A tropical view on the Bruhat-Tits building of SL and its compactifications
aa r X i v : . [ m a t h . C O ] M a y A tropical view on the Bruhat-Tits building of SL and itscompactifications
Annette Werner
Institut f¨ur MathematikGoethe-Universit¨at FrankfurtRobert-Mayer-Strasse 8D- 60325 Frankfurtemail: [email protected]
Abstract:
We describe the stabilizers of points in the Bruhat-Tits building of the group SL with tropical geometry. There are several compactifications of this building associ-ated to algebraic representations of SL . We show that the fans used to compactifyapartments in this theory are given by tropical Schur polynomials. Keywords:
Bruhat-Tits buildings, tropical hypersurfaces, Schur polynomials
Introduction
In this paper we relate tropical geometry to point stabilizers and compactificationsof the Bruhat-Tits building associated to the group SL .1et V be vector space of dimension n over a field K with a non-trivial non-Archimedeanabsolute value. Then the Bruhat-Tits building B ( SL ( V ) , K ) is a metric space with acontinuous SL ( V )( K )-action. It consists of apartments, i.e. real cocharacter spacesof maximal split tori. They are glued together with the help of certain subgroupsof SL ( V )( K ) which turn out to be the stabilizers of points in the building. We canregard the apartments which are real vector spaces from a tropical point of view.Every element in the special linear group over K defines a real matrix by taking thenegative valuation of all coefficients. Replacing addition and scalar multiplication bytropical addition and tropical scalar multiplication, such a matrix defines a tropicallinear map. In theorem 1.2 we prove that the tropical linear stabilizer of a point inan apartment coincides with its stabilizer with respect to the SL ( V )( K )-action onthe Bruhat-Tits building.Every geometrically irreducible algebraic representation of SL ( V ) induces a compact-ification of the building B ( SL ( V ) , K ) by [We3]. This compactification is defined byglueing together compactified apartments. Every apartment is compactified with thehelp of a fan defined by the combinatorics of the weights given by the representation.We assume that K has characteristic zero. Every representation of SL ( V ) is thengiven by a partition λ = ( λ ≥ . . . ≥ λ n ≥ S λ associated to this partition. We look at the tropical hypersurface defined bythe Schur polynomial which gives rise to a natural polyhedral complex. In Theorem2.4 we prove that the following fact. If the coefficients of the Schur polynomial allhave valuation zero, then the fan used to compactify one apartment is precisely thefan given by the tropical Schur polynomial.This result was motivated by the following example. Consider the identical repre-2entation of SL ( V ). The corresponding partition is (1 , . . . , S (1 , ,..., ( z , . . . , z n ) = z + . . . + z n . In this case the fan used tocompactify the apartment plays an important role in tropical convexity defined, asdefined by Develin and Sturmfels in [De-St].This paper was inspired by [JSY] who took another look the Bruhat-Tits buildingfor SL from a tropical point of view by describing and computing convex subsets ofthe building via tropical geometry. Acknowledgements:
I thank Michael Joswig and Thorsten Theobald for usefuldiscussions on tropical geometry.
Tropical geometry is based on the tropical semiring ( R , ⊕ , ⊙ ) with a ⊕ b = max { a, b } and a ⊙ b = a + b . Some authors use the (min , +)-version of the tropical semiringinstead of the (max , +) version.The space R n together with componentwise addition ⊕ is a semimodule under thesemiring ( R , ⊕ , ⊙ ), if we put a ⊙ ( x , . . . , x n ) = ( a + x , . . . , a + x n ) for a ∈ R and( x , . . . , x n ) ∈ R n . Let T n − = R n / R (1 , . . . , R n after the following equivalence relation: ( x , . . . , x n ) ∼ ( y , . . . , y n )if and only if there exists some a ∈ R such that x i = a ⊙ y i = a + y i for all i . We call T n − the tropical torus of rank n − T n − with the quotient3opology. In the literature, the space T n − is often called tropical projective space.We avoid this terminology, since T n − is not compact. In section 2.2, we considera compactification of T n − , which could be viewed as a tropical analogue of theprojective space.Let K be a field with a valuation map v : K × → R . We put v (0) = ∞ . Consider thenegative valuation map − v : K → R −∞ = R ∪ {−∞} . We extend the operations ⊙ and ⊕ to R −∞ by a ⊙ −∞ = −∞ for all a ∈ R −∞ and a ⊕ −∞ = a for all a ∈ R −∞ . Definition 1.1
