Dissimilarity Vectors of Trees and Their Tropical Linear Spaces

Transcription

1 Dissimilarity Vectors of Trees and Their Tropical Linear Spaces Benjamin Iriarte Giraldo Massachusetts Institute of Technology FPSAC 0, Reykjavik, Iceland

2 Basic Notions A weighted n-tree T : A trivalent tree with set of leaves [n] = {,..., n} and s.t. every edge e is assigned a weight w(e), where (F. Cools) Let C m S m be the set of cyclic permutations. ( Take ) {i, i,..., i m} [n]. Assume m WLOG that = i, = i,...,m = i m. Then, d...m w(e) R >0 if e is internal. = min {d σ() + d σ()σ () + σ Cm d R m and d σ is the total weight of the subtree of T spanned by the set of leaves σ. d σ ()σ () +d σ ()σ 4 () + +d σ m ()σ m () }. In particular, we have the nice equation d ijk = (d ij + d ik + d jk ) for all i < j < k n, from which we obtain (I.G.) that ( for ) all {i, j, k, l, p} [n], d R m is called the m-dissimilarity vector of T. d ij = (d ijk + d ijl + d ijp d ipl d ikl d ikp d jpl d jkl d jkp + d plk ).

3 This is shown for a -tree, for m = and m =. 00 e d = π d = + + e + 00 d 4 = + e d 4 = π e. π e 4 ( ) [] We obtain a vector d R, the -dissimilarity vector.

4 00 e d = π + d = e + 00 d 4 = + e + 00 d 4 = π e + e π e. 4 ( ) [] We obtain a vector d R, the -dissimilarity vector.

5 d...(0)()() = d + d d (0)() + d ()() + d ()

6 Tropical Algebraic Geometry Let K be the a.c. field of elements k=p ω = c k t k/q p Z, c p 0, q Z + and c k C Consider the valuation val: K Q { } by which val(ω) = p/q and val(0) = Let R = R { }. Tropical semiring (R,, ). a b = max{a, b} a b = a + b Tropicalization of a Polynomial f K[X,..., X N ]. Let A Z N 0 be finite. f = α A c αx α... X α N N Example ω = t + t + t and val(ω) = ω = (+i)t / (i)t / +()t / +( i)t / and val(ω) = / ω = t / ++t / +t +t / +t trop(f ) = α A Example val(c α) x α x α N N f K[X, Y, Z] f = ixy + (t + t + i)xyz Z 4 trop (f ) = max{x+y, +x+y+z, 4z} +t /... and val(ω) = /

7 Tropical Hypersurface defined by trop(f ). T ( trop(f )) := {x R N the maximum of trop(f ) is attained at least twice} Example (Cont.) T ( trop(f )) = {x, y, z x + y = + x + y + z 4z} The Tropical Variety of an ideal I of K[X,..., X N ]: T (I ) = T ( trop(f )) f I Fundamental Theorem of Tropical Algebraic Geometry: T (I ) = closure of {(val(c ),..., val(c N )) (C,..., C N ) V (I ) K N } {x, y, z x + y = 4z + x + y + z} {x, y, z + x + y + z = 4z x + y} T ( trop(f )) and T (I ) often considered in TP N = R N /(,,..., )R

8 Tropical Linear Spaces ( ) Plücker ideal I m,n in K[P i i...im {i, i,..., i m} [n] ], consisting of the algebraic relations-syzygies m among the maximal minors of a generic matrix in K m n. Elements of I m,n are called Plücker relations. Quadrics form a Gröbner basis for I m,n, among which we find the three-term Plücker relations: The tropical Grassmannian-(m, n): P Sij P Skl P Sik P Sjl + P Sil P Sjk for all S and {i, j, k, l} m G m,n = T (I m,n) A vector p R m satisfies the tropical Plücker relations if and only if: \S p T ( trop (P Sij P Skl P Sik P Sjl + P Sil P Sjk ) ) S m \S {i,j,k,l} 4 Assume p R m R m satisfies the tropical Plücker relations. 4

9 Notation: Canonical basis of R n : {e,..., e n}. For I, e m I = i I e i. (D. Speyer) Every face of the regular subdivision of the m-hypersimplex H m := chull{e I I induced by p is matroidal. Three objects: Facts (D. Speyer): I. L(p) := T {i <i < <im<i m+ } m+ m+ (p i i r=...îr...imi x m+ ir ) } R n m II. P p := {x R n x i + x i + + x im p i i...im for all {i, i,..., i m} [n] m} III. P p := {x Pp If y x and y Pp, then y = x} i. L(p) = P p in TPn. ii. P p is an (m )-dimensional pure and contractible polyhedral complex. iii. Tropical linear space associated to p. The tight span T p is the complex of bounded faces of P p (or P p, or L(p) projectively). To define Tp, we can relax the requirement that p satisfies the tropical Plücker relations. Face enumeration of T p and the f -vector conjecture (Speyer).

