satisfy the following bilinear identities that characterize the tau functions of the (p + q)-KP hierarchy (see [13]): ∗ ∗ ∗ ⊤ Wmn(z; s, t)Wm∗n∗ (z; s ,t ) dz =0, ∞ which is equivalent to the single identity p ∗ ∗ σβ (n) −1 ∗ −1 ξ(tβ −t ,z) nβ−n −2 ∗ ∗ β β ( 1) τm,n−eβ (tβ z )τm ,n +eβ (tβ + z )e z dz = ∞ − − β=1 q ∗ ∗ σα(m) −1 ∗ −1 ξ(sα−s ,z) m −mα−2 ( 1) τ (s z )τ ∗ ∗ (s + z )e α z α dz, (51) − m+eα,n α − m −eα,n α α=1 ∞ where α β ∗ ∗ σα(m)= (mα′ mα′ ) and σβ(n)= (nβ′ nβ′ ). (52) ′ − ′ − α=1 β=1 and m∗ = n∗ +1 and m = n 1. | | | | | | | |− Proof For the entry (β′,β′′) of the product ∗ −1 ⊤ ξ(t −t∗,z) ξ(t −t∗,z) ξ(s −s∗,z) ξ(s −s∗,z) ∗⊤ Y ∆(Y ∆ ) = Y diag(e 1 1 , ...,e p p ,e 1 1 , ...,e q q )Y , we need to prove that p q ′ ′′ ∗ ′ ′′ ∗ (β ,β) ∗ ∗(β ) ξ(tβ −tβ ,z) (β ) ∗(β ,α) ξ(sα−sα,z) Q (z) t∗ P ∗ ∗ (z) e dz = −s Q (z) P ∗ ∗ (z) e dz mn ϕ n m ψα mn n m C β C ∞ α=1 ∞ β=1 (53) where it is understood that all polynomials P ∗ go with starred times s∗ and t∗. Also, the integral stands for (minus) the residue at infinity, and can be ∞ i computed using the following formal residue identities, with f(z)= i=0 aiz , 1 f(z) ψg(z) dz = f(x)ψ(x) g(y) , (54) 2π√ 1 C | − ∞ 1 ∗ f(z) g(z) dz = f(x) ϕ(y)g(y) , (55) 2π√ 1 Cϕ | − ∞ whose proof we defer until the end. Using this, and Definition (47) of the ∗(β) functions Pnm (x), the left hand side in (53) becomes (up to a factor 2π√ 1) − p p ′′ ′ ∗ ∗ ′′ ′ ∗(β ) (β ,β) t ξ(tβ −tβ ,y) ∗(β ) (β ,β) t Pn∗m∗ (x) Qmn (y) ϕβ (y) e = Pn∗m∗ (x) Qmn (y) ϕβ(y) β=1 ! " β=1 ! " ′′ ′ ∗(β ) (β ) = Pn∗m∗ (x) Qmn (y) . ! " 21 Similarly, the right hand side in (53) becomes (up to a factor 2π√ 1) − q q ′′ ∗ ′ ′′ ∗ ′ ∗(β ,α) −s ξ(sα−sα,x) (β ) ∗(β ,α) −s (β ) Pn∗m∗ (x) ψα (x) e Qmn (y) = Pn∗m∗ (x) ψα (x) Qmn (y) α=1 ! " α=1 ! " ′′ ∗(β ) (β′) = Pn∗m∗ (x) Qmn (y) . ! " The three other identities are obtained in the same way. In terms of tau functions, it means that we have shown that for any m, n, m∗,n∗,β′ and β′′,with m = n and m∗ = n∗ the following bilinear identities hold: | | | | | | | | p ∗ −1 ∗ −1 ξ(tβ −t ,z) ∗ ∗ β τm,n+eβ′ −eβ (tβ z )τm ,n +eβ −eβ′′ (tβ + z )e dz = ∞ ♣ − β=1 q ∗ −1 ∗ −1 ξ(sα−s ,z) τ (s z )τ ∗ ∗ (s + z )e α dz, ♠ m+eα,n+eβ′ α − m −eα,n −eβ′′ α α=1 ∞ where ∗ ∗ nβ −n −2+δ ′ +δ ′′ = εβ′β(n)εβ′′β(n )z β