aa r X i v : . [ m a t h . A T ] M a y Tropicalisation for topologists
Hadi Zare
Abstract
We consider the problem of translating notions from classical topology to tropicallanguage. We consider the tropical projective and Grassmannian spaces. We give afairly easy classification of the projective gadget whereas the Grassmannians seemsrather more difficult. We also consider the notion of the tropical matrices, and definea variant of tropical orthogonal matrices. We completely determine Gl ( R > , n ) and O ( n ) trop . We also give some results on the idempotent elements of the tropical matrixalgebra M n,n ( R > ). Such a notion will be important in the related bundle theory. Wenote that tropical phenomena have been studies by algebraic geometers, and our workhere may overlap with them and there is no claim on the originality of these notes,neither no claim on them being unoriginal. Contents R trop ) n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 A Tropical Structure on R n > . . . . . . . . . . . . . . . . . . . . . 52.1.2 Flows on R n > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Tropical Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Subspaces in Tropical Euclidean Spaces . . . . . . . . . . . . . . . . . . 82.3.1 Linear independence in the tropical sense . . . . . . . . . . . . . 82.3.2 Relating G k ( R n > ) to configuration spaces . . . . . . . . . . . . . . 102.4 R n > as an R trop -module . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.1 The space G Trop k ( R n > ) . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.2 k -subspaces of ( R trop ) n . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Tropical Matrices, Tropical Orthogonal Matrices . . . . . . . . . . . . . 162.5.1 Idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.2 Stablisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.3 Comments on Tropical Bundle Theory . . . . . . . . . . . . . . . 19 Motivation, Basics
The min -algebra, N ∞ = N ⊔ { + ∞} together with the binary operation of takingminimum is our main motivation. Definition 1.1.
By a tropical space, we mean a pointed set ( T, ∞ ) together with a minimum -function min = min T : T × T → T satisfying min( t, ∞ ) = min( ∞ , t ) = t for all t ∈ T, and ( t, t ) = t for all t ∈ T . Notice that under this operation, T turns into a monoid, whose each element isidempotent. The tropical space T is called commutative if min( t, t ′ ) = min( t ′ , t ) for all t, t ′ ∈ T . Next, we make it clear what we mean by a tropical module, tropical algebra,etc. Let R be a commutative ring. By a left tropical R -module T we mean a tropicalset T of which has the structure of a left R -module, and the R -action fixes the ∞ ,i.e. r ∞ = ∞ . Similarly, we can define a tropical algebra T to be a tropical space T together with a multiplication which is compatible with the tropical structure.Notice that we may consider the category of these objects, say category of tropical R -modules together with the forgetful functorRMod trop −→ RMod . Next, we show how it is possible to construct examples of these spaces. This ofcourse reduces to find reasonable ways of defining the min -function.
Note . We note that it is possible to have a dual tropical structure given by taking maximum of two given elements. We will use this in some of our important examples.
Let us write R > for the half line of nonnegative real numbers. Let ( M, | | ) be a spacewith a norm function | | : M → R > . Notice that any norm gives rise to a metric, andany metric space ( M, d ) with a chosen point 0 ∈ M can be turned into a normed spaceby setting | v | = d ( v,
0) for any v ∈ M . Having a norm define m : M → R > by m ( a, b ) = min( | a | , | b | )where the right hand side is the minimum function of the real line. Notice that if wechoose M = R with the usual the metric on R , then m ( a, b ) = min( a, b ). Now, we maydefine min = min M : M × M → M by min( a, a ) = a andmin( a, b ) = (cid:26) a if m ( a, b ) = | a | ,b if m ( a, b ) = | b | . or a given normed space M , we define tropicalisation of M by M trop = M ⊔ {∞} withgeneralised min -function satisfyingmin( m, ∞ ) = min( ∞ , m ) for all m ∈ M and min( ∞ , ∞ ) = ∞ . Hence we obtain, a tropicalisation functor T rop : Normed-Space −→ Trop-Space . Notice that category of normed spaces contains the category of finite dimensional vectorspaces inside it.The fact that the tropicalisation of M depends on the norm, or say the metric, showsthat this can be used as an invariant of metric spaces, and not necessarily as aninvariant of topological spaces. The reason is that it is possible to choose differentmetrics, yielding the same topology on a space. Note . We note that the tropicalisation functor in fact is defined from the categoryof totally ordered sets into the category of tropical sets, as the notion of order naturallytells how to choose smaller element among two given ones.
