1 Introduction

Let K be a field with a surjective discrete valuation \({{\,\mathrm{val}\,}}: K \rightarrow {\mathbb {Z}}\cup \{\infty \} \). We fix \(p \in K\) satisfying \(\mathrm{val}(p) = 1\). The valuation ring \({\mathcal {O}}_K\) is the set of elements in K with non-negative valuation. This is a local ring with maximal ideal \(\langle p \rangle = \{ x \in {\mathcal {O}}_K : {{\,\mathrm{val}\,}}(x) > 0\}\). In our examples, \(K = {\mathbb {Q}}\) is the field of rational numbers, with the p-adic valuation for some prime p.

We write \(K^{d \times d}\) for the ring of \(d \times d\) matrices with entries in K. The map \(\mathrm{val}\) is applied coordinatewise to matrices and vectors. For example, if \(K = {\mathbb {Q}}\) with \(p=2\), then the vector \(x = ( 8/7 , 5/12 , 17)\) has \(\mathrm{val}(x) = (3 , -2 , 0)\). In what follows, we often take \( X = (x_{ij})\) to be a \(d \times d\) matrix with nonzero entries in K. In this case, \(\mathrm{val}(X) = (\mathrm{val}(x_{ij}))\) is a matrix in \({\mathbb {Z}}^{d \times d}\).

Fix any square matrix \(M = (m_{ij})\) in \({\mathbb {Z}}^{d\times d}\). This paper revolves around the interplay between the following two objects associated with M, one algebraic and the other geometric:

  1. 1.

    the set \(\Lambda _M=\{ X \in K^{d \times d} : {{\,\mathrm{val}\,}}(X) \ge M \}\), an \({\mathcal {O}}_K\)-lattice in the vector space \(K^{d \times d}\);

  2. 2.

    the set \(Q_M = \{ u\in {\mathbb {R}}^d/{\mathbb {R}}\mathbf{1} : u_i - u_j \le m_{ij} for 1 \le i,j \le d \} \), where \(\mathbf{1} = (1,\ldots ,1)\).

This interplay is strongest and most interesting when \(\Lambda _M\) is closed under multiplication. In this case, \(\Lambda _M\) is a non-commutative ring of matrices. Such a ring is called an order in \(K^{d\times d}\). The quotient space \( {\mathbb {R}}^d/{\mathbb {R}}\mathbf{1} \simeq {\mathbb {R}}^{d-1}\) is the usual setting for tropical geometry (Joswig 2022; Maclagan and Sturmfels 2015). 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 (Joswig and Kulas 2010; Tran 2017), we use the term polytrope for \(Q_M\).

Example 1

For \(d\!=\!4\), fix the matrix with diagonal entries 0 and off-diagonal entries 1:

$$\begin{aligned} M \, = \, \begin{bmatrix} \, 0 &{} 1 &{} 1 &{} 1 \, \\ \, 1 &{} 0 &{} 1 &{} 1 \, \\ \, 1 &{} 1 &{} 0 &{} 1 \,\\ \, 1 &{} 1 &{} 1 &{} 0 \, \end{bmatrix}. \end{aligned}$$
(1)

The polytrope \(Q_M\) is the set of solutions to the 12 inequalities \(u_i - u_j \le 1\) for \(i \not = j\). It is the 3-dimensional polytope shown in Fig. 1. Namely, \(Q_M\) is a rhombic dodecahedron, with 14 vertices, 24 edges and 12 facets. The vertices are the images in \({\mathbb {R}}^4 / {\mathbb {R}}\mathbf{1}\) of the 14 vectors in \(\{0,1\}^4 \backslash \{ \mathbf{0},\mathbf{1}\}\). Vertices \(e_i \) are blue, vertices \(e_i + e_j\) are yellow, and vertices \(e_i+e_j+e_k\) are red.

The order \(\Lambda _M\) consists of all \(4 \times 4\) matrices with entries in the valuation ring \({\mathcal {O}}_K\) whose off-diagonal elements lie in the maximal ideal \(\langle p \rangle \). We shall see in Theorem 16 that the blue and red vertices encode the injective modules and the projective modules of \(\Lambda _M\) respectively.

Fig. 1
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 \(\Lambda _M\) (colour figure online)

Full size image

The connection between algebra, geometry and combinatorics we present was pioneered by Plesken and Zassenhaus. Our primary source on their work is the book (Plesken 1983). One objective of this article is to give an exposition of their results using the framework of tropical geometry (Joswig 2022; Maclagan and Sturmfels 2015). But we also present a range of new results. Our presentation is organized as follows.

Section 2 concerns graduated orders in \(K^{d \times d}\). 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 Sect. 3. Theorem 10 gives a tropical matrix formula for the Plesken-Zassenhaus order of a collection of diagonal lattices.

In Sect. 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 \(\Lambda _M\). These are parametrized by the ideal class polytrope \({\mathcal {Q}}_M\). In Sect. 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 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.

2 Graduated orders

By a lattice in \(K^d\) we mean a free \({\mathcal {O}}_K\)-submodule of rank d. Two lattices L and \(L'\) are equivalent if \(L' = p^n L \) for some \(n \in {\mathbb {Z}}\). We write \([L] = \{ p^n L : n \in {\mathbb {Z}}\}\) for the equivalence class of L. An order in \(K^{d\times d}\) is a lattice in the \(d^2\)-dimensional vector space \(K^{d\times d}\) that is also a ring. Thus, every order contains the identity matrix. An order \(\Lambda \) is maximal if it is not properly contained in any other order. One example of a maximal order is the matrix ring

$$\begin{aligned} {\mathcal {O}}_K^{d\times d} \,\,:=\,\, \{ X \in K^{d\times d} : {{\,\mathrm{val}\,}}(x_{ij}) \ge 0 \text{ for } \text{ all } 1\le i, j \le d \} . \end{aligned}$$

This is spanned as an \({\mathcal {O}}_K\)-lattice by the matrix units \(E_{ij} \) where \(1\le i, j \le d \). It is multiplicatively closed because \(E_{ij} E_{jk} = E_{ik}\). We begin with some standard facts found in Plesken (1983). The first is a natural bijection between lattice classes [L] in \(K^d\) and maximal orders in \(K^{d\times d}\).

Proposition 2

Any order \(\Lambda \) in \(K^{d\times d}\) is contained in the endomorphism ring of a lattice \(L\subset K^d\). The maximal orders in \(K^{d\times d}\) are exactly the endomorphism rings of lattices L:

$$\begin{aligned} {{\,\mathrm{\mathrm {End}}\,}}_{{\mathcal {O}}_K}(L) \,:=\, \{ X \in K^{d\times d} : X L \subseteq L \} . \end{aligned}$$

Two lattices L and \(L'\) in \(K^d\) are equivalent if and only if \(\,{{\,\mathrm{\mathrm {End}}\,}}_{{\mathcal {O}}_K}(L) = {{\,\mathrm{\mathrm {End}}\,}}_{{\mathcal {O}}_K}(L') \).

Proof

Let \(\Lambda = \bigoplus _{j=1}^{d^2} {\mathcal {O}}_K X_j \) be an order in \(K^{d \times d}\). If we apply the matrices \(X_j\) to the standard lattice \(L_0 = {\mathcal {O}}_K^d = \bigoplus _{i=1}^d {\mathcal {O}}_K e_i \), then we obtain the following lattice in \(K^d\):

$$\begin{aligned} L\,\,:=\,\,\sum _{j=1}^{d^2} X_j L_0 \, \,= \,\, \sum _{i=1}^d \sum _{j=1}^{d^2}{{\mathcal {O}}_K} X_j \,e_i . \end{aligned}$$

Since \(\Lambda \) is multiplicatively closed, we have \(X_j L \subseteq L\) for all j. Therefore \(\Lambda \subseteq {{\,\mathrm{\mathrm {End}}\,}}_{{\mathcal {O}}_K}(L)\).