ββ ββ , ♣ ∗ ∗ ∗ m −mα−2 = ǫ ′ (m, n)ǫ ′′ (n ,m )z α . ♠ αβ β α For different values of β′ and β′′ this yields the same identity, up to a relabeling ∗ ∗ of n and n . Namely, replace in the bilinear identity n+eβ′ by n and n eβ′′ by n∗+···+n∗ − n∗ and multiply by ( 1)n1+···+nβ′ ( 1) 1 β′′ to find the following symmetric expression for the identity,− that is− independent of β′ and β′′: p ∗ ∗ σβ (n) −1 ∗ −1 ξ(tβ −t ,z) nβ−n −2 ∗ ∗ β β ( 1) τm,n−eβ (tβ z )τm ,n +eβ (tβ + z )e z dz = ∞ − − β=1 q ∗ ∗ σα(m) −1 ∗ −1 ξ(sα−s ,z) m −mα−2 ( 1) τ (s z )τ ∗ ∗ (s + z )e α z α dz, − m+eα,n α − m −eα,n α α=1 ∞ where σα(m)andσβ(n) are given by (52). Notice that, due to the shift, one must have in this symmetric form that m = n 1and m∗ = n∗ +1.The other three identities also yield the above| identity,| | |− up to r|elabeling.| | | Finally, to prove (54), compute ∞ ∞ 1 ψ(x) 1 1 f(z) g(y) dz = a zi xj ψ(x) g(y) √ z x | √ i zj+1 | 2π 1 ∞ 2π 1 ∞ i=0 j=0 − $ − % − ∞ ∞ = a xiψ(x) g(y) = a xi ψ(x) g(y) = f(x)ψ(x) g(y) , i | i | | i=0 i=0 and similarly for (55), completing the proof. ✷ 22 6 Consequences of the bilinear identities In this section we will derive from the bilinear identities (51) a series of PDE’s for the tau functions τmn. In order to keep the formulas transparant we will use the following simplification in the notation. Recall that we have time variables sα =(sα1,sα2,...)andtβ =(tβ1,tβ2,...), where α =1,...,q and β =1,...,p. In the bilinear identities (51) we consider in each term a shift in tα, for a single α,orinsβ, for a single β; we will denote this tα or sβ by v (so v is an infinite vector v =(v ,v ,...) and we assemble all the other r := p + q 1series 1 2 − of time variables in w =(w1,w2,...,wr), where w1 =(w11,w12,...)andso on. Moreover, precisely like in the bilinear identities we will want to consider an independent collection of all these variables, in fact we will consider here (v′,w′) and (v′′,w′′) besides (v,w). We use the Hirota symbol, which takes in our case the following form ′ ′ ′ ′ P (∂v,∂w) F G = P (∂v′ ,∂w′ ) F (v + v ,w+ w ) G(v v ,w w ) . (56) ◦ − − v′=w′=0 The elementary Schur polynomials Sℓ(v) are defined by ∞ ∞ k vkz k e k=1 = Sk(v)z , (57) P k=0 for ℓ 0andS (v) := 0 otherwise. In particular, if we put degree v := i,then ≥ ℓ i S0 =1,S1(v)=v1,Sℓ(v)=vℓ +degreeℓ in v1,...,vℓ−1. (58) We also use the standard notation ∂ 1 ∂ 1 ∂ ∂˜ = , , ,... . v ∂v 2 ∂v 3 ∂v 1 2 3 We first give an identity which will allow us to compute the formal residues which appear in (51) in terms of derivatives of the tau function. Lemma 6.1 For any n Z we have the following formal residue identity ∈ ′′ −1 ′′ ′ −1 ′ ∞ (v′ −v′′)zℓ n dz F (v + z ,w ) G(v z ,w ) e ℓ=0 ℓ ℓ z − 2π√ 1 ∞ P − ∞ ∂ r ∂ ℓ=1(aℓ ∂v + γ=1 bγℓ ∂w ) = Sj−1−n( 2a) Sj(∂˜v) e ℓ γℓ F (v,w) G(v,w), − P P ◦ j≥0 (59) where v′ = v a, v′′ = v + a, w′ = w b ,w′′ = w + b , − i i − i i i i a =(a1,a2,a3,...),bi =(bi1,bi2,bi3,...), for 1 i r. ≤ ≤ 23 Proof The proof is an immediate, but tricky, consequence of Definition (57) of the Schur functions and of the following two properties of the Hirota symbol: ∞ F (v + z−1 ,w) G(v z−1 ,w)= z−jS (∂˜ ) F G, − j v ◦ j=0 ∞ ∂ r ∂ (aℓ + bγℓ ) F (v + a,w + b) G(v a,w b)=e ℓ=0 ∂vℓ γ=1 ∂wγℓ F G. − − P P ◦ ✷ Proposition 6.2 The bilinear equations imply, upon specialization, that the tau functions τmn,with m = n satisfy the following PDE’s expressed in terms of the Hirota symbol: | | | | 2 2 ∂ ˜ τmn ln τmn = Sℓ+2δββ′ (∂tβ )τm,n+eβ −eβ′ τm,n+eβ′ −eβ (60) ∂tβ,ℓ+1∂tβ′,1 ◦ 2 2 ∂ ˜ τmn ln τmn = Sℓ+2δαα′ (∂sα )τm+eα′ −eα,n τm+eα−eα′ ,n(61) ∂sα,ℓ+1∂sα′,1 ◦ 2 2 ∂ ˜ τmn ln τmn = Sℓ(∂tβ )τm+eα,n+eβ τm−eα,n−eβ (62) − ∂sα,1∂tβ,ℓ+1 ◦ 2 2 ∂ ˜ τmn ln τmn. = Sℓ(∂sα )τm−eα,n−eβ τm+eα,n+eβ (63) − ∂tβ,1∂sα,ℓ+1 ◦ Equations (60) resp. (61) for β′ = β (resp. for α′ = α) yield a solution to the ′ ′ KP hierarchy in tβ (resp. in sα), while for β = β and α = α, (60) — (63) yields 2 ∂ τm,n+eβ −eβ′ τm,n+eβ′ −eβ ln τmn = 2 (64) ∂tβ,1∂tβ′,1 τmn 2 ∂ τm+eα′ −eα,nτm+eα−eα′ ,n ln τmn = 2 (65) ∂sα,1∂sα′,1 τmn 2 ∂ τm+eα,n+eβ τm−eα,n−eβ ln τmn = 2 (66) ∂sα,1∂tβ,1 − τmn ∂2 ln τmn ∂ τm,n+eβ −eβ′ ∂tβ,2∂tβ′,1 ln = ∂2 (67) ∂tβ,1 τm,n+eβ′ −eβ ln τmn ∂tβ,1∂tβ′,1 ∂2 ln τmn ∂ τm−eα+eα′ ,n ∂sα,2∂sα′,1 ln = ∂2 (68) ∂sα,1 τm−eα′ +eα,n ln τmn ∂sα,1∂tα′,1 ∂2 ln τmn ∂ τm+eα,n+eβ ∂tβ,2∂sα,1 ln = ∂2 (69) ∂tβ,1 τm−eα,n−eβ ln τmn ∂tβ,1∂sα,1 ∂2 ln τmn ∂ τm−eα,n−eβ ∂sα,2∂tβ,1 ln = ∂2 . (70) ∂sα,1 τm+eα,n+eβ ln τmn ∂sα,1∂tβ,1 24 p+q It leads to the following 2 PDE’s for ln τmn involving not just one sα or tβ, but a few of them ∂2 ∂2 ln τmn ln τmn ∂ ∂tβ,2∂tβ′,1 ∂ ∂tβ′,2∂tβ,1 ∂2 + ∂2 =0, (71) ∂tβ′,1 ln τmn ∂tβ,1 ln τmn ∂tβ,1∂tβ′,1 ∂tβ′,1∂tβ,1 ∂2 ∂2 ln τmn ln τmn ∂ ∂sα,2∂sα′,1 ∂ ∂sα′,2∂sα,1 ∂2 + ∂2 =0, (72) ∂sα′,1 ln τmn ∂sα,1 ln τmn ∂sα,1∂sα′,1 ∂sα′,1∂sα,1 ∂2 ∂2 ln τmn ln τmn ∂ ∂tβ,2∂sα,1 ∂ ∂sα,2∂tβ,1 ∂2 + ∂2 =0. (73) ∂sα,1 ln τmn ∂tβ,1 ln τmn ∂tβ,1∂sα,1 ∂sα,1∂tβ,1 Proof Let us denote for a =(a1,...,aq)andb =(b1,...,bq)byΩ(a, b)the differential operator ∞ q p ∂ ∂ Ω(a, b):= aα′ℓ + bβ′ℓ . (74) ′ ′ ′ ∂sα ℓ ′ ∂tβ ℓ ℓ=1 α=1 β=1 Using Lemma (6.1), rewrite the bilinear identity12 (51): p ∞ σβ (n) Ω(a,b) ( 1) Sn∗ −n +1+k( 2bβ)Sk(∂˜t )e τm∗,n∗+e τm,n−e − β β − β β ◦ β β=1 k=0 q ∞ σα(m) Ω(a,b) ( 1) Sm −m∗ +1+k( 2aα)Sk(∂˜s )e τm∗−e ,n∗ τm+e ,n =0. − − α α − α α ◦ α α=1 k=0 (75) Note that all infinite vectors aα and bβ can be chosen completely arbitrary. We set all components of a and b equal to zero, except bβ,ℓ+1 = B =0(forsome ∗ ∗ ′ fixed β and ℓ), and we set m = m and n n = 2e ′ (for some fixed β ). − − β Then only the first term in (75) survives, the signs σβ(n) are all 1 (see (52)) and, in view of (58), the identity (75) becomes p ∞ ∂ B ∂t ′′ ˜ β,ℓ+1 0= S1+k−2δβ′β′′ ( 2bβ )Sk(∂tβ′′ )e τm,n+eβ′′ −2eβ′ τm,n−eβ′′ ′′ − ◦ β=1 k=0 2 ˜ ∂ 2 = B 2Sℓ+2δββ′ (∂tβ )τm,n+eβ −2eβ′ τm,n−eβ + τm,n−eβ′ τm,n−eβ′ + O(B ). − ◦ ∂t ′ ∂t ◦ β ,1 β,ℓ+1 Expressing that the coefficient of B in this expression must vanish we get (60), upon relabeling n eβ′ n and upon using the following property of the Hirota symbol, valid for f− depending→ on (time-) variables s and t: ∂2 ∂2 F F =2F 2 ln F. (76) ∂t∂s ◦ ∂t∂s 12Recall that in this form of the bilinear identity m∗ = n∗ +1and m = n 1. | | | | | | | |− 25 (61) follows from (60) by duality, using P ( ∂ ) F G = P (∂ ) G F .Inorder − s ◦ s ◦ to obtain (62) we consider again (75), with bβ,ℓ+1 = B = 0 and all other ∗ ∗ components of a and b equal to zero, but we set now n = n and m m = 2eα. Then (75) becomes − − ∞ B ∂ 0= S ( 2b )S (∂˜ ) e ∂tβ,ℓ+1 τ τ k+1 − β k β m+2eα,n+eβ ◦ m,n−eβ k=0 ∞ B ∂ S (0)S (∂˜ ) e ∂tβ,ℓ+1 τ τ − k−1 k sα m+eα,n ◦ m+eα,n k=0 ∂2 = B 2S (∂˜ )τ τ + τ τ + O(B2). − ℓ tβ m+2eα,n+eβ ◦ m,n−eβ ∂s ∂t m+eα,n ◦ m+eα,n α,1 β,ℓ+1 The nullity of the coefficient of B in this expression, rewritten by using (76), leads at once to (62), upon doing the relabeling m + eα