Toeplitz minors and specializations of skew Schur polynomials
aa r X i v : . [ m a t h . C O ] N ov TOEPLITZ MINORS AND SPECIALIZATIONS OF SKEW SCHURPOLYNOMIALS
DAVID GARCIA-GARCIA AND MIGUEL TIERZ
Abstract.
We express minors of Toeplitz matrices of finite and large dimension, includingthe case of symbols with Fisher-Hartwig singularities, in terms of specializations of symmetricfunctions. By comparing the resulting expressions with the inverses of some Toeplitz matrices,we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skewSchur polynomials.
Keywords:
Toeplitz minor, skew Schur polynomial, Fisher-Hartwig singularity, Toeplitz inverse. Introduction
Let f ( e iθ ) = P k ∈ Z d k e ikθ be an integrable function on the unit circle. The Toeplitz matrixgenerated by f is the matrix T ( f ) = ( d j − k ) j,k ≥ . That is, T ( f ) is an infinite matrix, constant along its diagonals, which entries are the Fouriercoefficients of the function f . We denote by T N ( f ) its principal submatrix of order N , and D N ( f ) = det T N ( f ) . This determinant has the following integral representation D N ( f ) = Z U ( N ) f ( M ) dM = 1 N ! 1(2 π ) N Z π ... Z π N Y j =1 f ( e iθ j ) Y ≤ j Symmetric functions. Let us recall some basic results involving symmetric functionsthat can be found in [29, 35], for example. We denote z = e iθ in the following, and treat z as aformal variable. A partition λ = ( λ , . . . , λ l ) is a finite and non-increasing sequence of positiveintegers. The number of nonzero entries is called the length of the partition and is denoted by l ( λ ), and the sum | λ | = λ + · · · + λ l ( λ ) is called the weight of the partition. The entry λ j isunderstood to be zero whenever the index j is greater than the length of the partition. Thenotation ( a b ) stands for the partition with exactly b nonzero entries, all equal to a . A partitioncan be represented as a Young diagram, by placing λ j left-justified boxes in the j -th row of thediagram. The conjugate partition λ ′ is then obtained as the partition which diagram has asrows the columns of the diagram of λ (see figure 2.1 for an example). The following proceduredescribes how to obtain the Toeplitz minor D λ,µN ( f ) from the underlying Toeplitz matrix (we We abuse notation here; we assume it is clear when the expression f ( M ) should be read as Q j f ( e iθ j ) (i.e.when f is a function on the unit circle) and when it should be read as f ( e iθ , . . . , e iθ N ) (i.e. when f is a symmetricfunction in several variables). See [29] and section 2.1 for definitions of Schur polynomials. OEPLITZ MINORS AND SPECIALIZATIONS OF SKEW SCHUR POLYNOMIALS 3 Figure 1. The partition (3 , , 2) and its conjugate (3 , , λ and µ is less than or equal to N , thesize of the matrix under consideration): • Strike the first | λ − µ | columns or rows of T N +max { λ ,µ } ( f ), depending on whether λ − µ is greater or smaller than zero, respectively. • Keep the first row of the matrix, and strike the next λ − λ rows. Keep the next row,and strike the next λ − λ rows. Continue until striking λ l ( λ ) − λ l ( λ )+1 = λ l ( λ ) rows. • Repeat the previous step on the columns of the matrix with µ in place of λ . The resultingmatrix is precisely T λ,µN ( f ).If x = ( x , x , ... ) is a set of variables, the power-sum symmetric polynomials p k are definedas p k ( x ) = x k + x k + . . . for every k ≥ 1, and p ( x ) = 1. They are related to the elementarysymmetric polynomials e k ( x ) and the complete homogeneous polynomials h k ( x ) by the formulasexp ∞ X k =1 p k ( x ) k z k ! = ∞ X k =0 h k ( x ) z k = ∞ Y j =1 − x j z = H ( x ; z ) , exp ∞ X k =1 ( − k +1 p k ( x ) k z k ! = ∞ X k =0 e k ( x ) z k = ∞ Y j =1 (1 + x j z ) = E ( x ; z ) . (2)We also set p k ( x ) = h k ( x ) = e k ( x ) = 0 for negative k . The families { h k } k ≥ and { e k } k ≥ consistof algebraically independent functions. Hence, we will see H and E as arbitrary functions onthe unit circle depending on the parameters x , and we will use indistinctly their infinite productexpression. The classical Jacobi-Trudi identities express Schur polynomials as Toeplitz minorsgenerated by the above functions s µ ( x ) = det (cid:0) h j − k + µ k ) ( x ) (cid:1) Nj,k =1 = D ∅ ,µN ( H ( x ; z )) ,s µ ′ ( x ) = det (cid:0) e j − k + µ k ) ( x ) (cid:1) Nj,k =1 = D ∅ ,µN ( E ( x ; z )) , where l ( µ ) , l ( µ ′ ) ≤ N , respectively, and ∅ denotes the empty partition. More generally, skewSchur polynomials can be expressed as the minors s µ/λ ( x ) = D λ,µN ( H ( x ; z )) , s ( µ/λ ) ′ ( x ) = D λ,µN ( E ( x ; z )) , (3)where l ( µ ) , l ( µ ′ ) ≤ N respectively. A skew Schur polynomial vanishes if λ * µ , which canbe seen as a consequence of its Toeplitz minor representation and the fact that the Toeplitzmatrices above are triangular. A central result in the theory of symmetric functions is theCauchy identity, and its dual form X ν s ν ( x ) s ν ( y ) = ∞ Y j =1 ∞ Y k =1 − x j y k , X ν s ν ( x ) s ν ′ ( y ) = ∞ Y j =1 ∞ Y k =1 (1 + x j y k ) , where y = ( y , y , . . . ) is another set of variables and the sums run over all partitions ν . DAVID GARCIA-GARCIA AND MIGUEL TIERZ Gessel [23] obtained the following expression for the Toeplitz determinant generated by thefunction f ( z ) = H ( y ; z − ) H ( x ; z ) D N ∞ Y k =1 − y k z − ∞ Y j =1 − x j z = X l ( ν ) ≤ N s ν ( y ) s ν ( x ) , (4)where the sum runs over all partitions ν of length l ( ν ) ≤ N . If one of the sets of variables x or y is finite, say y = ( y , . . . , y d ), comparing the right hand side above with the sum in Cauchyidentity and recalling that the Schur polynomial s ν ( y , ..., y d ) vanishes if l ( ν ) > d one obtains awell known identity of Baxter [7] D N d Y k =1 − y k z − ∞ Y j =1 − x j z = d Y k =1 ∞ Y j =1 − x j y k , (5)valid when N ≥ d . Note that the right hand side above is independent of N . An analogousidentity follows if the factor H ( x ; z ) is replaced by E ( x ; z ), using the dual Cauchy identityinstead. However, no such identity is available for Toeplitz determinants generated by symbolsof the type E ( y ; z − ) E ( x ; z ); this will be relevant later.2.2. Toeplitz determinants and minors generated by smooth symbols. We record nowprecise statements of the strong Szeg˝o limit theorem and of its generalization to Toeplitz minors. Theorem (Szeg˝o) . Let f ( e iθ ) = P k ∈ Z d k e ikθ be a function on the unit circle, and suppose itcan be expressed as f ( e iθ ) = exp( P k ∈ Z c k e ikθ ) , where the coefficients c k verify X k ∈ Z | c k | < ∞ , X k ∈ Z | k || c k | < ∞ . Then, as N → ∞ , D N ( f ) ∼ exp N c + ∞ X k =1 kc k c − k ! . A function f satisfying the hypotheses of this theorem is continuous, nonzero, and has windingnumber zero [10]. Functions with Fisher-Hartwig singularities need not verify these properties(see section 4.2). Under these same conditions, the following theorem holds. Theorem (Bump, Diaconis [13]) . Let f verify the hypotheses in the previous theorem, andsuppose λ and µ are partitions of weights n and m respectively. Then, as N → ∞ D λ,µN ( f ) ∼ D N ( f ) X φ ⊢ n X ψ ⊢ m χ λφ χ µψ z − φ z − ψ ∆( f, φ, ψ ) , (6) where the sum runs over all the partitions φ of n and ψ of m , the terms z φ , z ψ are the ordersof the centralizers of the equivalence classes of the symmetric groups S n , S m indexed by φ and ψ respectively, the functions χ λ , χ µ are the characters associated to the irreducible representationsof S n and S m indexed by λ and µ respectively, and ∆( f, φ, ψ ) = ∞ Y k =1 ( k n k c n k − m k − k m k ! L ( n k − m k ) m k ( − kc k c − k ) , if n k ≥ m k k m k c m k − n k k n k ! L ( m k − n k ) n k ( − kc k c − k ) , if n k ≤ m k . Above, the coefficients n k , m k correspond to the partitions φ = (1 n n . . . ) and ψ = (1 m m . . . ) in their frequency notation, and L ( a ) n are the Laguerre polynomials [36] . OEPLITZ MINORS AND SPECIALIZATIONS OF SKEW SCHUR POLYNOMIALS 5 λ µ lim N →∞ D λ,µN ( f ) /D N ( f ) λ µ lim N →∞ D λ,µN ( f ) /D N ( f ) ∅ c ∅ c + c ∅ c − c ∅ c + c c + c ∅ c − c c + c ∅ c − c c + c λ µ lim N →∞ D λ,µN ( f ) /D N ( f ) c − c + c − c − c − c − c − c + c − c + 1 c − c + c + c − c c + c + c − c Table 1. Some values of the formula (6).Note that the product in the factor ∆( f, φ, ψ ) is actually finite, since only a finite number of n k ’s and m k ’s are distinct from zero for each pair φ, ψ . As mentioned before, we see that in the N → ∞ limit the Toeplitz minor generated by a regular symbol factors as the correspondingToeplitz determinant times a sum depending