Permutation Matrices, Alternating Sign Matrices, and Generalizations

Abstract: The study of permutations is both ancient and modern. One can view them as the integers 1,2, … , n in some order or as n x n permutation matrices. They can be regarded as data which is to be sorted. The explicit definition of the determinant uses permutations. An inversion of a permutation occurs when a larger integer precedes a smaller integer. Inversions can be used to define two partial orders on permutations, one weaker than the other. Partial orders have a unique minimal completion to a lattice, the Dedekind-MacNeille completion. Generalizations of permutation matrices determine related matrix classes, for instance, alternating sign matrices (ASMs) which arose independently in the mathematics and physics literature. Permutations may contain certain patterns, e.g. three integers in increasing order; avoiding such patterns determines certain permutation classes. Similar restrictions can be placed more generally on (0,1)-matrices.

There are continuous analogs of permutation matrices and alternating sign matrices, and even higher dimension analogs. We shall explore these and other ideas and their connections.

Mathematics

Audience: researchers in the topic

( slides | video )

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ITB Mathematics Distinguished Lecture Series

Series comments: We aim to bring prominent mathematicians exploring the role of mathematics from various fields, in an intriguing style. The online lectures address quite broad audience, from mathematics students (advanced undergraduate to graduate ones), as well as mathematicians in Indonesia, and our neighboring countries, or even beyond our region. With this lecture series we hope to foster and promote research culture, as well as to highlight the prominent role of mathematics in shaping future society.

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