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A topological structure on certain initial algebras
Topology and its Applications 180 (2015) 199–208
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Topology and its Applications www.elsevier.com/locate/topol
A topological structure on certain initial algebras ✩ Sarah Anderson a , Andrew Smith b , Peter Stewart a , Mohammed Tesemma c,∗ , Jeremy Usatine d a b c d
Department Department Department Department
of of of of
Mathematical Mathematical Mathematics, Mathematics,
a r t i c l e
Sciences, Clemson University, United States Sciences, Carnegie Mellon University, United States Spelman College, United States Harvey Mudd College, United States
i n f o
Article history: Received 8 October 2013 Received in revised form 31 October 2014 Accepted 15 November 2014 Available online 10 December 2014 MSC: 06F15 13P10 16W22 54Cxx
a b s t r a c t There is a well-known natural topology on the set of compatible total orders on a group, and recent results of Clay [2], Dabkowska et al. [4], and Sikora [12] have characterized this topology for certain groups. We consider a similar topology on the set of distinct monomial algebras in polynomial and Laurent polynomial rings. We study the latter topological structure for monomial algebras that come from rings of multiplicative invariants and show that they are either finite discrete spaces or homeomorphic to the Cantor set. © 2014 Elsevier B.V. All rights reserved.
Keywords: Cantor space Orderable group Initial algebra Multiplicative invariants
1. Introduction A group G is called left (resp. right) orderable if there is a total order on G compatible with the group operation; that is, a b implies ca cb (ac bc) for all a, b, c ∈ G. Not every group admits such an order; for example, finite groups do not. The study of orderable groups is not new, but in recent years there has been considerable attention to the field due to the discovery of deep connections with topology, and dynamics of group action on a circle. Orderability is also a proven useful tool in 3-manifold theory. For details and other recent developments on orderable groups, we refer to [2,4,7,12]. An example of orderable ✩
This research was supported by NSF grant 1156761.
* Corresponding author. E-mail addresses: [email protected] (S. Anderson), [email protected] (A. Smith), [email protected] (P. Stewart), [email protected] (M. Tesemma), [email protected] (J. Usatine). http://dx.doi.org/10.1016/j.topol.2014.11.008 0166-8641/© 2014 Elsevier B.V. All rights reserved.
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group is the free abelian group Zn with the lexicographic (or dictionary) order, lex , given by a lex b if and only if a = b or the first nonzero entry of a − b is positive. Our investigation in this paper is based on the set of all compatible orders on Zn , denoted Ω. Sikora [12] defined a natural topology on Ω with subbasis Ua,b = { ∈ Ω | a b} for each a, b ∈ Zn and showed that this space is a totally disconnected, compact metric space with no isolated points. Such spaces are homeomorphic to the Cantor set due to Brouwer [1]. ±1 On the other hand, let’s consider the Laurent polynomial ring, k[x±1 ] = k[x±1 1 , . . . , xn ], in n variables. A topological structure on the set V = V : V is a k-subspace of k x±1 spanned by monomials is defined by Kuroda [6]. Further details on a special subspace of V that come from initial algebras will be presented in Section 3. Now for ∈ Ω, the initial exponent of a nonzero polynomial f ∈ k[x±1 ] with respect to is in (f ) = max a ∈ Zn : xa occurs in f with nonzero coefficient .
The initial algebra of a subalgebra R of k[x±1 ] with respect to ∈ Ω is the monomial algebra in (R) = k[xin (f ) : f ∈ R \ {0}]. Note that in (R) ∈ V. Initial algebras play an important role in SAGBI theory, analogous to the role play by initial ideals in the theory of Gröbner basis. Here ‘SAGBI” stands for “Subalgebra Analogue to Gröbner Bases for Ideals.” This notion was pioneered by Robbiano and Sweedler [11] and independently by Kapur and Madlener [5]. Let G be a subgroup of the symmetric group Sn . G acts on k[x±1 ] by k-algebra automorphism permuting the variables, σ(xi ) = xσ(i) for each σ ∈ G. The subalgebra R of permutation invariants is the set of all Laurent polynomials f ∈ k[x±1 ] that are fixed under the action of G. Kuroda [6] showed that the set of distinct initial algebras {in (R) : ∈ Ω} is either finite or uncountable. Kuroda’s argument is a careful analysis of the subspace topology {in (R) : ∈ Ω} ⊆ V. Tesemma [13] generalized Kuroda’s result on the cardinality of distinct initial algebras for subalgebras R that are invariant under the action of an arbitrary finite subgroup of GLn (Z). These subrings are called rings of multiplicative invariants. Note that permutation invariants are special cases of multiplicative invariants. Tesemma’s approach did not use a topology on the space of initial algebras and instead involved counting a family of convex polyhedral cones associated to these initial algebras. Reichstein [9] also studied these polyhedral cones to show that the ring of multiplicative invariants has a SAGBI basis if and only if the group G is generated by reflections. Initial algebras play an important role in SAGBI bases theory, analogous to the role played by initial ideals in the theory of Gröbner bases. Here SAGBI is an acronym for Subalgebra Analogue to Geöbner Bases for Ideals. This notion was pioneered by Robbiano and Sweedler [11]. Our interest in this paper is to study the topological properties of the space of initial algebras that come from rings of multiplicative invariants. More precisely, let R be an arbitrary ring of multiplicative invariants. We study {in (R) : ∈ Ω} ⊆ V as a quotient space of Ω via the map Ω → V :→ in (R). Our main result is the following analogue to the topology on Ω. Theorem 1.1. Let R be a ring of multiplicative invariants of k[x±1 ] under action of a finite group G GLn (Z). Denote the set of initial algebras {in (R) : ∈ Ω} as Ψ . (1) If G is generated by reflections, then Ψ is a finite set with a discrete topology. (2) If G is not generated by reflections, then Ψ is homeomorphic to the Cantor set.
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2. Preliminaries 2.1. Basics on ring of multiplicative invariants Let G be a finite subgroup of GLn (Z) and k[x±1 ] be the Laurent polynomial ring in n variables. G acts on the multiplicative group of monomials {xa , a ∈ Zn } by σ(xa ) = xσ·a , where σ · a is multiplication of the n × n matrix σ in G by the vector a in Zn . This action extends to the Laurent polynomial ring k[x±1 ] by linearity. The subalgebra G k x±1 = f ∈ k x±1 σ(f ) = f, ∀σ ∈ G is called an algebra or ring of multiplicative invariants. The orbit sum of each monomial, ϑ(xa ) = is an invariant polynomial. Moreover, the set
σ∈G
xσa ,
a ϑ x : a ∈ Zn is a k-basis of the ring of multiplicative invariants. For details on the theory of multiplicative invariants, we direct the reader to [8]. Definition. Let σ ∈ GLn (Z) be a linear transformation on Rn . We call σ a reflection if σ fixes a hyperplane and σ 2 = I, where I is the identity matrix. A group G GLn (Z) is called a reflection group if it is generated by reflections. 2.2. Orders on Zn determined by weight vectors In this section, we study a class of orders on Zn defined by vectors in Rn of rational dimension n. Note that the rational dimension of a vector w = (w1 , . . . , wn ) ∈ Rn , denoted dimQ (w), is the dimension of the Q-vector space spanned by the set {w1 , . . . , wn }. Robbiano [10] gave a classification of the set Ω of all orders on Zn which plays an important role in Gröbner and SAGBI basis theories. Given a vector w in Rn , define the map Zn → R : a → w · a, where w · a is the usual dot product on Rn . This map is one-to-one if and only if dimQ (w) = n. Hence, each vector w of rational dimension n determines an ordering, w , on Zn by aw b
⇔
a · w b · w;
∀a, b ∈ Zn .