m. From it, (63) follows by duality. Equations (64) — (66) follow from (60)→ — (62) by setting β′ = β, α′ = α and ℓ = 0. Equations (67) — (70) follow from (60) — (63) by setting β′ = β, α′ = α and forming in each equation the ratio of the cases ℓ =0 and ℓ = 1, and using the following property of the Hirota symbol, valid for F and G depending on a (time-) variable t: ∂ ∂ F F G = FG ln . ∂t ◦ ∂t G Equations (71) — (73) are just respectively the compatibility equations between (67) and (67)β↔β′ , between (68) and (68)α↔α′ , and between between (69) and (70). ✷ ′′ ′′ ∗(β ) ∗(β ) ∗ ∗ Corollary 6.3 The tau functions τmn and the polynomials Pn∗m∗ (x)=Pn∗m∗ (x,s ,t ), ′ ′ (β ) (β ) ∗ Qmn (y)=Qmn (y,s,t) appearing in Y and Y respectively, satisfy the following ∗ ∗ ∗ ∗ n1+n2+···+n ′ +n +n +···+n ′′ 4 formal series identities (δ ′ ′′ (n, n )=( 1) β 1 2 β ): β β − ′′ ∗ ∗ ∗ ∗(β ) (β′) 0=δ ′ ′′ (n, n )τ (s, t)τ ∗ ∗ (s ,t ) P ∗ ∗ (x) Q (y) β β mn m n n m mn m∗ m → ∗ ˛ ′ ′′ ! "˛ n n eβ + eβˆ,n n + eβ eβˆ ˛ ˛ ∗→ − → ∗ − ˛ sα sα,tβ tβ bβ,t tβ + bβ ˛ β ˛ → → − → ˛ p 2S (∂˜ )τ τ ˛ ∞ ℓ+2δββˆ tβ m,n+eβ −eβˆ m,n−eβ +eβˆ − 2 ◦ 2 = bβ,ℓ+1 ∂ + O(b ), + τmn τmn β=1 ℓ=0 ∂tˆ ∂tβ,ℓ+1 ◦ β,1 ′′ ∗ ∗ ∗ ∗(β ) (β′) 0=δβ′β′′ (n, n )τmn(s, t)τm∗n∗ (s ,t ) Pn∗m∗ (x) Qmn (y) ∗ m m eαˆ,m m + eαˆ → − ∗ → ! "˛ ′ ′′ ˛ n n eβ ,n n + eβ ˛ → − ∗→ ∗ ˛ s s a ,s s + a ,t t ˛ α α α α α α β β ˛ → − → → ˛ q ∞ ˜ 2Sℓ+2δααˆ (∂sα )τm+eαˆ −eα,n τm−eαˆ +eα,n˛ 2 ◦ 2 = aα,ℓ+1 ∂ + O(a ), τmn τmn α=1 ℓ=0 −∂sα,ˆ 1∂sα,ℓ+1 ◦ 26 ′′ ∗ ∗ ∗ ∗(β ) (β′) δβ′β′′ (n, n )τmn(s, t)τm∗n∗ (s ,t ) Pn∗m∗ (x) Qmn (y) ∗ m m eαˆ,m m + eαˆ → − ∗ → ! "˛ ′ ′′ ˛ n n eβ ,n n + eβ ˛ ∗→ − → ∗ ˛ s s ,t t b ,t t + b ˛ α α β β β β β β ˛ → → − → p ∞ ˛ ˛ = 2 b S (∂˜ )τ τ + O(b2) − β,ℓ+1 ℓ tβ m+eαˆ ,n+eβ ◦ m−eαˆ ,n−eβ β=1 ℓ=0 and p ∞ 2 ∂ 2 = bβ,ℓ+1 τmn τmn + O(b ), ∂sα,ˆ 1∂tβ,ℓ+1 ◦ β=1 ℓ=0 ′′ ∗ ∗ ∗ ∗(β ) (β′) δ ′ ′′ (n, n )τ (s, t)τ ∗ ∗ (s ,t ) P ∗ ∗ (x) Q (y) β β mn m n n m mn m∗ m ˛ → ∗ ! "˛ n n eβ′ + e ˆ,n n + eβ′′ e ˆ ˛ β β ˛ → − ∗ → ∗ − ˛ sα sα aα,sα sα + aα,tβ tβ ˛ ˛ → − → → q ∞ 2 ˛ ∂ ˛ = a τ τ + O(a2), α,ℓ+1 ∂t ∂s mn ◦ mn α=1 β,ˆ 1 α,ℓ+1 ℓ=0 and q ∞ 2 = 2 aα,ℓ+1Sℓ(∂˜s )τm−e ,n−e τm+e ,n+e + O(a ). − α α βˆ ◦ α β α=1 ℓ=0 Proof ¿From the proof of Theorem 5.1 and (59) it follows that ′′ ∗ ∗ ∗ ∗(β ) (β′) δβ′β′′ (n, n )τmn(s, t)τm∗n∗ (s ,t ) Pn∗m∗ (x) Qmn (y) ∗ ∗ n n eβ′ ,n n + eβ′′ → − ∗→ ! "˛ ˛ sα sα aα,sα sα + aα ˛ → − ∗ → ˛ t t b ,t t + b ˛ β β β β β β ˛ → − → p ∞ ˛ ˛ σβ (n) Ω(a,b) = ( 1) Sn∗ −n +1+k( 2bβ)Sk(∂˜t )e τm∗,n∗+e τm,n−e − β β − β β ◦ β β=1 k=0 and q ∞ σα(m) Ω(a,b) = ( 1) S ∗ ( 2a )S (∂˜ )e τ ∗ ∗ τ − mα−mα+1+k − α k sα m −eα,n ◦ m+eα,n α=1 k=0 and so if we just follow the 4 specializations leading to (60) – (63), in order, we find the 4 equations of the corollary, in their given order. ✷ 7Examples 7.1 Biorthogonal polynomials (p = q =1) Given the (not necessarily symmetric) inner product with regard to the weight ρ(x, y)onR2, f(x) g(y) := f(x)g(y)ρ(x, y) dx dy | R2 27 and the deformed weight ∞ k k (tky −sk x ) ρt,s(x, y)=e 1 ρ(x, y). P Setting p = q =1,m= m1,n= n1,withm = n, implies that the indices m, n in τmn can be replaced by one single index; namely, set τn := τmn,where ∞ k ∞ k i − sk x j tk y τn(t, s)=det x e 1 y e 1 . P P 0≤i,j≤n−1 ! " (1) Moreover, set ψ1 = ϕ1 = 1 and define the monic polynomials pn (y):= (1) (2) (2) −1 pn (t, s; y)andpn (x):=pn (t, s; x)(withhn−1 the leading coefficient of ∗(1,1) Pnm (x)) by p(1)(y):=Q(1,1)(y)=yn + n mn ··· h−1 p(2) (x):=P ∗(1,1)(x)=h−1 xn−1 + n−1 n−1 nm n−1 ··· The orthogonality conditions (4) and (6) imply ∞ k ∞ k i − 1 sk x (1) 1 tk y x e pn (y)e =0for0 i n 1 P P ≤ ≤ − ! " ∞ k ∞ k −1 (2) − 1 sk x j 1 tk y hn pn (x)e y e =0for0 j n 1 P P ≤ ≤ − ! " =1forj = n. for all n 0, from which the bi-orthogonality can be deduced13 ≥ (2) (1) pn (x)pm (y)ρt,s(x, y)dxdy = δnmhn. R2 ¿From (3), (5), (7) and (8) and from hn = τn+1/τn, it follows that −1 n τn(t [z ],s) (1) z − = pn (z) τn(t, s) −1 n τn(t, s +[z ]) (2) z = pn (z) τn(t, s) −1 (2) −n−1 τn+1(t +[z ],s) pn (x) z = ρt,s(x, y)dxdy τ (t, s) R2 z y n − −1 (1) −n−1 τn+1(t, s [z ]) pn (y) z − = ρt,s(x, y)dxdy. (77) τ (t, s) R2 z x n − and from (51), the bilinear identity becomes −1 ′ −1 ′ ∞(t −t′ )zi n−m−2 τ (t [z ],s) τ (t +[z ],s) e 1 i i z dz n−1 − m+1 z=∞ P −1 ′ ′ −1 ∞(s −s′ )zi m−n = τ (t, s [z ]) τ (t ,s +[z ]) e 1 i i z dz, n − m z=∞ P 13 It turns out that hn = τn+1(t, s)/τn(t, s). 