SPECTRAL PROPERTIES OF SOME MATRICES CLOSE TO THE TOEPLITZ TRIANGULAR FORM

We consider a class of matrices, that we call nearly Toeplitz, and show that they have interesting spectral properties. More precisely, we show that the eigenvectors of certain nearly Toeplitz matrices give complete information about the structure of the symmetric group S(k), i.e., the group of permutations of the integers 1,...,k. Obtaining this kind of information is a central task in two seemingly unrelated branches of mathematics, namely the Character Theory of the Symmetric Group and the Polya's Theory of Counting.