Endomorphism rings of lattices are orders. Indeed, if \(L=g L_0\) for \(g \in {{\,\mathrm{GL}\,}}_d(K)\), then

$$\begin{aligned} {{\,\mathrm{\mathrm {End}}\,}}_{{\mathcal {O}}_K}(L)\,=\, g {{\,\mathrm{\mathrm {End}}\,}}_{{\mathcal {O}}_K}(L_0) g^{-1}\,=\,\,g \,{\mathcal {O}}_K^{d\times d} g^{-1}. \end{aligned}$$
(2)

This is a ring, and it is spanned as an \({\mathcal {O}}_K\)-lattice by \(\{ gE_{ij}g^{-1} : 1\le i,j\le d \}\). This allows to conclude that the maximal orders are exactly the endomorphism rings of lattices. \(\square \)

For \(u \in {\mathbb {Z}}^d\) we abbreviate \(\, g_u = \mathrm{diag}( p^{u_1},p^{u_2}, \ldots ,p^{u_d})\). This diagonal matrix transforms the standard lattice \({\mathcal {O}}_K^d\) to \(L_u = g_u {\mathcal {O}}_K^d\). The endomorphism ring \(\mathrm{End}_{{\mathcal {O}}_K}(L_u)\) is the maximal order in (2). Let M(u) denote the \(d \times d\) 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:

$$\begin{aligned} \mathrm{End}_{{\mathcal {O}}_K}(L_u) \,\, = \,\, \Lambda _{M(u)}\,= \, \{\, X \in K^{d \times d}\,: \, \mathrm{val}(X) \ge M(u)\} . \end{aligned}$$
(3)

Proof

The elements of \(\mathrm{End}_{{\mathcal {O}}_K}(L_u)\) are the matrices \(X = g_u Y g_u^{-1}\) where \(Y \in {\mathcal {O}}_K^{d \times d}\). Writing \(X = (x_{ij})\) and \(Y = (y_{ij})\), the equation \(X = g_u Y g_u^{-1}\) means that \(x_{ij} = p^{u_i-u_j} y_{ij} \) for all i, j. The condition \(\mathrm{val}(y_{ij}) \ge 0\) is equivalent to \(\mathrm{val}(x_{ij}) \ge u_i - u_j\). Taking the conjunction over all (i, j), we conclude that \(\mathrm{val}(Y) \ge 0\) is equivalent to the desired inequality \(\mathrm{val}(X) \ge M(u)\). \(\square \)

The matrices M(u) are characterized by the following two properties. All diagonal entries are zero and the tropical rank is one, cf. (Maclagan and Sturmfels 2015, Section 5.3). What happens if we replace M(u) in (3) by an arbitrary matrix \(M\in {\mathbb {Z}}^{d \times d}\)? Then we get the set \(\Lambda _M\) from the Introduction.

Remark 4

For any matrix \(M \in {\mathbb {Z}}^{d \times d}\), the set \(\Lambda _M\) is a lattice in \(K^{d \times d}\). It is generated as an \({\mathcal {O}}_K\)-module by the matrices \( p^{m_{ij}} E_{ij} \) for \(1 \le i,j \le d\). The lattice \(\Lambda _M\) may not be an order.

Write \({\mathbb {Z}}^{d \times d}_0\) for the set of integer matrices M with zeros on the diagonal, i.e. \(m_{ii} = 0\) for all i. If M lies in \({\mathbb {Z}}^{d \times d}_0 \) then \(\Lambda _M\) contains the identity matrix, but may still not be an order.

Example 5

Let \(K = {\mathbb {Q}}\) with the p-adic valuation, for some prime \(p \ge 5\). For \(d=3\), set

Since \(\mathrm{val}(X) = M\) and \(\mathrm{val}(X^2) = 0\), we have \(X \in \Lambda _M\) but \(X^2 \not \in \Lambda _M\). So \(\Lambda _M\) 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 (1983). He states them in Plesken (1983, Definition II.2) and (Plesken, 1983, Definition II.4). The orders \(\Lambda _M\) in Proposition 6 are called graduated orders in (Plesken, 1983, Remark II.4). They are also known as tiled orders (Dokuchaev et al. 2017; Jategaonkar 1974), split orders (Shemanske 2010) or monomial orders (Yang and Chia-Fu 2015). A graduated order \(\Lambda _M\) is in standard form if \(M \ge 0\) and \(m_{ij} + m_{ji} > 0\) for \(i \not = j\).

Proposition 6

Given \(M = (m_{ij})\) in \({\mathbb {Z}}^{d \times d}_0\), the lattice \(\Lambda _M\) is an order in \(K^{d \times d}\) if and only if

$$\begin{aligned} m_{ij} + m_{jk} \ge m_{ik} \quad \hbox {for all} \,\, \,1 \le i,j,k \le d. \end{aligned}$$
(4)

Proof

To prove the if direction, we assume (4). Our hypothesis \(m_{ii} = 0\) ensures that \(\Lambda _M\) contains the identity matrix, so \(\Lambda _M\) has a multiplicative unit. Suppose \(X,Y \in \Lambda _M\). Then the (i, k) entry of XY equals \(\sum _{j=1}^d x_{ij} y_{jk}\). 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 \(\Lambda _M\). However, \(XY = p^{m_{ij}+m_{jk}} E_{ik}\) is not in \(\Lambda _M\) because its entry in position (i, k) has valuation less than \(m_{ik}\). Hence \(\Lambda _M\) is not multiplicatively closed, so it is not an order. \(\square \)

Fix M that satisfies (4). The graduated order \(\Lambda _M\) is an \({\mathcal {O}}_K\)-subalgebra of \(K^{d\times d}\). It is therefore natural to ask which lattices in \(K^d\) are \(\Lambda _M\)-stable.

Proposition 7

A lattice L is stable under \(\Lambda _M\) if and only if \(L = L_u\) with \(u \in {\mathbb {Z}}^d\) that satisfies

$$\begin{aligned} u_i - u_j \,\le \, m_{ij} \quad \mathrm{for} \quad 1 \le i,j \le d . \end{aligned}$$
(5)

Moreover, if \(u,u' \in {\mathbb {Z}}^d\) satisfy (5), then the diagonal lattices \(L_u\) and \(L_{u'}\) are isomorphic as \(\Lambda _M\)-modules if and only if they are equivalent, i.e. \(u = u'\) in the quotient space \({\mathbb {R}}^d/{\mathbb {R}}\mathbf{1}\).

Proof

Fix a lattice L and let \(u = (u_1,\ldots ,u_d)\) be defined by \(u_i = \mathrm{min}\{ \mathrm{val}(b_i) : b \in L \}\). Then \(L \subseteq L_u\) because every \(b \in L\) is an \({\mathcal {O}}_K\)-linear combination of the standard basis of \(L_u\), namely \(\,b = \sum _{i=1}^d b_i e_i = \sum _{i=1}^d (b_i \,p^{-u_i}) \,p^{u_i} e_i \). Suppose that L is \(\Lambda _M\)-stable. Since \(m_{ii} = 0\), we have \(E_{ii} \in \Lambda _M\). Hence \(E_{ii}\, b = b_i e_i \in L\) for every \( b \in L\). This implies \(L_u \subseteq L\) and hence \(L = L_u\). Applying \(p^{m_{ij} } E_{ij} \in \Lambda _M\) to \(p^{u_j} e_j \in L_u\), we see that \(p^{m_{ij} + u_j} e_i\) lies in \(L_u\), and this implies \(m_{ij} + u_j \ge u_i\). Hence (5) holds. Conversely, suppose that (5) holds. Then the generator \(p^{m_{ij}} E_{ij}\) of \(\Lambda _M\) maps each basis vector \(p^{u_k} e_k\) of \(L_u\) either to zero (if \(j \not = k)\), or to \(p^{m_{ik}+u_k}e_i\in L_u\). This proves the first assertion.