Notice that on the real line one hasmin( a, b ) = a b − | a − b | . We use this as analogy to define min on a class of modules. Suppose we have a ring R which includes , say R = Z [ ] , Q , R , C , etc. Let M be an R -module together with aprojection, i.e. a mapping τ : M → M such that τ = τ . We then definemin τM ( a, b ) = 12 ( a + b − τ ( a − b )) . This then defines a min -function on M . Note that one may restrict to choose τ as anautomorphism of R -modules. But for the moment we shall not put this restriction. Wecan consider the tropicalisation of M trop τ = M ⊔ {∞} with generalised min -functiondefined as min M trop τ ( m, ∞ ) = min M trop τ ( ∞ , m ) = m for all m ∈ M, and min M trop τ ( ∞ , ∞ ) = ∞ . This then defines the tropicalisation functor T rop : Rmod pro −→ Trop-Spacewhere the left hand side is the category of R -modules with projections. If we givestructure of a monoid to M trop τ by min M trop τ and define the multiplication on M trop τ by generalising the addition operation of M , then we obtain an example of a tropicalalgebra. More precisely, we define ⊕ , ⊙ by m ⊕ m ′ = min M trop τ ( m, m ′ ) for all m, m ′ ∈ M trop τ ,m ⊙ m ′ = (cid:26) m + m ′ if m, m ′ ∈ M ∞ if m = ∞ , or m ′ = ∞ . otice that here ⊙ is commutative. We usually will drop ⊙ from our notation. Thisthen gives structure of a semi-ring to ( M trop τ , ⊕ , ⊗ ). Observe that the inclusion M → M trop τ acts like the exponential map as it sends addition to multiplication. If we choose τ tobe an automorphism, then we can give structure of an R -module to to M trop τ by requir-ing that the action fixes ∞ , and on the other points of M trop τ \ {∞} acts in the sameway as on M . This then makes the inclusion map M → M trop τ into a map of R -modules.Finally, we note that it is possible to define a “dual” tropical structure on M . Formotivation, notice that in R , we havemax( a, b ) = 12 ( a + b + | b − a | ) . One then may fix τ : M → M with τ = 1 and definemax τM ( a, b ) = 12 ( a + b + τ ( a − b )) . It is then possible to perfom as we did with the min -function.
I like to approach these topics with hat of a topologist on. These are quite familiarobjects, and it will be interesting how they look like geometrically.In topology, there are different approaches to define Grassmannian spaces. Let us towork with R for a moment. We consider R with its usual addition and multiplication.The Grassmannian G k ( R n + k ) may be viewed as the space of k -dimensional subspacesof R n + k which can be described as quotient of another space, namely the Steifel space V k ( R n + k ) of k -frames in R n + k together the quotient map V k ( R n + k ) → G k ( R n + k ) . Another way of defining G k ( R n + k ) is to consider it as the quotient O ( n + k ) O ( n ) × O ( k )where O ( n ) is the orthogonal group, the group of isometries of R n .The key element, in any of these approaches, is the type of algebraic structure of R n is a “set with addition” together with its structure as an R -module. On the otherhand, the standard R -module structure on R n is obtained by using the multiplicationoperation R × R → R .We consider the tropicalisation of the real line and determine a model for it, whichis more familiar to a topologist. We then consider the ways that one can define themultiplication operation on the tropical line, have granted that the addition is givenby the tropical structure. .1 A Model for ( R trop ) n To begin with, notice that R trop as a set is in a one to one correspondence with R > whereby the latter we mean the nonnegative real numbers. We fix such a correspondence asfollowing. First, notice that there is a homeomorphism of topological spaces f : R > → R defined by f ( x ) = − ln x where R > stands for the open half line of positive real numbers. We extend this to f : R > → R trop by setting f (0) = ∞ . This then enables us to get a one to one correspondence f × n = ( f, . . . , f ) : R n > → ( R trop ) n where X n denotes n -fold Cartesian product of X with itself. Note that it is quitestraightforward to see that the category of tropical spaces is closed under the Cartesianproduct. Note . For a topologist the object R n > is a familiar one. This is the space that isused as the model space to define manifolds with corners , where ∂ R n > = R n > − interior( R n > )corresponds to different types of (potential) singularities that such a manifold can have,whereas the interior corresponds to the smooth parts of such manifolds. Remark . Notice that the set R trop has two components, hence the set ( R trop ) n willhave 2 n components. We may write this as( R trop ) n = a k n ( R k ) ⊔ ( nk ) , where by ( R k ) ⊔ ( nk ) we mean (cid:0) nk (cid:1) -fold ` -product of R k with itself. We can make thismore precise. For instanse, in ( R trop ) we have one copy of R , two copies of R and onecopy of R in the following way: R corresponds to itself, one copy of R corresponds to {∞} × R , another copy of R corresponds to R × {∞} , and R corresponds to ( ∞ , ∞ ).Under the correspondence of ( R trop ) with R > we see that R corresponds to the interiorof R > , R × {∞} corresponds to the x -axis without the origin, {∞} × R corresponds tothe y -axis, and ( ∞ , ∞ ) corresponds to the origin.Our next objective, is to define an action on R trop and make sense of k -dimensionalsubspaces of ( R trop ) n + k . Here, we have to choose which object we like to act on R trop .We may choose R to acts on R trop , or we may choose R trop to act on R trop . R n > We consider the “dual” tropical structure on R > by using the taking the maximum oftwo given values. This operation has a min element 0 ∈ R > as its minimum. This theninduces a tropical structrue on R n > given v ⊕ v ′ = (max( v , v ′ ) , . . . , max( v n , v ′ n )) here v, v ′ ∈ R n > . In this case 0 ∈ R n > correspond to the minmum element where itsimage under f × n : R n > → ( R trop ) n corresponds to maximum element, i.e. ∞ . Noticethat g and f both are decreasing functions, hence they respect the tropical structure.Observe that the positive real line R > is a group under multiplication. In fact R > isthe tropicalisation of R > viewed as a group under multiplication. This then enablesus define the multiplication R > × R > → R > to be the usual multiplication of two realnumbers, i.e. ( r, s ) rs. It is straightforward to see that f and g become maps of tropical (semi)rings.Our next goal is to make it clear what we mean by the action of real numbers onthese spaces. R n > The one to