On Complexes of Products and Products of Complexes

(1)

Thesis

On Products of Complexes and

Complexes of Products

Vom Fachbereich Mathematik und Informatik

der Universit¨at Bremen

zur Verleihung des akademischen Grades

Doktor der Naturwissenschaften

(Doctor rerum naturalium, Dr. rer. nat.)

genehmigte Dissertation.

Author

Dipl.-Math. Martin Dlugosch

Advisor

Prof. Dr. Eva Maria Feichtner

Referees

Prof. Dr. Eva Maria Feichtner

Prof. Dr. Laura Anderson

Abgabe am 22.08.2013

Kolloquium am 24.09.2013

(2)

Acknowledgments

I would like to thank

• my advisor Prof. Dr. Eva Maria Feichtner for her constant mentoring and support during the last years,

• my coauthor PD. Dr. Emanuele Delucchi for the opportunity to work with him as well as his counsel,

• my coauthor and office colleague Dipl. Phys. Roman Bruckner,

• several anonymous referees for improving the articles which made it into this thesis,

• Nils Przigoda for his LA_{TEX-expertise,}

(4)

1 Introduction

A common method in geometric combinatorics is to assign geometric/topological complexes to given combinatorial data. It is natural to ask how those complexes behave under possible products of the data. Since the assigned complexes are often simplicial, the wish ”the complex of the combinatorial product is the geo-metric product of the complexes”, most times, remains unfulfilled. A particular example, where the assigned complex is not simplicial but only polyhedral, shows this wish can become true. The talk is of Bergman complexes of matroids - the main character of this thesis.

Let V be an r-dimensional subspace of the n-dimensional vector space Cn_{. The}

set of vectors

plog |v1|, . . . , log |vn|q P Rn,

for v1, . . . , vnrunning through all non-zero elements of V , is called the amoeba of V .

The limit set of these amoebas, for bases of the logarithm approaching zero, is a polyhedral fan called the Bergman fan of V . It first appeared in the original paper of Bergman [2] as logarithmic limit set of V . The Bergman fan is invariant under translation along Rp1, . . . , 1q and positive scaling. Hence we lose no information when restricting to the sphere S tω P Rn _:°n

i1ωi 0 ,

°n i1ω

2

i 1u. This

restriction is a spherical, polyhedral complex called Bergman complex.

(5)

via valuation theory [3]. Matroid theory comes in when one assignes a matroid to V by taking as circuits C the minimal sets for which there are linear forms of the form °_i_PCaixi vanishing on V . In 2002 Sturmfels showed that the Bergman

fan of V only depends on the associated matroid [27]. A related characterization states that the Bergman fan is the set of all vectors ω pω1, . . . , ωnq P Rn such

that for every circuit C the minimum of the set tωi|i P Cu is attained at least

twice. From this point on it was possible to assign Bergman fans and complexes to any matroid. For more background on matroid theory see [25].

In 2004 Ardila and Klivans compared the space of phylogenetic n-trees and the order complex of the proper part of the partition lattice Πn since it had been

known that both have the homotopy type of a wedge of pn 1q! many n 3-dimensional spheres. Bergman fans of the graphic matroids of complete graphs turned out to be the key structure to understand this phenomenon.

Definition 1.1 (Phylogenetic Tree). A phylogenetic n-tree is a rooted tree with n leaves and an assignment of weights to its edges such that the distance between the root and any of its leaves is constant.

Example 1.2. Figure 1.1 shows phylogenetic 4-trees. The roots are the top nodes, depicted as thick points.

A phylogenetic tree can be identified with the vector ω of length n₂in which the distances between any two leaves are listed. This data can be seen as a distance function of the set of leaves. Distance functions which arise from phylogenetic trees are precisely the ultrametrics on finite sets.

Definition 1.3 (Ultrametric). An ultrametric δ is a metric onrns : t1, . . . , nu such that for all 1¤ X, Y, Z ¤ n two of the values δpX, Y q, δpX, Zq and δpY, Zq are equal and not less than the third.

(6)

1

2

3 A

B

C

D

A

B

C

D

2

3

2

2 A

B

C

D

2

3

5

2

5

1

2 A

B

C

D

Figure 1.1: Phylogenetic trees with root as top node

Example 1.4. Ultrametrics onrns can be thought of as assignments of weights to the edges of Kn. Figure 1.2 shows a phylogenetic tree and its induced ultrametric,

illustrated as an assignment of weights the edges of K4.

2

1

3 A

B

C

D

A

B

C

D

6

4

2

Figure 1.2: A phylogenetic tree and its induced ultrametric

The surprising connection between phylogenetic trees/ultrametrics on the one side and matroid theory on the other side is given by the following theorem. Theorem 1.5 ( [1]). The distance vector ω P Rpn2q is an ultrametric if and only

(7)

The partition lattice Πn is the lattice of flats of the graphic matroid of the

complete graph Kn. So along with Theorem 1.5 comes a stratification of the

Bergman fan/space of phylogenetic ntrees Tn via the already mentioned order

complex ∆pΠn tˆ0, ˆ1uq and the map

ψ : Tn Ñ ∆pΠnztˆ0, ˆ1uq

T ÞÑ tπ P Πnztˆ0, ˆ1u | for some 0 ¤ x

X Y in π ô δTpX, Y q ¤ xu,

where δT is the ultrametric induced by the phylogenetic tree T . In [1] this

strat-ification is called the fine subdivision of the Bergman fan. Note that the well definedness of this map (the transitivity of the definition of equivalence) is a con-sequence of the ultrametric and does not hold for other metrics.

Although this stratification does not consider the finest partition ˆ0 and the coarsest partition ˆ1, it is more natural to allow those, too: For any tree and x¤ 0 we always get the partition ˆ0 since there are no distinct leaves with distance 0. For x sufficiently large (twice the distance from the leaves to the root) we always get the partition ˆ1. So for later use instead of claiming the chains in Πnztˆ0, ˆ1u

stratify the Bergman fan, let us say that the chains in Πn that do contain ˆ0 and ˆ1

stratify it.

Example 1.6. The chain of partitions, which is assigned to a phylogenetic tree, can be seen as developed by increasing x. The bigger x gets the coarser the assigned partition becomes. Figure 1.3 shows a phylogenetic tree and the partitions that different values of x induce. The total chain of this tree is

ˆ

(8)

1 3 1 3 1 3 1 3 2 3

1 A

B

C

D

x 2 3 ÝÑ A|B|C|D ˆ0 2 3 ¤ x 4 3 ÝÑ AB|C|D 4 3 ¤ x 2 ÝÑ ABC|D 2¤ x ÝÑ ABCD ˆ1 Figure 1.3: A phylogenetic tree and its assigned chain of partitions

In evolutionary biology phylogenetic trees are mathematical models of evolu-tionary branches. Thus biologists are most interested in the combinatorial type of the phylogenetic trees.

