Symmetric functions: a beginner's course (Q6541434)

This book is a text on symmetric functions aimed at intermediate to advanced undergraduates. The focus is on combinatorial aspects of the theory and as such the book only expects prerequisite material from basic group theory, linear algebra, and enumerative combinatorics. In particular, the book avoids direct mention of applications of symmetric functions to fields such as representation theory and algebraic geometry but would prepare students to study those connections.\N\NThe first section of the book establishes the combinatorics of Young tableaux, the ring of symmetric functions, and Schur polynomials in particular. This includes the Pieri rule, Jacobi-Trudi identities, and the Hall inner product on \(\Lambda\). This section avoids the RSK correspondence, instead relying on the Lindström-Gessel-Viennot lemma to prove Littlewood's theorem. The second section considers the ``arrays'' of Danilov and Koshevoy. These are used to establish the RSK-correspondence and the Littlewood-Richardson rule for Schur polynomials. The final section generalizes to Schubert polynomials and their combinatorial aspects. This includes the Bergeron-Billey-Fomin-Kirillov theorem, an analogue of Littlewood's theorem using ``pipe dreams'' as the combinatorial tool.\N\NEach chapter of the book ends with exercises, and each section includes a set of exercises that explore tangential topics related to that section. (These are, mainly, skew Schur polynomials, the classical combinatorial definition of the RSK correspondence, and an alternative proof of the B-B-F-K theorem via raising and lowering operators.)\N\NThe general treatment in the book is somewhat brisk but offers a valuable resource as a treatment that moves toward the major combinatorial results in the theory quickly, preparing students for further study of the subject or the application of their choice. In particular, arrays and Schubert polynomials are not well-represented in texts aimed at students, so become far more accessible after this publication.