10 Main Problems Given a vector d R m, is d the m-dissimilarity vector of an n-tree? Observation: For p R, p G,n if and only if p satisfies the tropical Plücker relations. Quadrics form a Tropical Basis for I,n. (P. Buneman) Four point condition theorem, modern statement: A vector d R is a -dissimilarity vector if and only if d G,n. (A. Dress) A vector d R is a -dissimilarity vector if and only if T d is a tree. (B. Sturmfels) In general, quadratics DO NOT form a tropical basis of I m,n for m >, only a Gröbner Basis. (L. Pachter, D. Speyer) Problem : If m >, decide whether the set of m-dissimilarity vectors of n-trees is contained in G m,n. (B. Sturmfels) Problem : Given an m-dissimilarity vector d R m, study the tropical linear space L(d), or equivalently P d. In particular, T d.

11 Ultrametrics Ultrametric n-tree: i. rooted ii. iii. iv. binary leaves [n] nonnegative real weight w(e) for each edge e v. λ-equidistant, for some λ > 0 induces a metric on [n]. Ultrametric: A metric space S with distance d : S S R 0 for which, d xz max{d xy, d yz } for all x, y, z S. Both definitions are equivalent. Theorem (I.G.) Let m >. If d R m is an m-dissimilarity vector, then d G m,n. Corollary If d R m is an m-dissimilarity vector, d satisfies the tropical Plücker relations, L(d) can be defined and has nice properties, in particular, it satisfies the f -vector conjecture of Speyer. Our approach produces a complete flag of tropical linear spaces.

12 r

13 The Tropical Linear Space Let < m < n. Fix a weighted n-tree T with m-dissimilarity vector d. We will describe the tropical linear space L(d). Let in(t ) be the internal subtree of T. Example We consider the combinatorial tree T of the figure, where in(t ) is shown in blue. Notation: st denotes the relation being a subtree of. Example In the figure, we have S st T, shown in red.

14 Definition For S st T, the extended tree S. The set of leaves of S is called the set of extended leaves of S. Example In the figure, we present S (green and red) corresponding to S (red). Any S st T determines two collections H and R of subsets of the set of leaves of T, L(T ) = [n] = {,,..., n}. i. Members of H are indexed by the leaves of S. We have i H l H if and only if the minimal path from i to S meets S at l. ii. Members of R are indexed by the extended leaves of S. We have i R l R if and only if the minimal path from i to S meets S at l. Observations: I. R is a partition of the set [n] of leaves of T. II. H is a collection of disjoint subsets of [n], but generally not a partition of [n]. III. Every member of H is the disjoint union of at least two elements of R. Definition Suppose S st T. If the collection H associated to S is a partition of [n], we say that S is a full subtree of T.

15 Example For the subtree S of the figure, we have H = {{}, {}, {}, {7, 8, 9, 0,, }}. Note the case of leaves, 4,. For the subtree S of the figure, we have R = {{}, {}, {}, {4}, {}, {}, {7, 8, 9, 0, }, {}}.

16 4 Definition Let i [m ] {0} and S st T. We say that (S, A) is an (m, i)-good pair of T if: i. i L(S) m, ii. iii. iv. m + i V(S), A L(S) with A = i, and L(S) L(T ) A. 7 Definition For all i [m ] {0}, define In this case, the subtree in red is a full subtree of our tree, and we have H = {{,,, 4, }, {}, {7, 8}, {9, 0, }, {}} T i := {(S, A) (S, A) is an (m, i)-good pair of T }, T in i := {(S, A) T i S st in(t )}, T := {S S is a full subtree of T with m leaves}, T in := {S T S st in(t )}.

17 Theorem (I.G.) Let i [m ] {0}. Then, the set of i-dimensional faces of P d is in bijection with the set T i, and the set of i-dimensional faces of T d is in bijection with the set T in i. The set of (m )-dimensional faces of P d is in bijection with the set T m T, and the set of (m )-dimensional faces of T d is in bijection with the set T in m Tin. In particular, the vertices of P d correspond to trees S st in(t ) with at most m leaves and at least m vertices. Theorem (I.G.) Let i [m ] {0}. The matroid M F of the face F of P d corresponding to an (m, i)-good pair (S, A) of T, ( ) has a collection of bases which consists of all A [n] such that: m A H = for all H H s.t. H = H l for some l A, A H for all H H s.t. H = H l for some l L(S)\A, A R for all R R. If F is bounded, then its face lattice is isomorphic to the face lattice ( of the i-dimensional cube in R n. Now, consider the (m )-dimensional pyrope P m := conv [, 0] m [, 0] m ), i.e. the convex hull of the 0- and 0-( ) cubes in R m. The matroid M F of an (m )-dimensional face F of P d corresponding to a full subtree S of T with m leaves has ( ) a collection of bases which consists of all A [n] such that: m A H = for all H H, so it is a transversal matroid. If F is bounded, then its face lattice is isomorphic to the face lattice of the (m )-dimensional pyrope.

18 Example (Matroid Inequalities) 4 4 = = 0 Inequalities describing a vertex of the tropical linear space Inequalities describing a -dimensional face of the tropical linear space

19 Example (Tight Spans) We present its tight span for m =. Consider the tree in the figure.

20 We present its tight span for m =. We present its tight span for m = 4.