Orders and Polytropes: Matrix Algebras from Valuations

We apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the graduated orders introduced by Plesken and Zassenhaus. These are classified by the polytrope region. We advance the ideal theory of graduated orders by introducing their ideal class polytropes. This article emphasizes examples and computations. It offers first steps in the geometric combinatorics of endomorphism rings of configurations in affine buildings.


Introduction
Let K be a field with a surjective discrete valuation val : K → Z ∪ {∞}. We fix p ∈ K satisfying val(p) = 1. The valuation ring O K is the set of elements in K with non-negative valuation. This is a local ring with maximal ideal p = {x ∈ O K : val(x) > 0}. In our examples, K = Q is the field of rational numbers, with the p-adic valuation for some prime p.
We write K d×d for the ring of d × d matrices with entries in K. The map val is applied coordinatewise to matrices and vectors. For example, if K = Q with p = 2, then the vector x = (8/7, 5/12, 17) has val(x) = (3, −2, 0). In what follows, we often take X = (x ij ) to be a d × d matrix with nonzero entries in K. In this case, val(X) = (val(x ij )) is a matrix in Z d×d .
Fix any square matrix M = (m ij ) in Z d×d . This paper revolves around the interplay between the following two objects associated with M , one algebraic and the other geometric: This interplay is strongest and most interesting when Λ M is closed under multiplication. In this case, Λ M is a non-commutative ring of matrices. Such a ring is called an order in K d×d . The quotient space R d /R1 R d−1 is the usual setting for tropical geometry [10,12]. Note that Q M is a convex polytope in that space. It is also tropically convex, for both the min-plus algebra and the max-plus algebra. Following [11,15], we use the term polytrope for Q M . (1) The polytrope Q M is the set of solutions to the 12 inequalities u i − u j ≤ 1 for i = j. It is the 3-dimensional polytope shown in Figure 1. Namely, Q M is a rhombic dodecahedron, with 14 vertices, 24 edges and 12 facets. The vertices are the images in R 4 /R1 of the 14 vectors in {0, 1} 4 \{0, 1}. Vertices e i are blue, vertices e i +e j are yellow, and vertices e i +e j +e k are red.
The order Λ M consists of all 4 × 4 matrices with entries in the valuation ring O K whose off-diagonal elements lie in the maximal ideal p . We shall see in Theorem 16 that the blue and red vertices encode the injective modules and the projective modules of Λ M respectively. Figure 1: The polytrope Q M on the left is a rhombic dodecahedron. The four blue vertices and the four red vertices, highlighted on the right, will play a special role for the order Λ M .
The connection between algebra, geometry and combinatorics we present was pioneered by Plesken and Zassenhaus. Our primary source on their work is the book [13]. One objective of this article is to give an exposition of their results using the framework of tropical geometry [10,12]. But we also present a range of new results. Our presentation is organized as follows.
Section 2 concerns graduated orders in K d×d . In Propositions 6 and 7 we present linear inequalities that characterize these orders and the lattices they act on. These inequalities play an important role in tropical convexity, to be explained in Section 3. Theorem 10 gives a tropical matrix formula for the Plesken-Zassenhaus order of a collection of diagonal lattices.
In Section 4 we introduce polytrope regions. These are convex cones and polyhedra whose integer points represent graduated orders. Section 5 is concerned with (fractional) ideals in an order Λ M . These are parametrized by the ideal class polytrope Q M . In Section 6 we turn to Bruhat-Tits buildings and their chambers. While the present study is restricted to Plesken-Zassenhaus orders arising from one single apartment, it sets the stage for a general theory.
Several of the results in this article were found by computations. The codes and all data are made available at https://mathrepo.mis.mpg.de/OrdersPolytropes/index.html.