Let g = ( g ij ) i,j be a matrix in GL n ( K ) . Then we define the as-sociated tropical matrix by g trop = ( − v ( g ij )) i,j ∈ Mat n × n ( R ) . For every vector x = t ( x , . . . , x n ) ∈ R n we define the vector g trop · x = t ( y , . . . , y n ) ∈ R n by thetropicalized linear action y i = − v ( g i ) ⊙ x ⊕ . . . ⊕ − v ( g in ) ⊙ x n = max j {− v ( g ij ) + x j } . Note that y i lies indeed in R , since at least one entry g ij in the i -th line must benon-zero, so that at least for one j the term − v ( g ij ) + x j is not equal to −∞ .Beware that this does not define an action of GL n ( K ) on R n , as the following exampleshows. Take n = 2, and put g = and h = − . Then( gh )trop · x x = x max { x , x } , but g trop · h trop · x x = max { x , x } max { x , x } . .2 Bruhat-Tits buildings Now we want to explain how the tropical torus T n − appears in the context ofBruhat-Tits buildings. Let us first introduce some notation. We fix a ground field K with a non-trivial non-Archimedean absolute value | | , i.e. there is a valuation map v : K × → R such that | x | = e − v ( x ) for all x ∈ K × . For example, K could be a localfield, i.e. a field which is complete with respect to discrete valuation and has finiteresidue field. In this case, K is either a finite extension of Q p or a field of formalLaurent series over a finite field. Alternatively, K could be C p , which denotes thecompletion of the algebraic closure of Q p , or the field of Puiseux series over C whichis quite popular in tropical geometry. By O K we denote the ring of integers in K ,i.e. O K = { x ∈ K : | x | ≤ } .Let G be a reductive group over K . In the two groundbreaking papers [Br-Ti1]and [Br-Ti2] Bruhat and Tits define a metric space B ( G, K ), now called Bruhat-Tits building, endowed with a continuous G ( K )-action, if G and K satisfy certainassumptions. These assumptions are for example fulfilled if the ground field is alocal field or if the group G is split, i.e. if G contains a maximal torus which is splitover K . If the valuation is discrete, the building B ( G, K ) carries a polysimplicialstructure. If the valuation is not discrete, there is still a notion of facets, see [Rou].Let us now recall the construction of B ( G, K ) in the special case G = SL ( V ) where V is a vector space of dimension n over K . Let T be a maximal split torus in SL ( V ), i.e.there is a basis e , . . . , e n of V such that T consists of all diagonal endomorphisms in SL ( V ) with respect to this basis. Let X ∗ ( T ) = Hom K ( T, G m ) denote the charactergroup. Here we write Hom K for morphisms of affine K -group schemes. Generally,we regard the algebraic groups T , SL ( V ) etc. as group schemes over K and write5 ( K ), SL ( V )( K ) etc. for the groups of K -rational points (i.e. for the correspondinggroups of matrices with coefficients in K ).We define a i ∈ X ∗ ( T ) by a i (diag( s , . . . , s n )) = s i , where diag( s , . . . , s n ) is the diagonal matrix with entries s , . . . , s n . Then X ∗ ( T ) = L ni =1 Z a i / Z ( a + . . . + a n ).Besides, let X ∗ ( T ) = Hom K ( G m , T ) denote the cocharacter group of T . Let η i : G m → GL ( V ) be the cocharacter of GL ( V ) mapping x to the diagonal matrix withentries t , . . . , t n , where t i = x and t j = 1 for j = i . Then X ∗ ( T ) = { m η + . . . + m n η n : m i ∈ Z with X i m i = 0 } . There is a natural pairing X ∗ ( T ) × X ∗ ( T ) → Z induced by a i ( η j ) = δ ij .The R -vector space A = X ∗ ( T ) ⊗ Z R = { P ni =1 x i η i : x i ∈ R with P i x i = 0 } is theapartment given by the torus T in the Bruhat-Tits building B ( SL ( V ) , K ). Mapping η , . . . , η n to the canonical basis of R n provides a homeomorphism A −→ T n − between the apartment A and the tropical space T n − . In the course of this paperwe always identify A with T n − using the map above.For every t = diag( t , . . . , t n ) ∈ T ( K ) with entries t , . . . , t n ∈ K × we define apoint in A by ν ( t ) = − v ( t ) η + . . . + − v ( t n ) η n . Then t ∈ T ( K ) acts on A bytranslation with ν ( t ). Besides, let N be the normalizer of T in SL ( V ). For everyelement n ∈ N ( K ) there is a permutation σ on { , . . . , n } such that n ( e i ) = t i e σ ( i ) for suitable t , . . . , t n ∈ K × . The Weyl group W = N ( K ) /T ( K ) can therefore be6dentified with the symmetric group S n . Hence W acts in a natural way on A bypermuting the coordinates of a given point. We can put both actions together to anaction of N ( K ) on A by affine-linear transformations. If A is endowed with its realtopology and N ( K ) is endowed with the topology given by the topology on K , thisaction is continuous.Bruhat-Tits theory provides a filtration on the K -rational points of the root groupsfor any reductive group. In our special case this structure can be described as follows.The root system Φ( T, SL ( V )) consists of all characters of the form a ij = a i /a j forall i, j ∈ { , . . . , n } with i = j . The root group U a corresponding to the root a = a ij consists of all u ∈ SL ( V ) such that u ( e k ) = e k for all k = i and u ( e j ) = e j + ωe i for some ω ∈ K .Then we have a homomorphism ψ a : U a ( K ) → Z ∪ {∞} mapping u to v ( ω ). The Bruhat-Tits filtration on U a ( K ) is given by the subgroups U a,l = { u ∈ U a ( K ) : ψ a ( u ) ≥ l } for all l ∈ Z .For all x ∈ A consider the group U x generated by U a, − a ( x ) for all a ∈ Φ( T, SL ( V )),and let N x be the subgroup of N ( K ) consisting of all elements stabilizing x . Thenthe group P x = N x U x = U x N x can be used to define the Bruhat-Tits building7 ( SL ( V ) , K ) as the quotient of SL ( V )( K ) × A after the following equivalence rela-tion: ( g, x ) ∼ ( h, y ) , if and only if there exists an element n ∈ N ( K )such that ν ( n ) x = y and g − hn ∈ P x . The quotient space B ( SL ( V ) , K ) is endowed with the product-quotient topologyand admits a natural continuous SL ( V )( K )-action via left multiplication in the firstfactor. For all x ∈ A , the group P x is the stabilizer of x in B ( SL ( V ) , K ). All subsetsof B ( SL ( V ) , K ) of the form gA for g ∈ SL ( V )( K ) are called apartments.If the valuation on K is discrete, then B ( P GL ( V ) , K ) can be identified with theGoldman-Iwahori space of all non-Archimedean norms on K n modulo scaling, see[Br-Ti3]. Here a map γ : K n → R ≥ is a non-Archimedean norm if γ ( λv ) = | λ | γ ( v )and γ ( v + w ) ≤ sup { γ ( v ) , γ ( w ) } for all λ ∈ K and v, w ∈ K n and γ ( v ) = 0 only for v = 0 holds. Via this identification, the apartment A consists of all norms (moduloscaling) of the form γ (( λ , . . . , λ n )) = sup {| λ | r , . . . , | λ n | r n } for some real vector ( r , . . . , r n ).In this case the simplicial complex B ( G, K ) can be described as a flag complexwhose vertex set consists of all similarity classes of O K -lattices in K n . Here twolattice classes are adjacent if and only if there are representatives M and N of thesetwo classes satisfying πM ⊂ N ⊂ M , where π is a prime element in the ring ofintegers O K . 8 .3 A tropical view on the stabilizer P x We will now show that the stabilizer groups P x defined above with Bruhat-Titstheory have a tropical interpretation. Namely, we show that P x coincides with theset of all g ∈ SL n ( K ) stabilizing x under tropical matrix multiplication. Recall thatthe map ( g, x ) g trop · x does not define an action of SL n ( K ) on R n . Hence thetheorem below also shows that the tropical stabilizer of a point in R n is a groupwhich is not a priori clear. We use the basis e , . . . , e n of V to identify SL ( V ) with SL n . Theorem 1.2
Let x = P i x i η i be a point in the apartment A , and let P x be thestabilizer of x with respect to the action of SL n ( K ) on B ( SL n , K ) . Then we have P x = { g ∈ SL n ( K ) : g trop · t ( x , . . . , x n ) = t ( x , . . . , x n ) } . Proof:
First we consider the case that every x i lies in the image of the valuationmap v : K × → R . Hence there exist elements t i ∈ K × satisfying x i = − v ( t i ). For g = ( g ij ) i,j in SL n ( K ) we have g trop · t ( x , . . . , x n ) = t (max j {− v ( g j t j ) } , . . . , max j {− v ( g nj t j ) } ) . Hence g trop t ( x , . . . , x n ) = t ( x , . . . , x n ) is equivalent to the fact that max j {− v ( g ij t j t − i ) } =0 for all i . Now g ij t − i t j is the entry at position ( i, j ) of the matrix t − gt , where t denotes the diagonal matrix with entries t , . . . , t n . Therefore the matrix t − gt lies in SL n ( O K ). Conversely, every g such that t − gt lies in SL n ( O K ) stabilizes t ( x , . . . , x n ). Therefore { g ∈ SL n ( K ) : g trop · t ( x , . . . , x n ) = t ( x , . . . , x n ) } = tSL n ( O K ) t − .