Consider the three trees and their induced chains in Π4 of Figure 1.4.

A

B C

D

ÞÝÑ

ˆ

₀

_{A|B|CD AB|CD ˆ1}

A

B C

D

ÞÝÑ

ˆ

₀

_{AB|CD ˆ1}

A

B C

D

ÞÝÑ

ˆ

₀

_{AB|C|D AB|CD ˆ1}

Figure 1.4: Phylogenetic trees and their induced chains

The combinatorial type is the same for all three trees. So it is natural to ask for a stratification in which the trees of Figure 1.4 are glued together. And indeed, the description of the Bergman fan, which is in terms of matroid theory, produces such a coarse subdivision of the Bergman fan [1]. Faces of this polyhedral subdivision

(9)

are in one-to-one correspondence with certain direct sums of minors of the initial matroid called matroid types. Whenever the polyhedral structure of Bergman fan-and complex is concerned, this subdivision is the one meant.

Example 1.7. The Bergman complex of the graphic matroid of K4 is the famous

Petersen graph. Figure 1.5 shows the graph K4 with an edge labeling. Next to it

the Hasse diagram of the lattice of flats of the graphic matroid of K4 is shown.

Remember that the latter is isomorphic to the partition lattice via an isomorphism sending 1ÞÑ AD|B|C, 125 ÞÑ ABD|C and 13 ÞÑ AD|BC. Figure 1.6 shows the fine subdivision of this Bergman complex. Its face poset is isomorphic to the (face poset of the) order complex ∆pLtˆ0, ˆ1uq. The induced stratification of the space of phylogenetic trees is hinted. The coarse subdivision can be constructed by removing the vertices 13, 24, 56 and gluing the respective adjacent line segments together.

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0 0 0 0 0

1 1 1 1 1 1

0 0 0 0 0 0

1 1 1 1 1 1

1 2 3 4 6 5

0 0

1 1

0 0 0 0 0 0 0

1 1 1 1 1 1 1

0 0 0 0 0

1 1 1 1 1

0 0

1 1

0 0

1 1

0 0 0 0

1 1 1 1

0 0 0 0 0 0 0

1 1 1 1 1 1 1

2 3 4 5 6 123456 125 13 24 345 56 236 1 A C B D 146

Figure 1.5: The graph K4and the lattice of flats L of its graphic matroid

(isomor-phic to the partition lattice Π4); Image from [1, Fig. 1]

Besides the fine and the coarse subdivision of the Bergman fan there is a whole class of subdivisions between those two. They are polyhedral realizations of simpli-cial complexes called the nested set complexes of the lattice of flats of the matroid. Nested sets specify the De Concini/Procesi compactication of the complement of

(10)

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

0

1

1 0 0

0 0

1 1

0 0

1 1

0

1

1 0 0 0

0 0 0

1 1 1

0 0 0

1 1 1

0 0 0

1 1 1

0 0 0 0

1 1 1 1

0 0 0

1 1 1

0 0

1 1

0 0 0 0 0

1 1 1 1 1

0 0 0 0 0 0

1 1 1 1 1 1

0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1

146 24 125 345 236 56 2 3 5 6 1 13 4

0 0

1 1

0 0

1 1

0 0

1 1

0 0

1 1

A D B C A B D C

0 0

1 1

0 0

1 1

0 0

1 1

1 1 0

0

1

1 0 0

0 0

1 1

0 0

1 1

0 0

1 1

1 1 0

0

1

1 0 0

0 0

1 1

0 0

1 1

0

1

1 0 0

0 0

1 1

0 0 0

1 1 1

0 0 0

1 1 1

0

1

1 0 0

0 0

1 1

B D C A B D C A D B C A B D C A D B C A A D B C

Figure 1.6: Fine subdivision of the Bergman complex resp. (link of the) space of phylogenetic trees; Image from [1, Fig. 5]

a hyperplane arrangement [15]. Despite their geometric origin nested sets are de-fined in terms of pure order theory. Feichtner/Sturmfels sharpened the result of Ardila/Klivans by showing that not only realizations of order complexes but even realizations of nested set complexes subdivide Bergman complexes [18].

This thesis is concerned with investigating Bergman complexes and comparing its various subdivisions. The matroid theoretic descriptions of matroid types are translated into order theoretic concepts. As a consequence we are able to assign coarser versions of order complexes to any poset, not just to geometric lattices.

(11)

These decomposition complexes generalize order complexes, nested set complexes as well as Bergman complexes. It is quite surprising that from this point of view everything revolves around intervals in the poset which can be decomposed into a product of subintervals. Thus in the end stands the (in this sense characterizing) result of closedness under products for decomposition complexes.

Chapter 2 is based on [14]. The results of this chapter (Sections 2.4 and 2.5) are partially from the period of writing the author’s diploma thesis. However, the presentation has since conceptually improved and the content fits perfectly into this thesis being the starting point of Chapter 3.

We start with an introduction to basic matroid theory, matroid polytopes and Bergman complexes. Order complexes as well as nested set complexes are intro-duced and the different subdivisions of Bergman complexes are compared. Given a chain in the lattice of flats Ardila/Klivans gave a formula [1, Prop. 2] which de-scribes its matroid type (supporting face in the coarse subdivision of the Bergman complex) as a direct sum of minors of the initial matroid. This formula was enhanced by Feichtner/Sturmfels [18, Thm. 4.2]. Their formula describes the sup-porting matroid types of nested sets (faces of nested set complexes). The resulting direct sum decomposition of the matroid types is finer than the one for chains. Both formulas only depend on the vertex set of the respective faces.

With a formula describing decompositions of matroid types for faces of Bergman complexes we improve this formula once more. We show that our formula is the finest decomposition one can get, i.e. the summands are connected.

Chapter 3 is based on [12] and is joint work with Emanuele Delucchi. Lattice path matroids were introduced by Bonin, de Mier and Noy [5] as a family of transversal matroids whose bases can be characterized by means of the lattice paths contained in the region of the plane bounded by two given lattice paths

(12)

p and q. They enjoy a host of nice enumerative and structural properties [7] and can be characterized among all matroids by a list of excluded minors [6, Theorem 3.1]. Recently, lattice path matroids appeared as a special case of the positroids used as indices of cells in Postnikov’s stratification of the totally non-negative Grassmannian [24, 26]: for these ‘special cells’ the computation of the corresponding Grassmann necklace is particularly tractable [24, Section 6].

Lattice path matroids have the advantage that vertices of Bergman complexes of lattice path matroids are geometrically intuitive as fundamental flats and non-land necks. In this chapter we determine the polyhedral structure of the Bergman complex of a given lattice path matroid, and give necessary and sufficient condi-tions for it to be simplicial. In the geometric spirit of lattice path matroids, our characterizations are in terms of the shapes of the bounding paths.