Graduated Orders
By a lattice in K d we mean a free O K -submodule of rank d. Two lattices L and L are equivalent if L = p n L for some n ∈ Z. We write [L] = {p n L : n ∈ Z} for the equivalence class of L. An order in K d×d is a lattice in the d 2 -dimensional vector space K d×d that is also a ring. Thus, every order contains the identity matrix. An order Λ is maximal if it is not properly contained in any other order. One example of a maximal order is the matrix ring This is spanned as an O K -lattice by the matrix units E ij where 1 ≤ i, j ≤ d. It is multiplicatively closed because E ij E jk = E ik . We begin with some standard facts found in [13]. The first is a natural bijection between lattice classes [L] in K d and maximal orders in K d×d . Proposition 2. Any order Λ in K d×d is contained in the endomorphism ring of a lattice L ⊂ K d . The maximal orders in K d×d are exactly the endomorphism rings of lattices L: then we obtain the following lattice in K d : Since Λ is multiplicatively closed, we have X j L ⊆ L for all j. Therefore Λ ⊆ End O K (L). Endomorphism rings of lattices are orders. Indeed, if L = gL 0 for g ∈ GL d (K), then This is a ring, and it is spanned as an O K -lattice by {gE ij g −1 : 1 ≤ i, j ≤ d}. This allows to conclude that the maximal orders are exactly the endomorphism rings of lattices.
is the maximal order in (2). Let M (u) denote the d × d matrix whose entry in position (i, j) equals u i − u j . Lemma 3. The endomorphism ring of the lattice L u is given by valuation inequalities: Proof. The elements of End O K (L u ) are the matrices X = g u Y g −1 u where Y ∈ O d×d K . Writing X = (x ij ) and Y = (y ij ), the equation X = g u Y g −1 u means that x ij = p u i −u j y ij for all i, j. The condition val(y ij ) ≥ 0 is equivalent to val(x ij ) ≥ u i − u j . Taking the conjunction over all (i, j), we conclude that val(Y ) ≥ 0 is equivalent to the desired inequality val(X) ≥ M (u).
The matrices M (u) are characterized by the following two properties. All diagonal entries are zero and the tropical rank is one, cf. [12,Section 5.3]. What happens if we replace M (u) in (3) by an arbitrary matrix M ∈ Z d×d ? Then we get the set Λ M from the Introduction.
Since val(X) = M and val(X 2 ) = 0, we have X ∈ Λ M but X 2 ∈ Λ M . So Λ M is not an order.
The inequalities derived in the next two propositions are the main points of this section. These results are due to Plesken [13]. He states them in [13, Definition II.2] and [13, Remark II.4]. The orders Λ M in Proposition 6 are called graduated orders in [13]. They are also known as tiled orders [7,9], split orders [14] or monomial orders [16]. A graduated order Λ M is in standard form if M ≥ 0 and m ij + m ji > 0 for i = j.
Proof. To prove the if direction, we assume (4). Our hypothesis m ii = 0 ensures that Λ M contains the identity matrix, so Λ M has a multiplicative unit. Suppose X, Y ∈ Λ M . Then the (i, k) entry of XY equals d j=1 x ij y jk . This is a scalar in K whose valuation is at least m ij + m jk for some index j. Hence it is greater than or equal to m ik since (4) holds.
For the only-if direction, suppose m ij + m jk < m ik . Then X = p m ij E ij and Y = p m jk E jk are in Λ M . However, XY = p m ij +m jk E ik is not in Λ M because its entry in position (i, k) has valuation less than m ik . Hence Λ M is not multiplicatively closed, so it is not an order.
Fix M that satisfies (4). The graduated order Λ M is an O K -subalgebra of K d×d . It is therefore natural to ask which lattices in K d are Λ M -stable.
Moreover, if u, u ∈ Z d satisfy (5), then the diagonal lattices L u and L u are isomorphic as Λ M -modules if and only if they are equivalent, i.e. u = u in the quotient space R d /R1.
Proof. Fix a lattice L and let u = (u 1 , . . . , u d ) be defined by we see that p m ij +u j e i lies in L u , and this implies m ij + u j ≥ u i . Hence (5) holds. Conversely, suppose that (5) holds. Then the generator p m ij E ij of Λ M maps each basis vector p u k e k of L u either to zero (if j = k), or to p m ik +u k e i ∈ L u . This proves the first assertion.
For the second assertion, let u, u ∈ Z d satisfy (5). Since multiplication by α ∈ K * is an isomorphism of O K -modules, the if-direction is clear. Conversely, if L u and L u are isomorphic, then there exists g ∈ GL d (K) such that L u = gL u and gX = Xg for all and therefore g is a multiple of the identity matrix.