9y Bruhat-Tits theory, tSL n ( O K ) t − is indeed the stabilizer of x = P i − v ( t i ) η i inthe apartment A .For a general point x = P i x i η i ∈ A there exists a non-Archimedean extensionfield L of K such that all x i are contained in the image of the valuation mapof L and a continuous SL n ( K )-equivariant embedding B ( SL n , K ) ֒ → B ( SL n , L ),see e.g. [RTW1], (1.2.1) and (1.3.4). We have just shown that { g ∈ SL n ( L ) : g trop · t ( x , . . . , x n ) = t ( x , . . . , x n ) } is equal to the stabilizer of x in the buildingover L . Intersecting with SL n ( K ) our claim follows. (cid:3) In this section we assume that our non-Archimedean field K has characteristic zero. Schur polynomials play an important role in the representation theory of the sym-metric group and of the general linear group. The polynomial representations of GL n over C were determined by Isaai Schur in his thesis. See [Pro] for a modernaccount. In [Gr], the theory is developped over arbitrary infinite ground fields. Werecall the following facts.Fix a natural number n ≥
2. We call λ = ( λ , . . . , λ n ) a partition (of λ + . . . + λ n )if λ ≥ . . . ≥ λ n ≥
0. 10he Schur polynomial S λ can be defined by the following formula: S λ ( z , . . . , z n ) = det(( z λ i + n − ij ) i,j =1 ,...n )det(( z n − ij ) i,j =1 ,...,n ) . For properties of these homogeneous symmetric polynomials see [Mac], I.3.
Example: S , ,..., ( z , . . . , z n ) = z + . . . + z n .Let V be a vector space over K of dimension n . For every partition λ there isa geometrically irreducible algebraic representation S λ ( V ) of SL ( V ) such that forevery g ∈ SL ( V ) with eigenvalues c , . . . , c n the trace of g on S λ ( V ) is equal to S λ ( c , . . . , c n ), where S λ is the Schur polynomial associated to λ . The partitions( λ , . . . , λ n ) and ( λ + m, . . . , λ n + m ) give the same representation for all integers m , and every geometrically irreducible algebraic representation of SL ( V ) arises inthis way.Let W be a K -vector space and let ρ : SL ( V ) → GL ( W ) be an irreducible algebraicrepresentation. Then the action of the torus T on W defines a decomposition of W in weight spaces. Put W µ = { w ∈ W : ρ ( t )( w ) = µ ( t ) w for all t ∈ T ( K ) } . Then every character µ of T with W µ = 0 is called a weight of ρ .A subset ∆ of the root system Φ( T, SL ( V )) is called a basis if it is linear independentand if every root a in Φ( T, SL ( V )) can be written as a = P c ∈ ∆ m c c where eitherall m c are non-negative or all m c are non-positive. The Weyl group W acts simplytransitively on the set of bases, see [Bou], chapter VI, § µ (∆) such that for all weights µ we have µ (∆) − µ = P a ∈ ∆ n a a with non-negative integral coefficients n a .11ince the Schur polynomial S λ is the trace of the irreducible representation S λ ( V ),we can write it as S λ ( z , . . . , z n ) = X µ dim( S λ ( V )) µ z µ with z µ = z µ · · · z µ n n .Hence for any µ , . . . , µ n ∈ N with P i µ i = P i λ i , the element µ a + . . . + µ n a n (modulo Z P i a i ) is a weight of the representation S λ ( V ) if and only if the monomial z µ · · · z µ n n occurs with a non-zero coefficient in the Schur polynomial S λ .The coefficients of the Schur polynomial S λ are given by Kostka numbers. To beprecise, we have S λ = M λ + X µ K λµ M µ , where µ runs over all partitions with P i µ i = P i λ i , and where M µ is the sum ofthe monomial z µ z µ · · · z µ n n and all the monomials we get from it by permuting thevariables. The Kostka number K λµ denotes the number of ways one can fill theYoung diagram associated to λ with µ µ K λµ = 0implies that P ki =1 λ i ≥ P ki =1 µ i for all k , it follows that the weight λ a + . . . + λ n a n is the highest weight of the representation S λ ( V ) with respect to the basis ∆ = { a , . . . , a n − n } . Definition 2.1