Chapter 4 is based on [13]. Order complexes of posets have proven to contain important information about the poset, see [20], [22] or [4]. An abstract simplicial complex can be identified with its face poset. In the case of order complexes we obtain the set of non-empty chains in P ordered by inclusion.

Though easy to handle order complexes have one defect. As for all (abstract) simplicial complexes there is no product structure since, thinking in terms of real-izations, even the product of two 1-simplices is a quadrangle, which is not simpli-cial any more. At least, realizations of order complexes of products can be chosen such that they subdivide products of realizations of order complexes in terms of polyhedral complexes [29].

To fix this issue we introduce decomposition complexes. They are generaliza-tions of order complexes of posets that are closed under taking direct products. Decomposition complexes describe face posets of objects called polytopal pseudo-complexes instead of simplicial pseudo-complexes, which is instrumental for their good

(13)

behavior towards products. In the end we show how decomposition complexes generalize both nested set complexes as well as Bergman complexes.

Chapter 5 is joint work in progress with Roman Bruckner. Acyclic categories, also known as loopfree categories or scwols (small categories without loops) are natural generalizations of posets. They occur as combinatorial datum in form of face categories of polyhedral complexes [9], stratified spaces [28] or get associated to complex toric arrangements [11] or Morse functions on manifolds [10].

Nerves of acyclic categories ∆pCq are regular trisps, which are generalizations of simplicial complexes. The main difference is that faces of regular trisps are not determined by their vertex sets. A standard reference for acyclic categories, the nerve functor and regular trisps is [23]. In the special case of posets, the nerve yields the same simplicial structure as the corresponding order complex.

In this chapter we combine the generalizations of acyclic categories instead of posets and decomposition complexes instead of order complexes and define decom-position complexes for acyclic categories. We determine which of the concepts of decomposition complexes can be adapted in the new setting. In the end, the term decomposition seems even more natural in the language of category theory than order theory.

(14)

2 Bergman Complexes

This chapter is organized as follows. Section 2.1 introduces basic matroid the-ory via matroid polytopes. In section 2.2 the main objects Bergman complexes of matroids are defined. Section 2.3 introduces order complexes and nested set complexes. The latter are used to explore the various subdivisions of Bergman complexes in 2.4. Starting with this section all results are original - with the ex-ception of Corollary 2.22. This is one of the main theorems in [18], the proof given here is substantially shorter than the original one.

Section 2.5 is concerned with establishing the matroid type decomposition for-mula central to this work. Note that in the special case of faces of the order com-plex (fine subdivision of the Bergman comcom-plex) this formula is a generalization of [1, Prop. 2]. For faces of the nested set complexes the formula is a generaliza-tion of [18, Thm. 4.2]. We show that the direct summands are connected when the formula is applied to faces of the coarse subdivision of the Bergman complex.

2.1 Matroid Polytopes

Let us start with a geometric approach to matroids due to Gel’fand, Goresky, MacPherson and Serganova [21]. Let M be a family of subsets of size r of a ground setta1, . . . , anu. Each subset B of ta1, . . . , anu can be represented by its

incidence vector in Rn_{, i.e. the j-th coordinate is 1 if a}

(15)

we can identify M with the convex hull of its elements as incidence vectors:

PM : convteB : BP Mu Rn.

This yields a convex polytope in Rn_{. Since the generating vertices all lie on the}

simplex ∆ : tpx1, . . . , xnq P Rn : 0¤ xi for all i,

°n

i1xi ru, the dimension

is limited by n 1.

Definition 2.1. Let M be a family of r-element subsets of ta1, . . . , anu. If every

edge of the polytope PM is parallel to ei ej for some 1¤ i, j ¤ n, the set M is

called a matroid. The elements of M are called bases of the matroid. The polytope PM is the matroid (base) polytope.

Let M be a matroid with ground set EpMq. A subset I EpMq is independent if I B for a basis B P M. Hence, bases are the maximal independent sets. If not independent, a subset of EpMq is dependent. The rank of a matroid is simply r, the cardinality of its bases. The rank of a subset of EpMq is the cardinality of its largest independent subset. Hence, a set is independent if and only if its cardinality equals its rank. With circuits we denote minimal dependent sets with respect to inclusion. A flat F of M is a subset such that there is no circuit C of M with |CzF | 1. This means flats are those sets for which one can not obtain a new circuit inside by adding a new element of EpMqzF . Another way to look at this is that every new element, we add to F , is increasing the rank. The span of a subset G is the intersection of all flats containing G. It is the smallest flat which contains G. We can construct it, by adding all the elements of EpMqzG to G which do not increase the rank of G.

The collection of flats of M can be ordered by inclusion. The resulting poset is a lattice by setting F1^ F2 : F1X F2 and F1_ F2 : spanpF1Y F2q. It is

(16)

of flats of some matroid, is a geometric lattice. A matroid M is called loopfree if

M _B_PMB EpMq. Let F and G be flats of M such that F is contained in G. We define a new matroid on the ground set GzF by setting:

MrF, Gs : tB X pGzF q : B P M, |B X F | rankpF q, |B X G| rankpGqu.

The lattice LMrF,Gs is isomorphic to the intervalrF, Gs in LM.

There are two special cases of this that we want to bring up. The first is F ˆ0, the minimal element of the lattice. Mrˆ0, Gs is the restriction of M to G. Notice that the rank of the matroid MrH, Gs equals rankpGq. The other special case is called contraction and describes the case of G ˆ1 EpMq. Note that MrF, EpMqs is loopfree if and only if F is a flat of M.

There is an equivalence relation on the ground set of a matroid defined as fol-lows: Two elements x and y are equivalent if x y or there is a circuit of M which contains both x and y. The proof that this is an equivalence relation is given in [25, 124-125], like many other helpful statements for matroids. Note that connected matroids are loopfree for ground sets of cardinality of at least 2. The equivalence classes are the connected components of M . Let cpMq denote the num-ber of connected components of M . We say a matroid is connected if and only if cpMq ¤ 1. The case of cpMq 0 belongs solely to the empty matroid which is the matroid with empty ground set.

We say a flat F of M is connected if its restriction Mrˆ0, F s is connected. Dual to this, it is called co-connected if its contraction MrF, ˆ1s is connected.

Proposition 2.2. [18] The dimension of the matroid polytope PM is n cpMq.

Proof. The linear space parallel to the affine space spanned by the matroid poly-tope is generated by vectors of the form ei ej. An edge of PM is parallel to

(17)

such a difference of unit vectors if and only if the vertices at the end of that edge represent bases B, B1 just differing in the two elements ai and aj. But this is

in fact an equivalent condition to the existence of a circuit containing both. So dimpPMq ¤ n cpMq.