Bi-tropical Convexity
We now develop the relationship between graduated orders and tropical mathematics [10,12]. Both the min-plus algebra ( R, ⊕ , ) and the max-plus algebra ( R, ⊕ , ) will be used. Its arithmetic operations are the minimum, maximum, and classical addition of real numbers: If M and N are real matrices, and the number of columns of M equals the number of rows of N , then we write M N and M N for their respective matrix products in these algebras. . We find that There are two flavors of tropical convexity [12, Section 5.2]. A subset of R d is min-convex if it is closed under linear combinations in the min-plus algebra, and max-convex if the same holds for the max-plus algebra. Thus convex sets are images of matrices under linear maps.
We are especially interested in bi-tropical convexity in the ambient space R d /R1. This is ubiquitous in [10, Section 5.4] and [12]. Joswig [10, Section 1.4] calls it the tropical projective torus. At a later stage, we also work in the corresponding matrix space Such a set is known as a polytrope in tropical geometry [11,12]. Other communities use the terms alcoved polytope and weighted digraph polytope. We note that Q M is both min-convex and max-convex [10,Proposition 5.30] and, being a polytope, it is also classically convex. Using tropical arithmetic, the linear inequalities in (4) can be written concisely as follows: Thus, M is min-plus idempotent. This holds for M in Example 8. Joswig's book [10, Section 3.3] uses the term Kleene star for matrices M ∈ R d×d 0 with (7). Propositions 6 and 7 imply: The lattice Λ M is an order in K d×d if and only if (7) holds. In this case, the integer points u in the polytrope Q M are in bijection with the isomorphism classes of Λ M -lattices L u . Here, by a Λ M -lattice we mean a Λ M -module that is also a lattice in K d .
Let Γ = {L 1 , . . . , L n } be a finite set of lattices in K d , which might be taken up to equivalence. The intersection of two orders in K d×d is again an order. Hence the intersection is an order in K d×d . We call PZ(Γ) the Plesken-Zassenhaus order of the configuration Γ.
In the following we assume that each L i is a diagonal lattice, i.e. L i = L u (i) for u (i) ∈ Z d . Our next result involves a curious mix of max-plus algebra and min-plus algebra.
Then its Plesken-Zassenhaus order P Z(Γ) coincides with the graduated order Λ M where This max-plus sum of tropical rank one matrices is min-plus idempotent, i.e. (4) and (7) hold.
Proof. We regard Γ as a configuration in R d /R1. By construction, M is the entrywise smallest matrix such that Γ is contained in the polytrope Q M . From [10, Lemma 3.25] the matrix M is a Kleene star, that is (4) and (7) hold. The intersection in (8) is defined by the conjunction of the n inequalities val(X) ≥ M (u (i) ), which is equivalent to val(X) ≥ M .
consists of the three red points in Figure 2.
The red diagram is their min-plus convex hull. This tropical triangle consists of a classical triangle together with three red line segments connected to Γ. This red min-plus triangle is not convex. The green shaded hexagon is the polytrope spanned by Γ. By [10, Remark 5.33], this is the geodesic convex hull of Γ. It equals Q M where M is computed by (9): The polytrope Q M is both a min-plus triangle and a max-plus triangle. Its min-plus vertices, shown in blue, are equal in R 3 /R1 to the columns of M . Its max-plus vertices, shown in red, are the points u (i) . These are equal in R 3 /R1 to the columns of −M t ; cf. Theorem 16. Moreover, the three green cells correspond to the collection of homothety classes of lattices contained in u (i) ⊕ u (j) and containing u (i) ⊕ u (j) , for each choice of i = j.
u (1) u (2) u ( Remark 12. All lattices L u for u ∈ Q M are indecomposable as Λ M -modules, cf. [13]. This is no longer true if R is enlarged to the tropical numbers R ∪ {∞}. The combinatorial theory of polytropes in [10] is set up for this extension, and it indeed makes sense to study orders Λ M with m ij = ∞. While we do not pursue this here, our approach would extend to that setting.