Let λ = ( λ ≥ . . . ≥ λ n ≥ be a partition. By T ( S λ ) we denotethe tropical hypersurface of R n given by the Schur polynomial S λ , i.e. T ( S λ ) is theclosure of the set { ( − v ( α ) , . . . − v ( α n )) ∈ R n : α , . . . , α n ∈ K × with S λ ( α , . . . , α n ) = 0 } . K of K . Since S λ ishomogeneous, ist also defines a tropical hypersurface in T n − , which is the image of T ( S λ ) under the quotient map R n → T n − .We write S λ ( z , . . . , z n ) = P N c N z N with N = ( ν , . . . , ν n ) and z N = z ν · · · z ν n n .(Here we write out all monomials.) Then by [EKL], theorem 2.1.1 we have T ( S λ ) = { ( x , . . . , x n ) ∈ R n :max N =( ν ,...,ν n ) {− v ( c N ) + ν x + . . . + ν n x n } is attained at least twice } . The tropical hypersurface T ( S λ ) gives rise to a polyhedral complex in R n . Let usrecall the definition which works for arbitrary tropical hypersurfaces.For basics and terminology concerning polyhedral complexes see [Zie]. Consider theextended Newton polytope P in R n +1 given by S λ = P N c N z N , i.e. P is the convexhull of all points ( − v ( c N ) , ν , . . . , ν n ) such that c N = 0 in R n +1 . The normal fan N P associated to P is the fan in R n +1 consisting of all cones N P ( F ) = { w = ( w , . . . , w n ) ∈ R n +1 : F ⊂ { x ∈ P : < w, x > = max y ∈ P < w, y > }} , where F runs over the non-empty faces of P and <, > denotes the canonical scalarproduct. Definition 2.2
We define a polyhedral complex C ( S λ ) in R n as the collection of allcells of the form N P ( F ) ∩ { w = − } . Since N P is a complete fan, this polyhedral complex has support R n . If we onlyconsider the cells of the form N P ( F ) ∩ { w = − } for dim( F ) ≥ ,
13e get a pure polyhedral complex with support T ( S λ ). In general there are different ways to compactify a Bruhat-Tits building B ( G, K )associated to a reductive group G over K . In [Bo-Se], Borel and Serre attach theso-called Tits building at infinity in order to derive finiteness results for arithmeticgroup cohomology. In [We2] a concrete compactification of the Bruhat-Tits buildingof P GL is studied, which can be identified with the space of isometry classes of non-Archimedean norms on K n . In [We1] we investigate a dual approach which fits intothe description of the building with lattices in K n .In [We3] we develop a general theory of compactifications. For every semisimplegroup G over a non-Archimedean local field K and every faithful, geometricallyirreducible algebraic representation ρ : G → GL ( W ) we define a compactificationof B ( G, K ). Its boundary can be identified with the union of Bruhat-Tits buildingsassociated to certain types of parabolics in G . The strategy is the following: Use thecombinatorics of the weights given by the representation ρ in order to compactifyone apartment. Then generalize Bruhat-Tits theory to define groups P x for all x in the compactified apartment and glue all compactified apartments together as inthe definition of B ( G, K ) (see 1.2). There is an even more general approach: In[RTW1] we realize the Bruhat-Tits building B ( G, K ) inside the Berkovich analyticspace G an and use the projection to analytical flag varieties of G to obtain a familyof compactifications of B ( G, K ). This fits together with the approach in [We3], see[RTW2].In this section we look at the finite family of compactifications of B ( SL ( V ) , K )14efined in [We3]. Note that the assumption in [We3] that the ground field K is alocal field is not needed in the specific examples we discuss here.Let ρ : SL ( V ) → GL ( W ) be a geometrically irreducible algebraic representation ofthe group SL ( V ) on a K -vector space W . For every basis ∆ of the root systemΦ( T, SL ( V )) we denote by µ (∆) the corresponding highest weight of ρ . Definition 2.3
We define a fan F ρ as the set of all faces of the cones C ∆ ( ρ ) = { x ∈ A : µ (∆)( x ) ≥ µ ( x ) for all weights µ of ρ } , where ∆ runs over the bases of Φ( T, SL ( V )) . Example 1.