Conversely, every pair of connected elements ai, aj grant the existence of two

bases B, B1 differing just in these elements. So there is an edge of PM parallel to

ei ej. Thus dimpPMq ¥ n cpMq.

From here on we are focusing on connected matroids. Every non-connected matroid can be decomposed into a direct sum of connected matroids. Then the matroid polytope of the original matroid is just the cartesian product of the ma-troid polytopes of the direct summands. Note that decomposing a mama-troid into direct summands is the same as factorizing LM into the product of intervals of the

formrˆ0, Gjs for some GjP L.

Proposition 2.3. [18] For a connected matroid M of rank r the matroid polytope has the form:

PM

#

px1, . . . , xnq P ∆ :

¸

iPF

xi ¤ rankpF q for all flats F of M

+ .

Proof. It is enough to consider the facets of PM since they are bounding the

polytope. Let us assume a facet defining inequality is °n_i₁aixi ¤ c, where n

is the cardinality of EpMq. What we know from the definition of matroids is that all the edges are parallel to vectors ei ej. We can compute the normal

vector of the facet by looking at the edges it has to be perpendicular to. The constraints the edges impose are all of the form ai aj. Since the normal vector

is uniquely determined by these constraints up to scalar multiples, we can think of the ai to be either 0 or 1. So the inequality reduces to

°

iPAxi ¤ c1 for some

(18)

can attain? As a linear form its maximum has to be attained at some face of PM.

Pick any vertex of this face and evaluate the linear form on it. We obtain that c1 max t|B X A| : B P Mu rankpAq. What is left is to show that we just have to pick the subsets A which are flats. Let spanpAq be the flat spanned by A. Of course A spanpAq and rankpAq=rankpspanpAqq hold. Thus

¸ iPA xi¤ ¸ iPspanpAq xi¤ rankpspanpAqq rankpAq.

So the inequality of spanpAq in the middle already implies the inequality of A. Remark 2.4. Every face of PM is a matroid polytope, too, because its edges are

still parallel to some difference of standard basis vectors.

2.2 Bergman Complexes

Definition 2.5. For ω P Rn, the linear form °n_i₁ωixi attains its maximum

in PM at a face whose vertices are the incidence vectors of the bases, which are

possible outputs of the greedy algorithm respective ω. This matroid Mω is the

matroid type of the face.

Definition 2.6. A flat F of M s.t. °_i_PFxi¤ rankpF q describes a facet of PM is

called a flacet of M .

Note that matroids are determined by their flacets because matroid polytopes determine matroids.

Proposition 2.7. [18] A flat F of M is a flacet if and only if it is connected and co-connected.

Proof. We already saw that for connected matroids the matroid polytope has dimension n 1. So facets of this pure polyhedral complex have dimension n 2.

(19)

We just have to look out for those flats F such that the matroid polytope of the matroid type MeF has dimension n 2. We know what the bases of MeF are.

They are exactly the bases B of M such that |B X F | rankpF q holds. We can express this matroid type through the constructions of restriction, contraction and direct sum:

MeF MrH, F s ` MrF, EpMqs tb P M : |b X F | rankpF qu M.

Since this matroid type is a direct sum, its matroid polytope, which is a facet of PM, is a product of the matroid polytopes of MrH, F s and MrF, EpMqs. So the

dimension of the face defined by F must equal the sum of the dimensions of the matroid polytopes which is

|F | cpMrH, F sq n |F | cpMrF, EpMqsq n cpMrH, F sq cpMrF, EpMqsq.

From this we can see that the dimension n 2 appears if and only if

cpMrH, F sq cpMrF, EpMqsq 2.

Assume cpMrH, F sq is zero. So MrH, F s has to be the empty matroid. Thus F H follows and MrF, EpMqs is the same matroid as M. But PM_eH is no

proper face of PM, it is PM itself because eH is constant. The symmetric case of

cpMrF, EpMqsq 0 belongs to F EpMq. Again the induced face is the whole matroid polytope and thus has dimension n 1. So both summands have to equal 1 and this means the flacets are exactly those flats for which both restriction and contraction are non-empty connected matroids.

Recall that the linear form°n_i₁ωixi attaining its maximum at a certain face

(20)

Remark 2.8. For cP R and c P R the linear forms n ¸ i₁ pωi c qxi and n ¸ i₁ pωi cqxi

induce the same matroid type as°n_i₁ωixi.

So we still get all the different matroid types when restricting ω to elements of the unit (n 2)-sphere contained in a hyperplane orthogonal to p1, . . . , 1q, which is S tω P Rn:°n_i₁ωi 0,

°n i₁ω

2 i 1u.

Remark 2.9. The matroid type of ω p0, . . . , 0q is simply M again. See this as the matroid type of the empty face of the Bergman complex.

Consider the following equivalence relation on S: ω ω1 if and only if the induced matroid types Mωand Mω1coincide. The equivalence classes are relatively

open, convex polyhedral cones. These cones form a complete fan in Rn_{. It is the}

normal fan of PM. The equivalence classes define a spherical subdivision of S. This

subdivision is isomorphic to the boundary of the polar dual P_M of the matroid polytope. For simplicity of notation, we will identify the face ofBPM just like its

dual inBP_M with its matroid type Mω.

Definition 2.10. The Bergman Fan rBpMq is the subfan of the normal fan of PM

consisting of the matroid types which are loopfree. The Bergman Complex BpMq is the intersection rBpMq X S where S tω P Rn_:°n

i1ωi 0 ,

°n i1ω

2 i 1u.

Since bases of loopfree matroid types are sets of bases of M , they come with the natural partial order of inclusion. The face poset of the Bergman complex FpBpMqq is the dual of this poset of matroid types. So the face Mω is contained

in the face Mω1 in BpMq if and only if the reversed inclusion holds for Mω and

(21)

Proposition 2.11. [18] Vertices of the Bergman complex are in one-to-one cor-respondence with flacets of M .

Remark 2.12. Since every face of PM is uniquely determined by the set of facets

of the polyhedral complex which contain it, every face of BpMq is uniquely deter-mined by its vertices.

Example 2.13. Let M be the matroid with ground sett1, 2, 3, 4, 5, 6u and circuits t1, 2, 3, 4u, t1, 2, 5, 6u and t3, 4, 5, 6u. Hence the bases are all subsets of size four except for the circuits. There are two types of flacets here. On the one hand there are the singletons, i.e. the subsets of size one. On the other hand there are the three circuits themselves. The Bergman complex is a pure polyhedral complex of dimension two. There are two kinds of facets of the Bergman complex. There are twenty triangles, but there are also three quadrangles, whose vertices are in short-hand notation

1, 2, 1234, 1256 3, 4, 1234, 3456 5, 6, 1256, 3456.