For the second assertion, let \(u,u' \in {\mathbb {Z}}^d\) satisfy (5). Since multiplication by \(\alpha \in K^*\) is an isomorphism of \({\mathcal {O}}_K\)-modules, the if-direction is clear. Conversely, if \(L_u\) and \(L_{u'}\) are isomorphic, then there exists \(g \in \mathrm{GL}_d(K)\) such that \(L_{u'} = g L_u\) and \(g X = X g\) for all \(X \in \Lambda _M\). Pick \(s \in {\mathbb {Z}}_{>0}\) such that \(p^s {\mathcal {O}}_K^{d \times d} \subset \Lambda _M\). Then g commutes with every matrix in \(p^s {\mathcal {O}}_K^{d \times d}\). This implies that g is central in \({\mathcal {O}}_K^{d \times d}\), and therefore g is a multiple of the identity matrix. \(\square \)

3 Bi-tropical convexity

We now develop the relationship between graduated orders and tropical mathematics (Joswig 2022; Maclagan and Sturmfels 2015). Both the min-plus algebra \( (\,{\mathbb {R}}, \,\underline{\oplus }\,,\odot )\) and the max-plus algebra \( (\,{\mathbb {R}}, \,\overline{\oplus }\,,\odot )\) will be used. Its arithmetic operations are the minimum, maximum, and classical addition of real numbers:

$$\begin{aligned} a \,\underline{\oplus }\,b \, = \, \mathrm{min}(a,b) \, , \,\,\, a \,\overline{\oplus }\,b \, = \, \mathrm{max}(a,b) \, , \,\,\, a \odot b\, = \, a + b \quad \,\, \mathrm{for} \quad a,b \in {\mathbb {R}}. \end{aligned}$$

If M and N are real matrices, and the number of columns of M equals the number of rows of N, then we write \(M \,\underline{\odot }\,N\) and \(M \,\overline{\odot }\,N\) for their respective matrix products in these algebras.

Example 8

Consider the \(2 \times 2\) matrices . We find that

$$\begin{aligned} \begin{matrix} M \,\underline{\odot }\,M = \begin{bmatrix} 0 &{} 1 \\ 2 &{} 0 \end{bmatrix} \, , &{} M \,\underline{\odot }\,N = \begin{bmatrix} 1 &{} 0 \\ 0 &{} 0 \end{bmatrix} \, , &{} N \,\underline{\odot }\,M = \begin{bmatrix} 1 &{} 0 \\ 0 &{} 0 \end{bmatrix} \, , &{} N \,\underline{\odot }\,N = \begin{bmatrix} 0 &{} 0 \\ 0 &{} 0 \end{bmatrix} \, , \\ M \,\overline{\odot }\,M = \begin{bmatrix} 3 &{} 1 \\ 2 &{} 3 \end{bmatrix} \, , &{} M \,\overline{\odot }\,N = \begin{bmatrix} 1 &{} 1 \\ 3 &{} 2 \end{bmatrix} \, ,&{} N \,\overline{\odot }\,M = \begin{bmatrix} 2 &{} 2 \\ 2 &{} 1 \end{bmatrix} \, ,&{} N \,\overline{\odot }\,N = \begin{bmatrix} 2 &{} 1 \\ 1 &{} 0 \end{bmatrix} . \end{matrix} \end{aligned}$$

There are two flavors of tropical convexity (Maclagan and Sturmfels 2015, Section 5.2). A subset of \({\mathbb {R}}^d\) 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 \({\mathbb {R}}^d/{\mathbb {R}}\mathbf{1}\). This is ubiquitous in (Joswig, 2022, Section 5.4) and (Maclagan and Sturmfels, 2015). Joswig (2022, Section 1.4) calls it the tropical projective torus. At a later stage, we also work in the corresponding matrix space \({\mathbb {R}}^{d \times d}/ {\mathbb {R}}\mathbf{1}\).

Let \({\mathbb {R}}^{d \times d}_0\) denote the space of real \(d \times d\) matrices with zeros on the diagonal, which is a real \((d^2-d)\)-dimensional vector space with lattice \({\mathbb {Z}}^{d \times d}_0\). For \(M= (m_{ij}) \) in \({\mathbb {R}}^{d \times d}_0\), we define

$$\begin{aligned} Q_M \,\, = \,\, \bigl \{ u \,\in {\mathbb {R}}^d/{\mathbb {R}}\mathbf{1} \,:\, u_i - u_j \,\, \le \, m_{ij} \,\,\, \hbox {for} \,\, 1 \le i,j \le d \,\bigr \}. \end{aligned}$$
(6)

Such a set is known as a polytrope in tropical geometry (Joswig and Kulas 2010; Maclagan and Sturmfels 2015). Other communities use the terms alcoved polytope and weighted digraph polytope. We note that \(Q_M\) is both min-convex and max-convex (Joswig 2022, 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

$$\begin{aligned} M \,\underline{\odot }\,M \,= \, M. \end{aligned}$$
(7)

Thus, M is min-plus idempotent. This holds for M in Example 8. Joswig’s book (Joswig 2022, Section 3.3) uses the term Kleene star for matrices \(M \in {\mathbb {R}}^{d \times d}_0\) with (7). Propositions 6 and 7 imply:

Corollary 9

The lattice \(\Lambda _M\) is an order in \(K^{d \times d}\) 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 \(\Lambda _M\)-lattices \(L_u\). Here, by a \(\Lambda _M\)-lattice we mean a \(\Lambda _M\)-module that is also a lattice in \(K^d\).

Let \(\Gamma = \{ L_1, \ldots , L_n \}\) be a finite set of lattices in \(K^d\), which might be taken up to equivalence. The intersection of two orders in \(K^{d \times d}\) is again an order. Hence the intersection

$$\begin{aligned} \mathrm{PZ}(\Gamma ) \,\, = \,\, \mathrm{End}_{{\mathcal {O}}_K}(L_1) \,\cap \,\cdots \, \cap \,\mathrm{End}_{{\mathcal {O}}_K}(L_n) \end{aligned}$$
(8)

is an order in \(K^{d \times d}\). We call \({{\,\mathrm{\mathrm {PZ}}\,}}(\Gamma )\) the Plesken-Zassenhaus order of the configuration \(\Gamma \).

In the following we assume that each \(L_i\) is a diagonal lattice, i.e. \(L_i = L_{u^{(i)}}\) for \(u^{(i)} \in {\mathbb {Z}}^d\). Our next result involves a curious mix of max-plus algebra and min-plus algebra.

Theorem 10

Let \(\,\Gamma = \{L_{u^{(1)}}, \ldots , L_{u^{(n)}}\}\,\) be any configuration of diagonal lattices in \(K^d\). Then its Plesken-Zassenhaus order \(PZ(\Gamma )\) coincides with the graduated order \(\Lambda _M\) where

$$\begin{aligned} M \,\,\, = \,\,\, M(u^{(1)}) \,\,\overline{\oplus }\,\, M(u^{(2)}) \,\,\overline{\oplus }\,\,\cdots \, \,\overline{\oplus }\,\, M(u^{(n)}) . \end{aligned}$$
(9)

This max-plus sum of tropical rank one matrices is min-plus idempotent, i.e. (4) and (7) hold.

