Multivariable Christoffel-Darboux Kernels and Characteristic Polynomials of Random Hermitian Matrices

We study multivariable Christoffel-Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of Pfaffian and determinant formulas, as well as Schur polynomial expansions, for such kernels. In subsequent work, these results are applied in combinatorics (enumeration of marked shifted tableaux) and number theory (representation of integers as sums of squares).


Introduction
The Christoffel-Darboux kernel plays an important role in the theory of one-variable orthogonal polynomials. In the present work, we study a multivariable extension, which can be viewed as a reproducing kernel for anti-symmetric polynomials. As is explained below, our original motivation came from a very special case, having applications in number theory (sums of squares) and combinatorics (tableaux enumeration). More generally, this kind of kernels occur in random matrix theory as correlation functions for products of characteristic polynomials of random Hermitian matrices. The purpose of the present note is to highlight a number of useful identities for such kernels. Although, as we will make clear, the main results can be found in the literature, they are scattered in work belonging to different disciplines, so it seems useful to collect them in one place. Moreover, our proofs, with the interpretation as reproducing kernels, are new and conceptually very simple.
We first recall something of the one-variable theory. Let ⋆ This paper is a contribution to the Vadim Kuznetsov Memorial Issue "Integrable Systems and Related Topics". The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html be a linear functional defined on polynomials of one variable. We denote by V n the space of polynomials of degree at most n − 1. Assuming that the pairing is non-degenerate on each V n (for instance, if it is positive definite), there exists a corresponding system (p k (x)) ∞ k=0 of monic orthogonal polynomials. We may then introduce the Christoffel-Darboux kernel which is the reproducing kernel of V n , that is, the unique function such that (y → K(x, y)) ∈ V n and The Christoffel-Darboux formula states that More generally, let V m n , 0 ≤ m ≤ n, denote the mth exterior power of V n . It will be identified with the space of antisymmetric polynomials f (x) = f (x 1 , . . . , x m ) that are of degree at most n − 1 in each x i . Writing we equip V m n with the pairing Equivalently, in terms of the spanning vectors det 1≤i,j≤m Every element of V m n is divisible by the polynomial The map f → f /∆ is an isometry from V m n to the space W m n , consisting of symmetric polynomials in x 1 , . . . , x m that are of degree at most n − m in each x i , equipped with the pairing We remark that, normalizing to a probability distribution, it defines an orthogonal polynomial ensemble (the term is sometimes used also for the more general weights |∆(x)| β ). Such ensembles are important in a variety of contexts, including the theory of random Hermitian matrices [12]. An orthogonal basis of V m n is given by (p S (x)) S⊆[n], |S|=m , where Here and throughout, we write [n] = {1, . . . , n}, and we assume that the columns are ordered in the natural way. Indeed, it follows from (1) that We denote by ∆(x)∆(y)K m (x, y) the reproducing kernel of V m n , that is, the unique element of V m n ⊗ V m n such that Equivalently, K m (x, y) is the reproducing kernel of W m n . It is easy to see that Indeed, it follows from (1) and (3) that both sides satisfy (4). The equality of the two expressions can also be derived using the Cauchy-Binet formula.
The following elegant integral formula was recently obtained by Strahov and Fyodorov [29]. In Section 2 we will give a simple proof, using the interpretation of the left-hand side as a reproducing kernel.
As a consequence, the obvious S m × S m × Z 2 symmetry of K m (x, y) extends to a non-trivial S 2m symmetry.
The motivation for the work of Strahov and Fyodorov is an interpretation of the integral (6) as a correlation function for the product of characteristic polynomials of random Hermitian matrices. The case x = y is of particular interest, since (6) may then be written as which exhibits ∆(x) 2 K m (x, x) as a correlation function for the measure (2). In that case, the determinant formula (5) is classical, see [5,Chapter 5]. In our opinion, our proof of Proposition 1 is more illuminating than the inductive proof usually given for the special case x = y. For applications of Proposition 1 and for further related results, see [1, 2,4]. An alternative determinant formula for the integral (6) may be obtained by combining two classical results of Christoffel [9, Theorem 2.7.1] and Heine [9, Theorem 2.1.2], see also [4]. We will obtain it as a by-product of the proof of Proposition 1.
Proposition 2. In the notation above, More generally, such a determinant formula holds for the integrals 1≤j≤m 1≤k≤n but we focus on the case when m is even. The fact that the "two-point" determinant in (5) and the "one-point" determinant in Proposition 2 agree can also be derived from the work of Lascoux [15,Propositions 8.4.1 and 8.4.3], see [15,Exercise 8.33] for the integral formula in this context. Note that the reproducing property mentioned by Lascoux, and given explicitly in [17,Proposition 3], is of a different nature from (4), pertaining to integration against one-variable polynomials.
The special case of Proposition 2 obtained by subtracting the kth row from the (m + k)th, for 1 ≤ k ≤ m, and then letting z m+k → z k , gives the following formula for the correlation function (8).
Corollary 2. In the notation above, Next, we give Pfaffian formulas for K m . As is explained below, they can be deduced from results of Ishikawa and Wakayama [8], Lascoux [16], and Okada [22]. Nevertheless, we will give an independent proof, using Corollary 1. Recall that the Pfaffian of a skew-symmetric even-dimensional matrix is given by Proposition 3. For any choice of square roots Moreover, for any choice of ζ i such that Note that (9) implies In the special case z = (x, x), choosing Proposition 3 reduces to special cases of the identity The general case seems to lie deeper. Proposition 3 is actually equivalent to Proposition 4 below, which can be deduced from known results. Indeed, rewriting (11a) using [22,Theorem 4.2] and (11b) using the case n = m of [22,Theorem 4.7], we can easily see the resulting expressions to agree. More explicitly, this identity appears in [16], with a simple proof. The Pfaffian (11c) can be treated similarly, or else shown to agree with (11b) by means of a result of Ishikawa and Wakayama [8, Theorem 5.1], see Remark 3 below.
Proposition 4. Let a i and z i , 1 ≤ i ≤ 2m, be free variables, and let ζ i be as in (9). Moreover, let S ⊆ [2m] be an arbitrary subset of cardinality m. Then, In [27], we need an elementary result on the expansion of the kernel K m (x, y) into Schur polynomials s λ (x)s µ (y). Since we have not found a suitable reference, we include it here. The proof is given in Section 4.