one correspondence f × n : R n > → ( R trop ) n motivates us, and provides us with a tool, to define an action of R > on R n > . Let uswrite g for the inverse of f , i.e. g : R trop → R > is given by g ( x ) = e − x g ( ∞ ) = 0 . Recall from previous section that we have tropicalisation of any module with projection.In this case, R trop is the same as tropicalisation of R with τ given by the norm function.This then shows that it is possible to have a multiplication ⊙ : R trop × R trop → R trop given by t ⊙ t ′ = (cid:26) t + t ′ if t, t ′ ∈ R , ∞ if t = ∞ , or t ′ = ∞ . We use this to define the action of the additive group R on R trop as R × R trop → R trop given by ( r, t ) r ⊙ t. We use g to define an action of R × R > → R > by requiring that g respects this action,i.e. g has to be a map of R -modules. This induces the action R × R > → R > given by( r, v ) e − r v. Notice that here we consider the action of ( R , +) on R > . It is again clear that themappings f, g respect these actions. Notice that it is then quite clear that how todefine the corresponding actions of R on ( R trop ) n and R n > respectively. This is doneby defining the action component-wise. We investigate this in the next part, where welook at analogous of k -planes in these spaces. ote . Finally we explain the tittle for this section. The word flow , for a differentialtopologist, reminds the action of the real numbers under addition on a (smooth) man-ifold. The above is only one particular flow that we use, and let call it the “standardflow” on R > .It is possible to consider a more general setting. Notice that we have defined the map-pings f and g in a way that they respect the tropical (semi)ring structures. Hence,having any action of the additive group ( R , +) on R > will determine a correspondingaction on R trop , hence on R n > and ( R trop ) n respectively. Note that in the case of thestandard flow the point 0 ∈ R n > corresponds to a singular point of the flow. The study of tropical projective spaces, i.e. the space of lines in the tropical Euclideanspaces, seems to be the first natural step in attempt to understand the tropical Grass-mannian spaces. It turns out that, using our model for ( R trop ) n , it is an easy taskto identify P n trop as a set where P n trop is the set of all lines in ( R trop ) n +1 which “pass”through the origin. We state the result. Lemma 2.4.
There is a one to one correspondence P n trop → ∆ n where ∆ n is the standard n -simplex, i.e. ∆ n = { ( x , . . . , x n +1 ) ∈ R n +1 : x i > , X x i = 1 } . This is quite straightforward to see, when we use R n +1 > as our model for ( R trop ) n +1 .For instance, let n = 1. Then we need to identify all lines in R n +1 > . Notice that in R n +1 > any point together with the “origin” determines a line. Let ( a, b ) ∈ R > . Then the linepassing through this point and “reaching” the origin is determined by the orbit of thispoint under the action of real line, i.e. all points e − r ( a, b ) = ( e − r a, e − r b ) where r ∈ R .Notice that e − r is nothing but a positive real number. Hence, the orbit of ( a, b ) is theline passing ( a, b ) and the origin. However, notice that this will never reach the origin,and the origin will be a limit point for this line when r tends to + ∞ . This then showsthat the set { ( a, b ) : a > , b > , a + b = 1 } is in one to one correspondence with theseline. We need to identify the lines that correspond to the cases with a = 0 and b = 0.The x -axis and the y -axis then give these two end points. Notice that this look liketwo point compactification of the real line.For cases n > ∂ R n +1 > where as its interior points correspondto interior( R n +1 > ). Remark . The tropical projective space does not seem to be a tropical space, howeverits correspondence an standard simplex may be an evidence that it is a kind of avariety(?!). .3 Subspaces in Tropical Euclidean Spaces We like to look at the tropical version of the Grassmannian spaces. We note that thereis some work on this from an algebric-geometric point of view such asDavid Speyer, Bernd Sturmfels
The tropical Grassmannian
Adv. Geom. Vol.4 No.3pp389–411, 2004But I am not aware of the contents of this work, so I don’t make any comment.We choose to work with our model R n > to study the Grassmannian objects. TheGrassmannian space G k ( R n + k ) is the set of k -planes in R n + k . We take this approachand look at the k -planes in R n + k > . In the Euclidean space, it is quite straightforward to identify the k -dimensional sub-spaces of R n + k , i.e. the space spanned by k linear independent vector { v , . . . , v k : v i ∈ R n + k } .In tropical space ( R trop ) n + k two vectors ( t , . . . , t n ) , ( t ′ , . . . , t ′ n ) are linearly depen-dent in the tropical sense if and only if there exist r ∈ R such that t + r = t ′ , . . . , t n + r = t ′ n . However, our model is much easier to use. More precisely, two vectors ( t , . . . , t n ) , ( t ′ , . . . , t ′ n ) ∈ R n + k > are linearly dependent if there is a real number r such that( t , . . . , t n ) = e − r ( t ′ , . . . , t ′ n )i.e. t = e − r t ′ , . . . , t n = e − r t ′ n . This tells that two vectors in R n + k > are linear independent if one is not multiple ofthe other one, similar to the notion of the linear independent in the Euclidean sense.However, the notion of linear combination is quite different in two cases.Next, we have two identify what is meant by the space spanned by k linear independentvectors v , . . . , v k ∈ R n + k > in the tropical. The addition in R n + k > defined in previoussections shows that spanning in tropical sense is given bySpan trop { v , . . . , v k } = interior(Cone { v , . . . , v k } )where { v , . . . , v k } is an arbitrary set of k vectors in R n + k > and the cone Cone { v , . . . , v k } is the cone taken in usual Euclidean sense. For instance, for k = 2 the cone on twovectors is the area between the two lines determined by two vectors. By a k -subspace C ⊂ R n + k > we mean a cone which is span of k independent vectors, i.e. C = Span trop { v , . . . , v k } where { v , . . . , v k } is a linearly independent set.Notice that there is not a precise notion of dimension here. The reason is that not very point in R n + k > is a linear combination of finite number of vectors. The reason forthis lies in the way that the tropical addition, and scalar multiplication on R n + k > aredefined. The following observation provides us with a framework to look at this. Lemma 2.6.