Proof

We regard \(\Gamma \) as a configuration in \({\mathbb {R}}^d/{\mathbb {R}}\mathbf{1}\). By construction, M is the entrywise smallest matrix such that \(\Gamma \) is, contained in the polytrope \(Q_M\). From (Joswig, 2022, 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 \(\mathrm{val}(X) \ge M(u^{(i)})\), which is equivalent to \(\mathrm{val}(X) \ge M\). \(\square \)

Fig. 2
figure 2

A polytrope with three min-plus vertices (blue) and three max-plus vertices (red) (colour figure online)

Full size image

Example 11

For \(d=n=3\), fix \(u^{(1)} = (-2 , -1,0) \), \( u^{(2)} = (2 , 1,0) \), \(u^{(3)} = (-1 , 3,0)\,\) in \({\mathbb {R}}^3/{\mathbb {R}}\mathbf{1}\). The configuration \(\Gamma = \{ u^{(1)}, u^{(2)},u^{(3)}\}\) consists of the three red points in Fig. 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 \(\Gamma \). This red min-plus triangle is not convex. The green shaded hexagon is the polytrope spanned by \(\Gamma \). By (Joswig, 2022, Remark 5.33), this is the geodesic convex hull of \(\Gamma \). 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 \({\mathbb {R}}^3/{\mathbb {R}}\mathbf{1}\) to the columns of M. Its max-plus vertices, shown in red, are the points \(u^{(i)}\). These are equal in \({\mathbb {R}}^3/{\mathbb {R}}\mathbf{1}\) 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 \(u^{(i)}\,\underline{\oplus }\,u^{(j)}\) and containing \(u^{(i)}\,\overline{\oplus }\,u^{(j)}\), for each choice of \(i\ne j\).

Remark 12

All lattices \(L_u\) for \(u \in Q_M\) are indecomposable as \(\Lambda _M\)-modules, cf. (Plesken, 1983). This is no longer true if \({\mathbb {R}}\) is enlarged to the tropical numbers \({\mathbb {R}}\cup \{ \infty \}\). The combinatorial theory of polytropes in (Joswig, 2022) is set up for this extension, and it indeed makes sense to study orders \(\Lambda _M\) with \(m_{ij} = \infty \). While we do not pursue this here, our approach would extend to that setting.

Example 13

Set \(d=4\). The rhombic dodecahedron in Example 1 was called the pyrope in (Joswig and Kulas, 2010, Figure 4) and can be seen as the unit ball with respect to the tropical metric, cf. (Cohen et al. 2004, §3.3). This \(Q_M\) is a tropical tetrahedron for both min-convexity and max-convexity. The respective vertices are shown in red and blue in Fig. 1. We have \(\Lambda _M = \mathrm{PZ}(\Gamma )\) where \(\Gamma \) is either set of four vertices. The \(\Lambda _M\)-lattices \(L_u\) correspond to the 15 integer points in \(Q_M\).

4 Polytrope regions

We next introduce a cone that parametrizes all graduated orders \(\Lambda _M\). Following (Tran, 2017), the polytrope region \({\mathcal {P}}_d \) is the set of all min-plus idempotent matrices \(M \in {\mathbb {R}}^{d \times d}_0\). Thus, \({\mathcal {P}}_d\) 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 \({\mathcal {K}}_d\). This is the lineality space of \({\mathcal {P}}_d\). Modulo this \((d-1)\)-dimensional space, the polytrope region \({\mathcal {P}}_d\) is a pointed cone of dimension \((d{-}1)^2\). We view it as a polytope of dimension \(d^2-2d\). Each inequality \(m_{ik} \le m_{ij} + m_{jk}\) is facet-defining, so the number of facets of \({\mathcal {P}}_d\) is \(d(d-1)(d-2)\).

It is interesting but difficult to list the vertices of \({\mathcal {P}}_d\) and to explore the face lattice. The same problem was studied by Avis (1980) for the metric cone, which is the restriction of \({\mathcal {P}}_d\) to the subspace of symmetric matrices in \({\mathbb {R}}^{d \times d}_0\). A website maintained by Antoine Deza (2021) 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 \([m_{ij}]\). Here orbit and representatives refer to the natural action of the symmetric group \(S_d\) on \({\mathcal {P}}_d\). The matrices \([m_{ij}]\) in \({\mathbb {Z}}^{3 \times 3}_0\) are written in the vectorized format \([m_{12}m_{13}m_{21}m_{23}m_{31}m_{32}]\).

Proposition 14

The polytope \({\mathcal {P}}_3\) is a bypramid, with f-vector (5, 9, 6). Its five vertices are

$$\begin{aligned} 3 \,,\, 4\ [001100]\ \ and \ \ 2\, ,\, 3 \ [001110]. \end{aligned}$$

The polytope \({\mathcal {P}}_4\) has the f-vector (37, 327, 1140, 1902, 1680, 808, 204, 24). Its 37 vertices are

$$\begin{aligned} \!\!\! \begin{matrix} 12, 10 \!\!\! &{} [1 1 1 0 1 1 0 0 1 0 0 1] &{} \quad 6 , 12 \!\!\! &{} [1 1 1 0 1 1 0 0 1 0 0 0] &{} \quad 12 , 14 \!\!\! &{} [0 1 1 0 1 1 0 0 1 0 0 0] \\ 3 , 16 \!\!\! &{} [0 1 1 0 1 1 0 0 0 0 0 0] &{} \quad 4 ,18 \!\!\! &{} [1 1 1 0 0 0 0 0 0 0 0 0] . \end{matrix} \end{aligned}$$

The corresponding polytropes \(Q_M\) are pyramid, tetrahedron, triangle, segment, and segment. The 15-dimensional polytope \({\mathcal {P}}_5\) has 2333 vertices in 33 symmetry classes. These classes are

$$\begin{aligned} \!\!\! \begin{matrix} 5 , 48 \!\!\! &{} [00000000000000001111] &{} 10 , 18 \!\!\! &{} [00001001211121111100] &{} 10 , 42 \!\!\! &{} [00000000000011101110] \\ 20 , 15 \!\!\! &{} [00002012323231012201] &{} 20 , 21 \!\!\! &{} [00001000110021112111] &{} 20 , 39 \!\!\! &{} [00000000000011101111] \\ 24 , 20 \!\!\! &{} [00001001210122111110] &{} 24 , 30 \!\!\! &{} [00001000110011101111] &{} 30 , 24 \!\!\! &{} [00001000110121111110] \\ 30 , 30 \!\!\! &{} [00000000110011111111] &{} 30 , 30 \!\!\! &{} [00000000110111111110] &{} 30 , 36 \!\!\! &{} [00000000110011001111] \\ 40 , 18 \!\!\! &{} [00002000221222212212] &{} 60 , 18 \!\!\! &{} [00001000210122112110] &{} 60 , 18 \!\!\! &{} [00001001210122121100] \\ 60 , 22 \!\!\! &{} [00001000110122111110] &{} 60 , 27 \!\!\! &{} [00001000110011102111] &{} 60 , 29 \!\!\! &{} [00000000110011102211] \\ 60 , 33 \!\!\! &{} [00000000110011101111] &{} 120 , 16 \!\!\! &{} [00001001220132122110] &{} 120 , 17 \!\!\! &{} [00001001210122122110] \\ 120 , 18 \!\!\! &{} [00001001210122112110] &{} 120 , 18 \!\!\! &{} [00001001210122122210] &{} 120 , 18 \!\!\! &{} [00001001210222122110] \\ 120 , 18 \!\!\! &{} [00001001220132213210] &{} 120 , 19 \!\!\! &{} [00001000210022103221] &{} 120 , 19 \!\!\! &{} [00001000210122122110] \\ 120 , 19 \!\!\! &{} [00001001210122212210] &{} 120 , 22 \!\!\! &{} [00001000110021102221] &{} 120 , 22 \!\!\! &{} [00001000110122121110] \\ 120 , 23 \!\!\! &{} [00001000110021102211] &{} 120 , 23 \!\!\! &{} [00001000110021102222] &{} 120 , 25 \!\!\! &{} [00001000110011102211] \end{matrix} \end{aligned}$$

Proof

This was found by computations with Polymake (Gawrilow and Joswig 2000); see our mathrepo site. \(\square \)

Remark 15

The integer matrices M in the polytrope region \({\mathcal {P}}_d\) represent the graduated orders \(\Lambda _M \subset K^{d \times d}\). The data above enables us to sample from these orders. A variant of \({\mathcal {P}}_d\) that assumes nonnegativity constraints was studied in (Deza et al. 2002), which offers additional data. We also refer to (Dokuchaev et al. 2017) 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 \(\Lambda _M\).

