Math 662: Seminar in Algebra
Title:  Symmetric Functions and Combinatorial Hopf Algebras

Instructor: Frank Sottile

Topics:  The first half will cover the classical theory of
 Symmetric functions and symmetric polynomials, in particular
 the topics:  Classical bases, including the Schur functions.
 Young Tableaux, Schensted insertion, Schutzenberger's
 jeux-de-taquin.  Pieri and Littlewood-Richardson rules.

 The later part of the course will study quasi-symmetric
 funtions and their role in combinatorial enumeration,
 specifically P-partitions and combinatorial Hopf algebras.
 We will also study generalizations of Schur functions, such
 as Schup P- and Q- functions and Hall-littlewood functions,
 and other Hopf algebras such as non-commutative symmetric
 functions, the Malvenuto-Reutenauer Hopf algebra of permutations
 and the Loday-Ronco Hopf algebra of trees.

 Because the later material is not found in any book,
 I will not assign a text for the course, but point out
 the main sources for the classical material, which include:
 Chapter 7 of Stanley's "Enumerative Combinatorics, Volume 2", 
 Chapter I  of Macdonald's "Symmetric functions and Hall 
 polynomials", and Chapter 4 (and part of chapter 3) of 
 Sagan's "The Symmetric group".   I will have extra copies of
 these books available for short-term loan.

 The prerequisite will be first semester of graduate algebra
 (but it would be good to be at least taking the second
  semester concurrently.)