Classification of external Zonotopal algebras

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.


Introduction
In this paper we work with power algebras, which are quotients of polynomial rings by power ideals. We consider zonotopal ideals, which are associated to zonotopes. These ideals were independently introduced in two different ways. There are three types of algebras. We will work with the definition from F. Ardila and A. Postnikov [2]; it comes from algebras generated by the curvature forms of tautological Hermitian linear bundles [3,28], see also papers [4,5,14,15,16,22,23,26,27,29], where people work with quotients algebras by these ideals. At the same time the definition and the name was established by O. Holtz and A. Ron [12]; it comes from Box-Splines and from Dahmen-Micchelli space [1,8,10], see also the papers [9,11,13,18,19,20,21,30].
Let A ∈ R n×m be a matrix of rank n. Denote by y 1 , . . . , y m ∈ R n the columns and by t 1 , . . . , t n ∈ R m the rows. For a matrix A, we define the zonotope A be the quotient algebra Key words and phrases. Commutative algebra, Power ideals, Zonotopes, Matroids, Lattice points.
where I

(k)
A is the zonotopal ideal generated by the polynomials p There are 3 main cases, where k = ±1 and 0; they were considered in [2,12].
A is the external zonotopal algebra for A; A is the central zonotopal algebra for A; is the internal zonotopal algebra for A.
Remark 1. The case k > 1 is not "zonotopal", because the idealÎ (k) generated by . They coincide only for the case where k ≤ 1.
In the case k ≤ −5, Hilbert series is not a specialization of the corresponding Tutte polynomial, see [2].
Theorem 1 (cf. [2,4,12,20], External [27], Central for graphs [26]). For a matrix A ∈ R n×m , the Hilbert series of zonotopal algebras are given by There are other definitions of external algebras from [27]. Let Φ m be the square-free commutative algebra generated by φ i , i ∈ [m], i.e., with relations . Theorem 2 (cf. [27]). The external algebra C Ex A is isomorphic to the subalgebra of Φ Ex A := Φ m generated by In papers [15,16] we constructed the analogue of Theorem 2 in the case of central and internal zonotopal algebras for totally unimodular matrices, see the definition below.
The main interesting examples of zonotopal algebras arise for totally unimodular matrices and for graphs. The matrix A is totally unimodular if any its minors is equal to ±1 or 0. In this case the total dimensions of the algebras have a nice interpretation.
Theorem 3 (cf. [12]). Let A ∈ R n×m be a totally unimodular matrix of rank n. Then the total dimension The main examples of totally unimodular matrices are graphs. Namely, let G be a graph on n vertices; then the incidence matrix of any orientation of G is totally unimodular. To construct the zonotopal algebra, we should forget exactly one row for each connected component of G. These algebras are independent (up to isomorphism) of the choice of orientations and rows.
These graphical algebras were considered in [14,15,22,23,27]. In the graphical case Theorem 3 can be written in graph theory terminology.
Theorem 4 (cf. [26]). Let G be a graph. Then the total dimension 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 [6,17] (Points of the zonotope correspond to score vectors).  The zonotope Z A has 6 facets. We need the set of its normals (note that parallel facets have the same normal up to a factor). There are 3 normals It is easy to check that m(η 1 ) = m(η 2 ) = 3 and m(η 3 ) = 2. Hence, It is easy to check that 10, 5, and 2 are exactly the number of lattice points, the area, and the number of interior lattice points of Z A , respectively. Furthermore, 10 and 5 are the number of forests and trees in G. In this case the Tutte polynomial is given by Anyone can check the formulas for the Hilbert series.
The following important property of external graphical algebras was proved in [23].
Theorem 5 (cf. [23]). Given two graphs G 1 and G 2 . Then the following are equivalent: • C Ex G 1 and C Ex G 2 are isomorphic as non-graded algebras; • C Ex G 1 and C Ex G 2 are isomorphic as graded algebras; • the graphical matroids M G 1 and M G 2 are isomorphic.
The following conjecture was formulated for the central case: Conjecture 1 (cf. [23]). Given two connected graphs G 1 and G 2 , the following are equivalent: • C C G 1 and C C G 2 are isomorphic as non-graded algebras; • C C G 1 and C C G 2 are isomorphic as graded algebras; • the bridge-free matroids M G 1 and M G 2 are isomorphic.
Here the bridge-free matroid of a graph is its graphical matroid after deleting all bridges.
In the paper [22], the K-theoretic filtration was considered, see definition there. Denote by K Ex G the K-theoretic filtration of C Ex.

G
Theorem 6 (cf. [22]). Given two graphs G 1 and G 2 without isolated vertices, the filtered algebras K Ex G 1 and K Ex G 2 are isomorphic if and only if G 1 and G 2 are isomorphic.
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.
Acknowledgments. This material is based upon work supported by the National Science Foundation under Grant DMS-1440140 while the author was staying at the Mathematical Sciences Research Institute in Berkeley, California, during the program "Geometric and Topological Combinatorics" in the fall 2017. He also would like to thank the participiants of the PA-seminar for their comments.

Main results
Definition 1. Two linear spaces V 1 ⊂ R m 1 and V 2 ⊂ R m 2 are called z-equivalent if m 1 = m 2 = m and there is an invertible diagonal matrix D ∈ R m×m and a permutation π ∈ S m such that The matrices A 1 ∈ R n 1 ×m 1 of rank n 1 and A 2 ∈ R n 2 ×m 2 of rank n 2 are called z-equivalent if the span of rows of A 1 is z-equivalent to the span of rows of A 2 .