only on f and the partitions λ, µ (and not on N ).The formula (6) can be implemented in MatLab for example, leading to quick evaluations forvalues of, say, | λ | , | µ | = 15. Table 1 shows some of these values for particular choices of λ and µ .3. Toeplitz minors generated by arbitrary symbols We now show that the asymptotic formula (6) is valid for Toeplitz minors generated byarbitrary symbols. Theorem 1. Let f ( e iθ ) = P k ∈ Z d k e ikθ be an integrable function on the unit circle, possibly withFisher-Hartwig singularities, and let λ and µ be partitions. Then, as N → ∞ , D λ,µN ( f ) ∼ D N ( f ) X ν s λ/ν ( y ) s µ/ν ( x ) , (7) where the variables y, x are such that f ( z ) = H ( y ; z − ) H ( x ; z ) , and the sum runs over allpartitions ν contained in λ and µ . An analogous result holds if one or both of the factors H in the factorization of f are replacedby factors of the type E , conjugating the corresponding indexing partitions in the sum on theright hand side of (7). Proof. We will use the following lemma, that has an elementary proof. Lemma. Let ν be a partition verifying ν ⊂ ( d N ) , and consider the partition ←− ν d = ( d − ν N , . . . , d − ν ) which is obtained by rotating 180 o the complement of ν in the diagram of therectangular partition ( d N ) . Then the Schur polynomial s ν verifies s ν ( x − , . . . , x − N ) = s ←− ν d ( x , . . . , x N ) N Y j =1 x − dj . If R, S are two strictly increasing sequences of natural numbers, we denote by det R,S M theminor of the matrix M obtained by taking the rows and columns of M indexed by R and S ,respectively. Using the above lemma with d = max { λ , µ } we see that D λ,µN ( f ) = Z U ( N ) s λ ( M ) s µ ( M ) f ( M ) dM = Z U ( N ) s ←− λ d ( M ) s ←− µ d ( M ) f ( M ) dM = det R,S T ( f ) , DAVID GARCIA-GARCIA AND MIGUEL TIERZ where the sequences R, S are given by R = ( r j ) Nj =1 = ( j + µ N +1 − j ) Nj =1 and S = ( s k ) Nk =1 =( k + λ N +1 − k ) Nk =1 . Since the Toeplitz matrices generated by each of the factors of f verify T ( f ( z )) = T ( H ( y ; z − )) T ( H ( x ; z )), the use of Cauchy-Binet formula gives det R,S T ( f ( z )) = X T det R,T T ( H ( y ; z − )) det T,S T ( H ( x ; z )) , where the summation is over all the strictly increasing sequences T = ( t , . . . , t N ) of length N ofpositive integers. There is a correspondence between such sequences and partitions ν of length l ( ν ) ≤ N , given by ν N +1 − j = t j − j , for j = 1 , ..., N . Thus, for each T we havedet T,S T ( H ( x ; z )) = det( h t j − s k ( x )) Nj,k =1 = det( h j + ν N +1 − j − ( k + λ N +1 − k ) ( x )) Nj,k =1 . Reversing the order of its rows and columns we see that the last determinant above is D λ,νN ( H ( x ; z )).According to (3) this is precisely the skew Schur polynomial s ν/λ ( x ), and an analogous derivationyields det R,T T ( H ( y ; z − )) = s ν/µ ( y ). We thus obtain D λ,µN ( f ) = X l ( ν ) ≤ N s ν/µ ( y ) s ν/λ ( x ) . (8)This gives the desired conclusion, after using the following identity (Ex. I.5.26 in [29]) X ν s ν/µ ( y ) s ν/λ ( x ) = X κ s κ ( y ) s κ ( x ) X ν s λ/ν ( y ) s µ/ν ( x ) , where the sums run over all partitions, and Gessel’s identity (4). (cid:3) Note that the factorization of the symbol f as the product f ( z ) = H ( y ; z − ) H ( x ; z ) is notessential here. As stated previously, the functions { h k ( x ) } k ≥ are algebraically independent, andhence one can specialize them to any values and “forget” about the variables x . This idea wasalready used in [37] to show that Szeg˝o’s theorem is formally equivalent to Cauchy’s identity.Moreover, if the function f is integrable (as are functions with Fisher-Hartwig singularities), thesum in the right hand side of (7) is finite, due to identity (3). Hence, in the N → ∞ limit,a Toeplitz minor factorizes as the corresponding Toeplitz determinant times a “combinatorial”sum, regardless of the regularity (or lack thereof) of the symbol f . Moreover, the sum in theright hand side of (7) is an equivalent form of the sum in the right hand side of (6), as was notedin [15]. Therefore, the asymptotics of a Toeplitz minor generated by a symbol with or withoutFisher-Hartwig singularities can be recovered from formulas (6) or (7) and the results in [16].Exact formulas are