1934/05/19 by D. E. Littlewood, A. R. Richardson · 360 citations
Engineering · Computer Science · Mathematics · #graph theory and CDMA systems #semigroups and automata theory #Advanced Combinatorial Mathematics #Symmetric group #Analogy #Mathematics #TRACE (psycholinguistics) #Group (periodic table) #Combinatorics #Order (exchange) #Partition (number theory) #Representation theory of the symmetric group #Character table #Character (mathematics) #Algebra over a field #Matrix (chemical analysis) #General linear group #Pure mathematics #Physics #Geometry #Linguistics
paper · doi:10.1098/rsta.1934.0015
published in Philosophical Transactions of the Royal Society of London Series A Containing Papers of a Mathematical or Physical Character 233(721-730), 99-141 (Royal Society)
openalex publication_date 1934/05/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/06/11
Abstract It has been known for some time* that the elements of a matrix of degree n may be arranged in sets which correspond to cycles of the symmetric group of order n !, and that there are relations connecting permanents and determinants, e.g. , /a* p y S a |3 y 8/ ( Further, MACMAHON and BRIOSCHI have pointed out the close analogy which exists between the threefold algebra of the symmetric functions an, hn and sn, and the theory of determinants, permanents, and the cycles of substitutions of the symmetric group. Here we trace the analogy to its source by fixing attention on the characters of the irreducible representations of the symmetric group of linear substitutions, as the centre of the whole theory. By this means divers theories of combinatory analysis and algebra are seen to be merely different aspects of the same theory. For the symmetric group of order n ! the characters are all integers, and we associate with each partition of n both a character of the group and a cycle of substitutions.