Polytrope Regions
We next introduce a cone that parametrizes all graduated orders Λ M . Following Tran [15], the polytrope region P d is the set of all min-plus idempotent matrices M ∈ R d×d 0 . Thus, P d is the (d 2 −d)-dimensional convex polyhedral cone defined by the linear inequalities in (4). The equations m ik = m ij + m jk define the cycle space of the complete bidirected graph K d . This is the lineality space of P d . Modulo this (d − 1)-dimensional space, the polytrope region P d is a pointed cone of dimension (d−1) 2 . We view it as a polytope of dimension d 2 − 2d. Each inequality m ik ≤ m ij + m jk is facet-defining, so the number of facets of It is interesting but difficult to list the vertices of P d and to explore the face lattice. The same problem was studied in [2] for the metric cone, which is the restriction of P d to the subspace of symmetric matrices in R d×d 0 . A website maintained by Antoine Deza [5] reports that the number of rays of the metric cone equals 3, 7, 25, 296, 55226, 119269588 for d = 3, 4, 5, 6, 7, 8. We here initiate the census for the polytrope region. The following tables report the size of the orbit, the number of incident facets, and a representative matrix  Proof. This was found by computations with Polymake [8]; see our mathrepo site.
Remark 15. The integer matrices M in the polytrope region P d represent the graduated orders Λ M ⊂ K d×d . The data above enables us to sample from these orders. A variant of P d that assumes nonnegativity constraints was studied in [6], which offers additional data. We also refer to [7] for a study of the cone of polytropes from the perspective of semiring theory.
Our next result relates the structure of a polytrope Q M to that of its graduated order Λ M .
Taking the direct sum of these d lattices gives the following identification of O K -modules: We see that L m (j) is a direct summand of the free rank one module Λ M , so it is projective. Conversely, let P be any indecomposable projective Λ M -module. Then P ⊕ Q ∼ = Λ r M for some module Q and some r ∈ Z >0 . The module Λ r M decomposes into r · d indecomposables, found by aggregating r copies of (10). By the Krull-Schmidt Theorem, such decompositions are unique up to isomorphism, and hence P is isomorphic to L m (j) for some j.
A Λ M -module P is projective if and only if Hom O K (P, O K ) is an injective Λ M -module, but now with the action on the right. The decomposition (10) dualizes gracefully. We derive the assertion for injective modules by similarly dualizing all steps in the argument above.
The 961 vertices come in 65 orbits under the S 4 -action. Among the simple vertices we find: The list of all vertices, and much more, is made available at our mathrepo site. Such data sets can be useful for comprehensive computational studies of O K -orders in K d×d . Figure 3: The regular hexagon has 36 extreme subpolytropes in ten symmetry classes.

Ideals
To better understand the order Λ M for M ∈ P d , we study its (fractional) ideals. By an ideal of Λ M we mean an additive subgroup I of Λ M such that Λ M I ⊆ I and Example 20. Fix X ∈ K d×d and consider the two-sided Λ M -module X = Λ M XΛ M = AXB : A, B ∈ Λ M . This is an ideal when X ∈ Λ M . If X ∈ Λ M then αX ∈ Λ M for some α ∈ K * . Hence, X is a fractional ideal. These are the principal (fractional) ideals of Λ M .
For all that follows, we assume that M ∈ P d is an integer matrix in standard form.
Proposition 21. The nonzero fractional ideals of the order Λ M are the sets of the form where N = (n ij ) is any matrix in Z d×d with N M = M N = N . This is equivalent to n ik ≤ n ij + m jk and n ik ≤ m ij + n jk for 1 ≤ i, j, k ≤ d.
Proof. The result is due to Plesken who states it in (viii) from [13,Remark II.4]. The min-plus matrix identity N M = N is equivalent to n ik ≤ n ij + m jk because m jj = 0.

Remark 22.
If N has zeros on its diagonal and satisfies (4) then I N = Λ N is an order, as before. However, among all lattices in K d×d , ideals are more general than orders. In particular, we generally have n ii = 0 for the matrices N in (12). Let Q M denote the set of matrices N in R d×d that satisfy the inequalities in (13). These inequalities are bounds on differences of matrix entries in N . We can thus regard Q M as a polytrope in R d×d /R1, where 1 = d i,j=1 E ij . The matrices N parameterizing the fractional ideals I N of Λ M (up to scaling) are the integer points of Q M . One checks directly that Q M is closed under both addition and multiplication of matrices in the min-plus algebra. Its product represents the multiplication of fractional ideals as the following proposition shows.