This shows that Bergman complexes can be non-simplicial which means they are more complicated than simplicial complexes.

2.3 Order Complexes and Nested Set Complexes

Let L be a geometric lattice.

Definition 2.14. The set of totally ordered subsets ∆pLq of L, ordered by inclu-sion, is the order complex of L.

Although usually order complexes are abstract simplicial complexes, we identify them with their face posets for later purpose.

(22)

There is a more general way of defining a simplicial complex for a geometric lattice due to Feichtner and Kozlov [16].

Let intervals in L be denoted by rX, Y s : tZ P L : X ¤ Z ¤ Y u. For any X P L and any subset S L write S_¤X : tY P S : Y ¤ Xu. The same way we can define S_X, S_¥X and S_¡X. Last but not least, the set of maximal elements in S L is denoted by max S.

Definition 2.15. For a finite lattice L a subset G in L_¡ˆ0 is a building set if for any XP L_¡ˆ0 with max G_¤X tG1, . . . , Gku there is an isomorphism

ΦX : Πkj₁rˆ0, Gjs ÝÑ rˆ0, Xs

such that ΦXp, . . . , ˆ0, Gj, ˆ0, . . . , ˆ0q Gj for all j.

There is always a maximal as well as a minimal building set. The maximal one is always the whole lattice except for ˆ0. In this case any X is decomposed into just one factor, X itself. The minimal building set Gmin consists of all connected

flats and, if L is not connected already, the top element ˆ1.

Definition 2.16. Let L be a finite lattice and G a building set containing the top element ˆ1. A subset S G is called nested if for any set of incomparable elements X1, . . . , Xk of at least two elements of S, the join X1_ . . . _ Xk does not lie in

the building set G again. Since subsets of S fulfill the same condition, this is an abstract simplicial complex. Topologically, it is a cone with apextˆ1u. The basis of the cone NpL, Gq is the nested set complex of L with respect to the building set G. Again, as with order complexes we identify nested set complexes with their face posets which are the sets of nested sets ordered by inclusion.

The case of the minimal building set Gmin is called the nested set complex

(23)

Remark 2.17. The nested set complex with respect to the maximal building set equals the order complex Lztˆ0, ˆ1u, i.e. the nested sets are just the totally ordered subsets.

In order to compare nested set complexes of different building sets, Feichtner and M¨uller [17] proved the following.

Remark 2.18. [17] For building sets G and GY tXu, the nested set complex respective the building set GY txu can be obtained by a stellar subdivision from the nested set complex of the smaller building set G at the simplex whose vertices are the factors of X respective G.

Recursively we can construct the order complex of the proper part Lztˆ0, ˆ1u, which is the nested set complex of the maximal building set, from the nested set complex of the minimal building set by a sequence of stellar subdivisions. The single steps correspond to adding elements to the building sets in a non decreasing order. In particular, the order complex of the proper part of L and the nested set complex are homeomorphic.

A lattice is atomic if every element is a join of atoms. Geometric lattices are such atomic lattices. For simple matroids these atoms are just the elements of the ground set EpMq. For arbitrary atomic lattices Feichtner and Yuzvinsky [19] proposed the following polyhedral realization of nested set complexes.

Remark 2.19. [19] Let L be an atomic lattice with atoms ta1, . . . , anu and G a

building set containing ˆ1. For any GP G let eGP Rn be the incidence vector of G

respective the set of atoms, i.e. the i-th coordinate is 1 if and only if a1 G and 0

otherwise. For a nested set S the set of incidence vectors of its elements is linearly independent. Hence with R¥0teG|G P Su, they span a simplicial cone. For nested

sets S and S1 their respective cones intersect exactly in the cone belonging to the nested set SX S1. Thus the set of cones from nested sets form a simplicial fan.

(24)

Just like the Bergman Fan this fan has the property that its cones are in-variant under the translation along the line Rp1, . . . , 1q. Again we do not loose any information when restricting the fan to the (n-2)-sphere S tω P Rn _:

°n

i1ωi 0 ,

°n i1ω

2

i 1u. The resulting spherical complex is a geometric

realization of the nested set complex.

Example 2.20. Consider the matroid of Example 2.13. The lattice L is the lattice of flats of the matroid M . Though this is not true in general, the minimal building set Gmin consists exactly of the set of flacets. The complexes coincide except for

the three squares which are each replaced by two triangles with vertices:

1, 2, 1234 1, 2, 1256 3, 4, 1234 3, 4, 3456 5, 6, 1256 5, 6, 3456.

2.4 Subdivisions of Bergman Complexes

Ardila and Klivans [1] first showed that a realization of the order complex of the proper part of the lattice of flats is a refinement of the Bergman complex. Feichtner and Sturmfels [18] proved that this is even true for a specific realization of the nested set complex. Though we can gain a lot of insight from their proof, here is a shorter proof:

Recall the geometric realizations of the order complex (nested set complex of the maximal building set), the nested set complex (with minimal building set) and the Bergman complex.

Theorem 2.21. For any of these complexes any element ω in a face with vertices V is of the form°_F_PV λF eF with λF 0 for all F P V . Then the bases of the

matroid type Mω are

(25)

Proof. Vertices of faces are normalized incidence vectors of flats in the case of the order complex, connected flats in the case of the nested set complexes and flacets in the case of the Bergman complex. The faces itself are just the spherical convex of the vertices, i.e. the normalization of the convex hull of the vertices. Thus w.l.o.g. we can assume ω°_F_PV eF.

A basis B of M maximizes the linear functional induced by ω on PM if and only

if for all bases B1 of M the inequality eB1 ω ¤ eB ω holds. First consider a basis

B satisfying|B X F | rankpF q for all F P V .

eB1 ω eB1 ¸ FPV eF ¸ FPV eB1 eF ¸ FPV |B1_{X F |} ¤ ¸ FPV rankpF q ¸ FPV |B X F | ¸ FPV eB eF eB ¸ FPV eF eB ω

This shows that B is a basis of Mω.

Conversely assume eB1 ω ¤ eb ω holds for all B1 P M. Choose B1 such that

|B X F | rankpF q holds for all F P V . Then the equations above teaches us that °

F_PV |B X F |

°

F_PV rankpF q. Since for all pairs of summands the inequality

|B X F | ¤ rankpF q holds, equality holds for them, too.

Corollary 2.22. For all building sets G the nested set complex NpLM, Gq is a

refinement of the Bergman fan.

Proof. Already Ardila and Klivans [1, Thm. 1] noticed that the ωP Rn _{for which}

Mωis loopfree are exactly the ones lying in the interior of polyhedral cones spanned

by incidence vectors of flats. Due to Theorem 2.21 the induced matroid types are the same for all elements of any face of the realizations of our complexes.

