Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 10 September 2013


In view of the many applications of the theory of symmetric functions to representation theory it seems desirable to have a theory of symmetric functions in the spirit of Macdonald's book [Mac1979] for Weyl groups other than the symmetric group. Our approach to this problem is to find an algebraic structure which motivates each statement in the classical symmetric function theory. If this algebraic notion occurs across the board then this should indicate what the proper generalization is for other types. Note that from this point of view there may be several useful generalizations of a given concept depending on what symmetric function properties are desirable.

The goal of this paper is to offer a suggestion for the analogue of the basis of homogeneous symmetric functions for Weyl group symmetric functions. In this case the definition is motivated by the theory of centralizer algebras. The idea motivating the generalization is that it is really the Frobenius image of the homogeneous symmetric function that is the useful object. It is clear from the double centralizer theory that an analogue of the Frobenius characteristic map is a feature of the double centralizer mechanism, see [Ram1991-3]. With this point of view one finds an analogue of the "Jacobi-Trudi" formula in the work of Verma [Ver1984], Zelevinsky [Zel1988], Akin [Aki1989] and Goodman-Wallach [GWa1990]. In this paper I simply offer a mechanism by which to transfer their results to Weyl group symmetric functions.

I would like to thank D.-N. Verma for so patiently explaining the many many things about Lie algebras and their representations which he considered useful for me to know. In particular, he showed me the representation theoretic result, Theorem (4.6) in this paper, which motivates the definition of the homogeneous symmetric functions. I would like to thank N. Wallach for further useful discussions with me on this topic. I would like to thank the Tata Institute of Fundamental Research for their hospitality during my visit.

I have tried to organize this paper to motivate the concepts of symmetric functions by facts from representation theory. My hope is that this may serve as an introduction to representation theory for algebraic combinatorists who do not already know the subject. The paper begins with a brief resume of the classical symmetric function theory. In section 2 this theory is repeated except in the context of a general Weyl group. Section 3 is an attempt to explain the representation theoretic motivations behind the definitions of the monomial symmetric functions, the Weyl characters (analogues of the Schur functions) and the elementary symmetric functions. In Section 4 we prove the representation theoretic results which motivate the definition of the homogeneous symmetric functions. Section 5 introduces a formalism which is analoguous to the mechanism of the Cauchy identity in the classical theory and closes with some examples.

Notes and references

This is an excerpt of a paper entitled Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras, written by Arun Ram.

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