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Representation Theory: A Combinatorial Viewpoint

Amritanshu Prasad
Publisher: 
Cambridge University Press
Publication Date: 
2015
Number of Pages: 
191
Format: 
Hardcover
Series: 
Cambridge Studies in Advanced Mathematics
Price: 
80.00
ISBN: 
9781107082052
Category: 
Textbook
[Reviewed by
Felipe Zaldivar
, on
04/2/2015
]

Every element of the symmetric group Sn in n letters has a unique decomposition as a finite product of disjoint cycles. The lengths of these cycles, arranged in decreasing sizes, is the cycle type of the given permutation, and thus the cycle type is a partition of the integer n. All permutations in the same conjugacy class have the same cycle type, and reciprocally. Partitions of an integer n correspond to Young diagrams of size n. Thus, the cycle type of a given permutation corresponds to a Young diagram.

It is well known that the number of irreducible representations of a finite group is the number of its conjugacy classes. Therefore, the number of irreducible representations of the symmetric group in n letters is the number of Young diagrams of size n, or equivalently, the number of partitions of n. Moreover, there is canonical way to construct all irreducible representations of the group Sn using these Young diagrams. More precisely, by some fillings of these diagrams with integers following some precise rules, either standard or semistandard Young tableaux, and more to the point, by equivalence classes of these tableaux under a natural action of the symmetric group, the so-called Young tabloids.

The main construction and result in this context is the Specht module associated to a given partition of the integer n and the proof that these Specht modules form a complete list of irreducible representations of the symmetric group Sn. The second part of this theorem is the computation of the complex dimension of the Specht module as the number of standard Young tableaux associated to the given partition. This computation can be done in several ways, but a nice combinatorial computation uses the hook-length formula of Frame, Robinson and Thrall.

This sketchy summary just highlights the close relation between representation theory of the symmetric group and algebraic combinatorics. We note that to avoid unpleasantries we are dealing here with the semisimple case, that is when the characteristic of the underlying field does not divide the order of given group.

The book under review devotes its first four chapters to this combinatorial approach to the representation theory of the symmetric group and some of its subgroups, for example for the alternating group, although the Specht modules are not explicitly mentioned. Chapter five is an introduction to the elementary theory of symmetric functions in n variables, obtaining bases for the algebra of symmetric polynomials, from the elementary symmetric polynomials to the Schur functions. Highlights here include the Jacobi-Trudi identities, the branching rules, and the hook-length formula obtained as a consequence of the Frobenius formula for the dimension of the irreducible representations of the symmetric group. Chapter six gives an introduction to the representation theory of the general linear group and of its diagonal subgroups.

This is an elementary textbook, with some overlap with the well-known book: The Symmetric Group. Representations, Combinatorial Algorithms, and Symmetric Functions, by Bruce Sagan (Springer, Second Edition, 2001). I might be biased but I like better the approach taken by W. Fulton in: Young Tableaux. With Applications to Representation Theory and Geometry. (Cambridge University Press, 1997) for the representation theory of the symmetric group and general linear groups — but, alas, that book is at a different level.


Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.

List of tables
Preface
1. Basic concepts of representation theory
2. Permutation representations
3. The RSK correspondence
4. Character twists
5. Symmetric functions
6. Representations of general linear groups
Hints and solutions to selected exercises
Suggestions for further reading
References
Index.

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