Abstract
In this paper we work with power algebras associated to hyperplane arrangements. There are three main types of these algebras, namely, external, central, and internal zonotopal algebras. We classify all external algebras up to isomorphism in terms of zonotopes. Also, we prove that unimodular external zonotopal algebras are in one to one correspondence with regular matroids. For the case of central algebras we formulate a conjecture.
Highlights
In this paper we work with power algebras, which are the quotients of polynomial rings by power ideals
Postnikov [2]; it originates from the algebras generated by the curvature forms of tautological Hermitian linear bundles [4, 29], see papers [5, 6, 15, 16, 17, 23, 24, 27, 28, 30], where the quotient algebras by these ideals were discussed by details
In papers [16, 17] we obtained the analog of the Theorem 3 in the case of internal zonotopal algebras for totally unimodular matrices, see the definition below
Summary
In this paper we work with power algebras, which are the quotients of polynomial rings by power ideals. In papers [16, 17] we obtained the analog of the Theorem 3 in the case of internal zonotopal algebras for totally unimodular matrices, see the definition below. Dim(CGC) is equal to the number of trees in G (in the connected case) It is well-known that the number of lattice points (volume) of the corresponding zonotope and the number of forests (trees) of a graph are the same, see for example [7, 18]. In the paper [23], the K-theoretic filtrations of external and central graphical algebras were considered, see definitions there. The structure of this paper is as follows: in § 2 we present a classification of external zonotopal algebras and a conjecture for the central case; in § 3 we prove our classification
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