available when the function f can be obtained as a specialization with afinite number of variables. There are two possibilities: • Case 1: There is a factor of the type H specialized to a finite set of variables. Suppose f is of the form f ( z ) = H ( y , . . . , y d ; z − ) H ( x ; z ). Then, in the same fashion as in Baxter’sidentity (5), the corresponding Toeplitz determinant stabilizes and we obtain the formula D λ,µN d Y k =1 − y k z − ∞ Y j =1 − x j z = d Y k =1 ∞ Y j =1 − x j y k X ν s λ/ν ( y ) s µ/ν ( x ) , that holds for every N ≥ d . An analogous result holds for symbols of the type f ( z ) = H ( y , . . . , y d ; z − ) E ( x ; z ). We are actually using the infinite dimensional generalization of the Cauchy-Binet formula that appears in[37]. OEPLITZ MINORS AND SPECIALIZATIONS OF SKEW SCHUR POLYNOMIALS 7 • Case 2: There is a factor of the type E specialized to a finite set of variables. We assume,without loss of generality, that f is of the form f ( z ) = E ( y , . . . , y d ; z − ) E ( x ; z ). Asmentioned above, no N -independent formula is available for these symbols. However, itfollows from (3) that s (( d N )+ µ/λ ) ′ ( y − , . . . , y − d , x ) = D λ,µN d Y k =1 y − k d Y k =1 (1 + y k z − ) ∞ Y j =1 (1 + x j z ) == d Y k =1 y − Nk D λ,µN ( E ( y , . . . , y d ; z − ) E ( x ; z )) , (9)and we see that in this case the Toeplitz minor can be expressed essentially as thespecialization of a single skew Schur polynomial, indexed by the shape µ ′ dNλ ′ A similar identity was obtained in [2]. Comparing with the analogous of equation (8)for this symbol we see that (9) coincides with d Y k =1 y − Nk X ν ⊂ ( N d ) s ν/µ ′ ( y , ..., y d ) s ν/λ ′ ( x ) , where the (finite) sum runs over all partitions ν satisfying l ( ν ) ≤ N and ν ≤ d .4. Inverses of Toeplitz matrices and skew Schur polynomials The usual formula for the inversion of a matrix in terms of its cofactors reads as follows forthe case of Toeplitz matrices (cid:0) T − N ( f ) (cid:1) j,k = ( − j + k D (1 k − ) , (1 j − ) N − ( f ) D N ( f ) . (10)Hence, whenever the inverse of a Toeplitz matrix is known explicitely, formula (10) yields explicitevaluations of the formulas appearing in section 3. In particular, if the function f is of the form f ( z ) = E ( y , ..., y d ; z − ) E ( x ; z ), the Toeplitz minor in the right hand side above has severalexpressions: in terms of the inverse of the corresponding Toeplitz matrix D (1 k ) , (1 j ) N ( f ) = ( − j + k D N +1 ( f )( T − N +1 ( f )) j +1 ,k +1 , (11)as a specialization of a skew Schur polynomial D (1 k ) , (1 j ) N ( f ) = s ( N,...,N | {z } d ,j ) / ( k ) ( y − , . . . , y − d , x ) d Y r =1 y Nr , (12)and as the multiple integral D (1 k ) , (1 j ) N ( f ) = (13)1 N ! 1(2 π ) N Z π ... Z π e k ( e − iθ , ..., e − iθ N ) e j ( e iθ , ..., e iθ N ) N Y j =1 f ( e iθ j ) Y ≤ j A simple example is given by the Toeplitz matrix generatedby the function f ( z ) = E ( y ; z − ) E ( x ; z ), where x and y are single (nonzero) variables T N ( E ( y ; z − ) E ( x ; z )) = xy yx xy . . .. . . . . . . (15)The inverse of a tridiagonal Toeplitz matrix has an expression in terms of Chebyshev polynomialsof the second kind [36]. These are defined by the recurrence relation ( U n +1 ( z ) = 2 zU n ( z ) − U n − ( z ) ( n ≥ ,U ( z ) = 1 , U ( z ) = 2 z. The determinant of the matrix (15) is then given by [21] D N ( E ( y ; z − ) E ( x ; z )) = ( xy ) N +1 − xy − xy ) N/ U N ( c ) (cid:18) c = 1 + xy √ xy (cid:19) , (16)and its inverse by( T − N ( E ( y ; z − ) E ( x ; z ))) j,k = ( − j + k y k − j ( xy ) ( k − j +1) / U j − ( c ) U N − k ( c ) U N ( c ) ( j ≤ k ) , ( − j + k x j − k ( xy ) ( j − k +1) / U k − ( c ) U N − j ( c ) U N ( c ) ( j > k ) . Inserting these expressions in equation (12) we obtain the following expression for an arbitraryskew Schur polynomial indexed by a shape of at most two rows and specialized to two variables s ( N,j ) / ( k ) ( x, y − ) = ( xy − ) ( N + j − k ) / U min ( j,k ) ( c ) U N − max ( j,k ) ( c ) == 1 x k y N + j − k min ( j,k ) X r =0 ( xy ) r N X r =max ( j,k ) ( xy ) r , for j, k = 0 , ..., N and N ≥ 1. It is well known that a Schur polynomial specialized to twovariables is equal to a Chebyshev polynomial [24]. We also obtain from formula (14) that as N → ∞ s ( N,j ) / ( k ) ( x, y − ) ∼ x j y k ( xy ) − min ( j,k ) − − xy ) − − y − N D N ( E ( y ; z − ) E ( x ; z )) , where the determinant is given by any of the two expressions appearing in equation (16). OEPLITZ MINORS AND SPECIALIZATIONS OF SKEW SCHUR POLYNOMIALS 9 The pure Fisher-Hartwig singularity. The so-called pure Fisher-Hartwig singularityis the function | − e iθ | α e iβ ( θ − π ) (0 < θ < π ) , (17)where the parameters α, β satisfy Re( α ) > − / β ∈ C . The factor | − e iθ | α may havea zero, a pole, or an oscillatory singularity at the point z = 1, while the factor e iβ ( θ − π ) has ajump if β is not an integer. Thus, depending on the different values of the parameters α and β , the symbol above may violate the regularity conditions in Szeg˝o’s theorem. It will be moreconvenient to work with the equivalent factorization [10](1 − e iθ ) γ (1 − e − iθ ) δ . This function coincides with (17) if γ = α + β and δ = α − β ; we will assume in the followingthat the parameters γ and δ are positive integers. We can then express this function as thespecialization f ( z ) = ϕ γ,δ ( z ) = E (1 , ..., | {z } δ ; z − ) E (1 , ..., | {z } γ ; z ) . (18)Functions with general Fisher-Hartwig singularities are obtained as the product of a functionverifying the regularity conditions in Szeg˝o’s theorem times a finite number of translated puresingularities of the form ϕ γ r ,δ r ( e i ( θ − θ r ) ). Each of this factors has a singularity with parameters γ r , δ r at the point e iθ r .The inverse of the Toeplitz matrix generated by the pure FH singularity can be computed bymeans of the Duduchava-Roch formula [18, 34, 9] T ((1 − z ) γ ) M γ + δ T ((1 − z − ) δ ) = Γ( γ + 1)Γ( δ + 1)Γ( γ + δ + 1) M δ T ( ϕ γ,δ ) M γ , where M a is the diagonal matrix with entries ( M a ) k,k = (cid:0) a + k − k − (cid:1) , for k ≥ 1. B¨ottcher andSilbermann [11] used this formula to give an explicit expression for the determinant of theToeplitz matrix generated by the pure FH singularity D N ( ϕ γ,δ ) = G ( N + 1) G ( γ + δ + N + 1) G ( γ + δ + 1) G ( γ + 1) G ( γ + N + 1) G ( δ + 1) G ( δ + N + 1) , (19)where G is the Barnes function [4]. Also the inverse of the corresponding Toeplitz matrix canbe computed explicitely by means of this formula [9]( T − N ( ϕ γ,δ )) j,k = ( − j + k Γ( γ + j )Γ( δ + k )Γ( j )Γ( k ) N X r =max ( j,k ) Γ( r )Γ( γ + δ + r ) (cid:18) γ + r − k − r − k (cid:19)(cid:18) δ + r − j − r − j (cid:19) . Inserting these expressions in equation (12) we obtain s ( N,...,N | {z } d ,j ) / ( k ) (1 M ) = G ( N + 2) G ( M + N + 2) G ( M + 1) G ( M − d + 1) G ( M − d + N + 2) G ( d + 1) G ( d + N + 2) × (20)Γ( M − d + j + 1)Γ( j + 1) Γ( d + k + 1)Γ( k + 1) N X r =max ( j,k ) Γ( r + 1)Γ( M + r + 1) (cid:18) M − d + r − k − r − k (cid:19)(cid:18) d + r − j − r − j (cid:19) , for j, k ≤ N and M > d (or M ≥ d , if j = 0). The above formula recovers known evaluationswhenever k = 0 and thus the function in the left hand side above is a Schur polynomial (these canbe computed by means of the hook-content formula [35], for instance). Explicit expressions forsuch specialization of skew Schur polynomials indexed by partitions of certain shapes have beenobtained recently in [31], and coincide with the above formula when the shapes are the same. The shapes covered by the above formula are not a subset nor a superset of those considered in[31].Using expression (17), we see that the integral form of a Toeplitz minor generated by the pureFisher-Hartwig generality D λ,µN ( ϕ γ,δ ) = s (( δ N )+ µ/λ ) ′ (1 γ + δ ) = (21)1 N ! 1(2 π ) N Z π ... Z π s λ ( e − iθ ) s µ ( e iθ ) N Y j =1 e iθ j ( γ − δ ) | e iθ j | γ + δ Y ≤ j In order to study the principal specialization x j = q j − in theabove formulas, we recall the well known method of Borodin for obtaining the inverse of themoment matrix of a biorthogonal ensemble. We follow the presentation in [8], where details andproofs can be found. The starting point is a random matrix ensemble of the form Z · · · Z det ( ξ j ( z k )) Nj,k =1 det ( η j ( z k )) Nj,k =1 N Y j =1 f ( z j ) dz j (up to a constant), for a weight function f supported on some domain and two families offunctions ( ξ j ) and ( η j ). If one is able to find two new families ( ζ j ) and ( ψ j ) that biorthogonalize