Theorem 16

Let \(M \in {\mathcal {P}}_d\) be in standard form. The \((d-1)\)-dimensional polytrope \(Q_M\) is both a min-plus simplex and a max-plus simplex. The min-plus vertices u are the columns of M. They represent precisely those modules \(L_u\) over the order \(\Lambda _M\) that are projective. The max-plus vertices v are the columns of \(-M^t\), and they represent the injective \(\Lambda _M\)-modules \(L_v\).

Proof

Thanks to (Joswig and Kulas, 2010, Theorem 7), full-dimensional polytropes are tropical simplices, with vertices given by the columns of the defining matrix M. We know from bi-tropical convexity (Joswig 2022, Proposition 5.30) that \(Q_M\) is both min-convex and max-convex, so it is a simplex in both ways. This duality corresponds to swapping M with its negative transpose \(-M^t\). Note its appearence in (Maclagan and Sturmfels, 2015, Theorem 5.2.21). The connection to projective and injective modules appears in parts (v) and (vii) of (Plesken, 1983, Remark II.4). For completeness, we sketch a proof.

Recall that a module is projective if and only if it is a direct summand of a free module. Let \(m^{(1)},\ldots ,m^{(d)}\) denote the columns of M. The lattice associated to the j-th column equals

$$\begin{aligned} L_{m^{(j)}} \,\,\,= \,\,\, \bigl \{\,x \in K^d \,:\, \mathrm{val}(x_i) \ge m_{ij} \,\,\, \hbox {for}\,\, i = 1,\ldots ,d \,\bigr \}. \end{aligned}$$

Taking the direct sum of these d lattices gives the following identification of \({\mathcal {O}}_K\)-modules:

$$\begin{aligned} \Lambda _M \,\, = \,\, L_{m^{(1)}} \,\oplus \,L_{m^{(2)}} \,\oplus \,\cdots \,\oplus \,L_{m^{(d)}} . \end{aligned}$$
(10)

We see that \(L_{m^{(j)}}\) is a direct summand of the free rank one module \(\Lambda _M\), so it is projective.

Conversely, let P be any indecomposable projective \(\Lambda _M\)-module. Then \(P \oplus Q \cong \Lambda _M^r\) for some module Q and some \(r \in {\mathbb {Z}}_{>0}\). The module \(\Lambda _M^r\) decomposes into \(r\cdot d\) 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 \(\Lambda _M\)-module P is projective if and only if \(\mathrm{Hom}_{{\mathcal {O}}_K}(P,{\mathcal {O}}_K)\) is an injective \(\Lambda _M\)-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. \(\square \)

In relation to Theorem 16 we remark that the columns and negative rows of M also have a natural interpretation as potentials in combinatorial optimization; cf. (Joswig 2022, Theorem 3.26).

Example 17

The columns of the matrix M in Example 1 are the negated unit vectors \(-e_i\). The columns of \(-M^t\) are the unit vectors \(e_i\). Our color coding in Fig. 1 exhibits the two structures of \(Q_M\) as a tropical tetrahedron in \({\mathbb {R}}^4/{\mathbb {R}}\mathbf{1}\). The four red points are the min-plus vertices, giving the projective \(\Lambda _M\)-modules. The four blue points are the max-plus vertices.

Given a min-plus idempotent matrix \(M\in \mathcal {P}_d\), its truncated polytrope region is

$$\begin{aligned} {\mathcal {P}}_d(M) \, = \, \{ N \in {\mathcal {P}}_d \,: \, N \le M \}. \end{aligned}$$
(11)

This polytope has dimension \(d^2-d\) if M is in the interior of \({\mathcal {P}}_d\). It parametrizes all subpolytropes of \(Q_M\), i.e. all the polytropes \(Q_N\) contained in \(Q_M\), as the following lemma shows.

Lemma 18

Given matrices M in \({\mathcal {P}}_d\) and N in \({\mathbb {R}}_0^{d\times d}\) such that \(Q_N\subseteq Q_M\), there exists a matrix C in the truncated polytrope region \({\mathcal {P}}_d(M)\) such that \(Q_N=Q_C\).

Proof

For each choice of i and j, we define \(c_{ij}=\max \{u_i-u_j : u\in Q_N\}\). The matrix \(C=(c_{ij})\) lives in \({\mathbb {R}}_0^{d\times d}\) and has the property that \(Q_N=Q_C\). Moreover, since \(Q_N\) is contained in \(Q_M\), we have \(C\le M\). The fact that \(C\,\underline{\odot }\,C=C\) follows from the definition of the \(c_{ij}\)’s and (4). In particular, C belongs to the truncated polytrope region \({\mathcal {P}}_d(M)\). \(\square \)

On the algebraic side, \({\mathcal {P}}_d(M)\) parametrizes all \({\mathcal {O}}_K\)-orders \(\Lambda _N\) that contain the given order \(\Lambda _M\). Here M and N are assumed to be integer matrices. In particular, the integer points u in \(Q_M\) correspond to maximal orders \(\Lambda _{M(u)} = {{\,\mathrm{\mathrm {End}}\,}}_{{\mathcal {O}}_K}(L_u) \) that contain \(\Lambda _M\); cf. Proposition 2.

Example 19

Let M be the \(d \times d\) matrix with entries 0 on the diagonal and 1 off the diagonal. Thus \(Q_M\) is the pyrope (Joswig and Kulas 2010, § 3). We consider two cases: the hexagon \((d=3)\) and Example 1\((d=4)\). The truncated polytrope region \({\mathcal {P}}_d(M)\) classifies subpolytropes of \(Q_M\).

\( {d=3}\): The 6-dimensional polytope \({\mathcal {P}}_3(M)\) has the f-vector (36, 132, 199, 151, 60, 12). Its 36 vertices come in ten symmetry classes. We list the corresponding \(3 \times 3\) matrices:

$$\begin{aligned} \!\!\! \begin{matrix} 1, 6\,&{} [1, \!1, \! 1, \!1, \!1,\! 1] &{}2, 6&{} [1, \! \frac{1}{2}, \!\frac{1}{2},\! 1,\! 1,\! \frac{1}{2}] &{}3, 8&{} [0, \!-1\!, 0,\! -\!1, 1, 1] &{} 3, 8&{} [1, 0, \!-\!1,\! -\!1, 0, 1] &{} 3, 8&{} [1, \!0, \!1,\! 1,\! 0,\! 1] \\ 3, 6&{} [1, \! 1,\! 1,\! 1,\! 0,\! 0] &{} 3, 6&{} [0,\! 1,\! 1,\! 1,\! 1,\! 0] &{}6, 7&{} [0, -1, 1, 0, 1, 1] &{}6, 7&{} [1, 1, 1, 1, 0, 1] &{} 6, 6&{} [0, \! 0,\! 1,\! 1,\! 1,\! 0] \end{matrix} \!\!\!\end{aligned}$$

These polytropes are shown in red in Fig. 3. Our classification into \(S_3\)-orbits is finer than that from symmetries of the hexagon \(Q_M\), which leads to only eight orbits. For us, this classification is more natural because it reflects algebraic properties of orders. It distinguishes min-plus vertices from max-plus vertices of \(Q_M\). The polytope \({\mathcal {P}}_3(M)\) has 41 integer points, so there are 41 orders containing \(\Lambda _M\). In addition to 34 integer vertices, there are seven interior integer points, namely [0, 0, 0, 0, 0, 0] and six like [0, 0, 0, 0, 1, 1], not seen in Fig. 3.