Clearly w = λw for λ ∈ R>0 . But on the other hand if w1 and w2 are non-parallel vectors, then the set {v ∈ Rn : v · w1 > 0 > v · w2 } is a non-empty open convex cone in Rn . Hence by density of rationals, it contains some a ∈ Zn . It follows that a w1 0 and 0w2 a, and hence w1 = w2 . Thus the assignment w → w is an injection from the set S := {w ∈ Rn : |w| = 1, dimQ (w) = n} to Ω. The elements of S are what we call the weight vectors. If we endow S with the subspace topology from the standard metric topology on Rn , we have the following result of Kuroda [6]. Proposition 2.1. The map ι : S → Ω : w → w is continuous. Moreover, the image ι(S) is dense in Ω. Finally, we prove a lemma about weight orders for later use. Lemma 2.2. Let w ∈ S and v ∈ Rn . Then for any nonempty open interval I ⊂ R, there exists δ ∈ I such that w − δv has rationally independent coordinates.
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Proof. Let w = (w1 , . . . , wn ) and v = (v1 , . . . , vn ). Suppose for any δ that there exist a1 , . . . , an ∈ Q, not all zero, such that n
ai (wi − δvi ) = 0.
i=1
Because the wi are rationally independent,
n i=1
ai wi = 0. Thus
n i=1
ai vi = 0. So
n ai w i δ = i=1 ∈ Q(w1 , . . . , wn , v1 , . . . , vn ). n i=1 ai vi Thus any δ outside of the field extension Q(w1 , . . . , wn , v1 , . . . , vn ) must yield w − δv with rationally independent coordinates. Since Q(w1 , . . . , wn , v1 , . . . , vn ) is countable, any nonempty open I ⊂ R contains such a δ. 2 2.3. Polyhedral geometry A significant part of Section 4 will rely heavily on basic facts from polyhedral geometry. Here we state some important definitions we will make use of. Definition. (1) A subset C of Rn is called a polyhedral cone if there is a finite subset X ⊆ C such that C = Cone(X) =
λv v λv ∈ R0 .
v∈X
(2) The dual of a polyhedral cone C is C ∨ = {w ∈ Rn | w · v 0 ∀v ∈ C}. (3) A cone C is strongly convex if {0} is a face of C. Equivalently, C is strongly convex if and only if there are no nonnegative coefficients {λv 0 | v ∈ X} λv v = 0. which are not all 0 such that Also, C is strongly convex if and only if C ∨ is n dimensional, i.e. it is not contained in any n − 1 dimensional subspace of Rn . In particular, if C is strongly convex then the interior of C ∨ relative to Rn is a dense subset of C ∨ , and thus C ∨ meets any dense subset of Rn in a dense subset of C ∨ . For further reference, see [3]. 3. The space of initial algebras of multiplicative invariants Recall from Section 1 that the set Ω of all compatible orders on Zn is endowed with a natural topology with subbasis of open sets given by Ua,b = { ∈ Ω : a b} for each a, b ∈ Zn . Let R = k[x±1 ]G denote the ring of multiplicative invariants under the action of a finite group G GLn (Z). Define the canonical map QR : Ω → V : → in (R).
It is known that the set of distinct initial algebras is finite when the group acting on the ring is generated by reflections, see for example [13]. Hence as Ω is infinite, the map QR is not injective. Even in the case where G is not a reflection group, the map QR is not injective in general, as the following examples demonstrates. Therefore Theorem 1.1 is not a trivial consequence of the fact that Ω is homeomorphic to the Cantor set.
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Example. Let G GLn (Z) and let R = k[x±1 ]G be its ring of invariants. For any vector w ∈ S, we have QR (w ) = xa ∀σ ∈ G, w · a − σ(a) > 0 , so QR (w ) = QR (v ) if v − w is orthogonal to every a − σ(a). The subspace in Rn spanned by vectors of the form a − σ(a) is contained in the kernel of the projection G 1
Rn → Rn : x → σ(x), |G| σ∈G
where (Rn )G is the subspace of Rn invariant under the action of G. If (Rn )G = 0, then vectors of the form a − σ(a) do not span Rn , so there exists nonzero u ∈ Rn that is orthogonal to every a − σ(a). Pick w ∈ S not a multiple of u, and use Lemma 2.2 to get a rationally independent vector v that is not a multiple of w and satisfies v − w = δu is orthogonal to every a − σ(a). Thus QR (w ) = QR (v ) and w = v , so QR is not injective. Therefore QR is not injective whenever (Rn )G = 0. This holds, for example, for any permutation group, which must fix (1, 1, . . . , 1). Note that this condition is the same as the scaled projection σ∈G σ not being the zero matrix. We now endow the image QR (Ω) of distinct initial algebras of the ring of multiplicative invariants with the quotient topology. Denote this space Ψ . Note that Ψ ∼ = Ω/∼R , where ∼R is the equivalence relation on Ω given by ∼R if and only if in (R) = in (R). Next we will show that Ψ is metrizable by constructing a metric on QR (Ω) and showing it induces the topology of Ψ . Consider a finite filtration on Zn , i.e. a map ρ : Zn → N, such that ρ−1 (s) is finite for each s ∈ N. An example of such a ρ is ρ : (a1 , . . . , an ) → 1 +
|ai |.
The filtration ρ induces a metric dρ on V defined by Kuroda [6] as 1 dρ (V, W ) =
r,
0,
if r = max{s ∈ N | xa ∈ V ⇔ xa ∈ W, ∀a ∈ ρ−1 ([s])} ∈ R if no such r exists,
for any V, W ∈ V, where [s] = {1, . . . , s}. Let Ψ be the topological space obtained by restricting the metric dρ to QR (Ω) ⊂ V. Lemma 3.1. Ψ = Ψ . Hence Ψ is metrizable. Proof. We first show that the map QR : Ω → Ψ is continuous. Let QR () ∈ Ψ , ε > 0, and let r ∈ N such that 1 < ε. r For any a, b ∈ Zn , consider the subbasic open set Ua,b
⎧ ⎨ Ua,b , if xa xb = Ub,a , if xb xa ⎩ Ω, if xa = xb .