Let u , . . . , u n ∈ R n denote the standard Euclidean basis elements, i.e. u = (1 , , . . . , , . . . , u n = (0 , . . . , , . We then have v ∈ interior( R n > ) ⇐⇒ v ∈ Span trop { u , . . . , u n } , where interior( R n > ) = R n > − ∂ R n > . The space ∂ R n > is characterised by ( x , . . . , x n ) ∈ ∂ R n > ⇐⇒ x t = 0 for some t n. Remark . We note that according to this lemma, the space R n > is not finitely gen-erated as an ( R , +)-set. The reason for this is that we don’t have ∞ in R . Later on,we will consider the action of R trop on this set, where R n > becomes an R trop -module.Notice that a vector v is in interior( R n > ) if all of its component, written in the usualEuclidean basis, are positive. Observe that in particular, if we choose any vector, u i such as in the above lemma, then under the correspondence R n > → ( R trop ) n we can seethat Span trop { u i } ≃ R . Moreover, let us write Span trop { b u i } ≃ {∞} . The above lemma together with the notation that just introduced our allows us toformally rewrite the decomposition of R n > corresponding to the one given by Remark2.2. The result reads as following. Corollary 2.8.
The space R n > has the following decomposition as Span trop { u , . . . , u n }⊔ Span trop { u , . . . , u n − , b u n } ⊔ · · · ⊔ Span trop { b u , u , . . . , u n }⊔ Span trop { u , . . . , u n − , b u n − , b u n } ⊔ · · · ⊔ Span trop { b u , b u , u , . . . , u n }⊔⊔ · · · ⊔ Span trop { u , b u , . . . , b u n } ⊔ · · · ⊔ Span trop { b u , b u , . . . , b u n − , u n }⊔{ (0 , , . . . , } . Here b u i means that the vector u i is not in the set. Moreover, under the correspondence R n > → ( R trop ) n the space Span trop { u , . . . , u i − , b u i , . . . , b u j , u j +1 . . . u n } maps to Span trop { u } × · · · × Span trop { u i − } × Span trop { b u i } · · · × Span trop { b u j } × Span trop { u j +1 } × · · · × Span trop { u n } hich is the same as R × · · · × R × {∞} × · · · × {∞} × R × · · · × R where for each u k in the spanning set we obtain a copy of R at k th position, and foreach b u i we obtain a copy of {∞} at the i th position. Finally, we have a little observation which will be important later on when weconsider the generalised tropical Grassmannians.
Lemma 2.9.