Remark 2.
It is easy to see that z-equivalence is an equivalence relation.
In the case when A 1 and A 2 do not have proportional columns, we can say that the matrix A 1 is equivalent to A 2 if and only if their zonotopes are equivalent (since we can reconstruct the "matrix" from the zonotope in this case).
This equivalence is weaker than that of matroids. Proposition 7. If two matrices A 1 and A 2 are z-equivalent, then the matroids M A 1 and M A 2 are isomorphic.

It is easy to check that C Ex
A 1 and C Ex A 2 are isomorphic if A 1 and A 2 are z-equivalent. The converse also holds. Theorem 8. Let A 1 ∈ R n 1 ×m 1 and A 2 ∈ R n 2 ×m 2 be two matrices of rank n 1 and n 2 respectively. Then the following are equivalent: • C Ex A 1 and C Ex A 2 are isomorphic as non-graded algebras; • C Ex A 1 and C Ex A 2 are isomorphic as graded algebras; • A 1 and A 2 are z-equivalent. Corollary 1. Let A 1 ∈ R n 1 ×m 1 and A 2 ∈ R n 2 ×m 2 be two matrices of ranks n 1 and n 2 respectively, with isomorphic external algebras C Ex Then the matroids M A 1 and M A 2 are isomorphic.
The following theorems shows that unimodular external zonotopal algebras are in one to one correspondence with regular matroids.
Theorem 9. Let A 1 ∈ R n 1 ×m 1 and A 2 ∈ R n 2 ×m 2 be two unimodular matrices of rank n 1 and n 2 respectively. Then the following are equivalent:

• C Ex
A 1 and C Ex A 2 are isomorphic as non-graded algebras; • C Ex A 1 and C Ex A 2 are isomorphic as graded algebras; • A 1 and A 2 are z-equivalent.
• the matroids M A 1 and M A 2 are isomorphic.
Since for graphs we have a totally unimodular matrix, all graphical matroids are regular; the converse is almost true. Every regular matroid may be constructed by combining graphic matroids, cographic matroids, and a certain ten-element matroid R 10 , see [25] or the book [24]. In the graphical case the last theorem says that the algebra remembers graph up to 2-isomorphism, see [31].
For the central case, we can extend Conjecture 1 for all matrices. For a matrix A, we say that a column is a bridge-column if after deleting it the rank decreases.

Conjecture 2.
Let A 1 ∈ R n 1 ×m 1 and A 2 ∈ R n 2 ×m 2 be two matrices of ranks n 1 and n 2 respectively. Then the following are equivalent: • C C A 1 and C C A 2 are isomorphic as non-graded algebras; • C C A 1 and C C A 2 are isomorphic as graded algebras; is the submatrix of A i resulting after deleting all k i bridge-columns and those k i rows such that rk(A ′ i ) = n i − k i .

Proofs
Let B be a finite dimension algebra over R. We say that an element r = k i=1 a 2i a 2i+1 is reducible if a i ∈ B, i ∈ [2k] are nilpotent elements. For a nilpotent element a ∈ B we define the length ℓ(a) as the maximal ℓ such that a ℓ = 0.
Proof of Theorem 8. Clearly, we have 1 ⇐= 2 ⇐= 3, so we will prove 1 =⇒ 3. Let C Ex A be our algebra. We will work with the square-free definition, i.e., C Ex A is a subalgebra of Φ m , where m = max(ℓ(a) : a ∈ C Ex A ). (note that we can work with Φ m only theoretically, i.e., we do not know this embedding). We know which element is the unit, so we can chose basis x 1 , . . . , x n of nilpotents of C Ex A with the following property:

for any reducible r ∈ C Ex
A and x ∈ span{x 1 , . . . , x n }. Since we can define the algebra via some matrix A, then there is such basis.
Any element has the representation where a i,k ∈ R and r i is reducible. Let A ′ = {a i,k : (i, k) ∈ [n] × [m]} be the corresponding matrix. Our goal is to reconstruct A ′ up to zequivalece.
Consider the projective space P n−1 over R.
Define the set S of all non-zeroes s ∈ R such that there are i = j ∈ [n] and a non-zero t ∈ R for which It is easy to see that S is exactly the set , and a i,k , a j,k = 0 .
Then S is a finite set. Define a theoretical set S of rows of X as Any element of A is an element of S, so it is enough to find the multiplicity of any s ∈ S.
Consider the following partial order on elements of P n−1 : Note that if, for any s, we know the summary multiplication of all s ′ ≥ s in A, then we can calculate multiplicity of all elements. Given s ∈ S, then the summary multiplication of all s ′ ≥ s is equal to is the support of s; • c i , i ∈ I are generic with the linear condition i∈I c i s i = 0. Let us check it: Similarly we have ℓ i∈I c i x i = #{i ∈ k : i∈I c i a i,k = 0}.
Since b i , i ∈ I and c i , i ∈ I are generic with one condition i∈I c i s i = 0, we have the following property: if i∈I b i a i,k = 0 then i∈I c i a i,k = 0 if and only if (a 1,k , . . . , a n,k ) ≥ s.
Hence, we can compute the multiplicity of any s.
3 ⇐= 4. Let A 1 and A 2 be two totally unimodular matrices which give the same regular matroid (we assume that the order of elements are the same).
Also if M is a regular matroid, then all orientations of M differ only by reorientations (see Corollary 7.9.4 [7]). Hence, we can multiply some columns of A 2 by −1 and get A ′ 2 such that A 1 and A ′ 2 have the same oriented matroid.
It is well-known that if we have a totally unimodular matrix A i , then all minimal linear dependents of its columns have coefficients ±1. We get that matrices A 1 and A ′ 2 have linear dependents with the same coefficients and, hence,