Suppose that ρ = id . The weights of the identical representation are { a , . . . , a n } . For the basis ∆ = { a , a , . . . , a n − n } the highest weight is µ (∆) = a . Hence C ∆ ( ρ ) = { x ∈ A : a ( x ) ≥ a i ( x ) for all i } = { n X i =1 x i η i ∈ A : x ≥ x i for all i } . Since the Weyl group (which is isomorphic to the symmetric group S n ) acts simplytransitively on the set of bases, every maximal cone is of the formΓ k = { n X i =1 x i η i ∈ A : x k ≥ x i for all i } for some k = 1 , . . . , n . Hence the fan F ρ consists of the cones Γ k and of all theirfaces. Example 2.
Suppose that ρ is a representation with highest weight µ (∆) = na + ( n − a + . . . + 2 a n − + a n . Then for every root a ii +1 there exists a weight µ such that µ (∆) − µ = a ii +1 . Hence for ∆ = { a , . . . , a n − n } we find C ∆ ( ρ ) = { x ∈ A : a ( x ) ≥ , . . . , a n − n ( x ) ≥ } = { n X i =1 x i η i : x ≥ x ≥ . . . ≥ x n } .
15n this case we write C ∆ ( ρ ) = C (∆). It is the Weyl cone associated to ∆, see [Bou],chapter VI, § F ρ is the Weyl fan consisting of all C (∆) and their faces.Note that for every ρ the cone C ∆ ( ρ ) contains the Weyl cone C (∆). Since the unionof all Weyl cones is the total space A , we deduce that every fan F ρ has support A . The Weyl cones for different bases are different. In general however, we have C ∆ ( ρ ) = C ∆ ′ ( ρ ), whenever µ (∆) = µ (∆ ′ ), and this may happen for ∆ = ∆ ′ .The fan F ρ can be used to define a compactification A ρ of A , see [We3], section 2. Infact, we put A ρ = S C ∈F ρ A/ h C i and we endow this space with a topology given bytubular neighbourhoods around boundary points. For a more streamlined definitionof this topology see [RTW1], appendix B.Note that the compactification associated to the fan in example 1 above is the onestudied in [We1], whereas the compactification to the fan in example 2 is Landvogt’spolyhedral compactification studied in [La].The fan F ρ and hence the compactification A ρ only depend on the Weyl chamberface containing the highest weight of ρ , see [We3]. Hence we obtain a finite family ofcompactifications of A in this way. Identifying A with T n − we also obtain a finitefamily of compactifications for the tropical torus. The compactification of T n − givenby example 1 could be regarded as a tropical analog of projective space, see also [Jos].Fix a geometrically irreducible representation ρ and let λ be a partition such that ρ is isomorphic to S λ ( V ). We will now compare F ρ with the polyhedral complex C ( S λ ) given by the tropical Schur polynomial S λ , cf. section 2. Theorem 2.4
Assume that all the coefficients of the Schur polynomial S λ have val-uation zero. Then the fan F ρ coincides with the image of C ( S λ ) under the map n → T n − ≃ A . Proof:
Note that our hypothesis implies that all cells in C ( S λ ) are invariant un-der ( x , . . . , x n ) ( x + a, . . . , x n + a ) for any a ∈ R . It suffices to show that theimages of the cones of maximal dimension in C ( S λ ) are precisely the cones of max-imal dimension in F ρ , i.e. that they are of the form C ∆ = { x ∈ A : µ (∆)( x ) ≥ µ ( x ) for all weights µ } for some basis ∆ of Φ( T, SL ( V )).The cones of maximal dimension in C ( S λ ) are by definition of the form N P ( v ) ∩{ w = − } , where v is a vertex of P . Since by assumption all coefficients of S λ havevaluation zero, the polygon P can be identified with the convex hull of all ( ν , . . . , ν n )with c N = 0 in R n . Fix a vertex v of this polygon. Then v = ( µ , . . . , µ n ) with M = ( µ , . . . , µ n ) such that c M = 0, and the corresponding cone in C ( S λ ) is equal to C ( v ) = { ( x , . . . , x n ) : X i µ i x i ≥ X i ν i x i for all N = ( ν , . . . , ν n ) with c N = 0 } . Since this is a cone of full dimension in R n , its topological interior is non-empty.Hence there exists a point x ∈ R n such that P i µ i x i > P i ν i x i for all N =( ν , . . . , ν n ) with c N = 