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Because G is finite, for any a ∈ Zn Ua =
Ua,ϕ(a)
ϕ∈G
is open in Ω and so is the finite intersection
a∈ρ−1 ([r])
Ua . Now for any ∈
a∈ρ−1 ([r])
Ua , we have
1 dρ QR , QR () < ε. r Therefore QR : Ω → Ψ is continuous. Since Ψ is a metric space, it is Hausdorff. From Sikora [12] the space Ω is compact. It follows that the map QR : Ω → Ψ is closed. More over it is continuous, and surjective. Therefore QR is a quotient map from Ω to Ψ . Hence Ψ = Ψ . 2 Corollary 3.2. Ψ is compact. Proof. Ω is compact and Ψ is a quotient of Ω, so Ψ is compact.
2
Corollary 3.3. Ψ is totally disconnected. Proof. This follows from the fact that the metric for Ψ constructed above takes only rational values. 2 4. Further properties of the space Ψ In this section, we will prove that if G is not generated by reflections, then Ψ has no isolated points. In [6], Kuroda proves a similar result for the special case when G is a subgroup of the symmetric group Sn that is not generated by transpositions. Our proof is inspired by Kuroda’s, but requires important changes to work in our more general setting. In particular, Kuroda finds a point whose orbit values are incomparable in some ordering, and constructs a sequence of rational orbit-maximal points approaching it by following a line on a polytope M=
(v1 , · · · , vn ) ∈
Rn0
n
vi = 1 , i=1
whose surface is closed under permutations. Using this polytope fails in our setting because M is not closed under arbitrary action of G GLn (Z). Instead, we show that the rational orbit-maximal points are dense in a polyhedral cone containing the desired limit. Definitions & Remarks. Let G GLn (Z). (1) For each σ ∈ G, define Iσ := ker(σ − 1), where 1 is the identity matrix. Let IG :=
Iσ .
σ∈G\{1}
(2) For each σ ∈ G and w ∈ S, define the linear functional fσw : Rn → R by fσw (x) = w · (x − σ(x)).
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(3) An inner product ·, · G on Rn is called G-invariant if x, y G = ϕ(x), ϕ(y) G for all x, y ∈ Zn and ϕ ∈ G. Such inner product exists for finite groups by defining a new inner product as follows: x, y G :=
ϕ(x) · ϕ(y).
ϕ∈G
(4) Note that dim Iσ = n − 1 if and only if σ is a reflection. In particular, Rn \ Iσ is disconnected if and only if σ is a reflection. (5) τ (IG ) = IG for each τ ∈ G. This follows from the observation that τ maps Iσ onto Iτ στ −1 . Lemma 4.1. Assume G is not a reflection group. Then for every a ∈ Rn \IG , every path-connected component of S G \ IG contains at least two points of the orbit ϑ(a) = σ(a) σ ∈ G . Proof. By Remark 5, any σ ∈ G preserves IG , so it must map Rn \IG onto itself. Since σ is a homeomorphism, it preserves components of Rn \ IG . If Rn \IG has only one path-connected component, then that component must contain ϑ(a). Since a ∈ / IG , ϑ(a) contains at least two elements, satisfying the claim. Otherwise, let C = D be two path-connected components of Rn \ IG . Consider a path ψ : [0, 1] → Rn between a point in C and a point in D. Further suppose that this path does not intersect any Iσ for non-reflections σ or any Iσ ∩ Iτ for σ = τ ∈ G and that
t ∈ [0, 1] ψ(t) ∈ Iτ for some reflection τ ∈ G
is finite. Denote these points t0 < · · · < tr , and for each ti , let τi be the reflection in G whose invariant subspace contains ψ(ti ). Since each τi is a homeomorphism, τr ◦ · · · ◦ τ1 is a bijection mapping C onto D. Since τr ◦ · · · ◦ τ1 is also a bijection on ϑ(a), C and D must contain the same cardinality of elements of ϑ(a). Now, suppose some component C has only one element of ϑ(a); then every component does. For each σ ∈ G, σ(a) = a because a ∈ / Iσ . Hence σ(a) lies in a different component D. Since σ(a) is D’s unique element of ϑ(a), the product of reflections τ mapping C onto D must also map a onto σ(a). Then στ −1 fixes a ∈ / IG , so we must have στ −1 = I. Hence σ = τ , and σ is a product of reflections in G. Hence G is a reflection group, contradicting the assumption. 2 We will now define three sets of points for a fixed weight order w. Let Ow denote the rational open polyhedral cone Ow = x ∈ Qn fτw (x) > 0 for all nonidentity τ ∈ G and let Cw denote the real closed cone Cw = x ∈ Rn fτw (x) 0 for all nonidentity τ ∈ G . Let Bw be boundary of Cw , i.e. the subset containing those points for which some σ ∈ G \{I} has fσw (x) = 0. Note that Cw is the set of points which are maximal under w in their G-orbits, so its integer points are precisely the exponents of inw (k[x±1 ]G ). We will show that we can find a rational point a ∈ Cw arbitrarily close to the border, so that certain arbitrarily small modifications to w will change whether some multiple ra ∈ Zn is in the initial algebra.
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Proposition 4.2. For any w ∈ S, Bw \ IG is nonempty. Proof. Choose any a ∈ Rn \ IG such that a w σ(a) for every σ ∈ G. By Lemma 4.1, we can find some σ ∈ G such that σ(a) is in the same connected component of Rn \ IG . Note that a ∈ Cw and σ(a) ∈ / Cw . n n Thus the set Cw \ IG , which is closed in R \ IG , cannot be a union of connected components of R \ IG , so it must have nontrivial boundary in Rn \ IG . 2 Proposition 4.3. For any w ∈ S, Ow is dense in Cw . Proof. For each ρ, write ρt for the transpose of ρ. Then observe that fρw (x) = w · (I − ρ)x = I − ρt w · x. Thus Cw is simply the dual cone to the cone generated by the vectors (I − ρt )w. Next note that because the coordinates of w are rationally independent, no rational combination of them can be 0, so w cannot be annihilated by any nonzero rational matrix. But each (I − ρt ) is rational and not zero unless ρ = I, so (I − ρt )w is nonzero. Hence the points {x ∈ Rn | fρw (x) 0} form a closed half-space, and the open half-space {x ∈ Rn | fρw (x) > 0} is a dense open subset. Hence the real open cone O = x ∈ Rn fτw (x) > 0 for all nonidentity τ ∈ G , which is the intersection of the open half-spaces is an intersection of dense open subsets of Cw , and thus itself a dense open subset. Now we will show that Cw is dual to a strongly convex cone. Suppose there are nonnegative scalars {rρ} such that
rρ I − ρt w = 0.
