Combinatorics and Algebra

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American Mathematical Society, 1984 - Mathematics - 318 pages
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This volume contains the Proceedings of the AMS-NSF Joint Summer Research Conference on Combinatorics and Algebra held at the University of Colorado during June 1983. Although combinatorial techniques have pervaded the study of algebra throughout its history, it is only in recent years that any kind of systematic attempt has been made to understand the connections between algebra and combinatorics. This Conference drew together specialists in both and provided an invaluable opportunity for them to collaborate. The topic most discussed was representation theory of the symmetric group and complex general linear group. The close connections with combinatorics, especially the theory of Young tableaux, was evident from the pioneering work of G.~Frobenius, I.~Schur, A.~Young, H.~Weyl, and D.~E.~Littlewood. Phil Hanlon gave an introductory survey of this subject, whose inclusion in this volume should make many of the remaining papers more accessible to a reader with little background in representation theory. Ten of the papers impinge on representation theory in various ways. Some are directly concerned with the groups, Lie algebras, etc., themselves, while others deal with purely combinatorial topics which arose from representation theory and suggest the possibility of a deeper connection between the combinatorics and the algebra. The remaining papers are concerned with a wide variety of topics. There are valuable surveys on the classical subject of hyperplane arrangements and its recently discovered connections with lattice theory and differential forms, and on the surprising connections between algebra, topology, and the counting of faces of convex polytopes and related complexes. There also appears an instructive example of the interplay between combinatorial and algebraic properties of finite lattices, and an interesting illustration of combinatorial reasoning to prove a fundamental algebraic identity. In addition, a highly successful problem session was held during the conference; a list of the problems presented appears at the end of the volume.

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