28 which characterizes the τ-functions for the 2-component KP hierarchy.Equa- tions (77) and the bilinear identity were obtained in [1]. Indicating the depen- dence on t, s in the polynomials, the following inner product can be computed in two different ways, leading to14 (2) ∞ k ′ k ′ ′ ′ ′ (1) 1 (tky −skx ) τn(t,s)τn+1(t ,s ) dxdy pn+1(t ,s; x)pn (t, s; y)e ρ(x,y) t t a R2 P ′→ ′− t t +a ′→ s = s ∞ = 2a S (∂˜ )τ τ + O(a2) − j+1 j t n+2 ◦ n j=0 ∞ 2 ∂ 2 = ak τn+1 τn+1 + O(a ) . (78) ∂tk∂s1 ◦ k=1 Identifying the coefficients of aj+1 in both expressions and shifting n n 1 yield a first identity; then redoing the calculation above for s s b, s′ → s′−+ b and t′ = t leads to a second one. All in all we find → − → 2 ˜ 2 ∂ Sj(∂t)τn+1 τn−1 = τn ln τn, ◦ − ∂s1∂tj+1 2 ˜ 2 ∂ Sj(∂s)τn−1 τn+1 = τn ln τn. ◦ − ∂t1∂sj+1 Specializing the identity (73) leads to an identity, which can be expressed as a sum of two Wronskians15 and which involves a single tau function: ∂2 ln τ ∂2 ln τ ∂2 ln τ ∂2 ln τ n , n + n , n =0. (79) ∂t1∂s2 ∂t1∂s1 ∂s1∂t2 ∂t1∂s1 &t1 &s1 The computation (2.2) was at the origin of the crucial argument (Theorem 5.1) in this paper. It illustrates in a simple way what is being done in this paper. These equations are used, when computing the PDE for the Dyson, Airy and Sine processes ([3]). 7.2 Orthogonal polynomials Given a weight ρ(z)onR, the symmetric inner product f(x) g(x) = f(x)g(x)ρ(x) dx, | R ∞ k tkz and the formal deformation by means of an exponential ρt(x):=ρ(x)e 1 . This is a special case of the previous example, where the deformationP only 14This integral is = 0, unless t = t′ 15 ∂f ∂g in terms of the Wronskian f, g t = g f . { } ∂t − ∂t 29 depends on t s;thust s can be replaced by t.Thenτ (t) is the determinant − − n of the moment matrix depending on t =(t1,t2,...), ∞ k i+j tk z τn(t):=det z e 1 ρt(z)dz . R P 0≤i,j≤n−1 Then, from (77), it follows at once that the orthogonal polynomials pn(x):= pn(t; x)aregivenby −1 n τn(t [z ]) z − = pn(z) τn(t) −1 −n−1 τn+1(t +[z ]) pn(x) z = ρt(x)dx. τ (t) R z x n − Moreover, the integral below can be computed in two different ways: on the one hand, it is automatically zero, because pn(z) is perpendicular to any polynomial of lower degree; on the other hand, for t and t′ close to each other, the integral can also be developed, using the technique of Proposition 6.2, in t′ t =2y, yielding the following formula − ′ ′ 0=τn(t)τn(t ) pn(t; z)pn−1(t ,z)ρt(z)dz R t t y ′→ − t t + y → ∞ ∂2 = y 2S ∂˜ τ τ + O(y2), k ∂t ∂t − k+1 t n ◦ n 3 1 k showing that τn(t) satisfies the KP hierarchy. 