Let v , . . . , v k ∈ R n > be linearly independent with k > . Let v ∈ Span trop { v α , . . . , v α j } where α , . . . , α j k and j < k . Then v Span trop { v , . . . , v k } .In particular, v i Span trop { v α , . . . , v α j } . This is easy to see, as if v ∈ Span trop { v α , . . . , v α j } then v has to belong to boundaryof the k -subspace determined by v , . . . , v k . The result then follows from Corollary 2.7. G k ( R n > ) to configuration spaces We like relate the Grassmannian space G k ( R n > ) to some configuration spaces. Themapping fails to be an isomorphism. But it at least provides a tool which presumablywill help to analyse, and calculate more sophisticated algebraic invariants of thesespaces.We start by recalling the analogous construction in homotopy. Let us fix an arbi-trary basis for R n . Let V ⊂ R n be a k -dimensional subspace. We then can choose abasis for it, say { v , . . . , v k } . The fact that there are linearly independent means thatthey give rise to k distinct lines, hence defines a mapping G k ( R n ) → F ( P n − , k ) , where given any set X we define the set of configuration of n point in X as F ( X, n ) = { ( x , . . . , x n ) : x i ∈ X, i = j = ⇒ x i = x j } . The above mapping fails to be a homeomorphism as it is possible to choose k distinctpoints in P n − , or say k distinct vectors, which are not necessarily linearly independent.We write G k ( R n > ) for the set of all k -subspaces of R n > . Assume that we have a k -subspace C = Span trop { v , . . . , v k } ⊂ R n > . Let l , . . . , l k be k distinct lines determinedby v , . . . , v k respectively. This determines a mapping G k ( R n > ) −→ F ( P n − , k ) . Notice that P n − is the same as ∆ n − . The fact that v , . . . , v k are linearly independentin the tropical sense, implies that non of these vectors falls into the cone generated bythe other ones. If we use v , . . . , v k to denote those k points in ∆ n − that correspondto these lines, we obtain a convex set, where here convex means convex as a subset of R n with its usual metric.On the other hand, if we choose any convex set in ∆ n − with k vertices we obtain k vector in R n > which are independent in the tropical sense. This completes the proof ofthe following observation. heorem 2.10. There is an isomorphism of sets G k ( R n > ) → F (∆ n − , k ) convex where F (∆ n − , k ) convex refers to a subset of F (∆ n − , k ) whose points are in one to onecorrespondence with convex subset of ∆ n − with k vertices. R n > as an R trop -module Recall from previous sections that R × R n > → R n > gives R n > structure of an ( R , +)-set.This action is not compatible when we consider the field of real lines, with its usualaddition and multiplication. However, it is possible to obtain structure of an R trop -module on R n > .First, define R trop × R > → R > by( r, t ) e − r t, (+ ∞ , t ) . Recall that R trop has a (semi)ring structure when regarded as ( R ∪ { + ∞} , min , +)whereas R > has the tropical structure when regarded as ( R > , max , · ) where · denotesthe usual product. We then define the action R trop × R n > → R n > to be the component-wise action, i.e. ( r, ( t , . . . , t n )) ( e − r t , . . . , e − r t n ) , (+ ∞ , ( t , . . . , t n )) (0 , . . . , . This definition is very similar to the previous one. It does not change the notion ofthe linear independence. However, there is slight difference in the notion of span. Wewrite Span
Trop to distinguish it from Span trop . Lemma 2.11.
Suppose v , . . . , v k ∈ R n > . Then Span
Trop { v , . . . , v k } = Cone { v , . . . , v k } . Hence, a slight change in the ground set acting on R n > , i.e. replacing R with R trop has the effect that it adds the limit points of a cone to it. As a corollary R n > becomesfinitely generated over R trop . We have the following. Corollary 2.12.
Suppose u , . . . , u n denote the usual basis elements for R n . We thenhave R n > = Span Trop { u , . . . , u n } . Here, any point on ∂ R n > will have tropical coordinates which are formed of real numbers,and + ∞ . Moreover, { u , . . . , u n } is the only set of vectors satisfying this property, i.e.if there is any set of vectors { v , . . . , v n } such that R n > = Span Trop { v , . . . , v n } , then each v i will be a re-scaling of of u j for unique j n . s an example, consider R > and consider the point (1 ,
0) which is on its boundary.The Corollary 2.7 implies that it can not be written as any linear combination of twovectors in R > when regarded as an R -set. However, as an R trop -module, we may write(1 ,
0) = e (1 ,
0) + ∞ (0 , R > the point (1 ,
0) may be written as the column vector (cid:20) ∞ (cid:21) . G Trop k ( R n > ) Likewise the space G k ( R n > ) we define G Trop k ( R n > ) to be the set of all k -subspaces in R n > when regarded as an R trop -module. We say C ⊆ R n > is a k -subspace if there are k linearly independent vector v , . . . , v k ∈ R n > such that C = Span Trop { v , . . . , v k } . We may refer to G Trop k ( R n > ) as the generalised tropical Grassmannian space. Noticethat there is a one to one correspondence G k ( R n > ) −→ G Trop k ( R n > )given by C C where C denotes the closure of C , i.e. C = C ∪ ∂C . The inverse mapping G Trop k ( R n > ) −→ G k ( R n > )given by C interior C = C − ∂C. Accordingly we obtain the following description of G Trop k ( R n > ). Theorem 2.13.