ρ∈G\{I}
Note that {ρt | ρ ∈ G} forms a finite group G t acting on Rn . Then taking a G t -invariant inner product ·, · G t , we have 0 = 0, w G t =
τ ∈G\{I}
rρ w, w G t − ρt w, w G t .
(1)
Because each ρt w, ρt w G t = w, w G t , we have that each ρt w, w G t is at most w, w G t with equality only if ρt w = w. Because all the rρ are nonnegative, this means each term of (1) is nonnegative, so the sum can only be 0 if each term is 0. But we know that w is not in the kernel of any I − ρt , so each w, w G t − ρt w, w G t > 0. Thus each rρ must be 0. Hence the dual to Cw is strongly convex. Then dim Cw = n, so rational points are dense in Cw . Hence for any open set in Cw , its intersection with O is a nonempty open set in Cw , and thus contains a rational point, which lies in Ow . 2 Theorem 4.4. Let R be a ring of multiplicative invariants under the action of a non-reflection subgroup of G GLn (Z). For any w ∈ S, any open ball B(w, ε) ⊆ S contains v such that QR (v ) = QR (w ). Proof. By Proposition 4.2, we can choose b ∈ Bw \ IG . Fix ρ ∈ G \ {I} such that fρw (b) = 0. Consider the open neighborhood consisting of those a ∈ Rn for which b − ρ(b) · b − ρ(b) − a − ρ(a) < ε
(2)
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and 0 < fρw (a) < ε .
(3)
Using Proposition 4.3, we may choose a ∈ Ow to be in this neighborhood. For each δ > 0, define wδ = w − δ(b − ρ(b)). For any ε > 0, we can find δ > 0 such that w − wδ < ε, |wδ | and by Lemma 2.2, we can choose δ such that wδ is rationally independent. For such δ, set ε =
2 δ b − ρ(b) . 1+δ
Using (2) and (3), we have wδ · ρ(a) − a = w − δ b − ρ(b) · ρ(a) − a = w · ρ(a) − a − δ b − ρ(b) · ρ(a) − a − ρ(b) − b + ρ(b) − b 2 > −ε − δε + δ b − ρ(b) = (−1 − δ)ε + (1 + δ)ε = 0. Now defining wδ =
wδ ∈S |wδ |
we see wδ · (ρ(a) − a) > 0 as well; that is, ρ(a) wδ a. Since a is rational, there is an integer r such that a = ra ∈ Zn , and ρ a = ρ(ra) = rρ(a) wδ ra = a .
Now any G-invariant polynomial containing xa must also contain xρ(a ) , so xa cannot be its initial term; thus xa ∈ / QR (wδ ). But the orbit sum
xσ(a )
σ∈G
is a G-invariant polynomial with initial term xa under w , so xa ∈ QR (w ). Thus QR (w ) = QR (wδ ), and by construction, wδ ∈ B(w, ε). 2 Tesemma has a similar result in [13, Theorem 3.5], but his approach does not use a topology on Ω. Moreover, his result make use of orders induced by all vectors w in Rn with the help of a fixed “tie breaker” in the case when dimQ (w) < n. Remark 4.5. If G is a reflection group, then by [13, Theorem 1.1] the space Ψ is finite. Since it is metrizable, then it is a finite discrete space.
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Proposition 4.6. If G GLn (Z) is a non-reflection group, then Ψ has no isolated points. Proof. Let A = in (R) for some ∈ Ω be an initial algebra and U ∈ Ψ be a neighborhood of A. Since ι and QR are continuous, the map ι
Q
R S −→ Ω −→ Ψ
is continuous. Now consider the open set (QR ◦ ι)−1 (U ). If A = QR (w ) for some rationally independent w, then by Lemma 4.4, the preimage has an element v satisfying QR (ι(v)) = QR (ι(w)). Hence the open set U contains the distinct element QR (ι(v)). If A = QR () but A = QR (w ) for any rationally independent w, then by the density of weight orders, any neighborhood of A contains a weight order, which cannot be A. In either case, A is a limit point, so all points of Ψ are limit points. 2 Now summarizing the results in Lemma 3.1, Corollary 3.2, Corollary 3.3 and Proposition 4.6, we have shown: Theorem 4.7. If G GLn (Z) is a non-reflection group, then Ψ is homeomorphic to the Cantor set. This theorem and Remark 4.5 give our main result, Theorem 1.1. Acknowledgements First, we would like to thank NSF for supporting this project under grant #1156761. This paper resulted from a 2013 REU project conducted at Clemson University, under the direction of Mohammed Tesemma and graduate advisor Sarah Anderson. We would like to thank the faculty at REU program, from whom we learned a lot during our stay. In particular we acknowledge Michael Burr for pointing us in the direction of polyhedral geometry. Finally we thank the anonymous referee for several helpful suggestions and comments. References [1] L.E.J. Brouwer, Over de structuur der perfekte puntverzamelingen, Amst. Akad. Versl. 18 (1910) 833–842. [2] A. Clay, Free lattice-ordered groups and the space of left orderings, Monatshefte Math. 167 (2012) 417–430. [3] D. Cox, J. Little, H. Shank, Toric Varieties, Grad. Stud. Math., vol. 124, American Mathematical Society, Providence, RI, 2011. [4] M.A. Dabkowska, M.K. Dabkowski, Compactness of the space of left orders, J. Knot Theory Ramif. 16 (3) (2007) 257–266. [5] D. Kapur, K. Madlener, A completion procedure for computing a canonical bases for a k-subalgebra, in: E. Kaltofen, S. Watt (Eds.), Proceedings of Computers and Mathematics, vol. 89, MIT, Cambridge, Mass, 1989, pp. 1–11. [6] S. Kuroda, The infiniteness of the SAGBI bases for certain invariant rings, Osaka J. Math. 39 (2002) 665–680. [7] P. Linnell, The space of left orders of a group is either finite or uncountable, Bull. Lond. Math. Soc. 43 (2011) 200–2002. [8] M. Lorenz, Multiplicative invariant theory, in: Invariant Theory and Algebraic Transformation Groups, in: Encyclopaedia Math. Sci., vol. 135, Springer-Verlag, Berlin Heidelberg, 2005. [9] Z. Reichstein, SAGBI bases in rings of multiplicative invariants, Comment. Math. Helv. 78 (1) (2003) 185–202. [10] L. Robbiano, Term orders on the polynomial ring, in: Proceedings of the EUROCAL 85, in: Lect. Notes Comput. Sci., vol. 204, 1985, pp. 513–517. [11] L. Robbiano, M. Sweedler, Subalgebra bases, in: Commutative Algebra, Salvador, 1988, in: Lect. Notes Math., vol. 1430, Springer, Berlin, 1990, pp. 61–87. [12] A. Sikora, Topology on the space of orderings of groups, Bull. Lond. Math. Soc. 36 (2004) 519–526. [13] M. Tesemma, On initial algebra of multiplicative invariants, J. Algebra 320 (2008) 3851–3865.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
A topological structure on certain initial algebras ✩ Sarah Anderson a , Andrew Smith b , Peter Stewart a , Mohammed Tesemma c,∗ , Jeremy Usatine d a b c d
Department Department Department Department
of of of of
Mathematical Mathematical Mathematics, Mathematics,
a r t i c l e
Sciences, Clemson University, United States Sciences, Carnegie Mellon University, United States Spelman College, United States Harvey Mudd College, United States
i n f o
Article history: Received 8 October 2013 Received in revised form 31 October 2014 Accepted 15 November 2014 Available online 10 December 2014 MSC: 06F15 13P10 16W22 54Cxx
a b s t r a c t There is a well-known natural topology on the set of compatible total orders on a group, and recent results of Clay [2], Dabkowska et al. [4], and Sikora [12] have characterized this topology for certain groups. We consider a similar topology on the set of distinct monomial algebras in polynomial and Laurent polynomial rings. We study the latter topological structure for monomial algebras that come from rings of multiplicative invariants and show that they are either finite discrete spaces or homeomorphic to the Cantor set. © 2014 Elsevier B.V. All rights reserved.
