Autor: Ashish Mishra
Bijections give an idea of “sameness” between two mathematical objects while preserving their underlying mathematical structure. We look at bijections in two different contexts: firstly in infinite sets and secondly in finite sets. In the first part of the minicourse, we will focus on Cantor’s use of the concept of bijective functions to study the cardinality of the infinite sets. His results on infinite sets were so much surprising that he exclaimed to De – dekind “I see it, but I don’t believe it.” After discussing these results for infinite sets, we will introduce the concepts of groups and group actions using the idea of bijections. Moreover, we will see an important example of bijective proof in the case of finite sets. The bijection, known as the Robinson—Schensted–Knuth correspondence, gives a bijective proof of the following identity in combinatorial representation theory: The total number of pairs of standard Young tableaux of the same shape, where the shape varies over Young diagrams of total n boxes, is equal to the order of the symmetric group on n letters.