There is a one to one correspondence G Trop k ( R n > ) −→ F (∆ n − , k ) convex . Remark . Before proceeding further, we like to draw the reader’s attention to anessential difference between G k ( R n > ) and G Trop k ( R n > ) in one hand and their Euclideananalogous G k ( R n ) on the other hand. In the Euclidean space R n any set of n -linearlyindependent set will span R n , however in R n > either as a ( R , +) or as an R trop -modulethe only option for such a choice is provided by the standard basis. Although, accordingto Lemma 2.6 as an ( R , +)-set this it is not possible to generate all of R n > in tropicalsense.For instance, consider R > and let C ∈ G k ( R > ) be defined as C = Span trop { (1 , , , (0 , , , (1 , , } . his is a 3-subspace in R > and yet it is not equal to R > . We note that all of those threevectors defining C are linearly independent. We can also consider to C ∈ G Trop3 ( R > )where C = R > .A consequence of this is that we may choose another vector v ∈ R > which does notbelong to C , i.e the set { (1 , , , (0 , , , (1 , , , v } is a linearly independent set in the tropical sense. This determines a cone formed by4 vectors which can not be generated by any 3 vectors. Hence we obtain a 4-subspacein R > giving rise to an element of G ( R > ).In general, we may choose k > n when we consider G k ( R n > ). In fact k can be anyarbitrary number.It is quite interesting to see how a k -subspace in R n > maps under the tropicalisomorphism R n > → ( R trop ) n . Recall from previous sections that ( R trop ) n is disjoint union of 2 n copies of R m with0 m n . Now assume that C ∈ G k ( R n > ), i.e. C = interior(Cone { v , . . . , v k } )where v , . . . , v k ∈ R n > are linearly independent(in the tropical sense). Notice that inthis case C is given by the interior of the cone, which implies that C ⊂ interior( R n > ).We now that interior( R n > ) maps to R n ⊂ ( R trop ) n . Moreover, note that the mapping f : R n > → ( R trop ) n is continuous when restricted to interior( R n > ). This implies that C also maps into R n ⊂ ( R trop ) n .Next, we consider C ∈ G Trop k ( R n > ) and its image under the f : R n > → ( R trop ) n .Notice that if C ∈ G Trop k ( R n > ) then it is a closed cone, where by being closed we meanclosed as a set in R n > viewed as a topological space with its topology inherited fromthe standard topology on R n .If C ⊂ interior( R n > ) then according to the previous case, all of C maps to only onecomponent of ( R trop ) n , namely to R n .Another possibility is that C ∩ ∂ R n > = φ . In this case then the image of C under f : R n > → ( R trop ) n will land in more than one factor of ( R trop ) n viewed as a disjointunion. The following provides us with an example. Example . Consider the space R > , together with vectors v = (1 , ,
1) and v =(0 , , C = Span Trop { v , v } as an element of G Trop2 ( R > ). Let L = { ( t, , t ) : t > } , L = { (0 , t, t ) : t > } , i.e. L i ∪ { (0 , , } is the line determined by v i . It is then clear that ∂C = L ∪ L . nder the correspondence f : R > → ( R trop ) we have f ( L ) ⊂ R × {∞} × R f ( L ) ⊂ {∞} × R × R f (0 , ,
0) = ( ∞ , ∞ , ∞ ) . The image of C under f is an example of a -subspace in ( R trop ) .In general, suppose C ∈ G Trop k ( R n > ) with C = Span Trop { v , . . . , v k } . If C ∩ ∂ R n > = φ then there are α i ∈ { , . . . k } with v α i ∈ ∂ R n > . Recall from Corollary 2.8 that R n > hasa decomposition asSpan trop { u , . . . , u n }⊔ Span trop { u , . . . , u n − , b u n } ⊔ · · · ⊔ Span trop { b u , u , . . . , u n }⊔ Span trop { u , . . . , u n − , b u n − , b u n } ⊔ · · · ⊔ Span trop { b u , b u , u , . . . , u n }⊔⊔ · · · ⊔ Span trop { u , b u , . . . , b u n } ⊔ · · · ⊔ Span trop { b u , b u , . . . , b u n − , u n }⊔{ (0 , , . . . , } . In this decomposition the first factor, i.e. Span trop { u , . . . , u n } corresponds to interior( R n > ),and the other factors correspond to ∂ R n > . Hence, each of v α i will belong to one, andonly one, of the factors in the above decomposition. For instance, assume v β , v β , . . . , v β j ∈ Span trop { u , . . . , b u i , . . . , b u j , . . . , u n } =: S. Notice that apart from { (0 , , . . . , } , any other factor in the above decompositionis an open set, when viewed as a subspace of R n with its usual topology. This thenimplies thatSpan Trop { v β , v β , . . . , v β j } ⊂ Span trop { u , . . . , b u i , . . . , b u j , . . . , u n } . Applying Corollary 2.8 shows that Span
Trop { v β , v β , . . . , v β j } maps into the factor of( R trop ) n corresponding to S . This then completely determines where C ∩ ∂ R n > mapsunder the tropical isomorphism f : R n > → ( R trop ) n . This concludes our notes on thegeneralised Grassmannian tropical spaces. ( R trop ) n . We like to analyse the those subspaces of ( R trop ) n which are in the images of G Trop k ( R n > )and map to more than one factor in the disjoint union decomposition for ( R trop ) n .We define what is meant by a k -subspace in ( R trop ) n . Recall that an example of a -subspace was given the previous section. n order to proceed, we need to fix an order on the disjoint union decompositionfor ( R trop ) n . Let I = ( i , . . . , i r ) with each of its entries belonging to { , } . By the set( R trop ) nI we mean a product of copies of the real line R , and the singleton {∞} in thefollowing way: if i j = 1 then we have a copy of R , and if i j = 0 then we have a copyof {∞} . For example, for n = 2, we have ( R trop ) , = R × {∞} . It is then clear that( R trop ) n = G I ∈{ , } n ( R trop ) nI . Moreover, let | I | = P i j + 1. We then will refer to ( R trop ) nI as the | I | -th factor of( R trop ) n . This also induces an order on these sequences (really the binary expansionof positive natural numbers) by I > J ⇐⇒ | I | > | J | . This is the same as the the lexicographic order on the sequences I and induces an orderon the factors of ( R trop ) n as following( R trop ) nI ( R trop ) nJ ⇐⇒ I J. Next, let k > k = ( k , . . . , k n ) ∈ Z n be a sequence of nonnegative integers,with the most left nonzero entry equal to k . Consider a collection of spaces K = { K i :1 i n } where K i is a subspace of the i -th factor of ( R trop ) n with K i = Span trop { v i , . . . , v ik i } and { v i , . . . , v ik i } being a linear independent set in the tropical sense. Moreover, weset the span of the empty set to be the empty set. We call K as k -subspace, if thereis a k -subspace C ∈ G Trop k ( R n > ) such that f ( C ) maps into ( R trop ) n with its image indifferent factors of ( R trop ) n being given by K i ’s. Remark . It is possible to give a more explicit account of the above calculations.In order to do this, we need to label different components of R n > . It is similar to whatwe did in above. Let I be a sequence of length n whose entries are either 1 or 0. Let u , . . . , u n denote the usual Euclidean basis for R n . Let ( R n > ) I ⊂ R n > be given by aproduct of copies of the open real half line R > and the singleton { } , where for i j = 1we have a copy of R > at j th position, and for i j = 0 we have have a copy of { } at the j th position. For example, in the case of n = 2 we have ( R > ) (1 , = R > × { } which isthe x -axis without { (0 , } . We refer to ( R n > ) I as the | I | -th component of R n > . Noticethat ( R n > ) I = Span trop { e , . . . , e n } where e j = u j if i j = 1 and e j = b u j if i j = 0. Hence,according to Corollary 2.8 we have R n > = [ I ∈{ , } n ( R n > ) I . It is now evident that the | I | -th component of ( R n > ) maps homeomorphically onto the | I | -th factor of ( R trop ) n .Now assume that C ∈ G Trop k ( R n > ) with C ∩ ∂ R n > = φ . Let C I = C ∩ ( R n > ) I be the | I | -th face of C . Then C I maps homeomorphically into the | I | -th factor of ( R trop ) n under the tropical isomorphism R n > → ( R trop ) n . .5 Tropical Matrices, Tropical Orthogonal Matrices An alternative description of Grassmannian spaces is provided by viewing them as theorbit space of two orthogonal groups acting on another one. This is our main motiva-tion study “tropical matrices”.Let M m,n ( R > ) be the set of all m × n matrix whose entries are elements in R > .This set admits structure of a semi-ring induces by the addition and multiplication in R > . More precisely, for A ∈ M m,n ( R > ) let A ij denote the ( i, j ) entry of A . Then for A, B ∈ M m,n ( R > ) we define the addition, denoted with ⊕ , by( A ⊕ B ) ij = A ij ⊕ B ij = max( A ij , B ij ) . If A ∈ M m,p ( R > ) and B ∈ M p,n ( R > ), then the multiplication, denotes with ⊙ , isdefined by ( A ⊙ B ) ij = ⊕ pk =1 A ik ⊙ B kj = max( A ik B kj : 1 k p ) . Under these operations, the identity element for ⊕ is the zero matrix, where as theidentity matrix for ⊙ is the identity matrix. Moreover, notice that M m,n ( R > ) is an R trop -module.One may ask whether if any matrix in M n,n ( R > ) is invertible. The answer is positive,and an example is provided by the set of all diagonal matrices whose diagonal doesnot have any nonzero element. This is a “universal example” of such matrices as thefollowing theorem confirms, by giving a complete classification of all such matrices. Theorem 2.17.
Let A ∈ M n,n ( R > ) be a matrix with right ⊙ -inverse with , i.e. thereexists B ∈ M n,n ( R > ) such that A ⊙ B = I . Then there exists σ ∈ Σ n and a diagonalmatrix D = ( D , . . . , D n ) ∈ M n,n ( R > ) whose diagonal entries are nonzero, and A = ( D σ − (1) , . . . , D σ − ( n ) ) . Here Σ n is the permutation group on n letters. Moreover, the matrix B is determinedby B ij = ( if A ji = 0 A ji if A ji = 0 . Notice that according to this theorem any matrix with right inverse also has a leftinverse and they are the same. This is again straight away to see this once we observethat ( A ⊙ B ) ii = max( A ik B ki : 1 > k > n ) = 1 , ( A ⊙ B ) ij = max( A ik B kj : 1 > k > n ) = 0 for i = j. For instance, let i = 1. The fact that ( A ⊙ B ) = 1 implies that there exists k such that A k = B k = 0. Combining A k = 0 together with ( A ⊙ B ) ij = 0 for i = j implies that B kj = 0 for j = i . Applying this for each row of A ⊙ B completes the proof.Next, we consider the problem of determining tropical orthogonal matrices. Recallthat in the Euclidean case, the orthogonal matrix O ( n ) is an n × n matrix whose ach column viewed has a vector in R n has unit norm, i.e. lies on the ( n − S n − = { x ∈ R n : || x || = 1 } , and is perpendicular to all other columns.We may define the tropical inner product h− , −i trop : R n > × R n > → R trop by h v, w i trop = ⊕ ni =1 v i ⊙ w i = max( v i w i : 1 > i > n )with v = ( v , . . . , v n ) and w = ( w , . . . , w n ). In particular, we have the tropical normon R n > given by || x || trop = (max( x i : 1 i n )) / . In particular, the tropical circle is given by S = { x ∈ R > : || x || trop = 1 } = { (1 , x ) ∈ R : x } ∪ { ( x ,
1) : x } . More generally, the tropical n -sphere is give by S n trop = { ( x , . . . , x n +1 ) ∈ R n +1 > : || x || trop = 1 } = S n +1 i =1 { ( x , . . . , x n +1 ) ∈ R n +1 > : x i = 1 , j = i = ⇒ x j } Moreover, these spaces admit a tropical structure.