We have only proved the validity of the formula for integer values of γ and δ . However, by Carlson’s theoremthe formula holds for any positive γ and δ . Note that we are actually considering biorthonormal functions; we stick to the original terminology of [8] hereand below and speak of biorthogonal functions for simplicity. OEPLITZ MINORS AND SPECIALIZATIONS OF SKEW SCHUR POLYNOMIALS 11 the former with respect to the weight f , that is ζ j ∈ Span { ξ , . . . , ξ j } , ψ j ∈ Span { η , . . . , η j } , Z ζ j ( z ) ψ k ( z ) f ( z ) dz = δ j,k , (23)then the matrix of coefficients of the kernel K N ( z, ω ) = N X r =1 ζ r ( z ) ψ r ( ω ) = N X j,k =1 c j,k ξ j ( z ) η k ( ω ) (24)satisfies (cid:2) ( c j,k ) Nj,k =1 (cid:3) − = (cid:18)Z ξ k ( z ) η j ( z ) f ( z ) dz (cid:19) Nj,k =1 . If the ensemble is an orthogonal polynomial ensemble, then the moment matrix on the righthand side above is a Hankel matrix, the functions ξ j and η j are the monomials z j − , and wehave that ζ j = ψ j = p j , the orthogonal polynomials with respect to the weight function f , thatis supported on the real line. The case where the moment matrix on the right hand side aboveis the Toeplitz matrix generated by a function f supported on the unit circle corresponds to thebiorthogonal ensemble with functions ξ j ( z ) = z − ( j − , η j ( z ) = z j − . Thus, the biorthogonalitycondition (23) amounts to finding two families of polynomials p j and q j such that12 π Z π p j ( e − iθ ) q k ( e iθ ) f ( e iθ ) dθ = δ j,k . (25)Let us remark that only when the Toeplitz matrix is hermitian (that is, when the function f isreal valued), these polynomials verify p j ( e − iθ ) = q j ( e iθ ), the q j are the orthogonal polynomialswith respect to f , and the kernel above is the usual Christoffel-Darboux kernel (see [7, 26] formore details). In general, one needs to consider a biorthogonal ensemble as above. Nevertheless,one can compute the polynomials ( p j ) and ( q j ) in a similar fashion to the orthogonal case. Lemma. Suppose the determinants D N ( f ) are nonzero for every N . Then, the polynomials p j and q j in (25) are given by p j ( z ) = 1( D j ( f ) D j +1 ( f )) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d d − . . . d − j d d . . . d − ( j − ... ... ... d j − d j − d − z . . . z j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,q j ( z ) = 1( D j ( f ) D j +1 ( f )) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d d − . . . d − ( j − d d . . . d − ( j − z ... ... ... ... d j d j − . . . d z j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Proof. The condition on the determinants implies the existence of the polynomials themselves(see proposition 2.9 in [8], for instance), and they are uniquely determined up to multiplicativeconstants. Hence, it suffices to verify the biorthogonality condition (25). We denote p j ( z ) = j X r =0 a ( j ) r z r , q k ( z ) = k X r =0 b ( k ) r z r . (26) Now, if j ≥ k in (25) we can rewrite this integral as the sum12 π Z π p j ( e − iθ ) q k ( e iθ ) f ( e iθ ) dθ = 1( D k ( f ) D k +1 ( f ) D j ( f ) D j +1 ( f )) / × k X r =0 b ( k ) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d d − . . . d − j d d . . . d − ( j − ... ... ... d j − d j − d − π R π e irθ f ( e iθ ) dθ π R π e i ( r − θ f ( e iθ ) dθ . . . π R π e i ( r − j ) θ f ( e iθ ) dθ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which vanishes if j > k and equals 1 if j = k , since the last row in the above determinants isprecisely ( d r , d r − , . . . , d r − j ). Analogously, if j < k in (25) the integral equals1( D k ( f ) D k +1 ( f ) D j ( f ) D j +1 ( f )) / j X r =0 a ( j ) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d d − . . . d − ( k − 1) 12 π R π e − irθ f ( e iθ ) dθd d . . . d − ( k − 2) 12 π R π e − i ( r − θ f ( e iθ ) dθ ... ... ... ... d k d k − . . . d π R π e − i ( r − k ) θ f ( e iθ ) dθ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and again all the determinants in the sum vanish. (cid:3) We now use this result to study the principal specialization of skew Schur polynomials indexedby the shapes considered earlier. We assume in the following that q is a new (real) variable. Wewill denote by Γ q and G q the q -Gamma and q -Barnes functions [32], that in