Example 2.23. Consider the matroid of Example 2.13 and 2.20. In Figure 2.1 we compare the excerpts of the order complex ∆pLztˆ0, ˆ1uq, the nested set complex

(26)

NpLM, Gminq and the Bergman complex BpMq which arise as subdivisions of each

other. The matroid type of the two-dimensional face is

Mω t1235, 1236, 1245, 1246u. 1234 2 1 12 1256 (a) excerpt of ∆pLMq 1234 2 1 1256 (b) excerpt of NpLM, Gminq 1234 2 1 1256 (c) excerpt of BpMq

Figure 2.1: The same excerpt of all polyhedral complexes

2.5 Matroid Type Decomposition

Starting here, M is a finite, connected, simple matroid with ground set EpMq. Let ω denote a vector of BpMq S tω P Rn _:°n

i1ωi 0,

°n i1ω

2

i 1u and

Mω is its induced, loopfree matroid type.

2.5.1 Decomposition Formula

Definition 2.24. Let A be a subset of EpMq. We say that A has full ω-rank if for all bases BP Mω,|A X B| rankpAq holds.

Note that the vertices of a face have full ω-rank by Theorem 2.21.

Proposition 2.25. For ωP S, the induced matroid Mω is loopfree if and only if

(27)

Proof. Assume A is not a flat, but it has full ω-rank. Then there exists a circuit C of M such that CzA txu. For all bases B P Mω:

|A X B| rankpAq rankpA Y txuq ¥ |pA Y txuq X B|.

So B can not contain x. Since this holds for all bases of Mω, this means x is a

loop of Mω. The same argument works for the other implication, too.

Proposition 2.26. Let A and A1 both having full ω-rank. Then AX A1, AY A1 as well as any connected component of a set with full ω-rank have full ω-rank, too. Proof. For all bases BP Mω:

rankpA Y A1q ¥|pA Y A1q X B|

|A X B| |A1_{X B| |pA X A}1_{q X B|q}

rankpAq rankpA1_{q |pA X A}1_{q X B|.}

With this inequality and submodularity of the rank function in mind it follows:

rankpAq rankpA1q ¤ rankpA Y A1q |pA X A1q X B|

¤ rankpA Y A1_{q rankpA X A}1_{q ¤ rankpAq rankpA}1_q.

From this we conclude that |pA X A1q X B| rankpA X A1q. So A X A1 has full ω-rank, too. Now we want to show the same for AY A1:

rankpA Y A1q ¥ |pA Y A1q X B|

|A X B| |pA X A1_{q X B| |A}1_{X B|}

(28)

Again, with this inequality and submodularity of the rank function in mind, we get

rankpAq rankpA1q ¤ rankpA Y A1q rankpA X A1q ¤ rankpAq rankpA1q.

The same way we conclude that|pA Y A1q X B| rankpA Y A1q. So A Y A1 has full ω-rank, too. Let A1, . . . , Atdenote the connected components of a set A with

full ω-rank. For BP Mω: t ¸ i1 |B X Ai| |B X A| rankpAq t ¸ i1 rankpAiq.

Together with the pairwise inequality|B X Ai| ¤ rankpAiq we can conclude that

|B X Ai| rankpAiq. So all the Ai have full ω-rank, too.

Corollary 2.27. The set of subsets with full ω-rank form a sublattice of LM.

The join-operation is already the union instead of just its closure. Additionally, it is closed under taking connected components, i.e. if an intervalrA, Bs in LM is

isomorphic torA, C1srA, C2s for A C1, C2 B and A, B have ω-full rank then

both C1 and C2 have this property, too. All information about Mω is contained

in this special sublattice. It is the smallest sublattice, which contains the elements ˆ

0, ˆ1 as well as the vertices (flacets) of the face Mω and is closed under taking

connected components simultaneously. These observations form the starting point for the theory of decomposition complexes (Chapter 3).

Here comes the generalized decomposition formula for supporting matroid types. Note that special instances of this formula include [1, Prop. 2] and [18, Thm. 4.2]. Theorem 2.28. Let Mω be the matroid type of ω lying in any face of either the

realization of the order complex ∆pLtˆ0, ˆ1uq, the nested set complex N pLM, Gminq

(29)

Additionally, let π :_F_PVpF | EpMqzF q be the partition of EpMq, which is the join in ΠE_pMq of the partitions consisting of just two blocks, the vertex and its

complement. For any block α in π let Vα denote the elements of V which contain

α as a subset. Then, Mω à αPπ M £Vα z α, £ Vα .

Example 2.29. Consider the matroid of Examples 2.13, 2.20 and 2.23. Let MC

be the matroid type of the support face of the chain C t12u (a face of the order complex). It can be computed by taking some weight vector inside the cor-responding face of the fine subdivision of the Bergman complex e.g. ω e12

p1, 1, 0, 0, 0, 0q. The induced partition of C is 12|3456, and the decomposition for-mula of Ardila/Klivans ( [1, Prop. 2]) yields

MC MrH, 12s ` Mr12, 123456s.

Let MS be the supporting matroid type of the nested set S t1, 2u (a face of

the nested set complex). Again, it can be calculated as MS Me1 e2. Note that

this face is the support of the chain C. Thus the matroid type is the same as above. The induced partition of S is 1|2|3456. The formula ( [18, Thm 4.2]) of Feichtner/Sturmfels yields

MC MS MrH, 1s ` MrH, 2s ` Mr12, 123456s.

This matroid type’s vertices, as a face of the Bergman complex are 1, 2, 1234 and 1256. Thus its induced partition is 1|2|34|56 and the complete decomposition of

(30)

the matroid type is

MC MS Mω MrH, 1s ` MrH, 2s ` Mr12, 1234s ` Mr12, 1256s.

Again, one can calculate the matroid type of a face with vertices t1, 2, 1234, 1256u as Mω Me1 e2 e1234 e1256 Mp3,3,1,1,1,1q.

Figure 2.2 illustrates the situation by showing excerpts of the different subdivi-sions. The red faces present the chain C (left), the nested set S (middle) and the matroid type Mω (right).

1234 2

1

12

1256

(a) Red: Chain C t12u

1234 2

1 1256

(b) Red: Nested set S t1, 2u

1234 2

1 1256

(c) Red: Matroid type Me1 e2

Figure 2.2: Excerpts of order complex, nested set complex and Bergman complex and the respective support of the red faces

Remark 2.30. A matroid theoretic decomposition of a matroid type corresponds to a geometric decomposition of the corresponding face of the matroid polytope into the product of the matroid polytopes of the factors.

Proof. First of all notice that the ground set of the summand corresponding to α is exactly α. So these ground sets form a disjoint union of the ground set EpMq. Notice that by construction of π a block α is either contained in or disjoint from

(31)

a flacet F of V . So denoting the elements of V disjoint to α as VC α we get: £ Vα X ¤ V_αC £Vαzα.