Keywords: Cantor space Orderable group Initial algebra Multiplicative invariants
1. Introduction A group G is called left (resp. right) orderable if there is a total order on G compatible with the group operation; that is, a b implies ca cb (ac bc) for all a, b, c ∈ G. Not every group admits such an order; for example, finite groups do not. The study of orderable groups is not new, but in recent years there has been considerable attention to the field due to the discovery of deep connections with topology, and dynamics of group action on a circle. Orderability is also a proven useful tool in 3-manifold theory. For details and other recent developments on orderable groups, we refer to [2,4,7,12]. An example of orderable ✩
This research was supported by NSF grant 1156761.
* Corresponding author. E-mail addresses: [email protected] (S. Anderson), [email protected] (A. Smith), [email protected] (P. Stewart), [email protected] (M. Tesemma), [email protected] (J. Usatine). http://dx.doi.org/10.1016/j.topol.2014.11.008 0166-8641/© 2014 Elsevier B.V. All rights reserved.
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group is the free abelian group Zn with the lexicographic (or dictionary) order, lex , given by a lex b if and only if a = b or the first nonzero entry of a − b is positive. Our investigation in this paper is based on the set of all compatible orders on Zn , denoted Ω. Sikora [12] defined a natural topology on Ω with subbasis Ua,b = { ∈ Ω | a b} for each a, b ∈ Zn and showed that this space is a totally disconnected, compact metric space with no isolated points. Such spaces are homeomorphic to the Cantor set due to Brouwer [1]. ±1 On the other hand, let’s consider the Laurent polynomial ring, k[x±1 ] = k[x±1 1 , . . . , xn ], in n variables. A topological structure on the set V = V : V is a k-subspace of k x±1 spanned by monomials is defined by Kuroda [6]. Further details on a special subspace of V that come from initial algebras will be presented in Section 3. Now for ∈ Ω, the initial exponent of a nonzero polynomial f ∈ k[x±1 ] with respect to is in (f ) = max a ∈ Zn : xa occurs in f with nonzero coefficient .
The initial algebra of a subalgebra R of k[x±1 ] with respect to ∈ Ω is the monomial algebra in (R) = k[xin (f ) : f ∈ R \ {0}]. Note that in (R) ∈ V. Initial algebras play an important role in SAGBI theory, analogous to the role play by initial ideals in the theory of Gröbner basis. Here ‘SAGBI” stands for “Subalgebra Analogue to Gröbner Bases for Ideals.” This notion was pioneered by Robbiano and Sweedler [11] and independently by Kapur and Madlener [5]. Let G be a subgroup of the symmetric group Sn . G acts on k[x±1 ] by k-algebra automorphism permuting the variables, σ(xi ) = xσ(i) for each σ ∈ G. The subalgebra R of permutation invariants is the set of all Laurent polynomials f ∈ k[x±1 ] that are fixed under the action of G. Kuroda [6] showed that the set of distinct initial algebras {in (R) : ∈ Ω} is either finite or uncountable. Kuroda’s argument is a careful analysis of the subspace topology {in (R) : ∈ Ω} ⊆ V. Tesemma [13] generalized Kuroda’s result on the cardinality of distinct initial algebras for subalgebras R that are invariant under the action of an arbitrary finite subgroup of GLn (Z). These subrings are called rings of multiplicative invariants. Note that permutation invariants are special cases of multiplicative invariants. Tesemma’s approach did not use a topology on the space of initial algebras and instead involved counting a family of convex polyhedral cones associated to these initial algebras. Reichstein [9] also studied these polyhedral cones to show that the ring of multiplicative invariants has a SAGBI basis if and only if the group G is generated by reflections. Initial algebras play an important role in SAGBI bases theory, analogous to the role played by initial ideals in the theory of Gröbner bases. Here SAGBI is an acronym for Subalgebra Analogue to Geöbner Bases for Ideals. This notion was pioneered by Robbiano and Sweedler [11]. Our interest in this paper is to study the topological properties of the space of initial algebras that come from rings of multiplicative invariants. More precisely, let R be an arbitrary ring of multiplicative invariants. We study {in (R) : ∈ Ω} ⊆ V as a quotient space of Ω via the map Ω → V :→ in (R). Our main result is the following analogue to the topology on Ω. Theorem 1.1. Let R be a ring of multiplicative invariants of k[x±1 ] under action of a finite group G GLn (Z). Denote the set of initial algebras {in (R) : ∈ Ω} as Ψ . (1) If G is generated by reflections, then Ψ is a finite set with a discrete topology. (2) If G is not generated by reflections, then Ψ is homeomorphic to the Cantor set.
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2. Preliminaries 2.1. Basics on ring of multiplicative invariants Let G be a finite subgroup of GLn (Z) and k[x±1 ] be the Laurent polynomial ring in n variables. G acts on the multiplicative group of monomials {xa , a ∈ Zn } by σ(xa ) = xσ·a , where σ · a is multiplication of the n × n matrix σ in G by the vector a in Zn . This action extends to the Laurent polynomial ring k[x±1 ] by linearity. The subalgebra G k x±1 = f ∈ k x±1 σ(f ) = f, ∀σ ∈ G is called an algebra or ring of multiplicative invariants. The orbit sum of each monomial, ϑ(xa ) = is an invariant polynomial. Moreover, the set
σ∈G
xσa ,
a ϑ x : a ∈ Zn is a k-basis of the ring of multiplicative invariants. For details on the theory of multiplicative invariants, we direct the reader to [8]. Definition. Let σ ∈ GLn (Z) be a linear transformation on Rn . We call σ a reflection if σ fixes a hyperplane and σ 2 = I, where I is the identity matrix. A group G GLn (Z) is called a reflection group if it is generated by reflections. 2.2. Orders on Zn determined by weight vectors In this section, we study a class of orders on Zn defined by vectors in Rn of rational dimension n. Note that the rational dimension of a vector w = (w1 , . . . , wn ) ∈ Rn , denoted dimQ (w), is the dimension of the Q-vector space spanned by the set {w1 , . . . , wn }. Robbiano [10] gave a classification of the set Ω of all orders on Zn which plays an important role in Gröbner and SAGBI basis theories. Given a vector w in Rn , define the map Zn → R : a → w · a, where w · a is the usual dot product on Rn . This map is one-to-one if and only if dimQ (w) = n. Hence, each vector w of rational dimension n determines an ordering, w , on Zn by aw b
⇔
a · w b · w;
∀a, b ∈ Zn .
