Toeplitz minors and specializations of skew Schur polynomials
Introduction
Let be an integrable function on the unit circle. The Toeplitz matrix generated by f is the matrix That is, is an infinite matrix, constant along its diagonals, which entries are the Fourier coefficients of the function f. We denote by its principal submatrix of order N, and This determinant has the following integral representation where dM denotes the normalized Haar measure on the unitary group . 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 . This can be realized, up to a sign, as the determinant of a matrix of the form where λ and μ are integer partitions that encode the particular striking considered (see section 2.1 for more details). We denote Toeplitz minors also have an integral representation [15], [1] where 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 in the following, and treat z as a formal variable. A partition 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 , and the sum is called the weight of the partition. The entry 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 Let 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 and are square summable. Then, for any two fixed partitions λ and μ we have 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 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 , 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)
- et al.
The Fisher-Hartwig conjecture and generalizations
Physica A
(1991) Polynomials defined by a difference system
J. Math. Anal. Appl.
(1961)- et al.
Toeplitz matrices and determinants with Fisher-Hartwig symbols
J. Funct. Anal.
(1985) - et al.
Toeplitz minors
J. Comb. Theory, Ser. A
(2002) Averages over classical Lie groups, twisted by characters
J. Comb. Theory, Ser. A
(2007)- et al.
Explicit inverses of some tridiagonal matrices
Linear Algebra Appl.
(2001) Symmetric functions and P-recursiveness
J. Comb. Theory, Ser. A
(1990)Inversion des matrices de Hankel
Linear Algebra Appl.
(1990)- et al.
Hook formulas for skew shapes I. q-analogues and bijections
J. Comb. Theory, Ser. A
(2018) Asymptotic behavior of block Toeplitz matrices and determinants. II
Adv. Math.
(1976)