Fig. 3
figure 3

The regular hexagon has 36 extreme subpolytropes in ten symmetry classes

Full size image

\( {d=4}\): The truncated polytrope region \({\mathcal {P}}_4(M)\) for (1) is 12-dimensional. Its f-vector is

$$\begin{aligned} (961, 17426, 103780, 304328, 517293, 549723, 377520, 168720, 48417, 8620, 894, 48 ). \end{aligned}$$

The 961 vertices come in 65 orbits under the \(S_4\)-action. Among the simple vertices we find:

$$\begin{aligned} \begin{matrix} 1, 12 &{} [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] \quad &{} \qquad &{} 8, 12&{} [1, 1, 1, 1, \frac{1}{2}, 1, 1, 1, \frac{1}{2}, 1, \frac{1}{2}, 1] \\ 4, 27&{} [1, 1, 1, -1, 0, 0, -1, 0, 0, -1, 0, 0] \quad &{} \qquad &{} 4, 27&{} [-1, -1, -1, 1, 0, 0, 1, 0, 0, 1, 0, 0]\\ \end{matrix} \end{aligned}$$

The list of all vertices, and much more, is made available at our mathrepo site. Such data sets can be useful for computational studies of \({\mathcal {O}}_K\)-orders in \(K^{d \times d}\).

5 Ideals

To better understand the order \(\Lambda _M\) for \(M \in {\mathcal {P}}_d\), we study its (fractional) ideals. By an ideal of \(\Lambda _M\) we mean an additive subgroup I of \(\Lambda _M\) such that \(\Lambda _M I \subseteq I\) and \(I \Lambda _M \subseteq I\). A fractional ideal of \(\Lambda _M\) is a (two sided) \(\Lambda _M\)-submodule J of \(K^{d \times d}\) such that \(\alpha J \subset \Lambda _M\) for some \(\alpha \in K^*\).

Example 20

Fix \(X \in K^{d \times d}\) and consider the two-sided \(\Lambda _M\)-module

$$\langle X \rangle =\, \Lambda _M X \Lambda _M \,= \,\big \{ A X B\, :A,B \in \Lambda _M \big \}.$$

This is an ideal when \(X \in \Lambda _M\). If \(X \not \in \Lambda _M\) then \(\alpha X \in \Lambda _M\) for some \(\alpha \in K^*\). Hence, \(\langle X \rangle \) is a fractional ideal. These are the principal (fractional) ideals of \(\Lambda _M\).

For all that follows, we assume that \(M \in {\mathcal {P}}_d\) is an integer matrix in standard form.

Proposition 21

The nonzero fractional ideals of the order \(\Lambda _M\) are the sets of the form

$$\begin{aligned} I_N \,\, = \,\, \bigl \{ \,X \in K^{d \times d} :{{\,\mathrm{val}\,}}(X) \ge N \,\bigr \}, \end{aligned}$$
(12)

where \(N = (n_{ij}) \) is any matrix in \( \,{\mathbb {Z}}^{d \times d }\) with \( N \,\underline{\odot }\,M = M \,\underline{\odot }\,N = N\). This is equivalent to

$$\begin{aligned} n_{ik} \le n_{ij} + m_{jk} \quad \mathrm{and} \quad n_{ik} \le m_{ij} + n_{jk} \qquad \mathrm{for} \quad 1 \le i,j,k \le d. \end{aligned}$$
(13)

Proof

The result appears in (viii) from (Plesken, 1983, Remark II.4). The min-plus matrix identity \(N \,\underline{\odot }\,M = N\) is equivalent to \(n_{ik} \le n_{ij} + m_{jk} \) because \(m_{jj} = 0 \). \(\square \)

Remark 22

If N has zeros on its diagonal and satisfies (4) then \(I_N = \Lambda _N\) is an order, as before. However, among all lattices in \(K^{d \times d}\), ideals are more general than orders. In particular, we generally have \(n_{ii} \not = 0\) for the matrices N in (12). A fractional ideal \(I_N\) is an ideal in \(\Lambda _M\) if and only if \(N \ge M\). If this holds then the polytrope \(Q_N\) is contained in \(Q_M\).

Example 23

The Jacobson radical of the order \( \Lambda _M\) is the ideal \(\mathrm{Jac}(\Lambda _M) = I_{M+\mathrm{Id}_d}\). Here \(\mathrm{Id}_d\) is the identity matrix. The quotient of \(\Lambda _M\) by its Jacobson radical is the product of residue fields \(\Lambda _M/\mathrm{Jac}(\Lambda _M)\cong ({\mathcal {O}}_K/\langle p \rangle )^d\). See (i) in (Plesken, 1983, Remark II.4) for more details.

Let \({\mathcal {Q}}_M\) denote the set of matrices N in \({\mathbb {R}}^{d \times d}\) that satisfy the inequalities in (13). These inequalities are bounds on differences of matrix entries in N. We can thus regard \({\mathcal {Q}}_M\) as a polytrope in \({\mathbb {R}}^{d \times d} / {\mathbb {R}}\mathbf{1}\), where \(\mathbf{1} = \sum _{i,j=1}^d E_{ij}\). The matrices N parameterizing the fractional ideals \(I_N\) of \(\Lambda _M\) (up to scaling) are the integer points of \({\mathcal {Q}}_M\). One checks directly that \({\mathcal {Q}}_M\) is closed under both addition and multiplication of matrices in the min-plus algebra. Its product \(\,\underline{\odot }\,\) represents the multiplication of fractional ideals as the following proposition shows.

Proposition 24

If \(M \!\in \! {\mathcal {P}}_d\) is in standard form and \(N,N' \!\in {\mathcal {Q}}_M\) then \(I_{N} I_{N'}\!=\! I_{N \,\underline{\odot }\,N'}\).

Proof

Let \(X \in I_N, Y\in I_{N'}\). The inequalities \({{\,\mathrm{val}\,}}(X) \ge N,\,{{\,\mathrm{val}\,}}(Y) \ge N'\) imply \({{\,\mathrm{val}\,}}(XY) \ge {{\,\mathrm{val}\,}}(X) \,\underline{\odot }\,{{\,\mathrm{val}\,}}(Y) \ge N \,\underline{\odot }\,N'\) and so \(X Y \in I_{N \,\underline{\odot }\,N'}\). This gives the inclusion \(I_N I_{N'} \subseteq I_{N \,\underline{\odot }\,N'}\). Let \(u_{ij} = \min \limits _{1 \le k \le d} (n_{ik} + n'_{kj})\) be the (i, j) entry of \(N \,\underline{\odot }\,N'\). For the inclusion \( I_{N \,\underline{\odot }\,N'} \subseteq I_N I_{N'} \), it suffices to show that \(p^{u_{ij}} E_{ij} \) is in \( I_N I_{N'}\) for all i, j. Fix i, j and let k satisfy \(u_{ij} = n_{ik} + n'_{kj}\). The matrices \(p^{n_{ik}} E_{ik}\) and \(p^{n_{kj}'} E_{kj}\) are in \(I_N\) and \(I_{N'}\). Their product \(p^{u_{ij}} E_{ij}\) is in \(I_{N} I_{N'}\). \(\square \)

We call \({\mathcal {Q}}_M\) the ideal class polytrope of M. The min-plus semigroup \(({\mathcal {Q}}_M, \,\underline{\odot }\,)\) plays the role of the ideal class group in number theory. Its neutral element is the given matrix M.