0 and N = M . The image of x in A lies in one of the Weylcones C (∆), where ∆ is a basis of the root system. Now the set of coefficients c N of S λ with N = ( ν , . . . , ν n ) and c N = 0 corresponds bijectively to the set of weights, ifwe map c N to ν = a ν + . . . + a n ν n . Let µ be the weight µ a + . . . + µ n a n . Then µ (∆) − µ is a linear combination of elements in ∆ with non-negative coefficients.Since all elements in ∆ are non-negative on the image of x , we find µ (∆) = µ .Therefore the image of C ( v ) in A is equal to C ∆ . Since every cone in F ρ is a face ofsome C ∆ , our claim follows. (cid:3) Let us have a look at the hypothesis that all Kostka numbers occuring as coefficients17f S λ have valuation zero. Note that this assumption is of course fulfilled if theresidue field of K has characteristic zero. Besides, for fixed λ , it is fulfilled for fieldsof almost all residue characteristics.The theorem implies that the polyhedral complex given by T ( S λ ) is in fact a fan,i.e. all faces are convex cones. This can also be seen directly from the description ofthe polyhedral complex in this special case. We consider the situation of example 1 in section 2.2. The identical representationof SL ( V ) corresponds to the partition λ = (1 , , . . . , A is given by the polyhedralcomplex associated to the Schur polynomial S (1 , ,..., ( z , . . . , z n ) = z + . . . + z n .Transferred to T n − it consists of all faces of the maximal conesΓ k = { ( x , . . . , x n ) + R (1 , . . . , ∈ T n − : x k ≥ x i for all i = 1 , . . . , n } . We have seen that this polyhedral complex arises in the compactification of thebuilding B ( SL ( V ) , K ) associated to the identical representation of SL ( V ). It alsoplays a role in the definition of tropical convexity.In [De-St], Develin and Sturmfels define tropical polytopes as tropical convex hulls offinitely many points in T n − . Every tropical polytope is the support of a polyhedralcomplex, see [De-St], section 3, which can be described as follows (see also [Jos]).Let M = { v , . . . , v r } be a finite subset of points in T n − . For every x ∈ T n − definethe type of x (depending on M ) astype( x ) = ( T , . . . , T n ) , T k = { i : v i − x ∈ Γ ′ k } with Γ ′ k = { ( x , . . . , x n ) + R (1 , . . . , ∈ T n − : x k ≤ x i for all i = 1 , . . . , n } . Then every type T = ( T , . . . , T n ) defines a polyhedron X T = { x ∈ T n − : x has type ( T , . . . , T n ) } . The union of all bounded polyhedra of this form is precisely the tropical convex hullof the finite set M .The cones Γ ′ k used in tropical convexity are the min-convex twins of the cones Γ k .To be precise, the map R → R given by x
7→ − x is an isomorphism between thetropical (min , +)-semiring and the tropical (max , +)-semiring, and we have − Γ ′ k =Γ k . Therefore a (max , +)-version of tropical convexity would use Γ k instead of Γ ′ k . We have seen that some features of the Bruhat-Tits building associated to SL ( V ) canbe interpreted in terms of tropical geometry. Every apartment in B ( SL ( V ) , K ) is atropical projective space. We see the stabilizers of points in the building as tropicalstabilizers via a tropical matrix action. Besides, we can identify the fans used tocompactify apartments with the polyhedral complex associated to tropical Schurpolynomials. It is a natural question if this can be generalized to other reductivegroups. 19esides, the compactification of B ( SL ( V ) , K ) described in example 1 in section 2.2is based on a fan which is also used in tropical convexity. What happens if we keepthe group SL ( V ), but use the fans associated to different compactifications, i.e. thefans associated to different tropical Schur polynomials? We can then define a versionof convexity adapting the definitions from [De-St]. What kind of geometry does thislead to? References [Bo-Se] A. Borel, J.-P. Serre:
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