Due to Proposition 2.26 the interval limits of the direct summands have full ω-rank and hence they are flats by Proposition 2.25.

For the first inclusion of bases let B be any basis of the matroid Mω. The

elements of V have full ω-rank by definition. By Proposition 2.26 an intersection of subsets with full ω-rank has full ω-rank, too. This yields|B XVα| rankp

Vαq. So BXVαis a basis of MrH, Vαs. By Proposition 2.26 even Vα X VC α

has full ω-rank. So |B X pVα X

VC αq| rankp Vα X VC αq holds and pB XVα X VC αq is a basis of MrH, Vα X VC αs. Therefore B X α is a

basis of the direct summand M Vα X

VC α, Vα and B_α_PπBX α is a basis ofÀ_α_PπMrVα X VC α , Vαs.

For the second inclusion of the matroids’ bases we have to show that any basis B1 ofÀ_α_PπMrVαzα,

Vαs is a basis of M, too. Afterwards, we will show that

the equation|B1X F | rankpF q holds for all F P V . Let B1 be a basis ofÀ_α_PπMrVαzα,

Vαs and B be a basis of Mω, then

rankpMq |B| ¸ αPπ |B X α| ¸ αPπ p|B X£Vα| |B X p £ Vαzαq|q ¸ αPπ prankpMrH,£Vαsq rankpMrH, £ Vαzαsqq ¸ αPπ rankpMr£Vαzα, £ Vαsq ¸ αPπ |B1_{X α| |B}1_|.

(32)

So we know the rank ofÀ_α_PπMrVαzα,

Vαs equals the rank of M. Now we

want to prove that B1 has full rank in terms of the matroid M . Assume this is not the case. Then choose a minimal block α for which there exists x P α such that by adding this to B1 we increase the rank in terms of the matroid M . Minimality is meant to be with respect to the order relation α¤ β if and only if Vα Vβ. Now, rankp Vαq rankp Vαzαq rankpMr Vαzα, Vαs) holds.

Since B1XVαzα has full rank in MrH,

Vαzα] by minimality and B1X Vαhas full rank in MrVαzα,

Vαs by construction of B1, we conclude that B1X

Vα

has full rank in MrH,Vαs, too. But this is a contradiction to the choice of α.

So B1has full rank in M and it has the size rankpMq. Therefore it is a basis of M. Moreover, for BP Mω and B1P

À αPπMr Vαzα, Vαs we get rankpF q |B X F | |B X ¤ α_F α| ¸ α_F |B X α| ¸ αF |B X p£Vαzp £ Vα X ¤ V_αCqq| ¸ αF p|B X£Vα| |B X p £ Vα X ¤ VαCq|q ¸ αF prankp£Vαq rankp £ Vα X ¤ V_αCqq ¸ α_F rankpMr£Vα X ¤ V_αC,£Vαsq ¸ αF |B1_{X α| |B}1_{X F |.}

Since|B1X F | rankpF q holds for all flacets F of V , B1 is a basis of Mω, too.

2.5.2 Connectedness of Summands

The following main theorem states that one gets the finest possible decomposition of matroid types from their vertex sets as faces of Bergman complexes.

(33)

Theorem 2.31. Let M be a connected matroid. For a face of its Bergman complex the direct summands of the decomposition of the matroid type in Theorem 2.28 are connected.

In order to show this, we will use the following propositions in the following way. We start with Proposition 2.32. Together with the latter and Proposition 2.33 one can show Proposition 2.34, which is the final ingredient for the proof of the theorem above.

Proposition 2.32. Let A be a connected flat of M and K1, . . . , Kt connected

components of MrA, EpMqs. Then Fi : A Y

jiKj EpMqzKi is a flacet

of M .

Proof. We want to show the following properties: 1. The Fi are connected.

2. The Fi are co-connected.

3. The Fi are flats.

The first condition, showing the connectedness of Fi, is straightforward making

use of the connectedness of M and the definition of circuits of the contraction MrA, EpMqs.

For the second part, in view of the Scum Theorem [25, 3.3.7], we get

MrFi, EpMqs MrA, A Y Kis.

So Fi is co-connected since Ki is a connected component.

Now the third part is less work. Since MrFi, EpMqs is even connected, this

(34)

Proposition 2.33. Let A EpMq and A1, . . . , An connected components of

MrH, As. We denote the connected components of MrA, EpMqs by K1, . . . , Kt.

If A has full ω-rank, then for any of the Ki and any connected component G of

MrH, A Y Kis, which intersects Ki, both AY Ki and G have full ω-rank, too.

Proof. Our goal is to show that |B X G| rankpGq holds for all bases B of Mω.

For any such B the equation|A X B| rankpAq holds because A has full ω- rank. By Proposition 2.26 all the Aiand any union of them have full ω-rank, too. Hence,

implies

|B X Ki| ¤ rankpKiY Aq |B X A| rankpKiY Aq rankpAq.

On the other hand BzA is a basis of MrA, EpMqs and thus:

t

¸

i₁

|B X Ki| |BzA| rankpMrA, EpMqsq

rank t à i₁ MrA, A Y Kis t ¸ i₁ prankpA Y Kiq rankpAqq.

In a nutshell, |B X Ki| ¤ rankpKiY Aq rankpAq and summing over all Ki we

obtain equality, so:

|B X Ki| rankpKiY Aq rankpAq ñ |B X pKiY Aq| rankpKiY Aq.

Again, by Proposition 2.26, all the connected components of KiY A, such as G,

(35)

Proposition 2.34. Every connected flat A of a matroid M is an intersection of flacets of M . Furthermore, if A has full ω-rank, then so do these flacets. This means they are vertices of the face Mω of the Bergman complex.

Proof. Let A be a connected flat of M and K1, . . . , Ktthe connected components

of MrA, EpMqs. Due to Proposition 2.32 the sets A Y_j_iKj are flacets for all i,

and A t £ i1 AY¤ ji Kj .

For the second part, consider Proposition 2.33. It states that AY Ki has full

ω-rank. And so by Proposition 2.26 _j_ipA Y Kjq A Y

j_iKj has full

ω-rank.

Now we can finally prove Theorem 2.31.

Proof. First we want to show that the matroid MrH,Vαs is connected. Let

A1, . . . , An be the connected components of MrH,

Γαs. The flat

Vα has full

ω-rank as an intersection of flacets with full ω-rank. The same is true for its individual connected components by Proposition 2.26. So due to Proposition 2.34 there exists Vi V for all i such that Ai

Vi. We know H α X Ai

αX pViq. This implies that α X F H for all F P Vi, which is indeed equivalent

to α F for all F P Vi. Therefore we know that α

Vi Ai.

Since the Ai are pairwise disjoint, α must be contained in exactly one of them.

