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Schur polynomials, banded Toeplitz matrices and Widom's formula
Stockholm University, Faculty of Science, Department of Mathematics.
2012 (English)In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 19, no 4, P22- p.Article in journal (Refereed) Published
Abstract [en]

We prove that for arbitrary partitions lambda subset of kappa, and integers 0 <= c < r <= n, the sequence of Schur polynomials S(kappa+k.1c)/(lambda+k.1r)(x(1), ... , x(n)) for k sufficiently large, satisfy a linear recurrence. The roots of the characteristic equation are given explicitly. These recurrences are also valid for certain sequences of minors of banded Toeplitz matrices. In addition, we show that Widom's determinant formula from 1958 is a special case of a well-known identity for Schur polynomials.

Place, publisher, year, edition, pages
2012. Vol. 19, no 4, P22- p.
Keyword [en]
Banded Toeplitz matrices, Schur polynomials, Widom's determinant formula, sequence insertion, Young tableaux, recurrence
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-83811ISI: 000310833500004OAI: oai:DiVA.org:su-83811DiVA: diva2:577130
Note

AuthorCount:1;

Available from: 2012-12-19 Created: 2012-12-14 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Combinatorial Methods in Complex Analysis
Open this publication in new window or tab >>Combinatorial Methods in Complex Analysis
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The theme of this thesis is combinatorics, complex analysis and algebraic geometry. The thesis consists of six articles divided into four parts.

Part A: Spectral properties of the Schrödinger equation

This part consists of Papers I-II, where we study a univariate Schrödinger equation with a complex polynomial potential. We prove that the set of polynomial potentials that admit solutions to the Schrödingerequation is connected, under certain boundary conditions. We also study a similar result for even polynomial potentials, where a similar result is obtained.

Part B: Graph monomials and sums of squares

In this part, consisting of Paper III, we study natural bases for the space of homogeneous, symmetric and translation-invariant polynomials in terms of multigraphs. We find all multigraphs with at most six edges that give rise to non-negative polynomials, and which of these that can be expressed as a sum of squares. Such polynomials appear naturally in connection to expressing certain non-negative polynomials as sums of squares.

Part C: Eigenvalue asymptotics of banded Toeplitz matrices

This part consists of Papers IV-V. We give a new and generalized proof of a theorem by P. Schmidt and F. Spitzer concerning asymptotics of eigenvalues of Toeplitz matrices. We also generalize the notion of eigenvalues to rectangular matrices, and partially prove the a multivariate analogue of the above.

Part D: Stretched Schur polynomials

This part consists of Paper VI, where we give a combinatorial proof that certain sequences of skew Schur polynomials satisfy linear recurrences with polynomial coefficients.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2013. 111 p.
Keyword
combinatorics, Schrödinger equation, Toeplitz matrix, sums of squares, Schur polynomials
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-88808 (URN)978-91-7447-684-2 (ISBN)
Public defence
2013-05-30, Lecture hall 14, House 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of doctoral defence the following papers were unpublished and had a status as follows: Paper 5: Manuscript; Paper 6: Manuscript

Available from: 2013-05-08 Created: 2013-03-30 Last updated: 2013-05-06Bibliographically approved

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