Toeplitz minors and specializations of skew Schur polynomials

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Abstract

We express minors of Toeplitz matrices of finite and large dimension in terms of symmetric functions. 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 skew Schur polynomials.

Introduction

Let f(eiθ)=kZdkeikθ be an integrable function on the unit circle. The Toeplitz matrix generated by f is the matrixT(f)=(djk)j,k1. That is, T(f) is an infinite matrix, constant along its diagonals, which entries are the Fourier coefficients of the function f. We denote by TN(f) its principal submatrix of order N, andDN(f)=detTN(f). This determinant has the following integral representationDN(f)=U(N)f(M)dM=1N!1(2π)N02π...02πj=1Nf(eiθj)1j<kN|eiθjeiθk|2dθ1...dθN, where dM denotes the normalized Haar measure on the unitary group U(N). This is known as Heine identity. A main result in the theory of Toeplitz matrices is the strong Szegő limit theorem, that describes the behaviour of these determinants as N grows to infinity, as long as the function f is sufficiently regular (see section 2.2 for a precise statement of the theorem).

A Toeplitz minor is a minor of a Toeplitz matrix, obtained by striking a finite number of rows and columns from T(f). This can be realized, up to a sign, as the determinant of a matrix of the formTNλ,μ(f)=(djλjk+μk)j,k=1N, where λ and μ are integer partitions that encode the particular striking considered (see section 2.1 for more details). We denoteDNλ,μ(f)=detTNλ,μ(f). Toeplitz minors also have an integral representation [15], [1]DNλ,μ(f)=U(N)sλ(M)sμ(M)f(M)dM=1N!1(2π)N02π...02πsλ(eiθ1,...,eiθN)sμ(eiθ1,...,eiθN)j=1Nf(eiθj)×1j<kN|eiθjeiθk|2dθ1...dθN, where sλ,sμ are Schur polynomials.1 Bump and Diaconis [15] described the asymptotic behaviour of Toeplitz minors generated by functions that are sufficiently regular, as in Szegő's theorem. They proved that in the large N limit, these minors can be expressed as the product of the corresponding Toeplitz determinant times a “combinatorial” factor, that depends only on the function f and the striking considered and is independent of N (see section 2.2 for a precise statement). Tracy and Widom obtained a similar result in [41], and they were compared in [17]. Further generalizations regarding the asymptotics of integrals of the type (2) were given in [16], [30]. Other works that study Toeplitz minors in relationship with Schur and skew Schur polynomials are [29], [2], [35], [32].

The asymptotics of Toeplitz determinants generated by symbols that do not verify the regularity conditions in Szegő's theorem have been long studied. In the seminal work [21], Fisher and Hartwig conjectured the asymptotic behaviour of Toeplitz determinants generated by a class of (integrable) functions that violate these conditions. The functions in this class are products of a function which is regular, in the sense of Szegő's theorem, and a finite number of so-called pure Fisher-Hartwig singularities. Their conjecture was later refined in [6] and [7], and only recently a complete description of the asymptotics of these determinants was achieved by Deift, Its and Krasovsky [18]. See [19] for a detailed historical account of the subject.

In this paper we exploit the formalism of symmetric functions to study Toeplitz minors. After section 2, where some known results are reviewed, we obtain an equivalent expression for the combinatorial factor of Bump and Diaconis in terms of skew Schur polynomials. This is done in section 3, where we also characterize (i) a class of Toeplitz minors for which an exact asymptotic expression can be obtained, and (ii) a class of Toeplitz minors that can be realized as the specialization of a single skew Schur polynomial. In section 4 we compute the inverses of some Toeplitz matrices, using the Duduchava-Roch formula and the kernel associated to two sets of biorthogonal polynomials on the unit circle. Comparing these matrices with their expressions in terms of Toeplitz minors we obtain explicit evaluations of a family of specialized skew Schur polynomials and of a Selberg-Morris type integral.

Section snippets

Symmetric functions

Let us recall some basic results involving symmetric functions that can be found in [31], [38], for example. We denote z=eiθ in the following, and treat z as a formal variable. A partition λ=(λ1,,λl) is a finite and non-increasing sequence of positive integers. The number of nonzero entries is called the length of the partition and is denoted by l(λ), and the sum |λ|=λ1++λl(λ) is called the weight of the partition. The entry λj is understood to be zero whenever the index j is greater than the

Toeplitz minors generated by symmetric functions

Let us now obtain an equivalent expression for the asymptotic formula (8) for the case of Toeplitz minors generated by formal power series.

Theorem 1

Letf(z)=H(x;z)H(y;z1), for some sets of variables x and y, where H is given by (3), and assume moreover that the sequences of complete homogeneous symmetric polynomials (hk(x)) and (hk(y)) are square summable. Then, for any two fixed partitions λ and μ we havelimNDNλ,μ(f)=νsλ/ν(y)sμ/ν(x)limNDN(f). Note that we understand f as a formal Laurent power

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 for the case of Toeplitz matrices(TN1(f))j,k=(1)j+kDN1(1k1),(1j1)(f)DN(f). Hence, whenever the inverse of a Toeplitz matrix is known explicitly, formula (17) yields explicit evaluations of the formulas appearing in section 3. In particular, if the function f is of the form f(z)=E(y1,...,yd;z1)E(x;z), the Toeplitz minor in the right hand side above has several expressions: in terms of the inverse of

Acknowledgements

We thank Jorge Lobera for a MatLab implementation of formula (8) and Alexandra Symeonides and Tânia Zaragoza for useful discussions. We also thank an anonymous referee for several helpful remarks. The work of DGG was supported by the Fundação para a Ciência e a Tecnologia through the LisMath scholarship PD/BD/113627/2015. The work of MT was partially supported by the Fundação para a Ciência e a Tecnologia through its program Investigador FCT IF2014, under contract IF/01767/2014. The work is

References (43)

  • M. Adler et al.

    Virasoro action on Schur function expansions, skew Young tableaux, and random walks

    Commun. Pure Appl. Math.

    (2005)
  • P. Alexandersson

    Schur polynomials, banded Toeplitz matrices and Widom's formula

    Electron. J. Comb.

    (2012)
  • R. Askey

    Remarks to G. Szegő, “Ein Beitrag zur Theorie der Thetafunktionen”

  • J. Baik et al.

    Algebraic aspects of increasing subsequences

    Duke Math. J.

    (2001)
  • E.W. Barnes

    The theory of the G-function

    Q. J. Pure Appl. Math.

    (1900)
  • E. Basor

    Asymptotic formulas for Toeplitz determinants

    Trans. Am. Math. Soc.

    (1978)
  • A. Borodin

    Biorthogonal ensembles

    Nucl. Phys. B

    (1999)
  • A. Borodin et al.

    A Fredholm determinant formula for Toeplitz determinants

    Integral Equ. Oper. Theory

    (2000)
  • A. Böttcher

    The Duduchava–Roch formula

  • A. Böttcher et al.

    Introduction to Large Truncated Toeplitz Matrices

    (1999)
  • A. Böttcher et al.

    Two elementary derivations of the pure Fisher-Hartwig determinant

    Integral Equ. Oper. Theory

    (2005)
  • Cited by (0)

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