Clearly w = λw for λ ∈ R>0 . But on the other hand if w1 and w2 are non-parallel vectors, then the set {v ∈ Rn : v · w1 > 0 > v · w2 } is a non-empty open convex cone in Rn . Hence by density of rationals, it contains some a ∈ Zn . It follows that a w1 0 and 0w2 a, and hence w1 = w2 . Thus the assignment w → w is an injection from the set S := {w ∈ Rn : |w| = 1, dimQ (w) = n} to Ω. The elements of S are what we call the weight vectors. If we endow S with the subspace topology from the standard metric topology on Rn , we have the following result of Kuroda [6]. Proposition 2.1. The map ι : S → Ω : w → w is continuous. Moreover, the image ι(S) is dense in Ω. Finally, we prove a lemma about weight orders for later use. Lemma 2.2. Let w ∈ S and v ∈ Rn . Then for any nonempty open interval I ⊂ R, there exists δ ∈ I such that w − δv has rationally independent coordinates.
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Proof. Let w = (w1 , . . . , wn ) and v = (v1 , . . . , vn ). Suppose for any δ that there exist a1 , . . . , an ∈ Q, not all zero, such that n
ai (wi − δvi ) = 0.
i=1
Because the wi are rationally independent,
n i=1
ai wi = 0. Thus
n i=1
ai vi = 0. So
n ai w i δ = i=1 ∈ Q(w1 , . . . , wn , v1 , . . . , vn ). n i=1 ai vi Thus any δ outside of the field extension Q(w1 , . . . , wn , v1 , . . . , vn ) must yield w − δv with rationally independent coordinates. Since Q(w1 , . . . , wn , v1 , . . . , vn ) is countable, any nonempty open I ⊂ R contains such a δ. 2 2.3. Polyhedral geometry A significant part of Section 4 will rely heavily on basic facts from polyhedral geometry. Here we state some important definitions we will make use of. Definition. (1) A subset C of Rn is called a polyhedral cone if there is a finite subset X ⊆ C such that C = Cone(X) =
λv v λv ∈ R0 .
v∈X
(2) The dual of a polyhedral cone C is C ∨ = {w ∈ Rn | w · v 0 ∀v ∈ C}. (3) A cone C is strongly convex if {0} is a face of C. Equivalently, C is strongly convex if and only if there are no nonnegative coefficients {λv 0 | v ∈ X} λv v = 0. which are not all 0 such that Also, C is strongly convex if and only if C ∨ is n dimensional, i.e. it is not contained in any n − 1 dimensional subspace of Rn . In particular, if C is strongly convex then the interior of C ∨ relative to Rn is a dense subset of C ∨ , and thus C ∨ meets any dense subset of Rn in a dense subset of C ∨ . For further reference, see [3]. 3. The space of initial algebras of multiplicative invariants Recall from Section 1 that the set Ω of all compatible orders on Zn is endowed with a natural topology with subbasis of open sets given by Ua,b = { ∈ Ω : a b} for each a, b ∈ Zn . Let R = k[x±1 ]G denote the ring of multiplicative invariants under the action of a finite group G GLn (Z). Define the canonical map QR : Ω → V : → in (R).
It is known that the set of distinct initial algebras is finite when the group acting on the ring is generated by reflections, see for example [13]. Hence as Ω is infinite, the map QR is not injective. Even in the case where G is not a reflection group, the map QR is not injective in general, as the following examples demonstrates. Therefore Theorem 1.1 is not a trivial consequence of the fact that Ω is homeomorphic to the Cantor set.
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Example. Let G GLn (Z) and let R = k[x±1 ]G be its ring of invariants. For any vector w ∈ S, we have QR (w ) = xa ∀σ ∈ G, w · a − σ(a) > 0 , so QR (w ) = QR (v ) if v − w is orthogonal to every a − σ(a). The subspace in Rn spanned by vectors of the form a − σ(a) is contained in the kernel of the projection G 1
Rn → Rn : x → σ(x), |G| σ∈G
where (Rn )G is the subspace of Rn invariant under the action of G. If (Rn )G = 0, then vectors of the form a − σ(a) do not span Rn , so there exists nonzero u ∈ Rn that is orthogonal to every a − σ(a). Pick w ∈ S not a multiple of u, and use Lemma 2.2 to get a rationally independent vector v that is not a multiple of w and satisfies v − w = δu is orthogonal to every a − σ(a). Thus QR (w ) = QR (v ) and w = v , so QR is not injective. Therefore QR is not injective whenever (Rn )G = 0. This holds, for example, for any permutation group, which must fix (1, 1, . . . , 1). Note that this condition is the same as the scaled projection σ∈G σ not being the zero matrix. We now endow the image QR (Ω) of distinct initial algebras of the ring of multiplicative invariants with the quotient topology. Denote this space Ψ . Note that Ψ ∼ = Ω/∼R , where ∼R is the equivalence relation on Ω given by ∼R if and only if in (R) = in (R). Next we will show that Ψ is metrizable by constructing a metric on QR (Ω) and showing it induces the topology of Ψ . Consider a finite filtration on Zn , i.e. a map ρ : Zn → N, such that ρ−1 (s) is finite for each s ∈ N. An example of such a ρ is ρ : (a1 , . . . , an ) → 1 +
|ai |.
The filtration ρ induces a metric dρ on V defined by Kuroda [6] as 1 dρ (V, W ) =
r,
0,
if r = max{s ∈ N | xa ∈ V ⇔ xa ∈ W, ∀a ∈ ρ−1 ([s])} ∈ R if no such r exists,
for any V, W ∈ V, where [s] = {1, . . . , s}. Let Ψ be the topological space obtained by restricting the metric dρ to QR (Ω) ⊂ V. Lemma 3.1. Ψ = Ψ . Hence Ψ is metrizable. Proof. We first show that the map QR : Ω → Ψ is continuous. Let QR () ∈ Ψ , ε > 0, and let r ∈ N such that 1 < ε. r For any a, b ∈ Zn , consider the subbasic open set Ua,b
⎧ ⎨ Ua,b , if xa xb = Ub,a , if xb xa ⎩ Ω, if xa = xb .
