Research Seminar in Mathematics - Spectra of transposition generated Cayley graphs

Speaker

Niklas Eriksen, institutionen för naturvetenskap och teknik, Örebro universitet.

Abstract

The symmetric group S_n consists of all permutations of length n and can be generated for instance by all transpositions, which are swaps of any two elements. Many group properties can be inferred from the transition matrices of Cayley graphs of S_n and especially its eigenvalues, which are all integers and can easily be computed using Young tableaux. A question of particular interest is determining which eigenvalues remain when we cluster permutations by pretending that some of their elements are identical (in algebraical terms, taking cosets with respect to a subgroup consisting of a product of smaller symmetric groups); this information is given by the Kostka numbers.

Our calculations strongly indicate that these results can be refined to groups generated by transposition with a fixed top number. Further, considering cosets closely related to the ones above, we conjecture that the eigenvalue multiplicities are given by a family of numbers with recursive properties very similar to the Kostka numbers. An application is the enumeration of a slightly altered version of de Bruijn sequences.

Welcome!