particular verifyΓ q ( k + 1) = Q kj =1 (1 − q j )(1 − q ) k = ( q ; q ) k (1 − q ) k , G q ( k + 1) = k − Y j =1 Γ q ( j + 1) . (27)whenever k is a natural number (we assume that an empty product takes the value 1). The q -binomial coefficient is then given by (cid:20) ωz (cid:21) q = Γ q ( ω + 1)Γ q ( z + 1)Γ q ( ω − z + 1) (Re( ω ) ≥ Re( z ) > . These functions coincide with their classical counterparts in the q → q → Γ q ( z ) = Γ( z ) , lim q → G q ( z ) = G ( z ) , lim q → (cid:20) ωz (cid:21) q = (cid:18) ωz (cid:19) , for all the ω and z such that the right hand sides above make sense. We consider the followingspecialization [20] f ( z ) = Θ γ,δ ( z ) = E (1 , q, . . . , q δ − ; z − ) E ( q, q , . . . , q γ ; z ) = γ X k = − δ (cid:20) δ + γδ + k (cid:21) q q k ( k +1) / z k , for some positive integers γ and δ . The Toeplitz determinant generated by this function equals D N (Θ γ,δ ) = G q ( N + 1) G q ( δ + γ + N + 1) G q ( δ + γ + 1) G q ( δ + 1) G q ( δ + N + 1) G q ( γ + 1) G q ( γ + N + 1) , and the biorthogonal polynomials p j , q j are given by p j ( z ) = (cid:18) ( q ; q ) δ + j ( q ; q ) γ + j ( q ; q ) j ( q ; q ) δ + γ + j (cid:19) / j X r =0 ( − j + r (cid:20) jr (cid:21) q ( q ; q ) γ + r ( q ; q ) γ + j ( q ; q ) δ + j − r − ( q ; q ) δ − z r ,q j ( z ) = (cid:18) ( q ; q ) δ + j ( q ; q ) γ + j ( q ; q ) j ( q ; q ) δ + γ + j (cid:19) / j X r =0 ( − j + r (cid:20) jr (cid:21) q ( q ; q ) γ + j − r − ( q ; q ) γ − ( q ; q ) δ + r ( q ; q ) δ + j q r z r , (28) OEPLITZ MINORS AND SPECIALIZATIONS OF SKEW SCHUR POLYNOMIALS 13 where ( q ; q ) k is as defined in (27). The last three identities can be proved directly from theirdeterminantal expressions. We do not include the computations here but point to the secondmethod of proof in [12], that can be generalized to the present setting. A similar computationis included in the appendix as an example. Recalling the notation (26), we have that the kernel(24) is then given by K N +1 ( z, ω ) = N X r =0 p r ( z ) q r ( ω − ) = N X j,k =0 N X r =max ( j,k ) a ( r ) j b ( r ) k z j ω − k = N X j,k =0 N X r =max j,k ( − j + k q j Γ q ( δ + j + 1)Γ q ( γ + k + 1)Γ q ( r + 1)Γ q ( j + 1)Γ q ( k + 1)Γ q ( δ + γ + r + 1) (cid:20) γ + r − k − r − k (cid:21) q (cid:20) δ + r − j − r − j (cid:21) q z j ω − k . Moreover, the coefficient of z j ω − k in the above sum is the ( j + 1 , k + 1)-th entry of the inverseof the matrix T N +1 (Θ γ,δ ). Inserting this into expression (12) we obtain s ( N,...,N | {z } d ,j ) / ( k ) (1 , q, . . . , q M − ) = q dj − ( d − k + d ( d − N/ G q ( N + 2) G q ( M + N + 2) G q ( M + 1) G q ( M − d + 1) G q ( M − d + N + 2) G q ( d + 1) G q ( d + N + 2) × N X r =max ( j,k ) Γ q ( M − d + j + 1)Γ q ( d + k + 1)Γ q ( r + 1)Γ q ( j + 1)Γ q ( k + 1)Γ q ( M + r + 1) (cid:20) M − d + r − k − r − k (cid:21) q (cid:20) d + r − j − r − j (cid:21) q , for j, k ≤ N and M > d (or M ≥ d , if j = 0). As expected, this expression coincides with (20)in the q → k = 0(and thus we have a Schur polynomial, comparing again with the hook-content formula [35], forinstance). Finally, it follows from (14) and the Cauchy identity that as N → ∞ s ( N,...,N | {z } d ,j ) / ( k ) (1 , q, . . . , q M − ) ∼ q dj − ( d − k + d ( d − N/ (1 − q ) d ( M − d ) G q ( d + 1) G q ( M − d + 1) G q ( M + 1) min ( j,k ) X r =0 q − r (cid:20) M − d + j − r − j − r (cid:21) q (cid:20) d + k − r − k − r (cid:21) q . Note that inverting a Toeplitz matrix by means of the kernel (24) is a general procedure thatcan be used to obtain explicit evaluations of other specializations of the skew Schur polynomialsof the shapes considered above, as long as the biorthogonal polynomials (25) are available. Inparticular, the results in subsection 4.2 for the pure Fisher-Hartwig singularity can be obtainedin such a way. The biorthogonal polynomials can be obtained as the q → f needs to befinite in equations (11)-(14), we can study the principal specialization of the above skew Schurpolynomials with an infinite number of variables. To do so, we consider the specialization f ( z ) = Θ δ ( z ) = E (1 , q − , . . . , q − ( δ − ; z − ) E ( q δ , q δ +1 , . . . ; z ) = ∞ X k = − δ q kδ + k ( k − / ( q ; q ) δ + k z k , In the hermitian case γ = δ , where the polynomials are a single