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Because G is finite, for any a ∈ Zn Ua =
Ua,ϕ(a)
ϕ∈G
is open in Ω and so is the finite intersection
a∈ρ−1 ([r])
Ua . Now for any ∈
a∈ρ−1 ([r])
Ua , we have
1 dρ QR , QR () < ε. r Therefore QR : Ω → Ψ is continuous. Since Ψ is a metric space, it is Hausdorff. From Sikora [12] the space Ω is compact. It follows that the map QR : Ω → Ψ is closed. More over it is continuous, and surjective. Therefore QR is a quotient map from Ω to Ψ . Hence Ψ = Ψ . 2 Corollary 3.2. Ψ is compact. Proof. Ω is compact and Ψ is a quotient of Ω, so Ψ is compact.
2
Corollary 3.3. Ψ is totally disconnected. Proof. This follows from the fact that the metric for Ψ constructed above takes only rational values. 2 4. Further properties of the space Ψ In this section, we will prove that if G is not generated by reflections, then Ψ has no isolated points. In [6], Kuroda proves a similar result for the special case when G is a subgroup of the symmetric group Sn that is not generated by transpositions. Our proof is inspired by Kuroda’s, but requires important changes to work in our more general setting. In particular, Kuroda finds a point whose orbit values are incomparable in some ordering, and constructs a sequence of rational orbit-maximal points approaching it by following a line on a polytope M=
(v1 , · · · , vn ) ∈
Rn0
n
vi = 1 , i=1
whose surface is closed under permutations. Using this polytope fails in our setting because M is not closed under arbitrary action of G GLn (Z). Instead, we show that the rational orbit-maximal points are dense in a polyhedral cone containing the desired limit. Definitions & Remarks. Let G GLn (Z). (1) For each σ ∈ G, define Iσ := ker(σ − 1), where 1 is the identity matrix. Let IG :=
Iσ .
σ∈G\{1}
(2) For each σ ∈ G and w ∈ S, define the linear functional fσw : Rn → R by fσw (x) = w · (x − σ(x)).
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(3) An inner product ·, · G on Rn is called G-invariant if x, y G = ϕ(x), ϕ(y) G for all x, y ∈ Zn and ϕ ∈ G. Such inner product exists for finite groups by defining a new inner product as follows: x, y G :=
ϕ(x) · ϕ(y).
ϕ∈G
(4) Note that dim Iσ = n − 1 if and only if σ is a reflection. In particular, Rn \ Iσ is disconnected if and only if σ is a reflection. (5) τ (IG ) = IG for each τ ∈ G. This follows from the observation that τ maps Iσ onto Iτ στ −1 . Lemma 4.1. Assume G is not a reflection group. Then for every a ∈ Rn \IG , every path-connected component of S G \ IG contains at least two points of the orbit ϑ(a) = σ(a) σ ∈ G . Proof. By Remark 5, any σ ∈ G preserves IG , so it must map Rn \IG onto itself. Since σ is a homeomorphism, it preserves components of Rn \ IG . If Rn \IG has only one path-connected component, then that component must contain ϑ(a). Since a ∈ / IG , ϑ(a) contains at least two elements, satisfying the claim. Otherwise, let C = D be two path-connected components of Rn \ IG . Consider a path ψ : [0, 1] → Rn between a point in C and a point in D. Further suppose that this path does not intersect any Iσ for non-reflections σ or any Iσ ∩ Iτ for σ = τ ∈ G and that
t ∈ [0, 1] ψ(t) ∈ Iτ for some reflection τ ∈ G
is finite. Denote these points t0 < · · · < tr , and for each ti , let τi be the reflection in G whose invariant subspace contains ψ(ti ). Since each τi is a homeomorphism, τr ◦ · · · ◦ τ1 is a bijection mapping C onto D. Since τr ◦ · · · ◦ τ1 is also a bijection on ϑ(a), C and D must contain the same cardinality of elements of ϑ(a). Now, suppose some component C has only one element of ϑ(a); then every component does. For each σ ∈ G, σ(a) = a because a ∈ / Iσ . Hence σ(a) lies in a different component D. Since σ(a) is D’s unique element of ϑ(a), the product of reflections τ mapping C onto D must also map a onto σ(a). Then στ −1 fixes a ∈ / IG , so we must have στ −1 = I. Hence σ = τ , and σ is a product of reflections in G. Hence G is a reflection group, contradicting the assumption. 2 We will now define three sets of points for a fixed weight order w. Let Ow denote the rational open polyhedral cone Ow = x ∈ Qn fτw (x) > 0 for all nonidentity τ ∈ G and let Cw denote the real closed cone Cw = x ∈ Rn fτw (x) 0 for all nonidentity τ ∈ G . Let Bw be boundary of Cw , i.e. the subset containing those points for which some σ ∈ G \{I} has fσw (x) = 0. Note that Cw is the set of points which are maximal under w in their G-orbits, so its integer points are precisely the exponents of inw (k[x±1 ]G ). We will show that we can find a rational point a ∈ Cw arbitrarily close to the border, so that certain arbitrarily small modifications to w will change whether some multiple ra ∈ Zn is in the initial algebra.
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Proposition 4.2. For any w ∈ S, Bw \ IG is nonempty. Proof. Choose any a ∈ Rn \ IG such that a w σ(a) for every σ ∈ G. By Lemma 4.1, we can find some σ ∈ G such that σ(a) is in the same connected component of Rn \ IG . Note that a ∈ Cw and σ(a) ∈ / Cw . n n Thus the set Cw \ IG , which is closed in R \ IG , cannot be a union of connected components of R \ IG , so it must have nontrivial boundary in Rn \ IG . 2 Proposition 4.3. For any w ∈ S, Ow is dense in Cw . Proof. For each ρ, write ρt for the transpose of ρ. Then observe that fρw (x) = w · (I − ρ)x = I − ρt w · x. Thus Cw is simply the dual cone to the cone generated by the vectors (I − ρt )w. Next note that because the coordinates of w are rationally independent, no rational combination of them can be 0, so w cannot be annihilated by any nonzero rational matrix. But each (I − ρt ) is rational and not zero unless ρ = I, so (I − ρt )w is nonzero. Hence the points {x ∈ Rn | fρw (x) 0} form a closed half-space, and the open half-space {x ∈ Rn | fρw (x) > 0} is a dense open subset. Hence the real open cone O = x ∈ Rn fτw (x) > 0 for all nonidentity τ ∈ G , which is the intersection of the open half-spaces is an intersection of dense open subsets of Cw , and thus itself a dense open subset. Now we will show that Cw is dual to a strongly convex cone. Suppose there are nonnegative scalars {rρ} such that
rρ I − ρt w = 0.
