Abstract
This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this theory.
Here we introduce a space of functions on a lattice which generalizes the space of quasipolynomials satisfying the difference equations associated to cocircuits of a sequence of vectors X, introduced by Dahmen and Micchelli [8]. This space \( \mathcal{F}(X) \) contains the partition function \( {\mathcal{P}_{(X)}} \). We prove a “localization formula” for any f in \( \mathcal{F}(X) \), inspired by Paradan's decomposition formula [12]. In particular, this implies a simple proof that the partition function \( {\mathcal{P}_{(X)}} \) is a quasi-polynomial on the Minkowski differences \( \mathfrak{c} - B(X) \), where c is a big cell and B(X) is the zonotope generated by the vectors in X, a result due essentially to Dahmen and Micchelli.
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Dedicated to Vladimir Morozov on the occasion of his 100th birthday
The first two authors are partially supported by the Cofin 40%, MIUR.
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De Concini, C., Procesi, C. & Vergne, M. Vector partition functions and generalized dahmen and micchelli spaces. Transformation Groups 15, 751–773 (2010). https://doi.org/10.1007/s00031-010-9102-9
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DOI: https://doi.org/10.1007/s00031-010-9102-9