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The group G ( m , 1 , n ) consists of n -by- n monomial matrices whose entries are m th roots of unity. It is generated by n complex reflections acting on C n . The reflecting hyperplanes give rise to a (hyperplane) arrangement G ⊂ C n . The internal zonotopal algebra of an arrangement is a finite dimensional algebra first studied by Holtz and Ron. Its dimension is the number of bases of the associated matroid with zero internal activity. In this paper we study the structure of the internal zonotopal algebra of the Gale dual of the reflection arrangement of G ( m , 1 , n ) , as a representation of this group. Our main result is a formula for the top degree component as an induced representation. We also provide results on representation stability, a connection to the Whitehouse representation in type A, and an analog of decreasing trees in type B. [ABSTRACT FROM AUTHOR]

Abstract: The action of the symmetric group on the set of parking functions of size n has received a great deal of attention in algebraic combinatorics. We prove that the action of on extends to an action of . More precisely, we construct a graded -module such that the restriction of to is isomorphic to . We describe the -Frobenius characters of the module in all degrees and describe the -Frobenius characters of in extreme degrees. We give a bivariate generalization of our module whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization. [Copyright &y& Elsevier]

Abstract: Let be a finite sequence of vectors from a vector space over any field. We consider the subspace of spanned by , where is a subsequence of . A result of Orlik and Terao provides a doubly indexed direct sum decomposition of this space. The main theorem is that the resulting Hilbert series is the Tutte polynomial evaluation . Results of Ardila and Postnikov, Orlik and Terao, Terao, and Wagner are obtained as corollaries. [ABSTRACT FROM AUTHOR]