Example 25

Fix \(M = \begin{bmatrix} 0 &{} 1\\ 1 &{} 0 \end{bmatrix} \in {\mathcal {P}}_2\). The polytrope \({\mathcal {Q}}_M\) is the octahedron with vertices

This octahedron contains 19 integer points N. These are in bijection with the equivalence classes of fractional ideals \(I_N\) in the order \(\Lambda _M\). The midpoint of \({\mathcal {Q}}_M\) corresponds to the Jacobson radical \(I_{M+\mathrm{Id}_2}\). The remaining 12 integer points are the midpoints of the edges.

One may ask whether the ideal class semigroup \(({\mathcal {Q}}_M, \,\underline{\odot }\,)\) is actually a group. To address this question, we define the pseudo-inverse of a fractional ideal I in the order \(\Lambda _M\) as follows:

$$\begin{aligned} (\Lambda _M: I) \,\,= \,\,\{ \,X\in K^{d \times d} \,:XI \subseteq \Lambda _M \, \text { and } \, IX \subseteq \Lambda _M\}. \end{aligned}$$

Lemma 26

The pseudo-inverse of a fractional ideal in \(\Lambda _M\) is a fractional ideal in \(\Lambda _M\).

Proof

Let \(A \in \Lambda _M\) and \(X \in (\Lambda _M : I)\), so that \( XI, IX \subseteq \Lambda _M\). Since I is a fractional ideal, we have \(AI \subseteq I\) and \(IA \subseteq I\). From these inclusions we deduce that XAI, IXA, AXI, IAX are all subsets of \( \Lambda _M\). This implies \(XA, AX \in (\Lambda _M : I)\). Hence \((\Lambda _M:I)\) is a fractional ideal. \(\square \)

Proposition 27

Let \(M \in {\mathcal {P}}_d\) in standard form and \(N \in {\mathcal {Q}}_M\). Then \((\Lambda _M: I_N) = I_{N'}\) where

$$\begin{aligned} \quad n'_{ij}\,\, =\,\, \max \limits _{ 1 \le \ell \le d } \bigl (\, \max (m_{\ell j} - n_{\ell i}, m_{i\ell } - n_{j\ell }) \, \bigr ) \qquad \mathrm{for} \,\,\, 1 \le i , j \le d. \end{aligned}$$
(14)

Proof

By Proposition 21 and Lemma 26, there exists \(N' \in {\mathcal {Q}}_M\) such that \(I_{N'} = (\Lambda _M :I_N)\). Then \(I_{N'} I_{N} \subseteq \Lambda _M\) and \(I_N I_{N'} \subseteq \Lambda _M\), and \(I_{N'}\) is the largest fractional ideal with this property. These two conditions are equivalent to \(\,p^{n'_{ij}} E_{ij} I_N \subseteq \Lambda _M\,\) and \(\,p^{n'_{ij}} I_N E_{ij} \subseteq \Lambda _M \) for all i, j. The first condition holds if and only if \(n'_{ij} + n_{j\ell } \ge m_{i\ell }\) for all \(\ell \). The second condition holds if and only if \(n_{\ell i} + n'_{ij} \ge m_{\ell j}\) for all \(\ell \). The smallest solution \(N' = (n_{ij}')\) is given by (14). \(\square \)

Passing from ideals to their matrices, we also call \(N'\) the pseudo-inverse of N in \({\mathcal {Q}}_M\).

Example 28

Let \(d=2\) and M as in Example 25. The 19 ideal classes N in \({\mathcal {Q}}_M\) have only three distinct pseudo-inverses: \(\,N' \in \bigl \{ \begin{bmatrix} 0 \! &{} \! 0 \\ 0 \! &{} \! 0 \end{bmatrix} ,\, \begin{bmatrix} 0 \! &{} \! 1 \\ 1 \! &{} \! 0 \end{bmatrix} ,\, \begin{bmatrix} 1 \! &{} \! 0 \\ 0 \! &{} \! 1 \end{bmatrix} \bigr \}\). For most ideal classes N, we have \(N \,\underline{\odot }\,N' \ne M\) and \( N' \,\underline{\odot }\,N \ne M\). This means that most N do not have an inverse in \(({\mathcal {Q}}_M, \,\underline{\odot }\,)\). In particular, the ideal class polytrope \({\mathcal {Q}}_M\) is a semigroup but not a group.

The semigroup \({\mathcal {Q}}_M\) has the neutral element M and each ideal class \(N \in {\mathcal {Q}}_M\) has a pseudo-inverse \(N'\) given by the formula (14). With this data, we define the ideal class group

$$\begin{aligned} {\mathcal {G}}_M \,\,= \,\, \bigl \{ \, N \in {\mathcal {Q}}_M \,:\, N \,\underline{\odot }\,N' \,=\, N' \,\underline{\odot }\,N \, =\, M \bigr \}. \end{aligned}$$

This is the maximal subgroup of the semigroup \({\mathcal {Q}}_M\). It would be interesting to understand how M determines the structure of \({\mathcal {G}}_M\). Note that \(\,{\mathcal {G}}_M = \bigl \{ \begin{bmatrix} 0 \! &{} \! 1 \\ 1 \! &{} \! 0 \end{bmatrix} ,\, \begin{bmatrix} 1 \! &{} \! 0 \\ 0 \! &{} \! 1 \end{bmatrix} \bigr \}\,\) in Example 28.

Example 29

Here are three examples of ideal class groups of graduated orders:

$$\begin{aligned} \begin{matrix} M_2 = \begin{bmatrix} 0 &{} 1 \\ 1 &{} 0 \end{bmatrix} &{} &{} M_3 = \begin{bmatrix} 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \end{bmatrix} &{} &{} M_4 = \begin{bmatrix} 0 &{} 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 &{} 0 \end{bmatrix} &{} &{}\\ {\mathcal {G}}_{M_2} \cong {\mathbb {Z}}/2{\mathbb {Z}}&{} &{} {\mathcal {G}}_{M_3} \cong {\mathbb {Z}}/ 6{\mathbb {Z}}&{} &{} {\mathcal {G}}_{M_4} \cong S_4 &{} &{} \end{matrix} \end{aligned}$$

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 (Joswig and Kulas 2010, §3) in higher dimensions.

We end this section with a conjecture about the geometry of \({\mathcal {G}}_M\) inside \({\mathcal {Q}}_M\).

Conjecture 30

For any integer matrix M in the polytrope region \( {\mathcal {P}}_d\), the elements in the ideal class polytrope \({\mathcal {G}}_M\) are among the classical vertices of the ideal class group \({\mathcal {Q}}_M\).

6 Towards the building

Affine buildings (Abramenko and Brown 2008; Zhang 2021) 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.

Definition 31

The affine building \({\mathcal {B}}_d(K)\) is an infinite simplicial complex. Its vertices are the equivalence classes [L] of lattices in \(K^d\). A configuration \(\{ [L_1], \ldots , [L_s] \} \) is a simplex in \({\mathcal {B}}_d(K)\) if and only if, up to some permutation, there exist representatives \({\tilde{L}}_i \in [L_i]\) satisfying \({\tilde{L}}_1 \supset {\tilde{L}}_2 \supset \cdots \supset {\tilde{L}}_s \supset p {\tilde{L}}_1\). The maximal simplices \(\{ [L_1] ,\ldots , [L_d] \} \) are called chambers. The standard chamber \(C_0 \) is given by the diagonal lattices \( L_i = L_{(\mathbf{1}_{i-1}, \mathbf{0}_{d-i+1})} = L_{(1,\dots ,1,0,\dots ,0)}\).