Lemma 2.18.
The tropical sphere S n trop together with the maximum operation, inher-ited from R n +1 > , is a tropical space with the identity element for this operation givenby (1 , , . . . , . It is quite tempting to see what the analogous of the orthogonal group will be. Itis quite easy to determine form of such matrices. The reason is provided with thefollowing lemma.
Lemma 2.19.
Let A = ( A , . . . , A n ) ∈ M n,n ( R > ) where A i denotes the i -th column of A . Suppose || A i || trop = 1 and h A i , A j i trop = 0 for i = j . Then A has the same columnsas the identity matrix. Let O ( n ) trop be the set of all matrices identifies by the above lemma, i.e. set of alltropical orthogonal matrices. Lemma 2.20.
Let Σ n denote the permutation group on n letters. Let the action Σ n × M m,n ( R > ) → M m,n ( R > ) be given by ( σ, ( A , . . . , A n )) ( A σ − (1) , . . . , A σ − ( n ) ) , where A = ( A , . . . , A n ) is an arbitrary m × n matrix written in a column form. Then O ( n ) trop is given by the orbit of the identity matrix under the action Σ n × M n,n ( R > ) → M n,n ( R > ) . otice that there is an inclusion, in fact a map of monoids, O ( n ) trop −→ Gl ( R > , n ) . The above lemma tells us that the action of O ( n ) trop × M n,n ( R > ) → M n,n ( R > )given by the tropical matrix multiplication, will be only the permutation of the rowsof a given matrix. Let us write GL ( R > , n ) for the set of all tropical n × n invertiblematrices. For A ∈ GL ( R > , n + k ) we write A = (cid:18) A nn BC A kk (cid:19) where A nn is the n × n and A kk is a k × k block. This allows one to define the actionof O ( n ) trop × O ( k ) trop on GL ( R > , n + k ). One then will guess that there should be aone to one correspondence Gl ( R > , n + k ) O ( n ) trop × O ( k ) trop −→ G k ( R n + k > ) . Finally, notice that O ( n ) trop is a monoid under the tropical matrix multiplication. Suppose M is an arbitrary monoid. An element a ∈ M is idempotent if a = a . Weconsider to the problem of determining the idempotent in the monoid M n,n ( R > ). Theresult reads as following. Lemma 2.21.
Let A be an idempotent n × n matrix entries from R > . Then A satisfiesthe following conditions A ii A ik A ki min( A ii , A kk ) if i = k. The proof is straightforward once we consider the diagonal elements. The equation A ⊙ A = A implies that max( A ik A ki : 1 k n ) = A ii . For instance, this implies that A ii A ii which means that A ii
1. The other inequalityis obtained in a similar fashion, by comparing the equations for A ii and A kk . We consider the idea of the infinite dimensional orthogonal tropical matrices. Noticethat for any m, n there is a mapping M m,n ( R > ) → M m +1 ,n +1 ( R > ) given by A (cid:18) A
00 1 (cid:19) . In particular, this induces a mapping O ( n ) trop → O ( n + 1) trop . We then define theanalogous of the infinite orthogonal group by O trop = colim O ( n ) trop . his object inherits a monoid structure induces from the monoid structure on thefinite dimensional tropical orthogonal monoids. One then hopes that this will give acharacterisation of the infinite dimensional Grassmannians. The algebraic topology of fibre bundles with a given topological space F as the fibre,is understood in terms of the classifying space of the groups of automorphisms of F .By analogy one may consider to fibre bundle theory of surjections E → B whose fibresare given by copies of the tropical space R n > . This then makes it quite reasonable toconsider the classifying space BO ( n ) trop of the tropical orthogonal monoids O ( n ) trop where these are monoids under the tropical matrix multiplication. The classifyingspace functor is defined for monoid (in fact for topological monoids which carry aweaker structure). Hence, one may observe that the R n > -bundles are classified in termsof mapping into BO ( n ) trop .The interest in such theory, and the theory of associated characteristic classes,seems to come from the theory of singularities. We note that a canonical example fora tropical bundle will be “tangent bundle” of a tropical manifold. This then motivatesone to claim that the classification of these singularities might be done in terms of thecharacteristic classes of the associated tangent bundle.Hadi Zare, The School of Mathematics, Manchester University, Manchester, UK M139PL, email: [email protected]: [email protected]