Proposition 5. One has
Finally, let us describe our original motivation, which comes from applications that seem completely unrelated to random matrix theory. Motivated by the theory of affine superalgebras, Kac and Wakimoto [11] conjectured certain new formulas for the number of representations of an integer as the sum of 4m 2 or 4m(m + 1) triangular numbers. These conjectures were first proved by Milne [19,20,21], and later by Zagier [31]. In [25], we re-derived and generalized the Kac-Wakimoto identities using elliptic Pfaffian evaluations. Extension of this analysis from triangles to squares leads to formulas involving Schur Q-polynomials [18] evaluated at the point (1, . . . , 1). (More precisely, these polynomials are normally labelled by positive integer partitions. Here, we need an extension to the case when some indices are negative.) Later, we realized that the resulting sums of squares formulas are equivalent to those of Milne [21]. Seeing this is far from obvious and requires an identification of the Schur Q-polynomials with kernels K m (z), where the underlying polynomials p k (x) are continuous dual Hahn polynomials. The key fact for obtaining this identification is the second part of Proposition 3. We refer to [26] for applications of the results above to Schur Q-polynomials and marked shifted tableaux, and to [27] for the relation to sums of squares.
A corresponding statement holds when V n is a general n-dimensional vector space and ∆ an element of the one-dimensional space (V * n ) n [3, § 8.5], the map φ : V m n → (V * n ) n−m often being called Hodge star or Poincaré isomorphism. For completeness, we provide a proof in the present setting.
Proof . The Vandermonde determinant evaluation ∆(x) = det 1≤i,j≤n (p j−1 (x i )) (12) gives where the sum is over bijections. By orthogonality, we may assume By iteration, it follows from Lemma 1 that By means of ∆(w, x) = (−1) m(n−m) ∆(x, w), x ∈ R m , w ∈ R n−m , this fact can be expressed as By the uniqueness of the reproducing kernel, it follows that This is equivalent to Proposition 1.
Remark 2. The equation (13) can also be obtained as the special case l = n of the contraction formula 0 ≤ m ≤ l ≤ n. Conversely, (14) follows easily from (13).

Proof of Proposition 3
Our main tool is the following elementary property of Pfaffians, which we learned from an unpublished manuscript of Eric Rains [24].
Lemma 2 (Rains). For arbitrary (a ij ) 1≤i,j≤2m , where χ(S) denotes the number of even elements in S.
For completeness, we sketch Rains' proof.
Since the map σ → (S, τ ) is m! to one, we obtain indeed The following identity appeared as [8, Theorem A.1].
Corollary 3 (Ishikawa and Wakayama). One has pfaff 1≤i,j≤2m This follows immediately from Lemma 2, by use of the Cauchy determinant det 1≤i,j≤m , which is reduced to its more well-known special case a = 1, b = 0, c = −1 through the elementary identity The proof of Corollary 3 given in [8] is more complicated.
We only need Corollary 3 in the case when x i = z i . Then, the Pfaffian is given by Indeed, one may use (15) to reduce oneself to the case a = 1, b = 0, c = −1, which is the Pfaffian evaluation in [ We are now ready to prove Proposition 3. By Lemma 2, we have in general pfaff 1≤i,j≤2m which, by Corollary 1, equals Consider first the case a i = √ z i . Then, by the case a = c = 0, b = 1, x i = √ z i of Corollary 4, the sum in (16) equals This yields the first part of Proposition 3. Similarly, letting a i = ζ i and using (10), we can compute the sum in (16) by the case a = 1, b = 0, c = −1, This completes the proof of Proposition 3.

Proof of Proposition 5
When (e k ) n k=1 is a basis of V n , let (e S ) S⊆[n], |S|=m be the corresponding basis of V m n defined by e S (x) = det 1≤i≤m, j∈S (e j (x i )).
On the one hand, reordering the rows gives ( e i , f j ) det i∈S c ,j∈T c ( e i , f j ).
This completes the proof.