Math 820, Topics in Algebra - The Symmetric Group

The group of all permutations of a finite set under composition is an elementary mathematical object. Yet, it is remarkably rich in terms of structure and connections with other areas of mathematics, such as representation theory, combinatorics, and even geometry. Furthermore, the study of the symmetric group is a good introduction to more complex and general objects such as reflection groups and Coxeter groups, Lie groups and Lie algebras, as well as their representations and geometry. In essence, learning about the symmetric group is very useful because it can lead to many different directions; and there are many beautiful things to discover with not much effort, given the fact that we are looking at an elementary object, which can be observed from many different perspectives.

Tentative syllabus. The structure of the symmetric group (the standard presentation, reduced words, the weak and strong Bruhat orders). Young tableaux and combinatorial algorithms related to the symmetric group (the Robinson-Schensted-Knuth correspondence, jeu de taquin, Knuth equivalence and the plactic monoid structure on words). Symmetric functions (various bases, a scalar product, Schur functions and their combinatorics, the Littlewood-Richardson rule for expanding the product of two Schur functions in the basis of Schur functions).

Prerequisites. The only prerequisite for this course is some elementary algebra (groups, linear algebra), without a specific course requirement.

Textbook. Sagan, Bruce E., The symmetric group. Representations, combinatorial algorithms, and symmetric functions. Second edition. Graduate Texts in Mathematics, 203. Springer-Verlag, New York, 2001.

Grading. There will be a number of problem sets, on which you are allowed to work together and ask for my help. Also, there will be a take-home final exam.

Pictures. The following pictures (taken from the paper "Structure of the Malvenuto-Reutenauer Hopf algebra of permutations" by M. Aguiar and F. Sottile) represent the permutahedron for n=4; the vertices correspond to permutations of 4 elements and the edges correspond to simple transpositions applied on the left.

Cristian Lenart, Department of Mathematics, ES 118, SUNY at Albany