family of orthogonal polynomials, one recoversthe family S an ( z ) introduced in [3] after substituting q by q / , z by q − / z and a by q γ . for some positive integer δ . The Toeplitz determinant generated by this function is D N (Θ δ ) = 1(1 − q ) δN G q ( δ + 1) G q ( N + 1) G q ( δ + N + 1) , and the biorthogonal polynomials on the unit circle with respect to this function are given by p j ( z ) = (cid:18) ( q ; q ) δ + j ( q ; q ) j (cid:19) / j X r =0 ( − j + r (cid:20) jr (cid:21) q ( q ; q ) δ + j − r − ( q ; q ) δ − q − ( δ − j − r ) z r ,q j ( z ) = (cid:18) q ; q ) j ( q ; q ) δ + j (cid:19) / j X r =0 ( − j + r (cid:20) jr (cid:21) q ( q ; q ) δ + r q δ ( j − r ) z r . Again, these expressions can be verified from their determinantal formulas. The kernel in thiscase is then K N +1 ( z, ω ) = N X j,k =0 N X r =max j,k ( − j + k q r +( δ − j − δk ( q ; q ) δ + k ( q ; q ) j (cid:20) rr − k (cid:21) q (cid:20) δ + r − j − r − j (cid:21) q z j ω − k . Inserting this in equation (12) we arrive at s ( N,...,N | {z } d ,j ) / ( k ) (1 , q, . . . ) = q ( d − j − dk + d ( d − N/ (1 − q ) d ( N +1) G q ( N + 2) G q ( d + 1) G q ( d + N + 2) ( q ; q ) d + k ( q ; q ) j N X r =max ( j,k ) q r (cid:20) rr − k (cid:21) q (cid:20) d + r − j − r − j (cid:21) q . Once again, this identity coincides with the one given by the hook-content formula for k = 0. Itfollows from (14) and the Cauchy identity that as N → ∞ s ( N,...,N | {z } d ,j ) / ( k ) (1 , q, . . . ) ∼ q dj − ( d − k + d ( d − N/ (1 − q ) d ( d − / G q ( d + 1)( q ; q ) d ∞ min ( j,k ) X r =0 q − r q ; q ) j − r (cid:20) d + k − r − k − r (cid:21) q , where ( q ; q ) ∞ = Q ∞ k =1 (1 − q k ) denotes the Euler function. Acknowledgements. We thank Jorge Lobera for a MatLab implementation of formula (6)and Alexandra Symeonides and Tˆania Zaragoza for useful discussions. The work of DGG wassupported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia through the LisMath scholarshipPD/BD/113627/2015. The work of MT was supported by the Funda¸c˜ao para a Ciˆencia e aTecnologia through its program Investigador FCT IF2014, under contract IF/01767/2014. Appendix: Direct computation of a minor generated by the pure FH singularity We now sketch a proof of identity (22). We follow the second of the two proofs given in [12]for the corresponding Toeplitz determinant. We include this computation to showcase how theToeplitz minor structure can be exploited to obtain evaluations of the more complicated objectsconsidered (i.e. multiple integrals, skew Schur polynomials), rather than for its mathematicalinsight.The Fourier coefficients of ϕ γ,δ are [10] d k = Γ( γ + δ + 1)Γ( γ − k + 1)Γ( δ + k + 1) ( − δ ≤ k ≤ γ ) . OEPLITZ MINORS AND SPECIALIZATIONS OF SKEW SCHUR POLYNOMIALS 15 After taking out the factors N Y j =1 Γ( γ + δ + 1)Γ( γ − µ N + N − j + 1) , N Y k =1 δ + µ k + N − k + 1) , coming from the rows and columns of D ∅ ,µN ( ϕ γ,δ ) respectively, we obtain the determinant (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( γ − µ N + N )Γ( γ − µ +1) Γ( δ + µ + N )Γ( δ + µ +1) Γ( γ − µ N + N )Γ( γ − µ +2) Γ( δ + µ + N − δ + µ ) . . . Γ( δ + µ N +1)Γ( δ + µ N − N +2)Γ( γ − µ N + N − γ − µ ) Γ( δ + µ + N )Γ( δ + µ +2) Γ( γ − µ N + N − γ − µ +1) Γ( δ + µ + N − δ + µ +1) . . . Γ( δ + µ N +1)Γ( δ + µ N − N +3) ... ... ... Γ( γ − µ N +1)Γ( γ − µ − N +2) Γ( γ − µ N +1)Γ( γ − µ − N +3) . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (29)Subtracting ( δ + µ N − N +1+ j ) times the ( j +1)-th row from the j -th row, for j = 1 , ..., N − , wecan make the last column vanish except for the 1 at the bottom, thus obtaining a determinantof order N − 1. After extracting the factor N − Y k =1 ( γ + δ + 1)( µ k − µ N + N − k )from the columns of the matrix, and N − Y j =1 Γ( γ − µ N + j )Γ( γ − µ N − + j )from the rows, we obtain a determinant with the same structure as (29), but with the followingchanges: N is replaced by N − δ is replaced by δ + 1 and µ is replaced by the partition( µ , . . . , µ N − ), that results from discarding the last part of µ . Making use of this recursivestructure and the well-known expression s µ (1 N ) = 1 G ( N + 1) Y ≤ j Gabor Szeg˝o: CollectedPapers, Volume 1, 1915-1927 (R. 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