Proof The semigroup Q M has the neutral element M and each ideal class N ∈ Q M has a pseudo-inverse N given by the formula (14). With this data, we define the ideal class group The isomorphism types of these groups were computed using GAP; the code is at our mathrepo site. We do not know how this list continues for pyropes [11, §3] in higher dimensions.
We end this section with a conjecture about the geometry of G M inside Q M .
Conjecture 30. For any integer matrix M in the polytrope region P d , the elements in the ideal class polytrope G M are among the classical vertices of the ideal class polytrope Q M .

Towards the building
Affine buildings [1,17] provide a natural setting for orders and min-max convexity. The objects we discussed in this paper so far are associated to one apartment in this building, namely, that corresponding to the diagonal lattices. The aim of this section is to present this perspective and to lay the foundation for a general theory that goes beyond one apartment.
Hence the apartment is where g ∈ GL d (K) is the matrix with columns b 1 , . . . , b d . The standard apartment is the one associated with the standard basis (e 1 , . . . , e d ) of K d . The vertices of the standard apartment are the diagonal lattice classes [L u ] for u ∈ Z d . We identify this set of vertices with Z n /Z1. The general linear group GL d (K) acts on the building B d (K). This action preserves the simplicial complex structure. In fact, the action is transitive on lattice classes, on apartments and also on the chambers. The stabilizer of the standard lattice L 0 is the subgroup Starting from the standard chamber C 0 , there exist reflections s 0 , s 1 , . . . , s d−1 in GL d (K) that map C 0 to the d adjacent chambers in the standard apartment. For i ≥ 1, define s i by s i (e i ) = e i+1 , s i (e i+1 ) = e i and s i (e j ) = e j when j = i, i + 1.
The map s 0 is defined by s 0 (e i ) = e i for i = 2, . . . , d−1 and s 0 (e d ) = pe 1 , s 0 (e 1 ) = p −1 e d . The reflections s 0 , . . . , s d−1 are Coxeter generators for the affine Weyl group W = s 0 , . . . , s d−1 . The group W acts regularly on the chambers C in the standard apartment [3, § 1.5, Thm. 2]: for every C there is a unique w ∈ W such that C = wC 0 . The elements of W are the matrices h σ g u where h σ = (1 i=σ(j) ) i,j for σ ∈ S d , and u ∈ Z d with u 1 + · · · + u d = 0. Thus W is the semi-direct product of S d and the group of diagonal matrices g u whose exponents sum to 0.
Our primary object of interest is the Plesken-Zassenhaus order PZ(Γ) of a finite configuration Γ in the affine building B d (K). This is the intersection (8) of endomorphism rings. In this paper we studied the case when Γ lies in one apartment. In Theorem 10 we showed that PZ(Γ) = Λ M where M is the matrix in P d that encodes the min-max convex hull of Γ. This was used in Sections 4 and 5 to elucidate combinatorial and algebraic structures in PZ(Γ). A subsequent project will extend our results to arbitrary configurations Γ in B d (K).
We conclude this article with configurations given by two chambers C, C in B d (K). We are interested in the their order PZ(C ∪ C ). A fundamental fact about buildings states that any two chambers C, C lie in a common apartment, cf. [3,1]. Also, since the affine Weyl group W acts regularly on the chambers of the standard apartment, we can then reduce to the case where the two chambers in question are C 0 and wC 0 for some w = h σ g u ∈ W .
Example 32. The standard chamber C 0 is encoded by M 0 = 1≤i<j≤d E ij . The polytrope Q M 0 is a simplex. The order PZ(C 0 ) = Λ M 0 consists of all X ∈ O d×d K with x ij ∈ p for i < j.
Let D u = val(g u ) denote the tropical diagonal matrix with u 1 , . . . , u d on the diagonal and +∞ elsewhere. We also write P σ := val(h σ ) for the tropical permutation matrix given by σ.
We may ask for invariants of the orders PZ(C 0 ∪ wC 0 ) in terms of w ∈ W . Clearly, not all polytropes in an apartment arise as the min-max convex hull of two chambers. Which graduated orders are of the form PZ(C 0 ∪ wC 0 )? Which other elements w in the affine Weyl group W give rise to the same Plesken-Zassenhaus order PZ(C 0 ∪ wC 0 ) up to isomorphism?