Given a basis \(\{b_1, \dots , b_d\}\) of \(K^d\), the apartment defined by this basis is the set of classes [L] of all lattices \(L= \bigoplus _{i=1}^d p^{u_i} {\mathcal {O}}_K b_i \) where \(u_1,\ldots , u_d \) range over \( {\mathbb {Z}}\). Hence the apartment is

$$\begin{aligned} \big \{ \left[ p^{u_1} {\mathcal {O}}_K b_1 \oplus \dots \oplus p^{u_d} {\mathcal {O}}_K b_d \right] :u_1, \dots , u_d \in {\mathbb {Z}}\,\big \} \,\,=\,\, \bigl \{\, [gL_u] \,:u \in {\mathbb {Z}}^d \,\bigr \}, \end{aligned}$$

where \(g \in {{\,\mathrm{GL}\,}}_d(K)\) is the matrix with columns \(b_1, \dots , b_d\). The standard apartment is the one associated with the standard basis \((e_1,\dots , e_d)\) of \(K^d\). The vertices of the standard apartment are the diagonal lattice classes \([L_u]\) for \(u \in {\mathbb {Z}}^{d}\). We identify this set of vertices with \({\mathbb {Z}}^n / {\mathbb {Z}}\mathbf{1}\).

The general linear group \({{\,\mathrm{GL}\,}}_d(K)\) acts on the building \({\mathcal {B}}_d(K)\). 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

$$\begin{aligned}{{\,\mathrm{GL}\,}}_d({\mathcal {O}}_K) \,\,= \,\, \{ \,g\in {\mathcal {O}}_K^{d\times d} \,:\, {{\,\mathrm{val}\,}}(\det (g)) = 0 \,\} \,\, \subset \,\, {{\,\mathrm{GL}\,}}_d(K). \end{aligned}$$

Starting from the standard chamber \(C_0\), there exist reflections \(s_0,s_1,\ldots , s_{d-1}\) in \({{\,\mathrm{GL}\,}}_d(K)\) that map \(C_0\) to the d adjacent chambers in the standard apartment. For \(i \ge 1\), define \(s_i\) by

$$\begin{aligned} s_i (e_i) = e_{i+1} \text {,} \quad s_i(e_{i+1}) = e_i \, \text { and } \, s_{i}(e_j) = e_j \text { when } j \ne i, i+1. \end{aligned}$$

The map \(s_0\) is defined by \(s_0(e_i ) = e_i\) for \(i=2,\ldots , d-1\) and \(s_0(e_d) = p e_1\), \(s_0(e_1) = p^{-1} e_d\). The reflections \(s_0,\ldots , s_{d-1}\) are Coxeter generators for the affine Weyl group \(\,W = \langle s_0 ,\ldots , s_{d-1 }\rangle \). The group W acts regularly on the chambers C in the standard apartment (Bourbaki 2002, § 1.5, Thm. 2): for every C there is a unique \(w\in W\) such that \(C=wC_0\). The elements of W are the matrices \(h_{\sigma } g_u\) where \(h_{\sigma } = (1_{i = \sigma (j)})_{i,j}\) for \(\sigma \in S_d\), and \(u \in {\mathbb {Z}}^{d}\) with \( u_1+\cdots +u_d = 0\). 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 \(\mathrm{PZ}(\Gamma )\) of a finite configuration \(\Gamma \) in the affine building \({\mathcal {B}}_d(K)\). This is the intersection (8) of endomorphism rings. In this paper we studied the case when \(\Gamma \) lies in one apartment. In Theorem 10 we showed that \(\mathrm{PZ}(\Gamma ) = \Lambda _M\) where M is the matrix in \({\mathcal {P}}_d\) that encodes the min-max convex hull of \(\Gamma \). This was used in Sections 4 and 5 to elucidate combinatorial and algebraic structures in \(\mathrm{PZ}(\Gamma )\). A subsequent project will extend our results to arbitrary configurations \(\Gamma \) in \({\mathcal {B}}_d(K)\).

We conclude this article with configurations given by two chambers \(C,C'\) in \({\mathcal {B}}_d(K)\). We are interested in the their order \({{\,\mathrm{\mathrm {PZ}}\,}}(C \cup C')\). A fundamental fact about buildings states that any two chambers \(C,C'\) lie in a common apartment, cf. (Bourbaki, 2002), (Abramenko and Brown, 2008). 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 \(w = h_{\sigma } g_u \in W \).

Example 32

The standard chamber \(C_0\) is encoded by \(M_0 = \sum _{1 \le i < j \le d} E_{ij}\). The polytrope \(Q_{M_0}\) is a simplex. The order \(\mathrm{PZ}(C_0) = \Lambda _{M_0}\) consists of all \(X \in {\mathcal {O}}_K^{d \times d}\) with \(x_{ij} \in \langle p \rangle \) for \(i < j\).

Let \(D_u = {{\,\mathrm{val}\,}}(g_u)\) denote the tropical diagonal matrix with \(u_1, \dots , u_d\) on the diagonal and \(+\infty \) elsewhere. We also write \(P_\sigma :={{\,\mathrm{val}\,}}(h_\sigma )\) for the tropical permutation matrix given by \(\sigma \).

Proposition 33

We have \( \mathrm{PZ}(C_0 \cup h_{\sigma }g_u C_0) = \Lambda _{M^{\sigma ,u}}\) where the matrix \(M^{\sigma ,u}\) equals

$$\begin{aligned} M^{\sigma ,u} \,\,=\,\, M_0\, \,\,\overline{\oplus }\,\, \left( P_\sigma \,\,\underline{\odot }\,\, D_{u} \,\,\underline{\odot }\,\, M_0 \,\,\underline{\odot }\,\, D_{-u} \,\,\underline{\odot }\,\, P_{\sigma ^{-1}} \right) . \end{aligned}$$

Proof

We have \({{\,\mathrm{\mathrm {PZ}}\,}}(C_0 \cup h_{\sigma }g_u C_0) = {{\,\mathrm{\mathrm {PZ}}\,}}(C_0) \cap {{\,\mathrm{\mathrm {PZ}}\,}}(h_{\sigma }g_u C_0)\) and \({{\,\mathrm{\mathrm {PZ}}\,}}(C_0) = \Lambda _{M_0}\) from Example 32. Suppose that \(M \in {\mathbb {Z}}_{0}^{d \times d}\) satisfies \({{\,\mathrm{\mathrm {PZ}}\,}}(h_{\sigma }g_u C_0) = \Lambda _{M}\). By Theorem 10, the order \(\Lambda _{M_0 \,\overline{\oplus }\,M}\) is equal to \({{\,\mathrm{\mathrm {PZ}}\,}}(C_0 \cup h_{\sigma }g_u C_0)\). To determine M, notice that \({{\,\mathrm{\mathrm {PZ}}\,}}(wC_0) = h_\sigma g_u {{\,\mathrm{\mathrm {PZ}}\,}}(C_0) g_{-u} h_{\sigma ^{-1}} \). This implies the stated formula \(\,M = P_\sigma \,\underline{\odot }\,D_{u} \,\underline{\odot }\,M_0 \,\underline{\odot }\,D_{-u} \,\underline{\odot }\,P_{\sigma ^{-1}} \). \(\square \)

We may ask for invariants of the orders \({{\,\mathrm{\mathrm {PZ}}\,}}(C_0 \cup w C_0)\) in terms of \(w \in W\). Clearly, not all polytropes in an apartment arise as the min-max convex hull of two chambers. Which graduated orders are of the form \({{\,\mathrm{\mathrm {PZ}}\,}}(C_0 \cup wC_0) \)? Which other elements \(w ' \) in the affine Weyl group W give rise to the same Plesken-Zassenhaus order \({{\,\mathrm{\mathrm {PZ}}\,}}(C_0 \cup wC_0) \) up to isomorphism?