Macdonald functions associated to complex reflection groups
ABSTRACT Let W be the complex reflection group . In the author's previous paper [J. Algebra 245 (2001) 650–694], Hall–Littlewood functions associated to W were introduced. In the special case where W is a Weyl group of type Bn, they are closely related to Green polynomials of finite classical groups. In this paper, we introduce a two variables version of the above Hall–Littlewood functions, as a generalization of Macdonald functions associated to symmetric groups. A generalization of Macdonald operators is also constructed, and we characterize such functions by making use of Macdonald operators, assuming a certain conjecture.
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ABSTRACT: Green functions associated to complex reflection groups G(e,1,n) were discussed in the author's previous paper. In this paper, we consider the case of complex reflection groups W=G(e,p,n). Schur functions and Hall–Littlewood functions associated to W are introduced, and Green functions are described as the transition matrix between those two symmetric functions. Furthermore, it is shown that these Green functions are determined by means of Green functions associated to various G(e′,1,n′). Our result involves, as a special case, a combinatorial approach to the Green functions of type Dn.Journal of Algebra - J ALGEBRA. 01/2002; 258(2):563-598.
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ABSTRACT: Green functions of classical groups are determined by the data from Weyl groups and by certain combinatorial objects called symbols. Generalizing this, we define Green functions associated to complex reflection groups G(e, 1, n) and study their combinatorial properties. We construct Hall–Littlewood functions and Schur functions in our scheme and show that such Green functions are obtained as a transition matrix between those two symmetric functions, as in the case of GLn.Journal of Algebra. 01/2001;
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