7.3 Orthogonal polynomials on the circle Consider the inner product on the circle between analytic functions on S1: dz f(z) g(z) = f(z−1)g(z) | 1 2π√ 1z S − and the determinant of moment matrices ∞ i ∞ i k − siz ℓ tiz τn(t, s):=detz e 1 z e 1 P P 0≤k,ℓ≤n−1 ! " dz −k+ℓ ∞(t zi−s z−i) =det z e 1 i i 1 2π√ 1z S − P 0≤k,ℓ≤n−1 Then it follows that −1 n τn(t [z ],s) (1) z − = pn (z) τn(t, s) −1 n τn(t, s +[z ]) (2) z = pn (z) τn(t, s) 30 −1 (2) −1 −n−1 τn+1(t +[z ],s) du pn (u ) ∞(t ui−s u−i) z = e 1 i i τn(t, s) 1 2π√ 1u z u S − − P −1 (1) τn+1(t, s [z ]) du pn (u) ∞ i −i −n−1 1 (tiu −siu ) z − = −1 e , τ (t, s) 1 2πiu z u n S − P (1) (2) −1 with pn (z)andpm (z ) monic orthogonal polynomials on the circle: dz (1) (2) −1 τn+1 pn (z)pm (z )=δnmhn, with hn = . 1 2πiz τ S n The nature of the inner product implies some extra-relationship between the orthogonal polynomials p(1) (z) zp(1)(z)=p(1) (0)znp(2)(z−1) n+1 − n n+1 n p(2) (z) zp(2)(z)=p(2) (0)znp(1)(z−1). n+1 − n n+1 n leading to (in the notation of footnote 8) h 2 h h n 1 n+1 1 m+1 h − h − h m+1 n m 1 ˜ ˜ = 2 2 Sn−m(∂t)τm+2 τn . Sn−m( ∂s)τm+2 τn . τm+2τn ◦ − ◦ In particular, for m = n 1, − h h ∂ ∂ 1 n+1 1 n = ln h ln h . − h − h −∂t n ∂s n n n−1 1 1 7.4 Non-intersecting Brownian motions Consider N non-intersecting Brownian motions x1(t),...,xN (t)inR,leaving from distinct points α1 <...<αN and forced to end up at distinct points β1 <...<βN . From the Karlin-McGregor formula (see [12]), the probability that all xi(t)belongtoE R can be expressed in terms of the Gaussian 2 ⊂ p(t, x, y)=e−(x−y) /2t/√2πt, as follows (0 Pβ (all x (t) E) α i ∈ Pβ (x1(0),...,xN (0)) = (α1,...,αN ) := α all xi(t) E ∈ (x1(1),...,xN (1)) = (β1,...,βN ) N 1 = det[p(t, αi,xj )]1≤i,j≤N det[p(1 t, xi,βj)]1≤i,j≤N dxi Z N − N E i=1 N 2 1 −xi αixj βixj 2t(1−t) t 1−t = ′ e dxi det e det e (80) Z N 1≤i,j≤N N E i=1 1≤i,j≤N ' ( ) * 31 The limiting case where several points α and β coincide has been the object of many interesting studies. It is obtained by taking appropriate limits of the formulae above. Just to fix the notation, consider m1 m2 mq (α1,...,αN )=(a1,a1,...,a1, a2,a2,...,a2,...,aq,aq,...,aq)