ρ∈G\{I}
Note that {ρt | ρ ∈ G} forms a finite group G t acting on Rn . Then taking a G t -invariant inner product ·, · G t , we have 0 = 0, w G t =
τ ∈G\{I}
rρ w, w G t − ρt w, w G t .
(1)
Because each ρt w, ρt w G t = w, w G t , we have that each ρt w, w G t is at most w, w G t with equality only if ρt w = w. Because all the rρ are nonnegative, this means each term of (1) is nonnegative, so the sum can only be 0 if each term is 0. But we know that w is not in the kernel of any I − ρt , so each w, w G t − ρt w, w G t > 0. Thus each rρ must be 0. Hence the dual to Cw is strongly convex. Then dim Cw = n, so rational points are dense in Cw . Hence for any open set in Cw , its intersection with O is a nonempty open set in Cw , and thus contains a rational point, which lies in Ow . 2 Theorem 4.4. Let R be a ring of multiplicative invariants under the action of a non-reflection subgroup of G GLn (Z). For any w ∈ S, any open ball B(w, ε) ⊆ S contains v such that QR (v ) = QR (w ). Proof. By Proposition 4.2, we can choose b ∈ Bw \ IG . Fix ρ ∈ G \ {I} such that fρw (b) = 0. Consider the open neighborhood consisting of those a ∈ Rn for which b − ρ(b) · b − ρ(b) − a − ρ(a) < ε
(2)
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and 0 < fρw (a) < ε .
(3)
Using Proposition 4.3, we may choose a ∈ Ow to be in this neighborhood. For each δ > 0, define wδ = w − δ(b − ρ(b)). For any ε > 0, we can find δ > 0 such that w − wδ < ε, |wδ | and by Lemma 2.2, we can choose δ such that wδ is rationally independent. For such δ, set ε =
2 δ b − ρ(b) . 1+δ
Using (2) and (3), we have wδ · ρ(a) − a = w − δ b − ρ(b) · ρ(a) − a = w · ρ(a) − a − δ b − ρ(b) · ρ(a) − a − ρ(b) − b + ρ(b) − b 2 > −ε − δε + δ b − ρ(b) = (−1 − δ)ε + (1 + δ)ε = 0. Now defining wδ =
wδ ∈S |wδ |
we see wδ · (ρ(a) − a) > 0 as well; that is, ρ(a) wδ a. Since a is rational, there is an integer r such that a = ra ∈ Zn , and ρ a = ρ(ra) = rρ(a) wδ ra = a .
Now any G-invariant polynomial containing xa must also contain xρ(a ) , so xa cannot be its initial term; thus xa ∈ / QR (wδ ). But the orbit sum
xσ(a )
σ∈G
is a G-invariant polynomial with initial term xa under w , so xa ∈ QR (w ). Thus QR (w ) = QR (wδ ), and by construction, wδ ∈ B(w, ε). 2 Tesemma has a similar result in [13, Theorem 3.5], but his approach does not use a topology on Ω. Moreover, his result make use of orders induced by all vectors w in Rn with the help of a fixed “tie breaker” in the case when dimQ (w) < n. Remark 4.5. If G is a reflection group, then by [13, Theorem 1.1] the space Ψ is finite. Since it is metrizable, then it is a finite discrete space.
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Proposition 4.6. If G GLn (Z) is a non-reflection group, then Ψ has no isolated points. Proof. Let A = in (R) for some ∈ Ω be an initial algebra and U ∈ Ψ be a neighborhood of A. Since ι and QR are continuous, the map ι
Q
R S −→ Ω −→ Ψ
is continuous. Now consider the open set (QR ◦ ι)−1 (U ). If A = QR (w ) for some rationally independent w, then by Lemma 4.4, the preimage has an element v satisfying QR (ι(v)) = QR (ι(w)). Hence the open set U contains the distinct element QR (ι(v)). If A = QR () but A = QR (w ) for any rationally independent w, then by the density of weight orders, any neighborhood of A contains a weight order, which cannot be A. In either case, A is a limit point, so all points of Ψ are limit points. 2 Now summarizing the results in Lemma 3.1, Corollary 3.2, Corollary 3.3 and Proposition 4.6, we have shown: Theorem 4.7. If G GLn (Z) is a non-reflection group, then Ψ is homeomorphic to the Cantor set. This theorem and Remark 4.5 give our main result, Theorem 1.1. Acknowledgements First, we would like to thank NSF for supporting this project under grant #1156761. This paper resulted from a 2013 REU project conducted at Clemson University, under the direction of Mohammed Tesemma and graduate advisor Sarah Anderson. We would like to thank the faculty at REU program, from whom we learned a lot during our stay. In particular we acknowledge Michael Burr for pointing us in the direction of polyhedral geometry. Finally we thank the anonymous referee for several helpful suggestions and comments. References [1] L.E.J. Brouwer, Over de structuur der perfekte puntverzamelingen, Amst. Akad. Versl. 18 (1910) 833–842. [2] A. Clay, Free lattice-ordered groups and the space of left orderings, Monatshefte Math. 167 (2012) 417–430. [3] D. Cox, J. Little, H. Shank, Toric Varieties, Grad. Stud. Math., vol. 124, American Mathematical Society, Providence, RI, 2011. [4] M.A. Dabkowska, M.K. Dabkowski, Compactness of the space of left orders, J. Knot Theory Ramif. 16 (3) (2007) 257–266. [5] D. Kapur, K. Madlener, A completion procedure for computing a canonical bases for a k-subalgebra, in: E. Kaltofen, S. Watt (Eds.), Proceedings of Computers and Mathematics, vol. 89, MIT, Cambridge, Mass, 1989, pp. 1–11. [6] S. Kuroda, The infiniteness of the SAGBI bases for certain invariant rings, Osaka J. Math. 39 (2002) 665–680. [7] P. Linnell, The space of left orders of a group is either finite or uncountable, Bull. Lond. Math. Soc. 43 (2011) 200–2002. [8] M. Lorenz, Multiplicative invariant theory, in: Invariant Theory and Algebraic Transformation Groups, in: Encyclopaedia Math. Sci., vol. 135, Springer-Verlag, Berlin Heidelberg, 2005. [9] Z. Reichstein, SAGBI bases in rings of multiplicative invariants, Comment. Math. Helv. 78 (1) (2003) 185–202. [10] L. Robbiano, Term orders on the polynomial ring, in: Proceedings of the EUROCAL 85, in: Lect. Notes Comput. Sci., vol. 204, 1985, pp. 513–517. [11] L. Robbiano, M. Sweedler, Subalgebra bases, in: Commutative Algebra, Salvador, 1988, in: Lect. Notes Math., vol. 1430, Springer, Berlin, 1990, pp. 61–87. [12] A. Sikora, Topology on the space of orderings of groups, Bull. Lond. Math. Soc. 36 (2004) 519–526. [13] M. Tesemma, On initial algebra of multiplicative invariants, J. Algebra 320 (2008) 3851–3865.