Let us say this connected component is Ai and let Aj with j i be a different

component disjoint to α. Then,

α Ai £ Vi ñ Vi Γα ñ H AjX Ai AjX £ Vi AjX £ Vα Aj.

So there must not exist any other connected component than Ai. Thus the matroid

(36)

Now we will see why we put all this work in the previous propositions. The requirements of Proposition 2.33 are satisfied for

• the matroid MrH,Vαs, • the setVα X VC α, which equals Vαzα, as A,

• any connected component Ki α of Mr

Vαzα,

Vαs,

• any connected component G of MrH, pVαzαq Y Kis intersecting Ki α.

Proposition 2.33 states that G has full ω-rank, too. In particular, G is a flat, due to Proposition 2.25, which is connected by construction as a connected component. So again with Proposition 2.34 we obtain GV1 for some V1 V . We know thatH KiXG αXG αXp

V1q. This implies that αXF H for all flacets F P V1, which is already equivalent to α F for all flacets F P V1. Therefore we know that V1 Vα, which yields to

Vα

V1. So we can conclude about G:

£ Vα £ V1 G p£Vα X ¤ V_αCq Y Ki £ Vα.

Thus G equalsVα and Ki equals α. Therefore Mr

Vαzα,

Vαs has only one

connected component.

Corollary 2.35. The decomposition of the matroid type in 2.28 for a face of the order complex is coarser than the decomposition of the matroid type of its supporting face in the nested set complex.

The decomposition of the matroid type for a face of the nested set complex is coarser than the decomposition of the matroid type of its supporting face in the Bergman complex.

(37)

Proof. We just have to compare the partitions_F_PΓpF |EpMqzF q for the different sets of vertices. Feichtner and M¨uller [17] showed that the nested set which is the support of the chain consists of the connected components of the chain elements. So the partition is becoming finer when passing to more vertices. Due to 2.31 the direct summands for faces of the Bergman complex are connected, so this must be the finest decomposition.

Remark 2.36. Note that the partition of the support of a face is strictly finer than the partition of the initial face if and only if the dimension of the faces is increasing while taking the support in the coarser complexes. For a maximal face the decompositions of the face and its support faces coincide. Using this we can easily see that the maximal faces of the Bergman complex correspond to transversal matroids, i.e. direct sums of matroids of rank 1.

(38)

3 Lattice Path Matroids

In this chapter we consider a special subclass of matroids - lattice path matroids. They are suited for describing their Bergman complexes in graphic terms. It is joint work with Emanuele Delucchi and is based on [12].

Section 3.1 introduces lattice path matroids. Additionally, we determine what geometric interpretations flacets of lattice path matroids have. In the following Section 3.2 we determine the relations among the constraints of the flacets. Thus for lattice path matroids we can give explicit descriptions of the vertex sets of faces of Bergman complexes and thereby the face posets themselves (Section 3.3). In the end we use those insights to determine the polyhedral structure of the faces.

3.1 Lattice Path Matroids and its Flacets

Let p, q be lattice paths in the plane with common starting (say at the origin p0, 0q) and ending point (say at a point pm, rq). We will assume that p never goes below q. We will write p and q as words

p p1. . . pm r q q1. . . qm r

where each letter is N or E, signaling a step Northp0, 1q or East p1, 0q.

(39)

go above p or below q. For any sP rp, qu write s s1. . . sm r and define

Bpsq : ti : si Nu.

Lemma 3.1 ( [7]). The set tBpsq : s P rp, quu is the set of bases of a matroid Mpp, qq on the ground set rm rs.

Definition 3.2. A lattice path matroid (LPM) is any matroid of the form Mpp, qq for two lattice paths p, q as above.

A lattice path matroid Mpp, qq is connected if and only if the paths p and q never touch except atp0, 0q and pm, rq [7, Theorem 3.5].

Unless otherwise stated, in the following we will consider only connected lattice path matroids.

Definition 3.3 (Fundamental flats, bays, land necks). We will say that a lattice pointpy1, y2q on the upper path p is a bay of p if py1 y2py1 y2 1 EN. Similarly,

a point pz1, z2q on q is a bay for q if qz1 z2qz1 z2 1 NE. Let Up, resp. Uq, be

the set of bays of p, resp. of q. The fundamental flats of M are sets of the form t1, . . . , y1 y2u for an py1, y2q P Up or of the form tz1 z1 1, . . . , m ru for

pz1, z2q P Uq.

We will say that iP rm rs is a land neck of Mpp, qq if the endpoint of p1. . . pi 1

lies one unit North of the endpoint of q1. . . qi. The set of land necks is Spp, qq.

The following is a rephrasing of Corollary 2.14 of [6] and Theorem 5.3 of [7]. Lemma 3.4. The flacets of a connected lattice path matroid Mpp, qq are

(a) the fundamental flats,

(40)

Figure 3.1 shows a lattice path matroid and some flacets. By Remark 2.11 the vertices of the Bergman complex correspond to flacets of the matroid. In the case of lattice path matroids, by Lemma 3.4 the flacets correspond to bays and non-land-necks as follows.

4 4

7 7

Figure 3.1: A lattice path matroid of rank 8 on the ground set r16s. Think of the lightly shaded region as an island: then the black dots mark the p-bays, the black squares mark the q-bays and the thick line shows that the singletont4u is a land neck (however, note that the singleton t7u is not).

Let M Mpp, qq be a connected lattice path matroid, and Mω a face of its

Bergman complex. Given BP M we will write ppBq for the corresponding lattice path. A node of the matroid type Mω is an integer lattice point that is visited by

every path ppBq with B P Mω.

Remark 3.5. Let F be a flacet of M and B P M. Then |B X F | rankpF q if and only if one of the following holds

(a) F is a fundamental flat and ppBq goes through the corresponding bay, (b) F tiu and ppBqi N.

We will then say that the path ppBq satisfies the constraint imposed by F . So on the one hand we can think of any face Mω of the Bergman complex of a lattice

(41)

path matroid M as the family of bases Mω which satisfy the constraints given by

some bays and some non-land-necks. On the other hand any set of such constraints determine the family of bases of M satisfying all those constraints. If this family is not only non-empty but loopfree we have found the a face of the Bergman complex. Example 3.6. On the left of Figure 3.2 we have marked some of the flacets of the lattice path matroid of Figure 3.1. The associated fundamental flats correspond to the p-bays p3, 3q, p4, 6q and the q-bay p2, 1q. The paths, that pass through these three points and go North at their 8th step, define the bases of a matroid M1, of which we give a lattice path representation on the right hand side (we remark that in order to get such a presentation of the matroid type, one has to change the initial order of the ground set).

8 8 (3, 3) (4, 6) (2, 1) 8 7 9

Figure 3.2: Some flacets and a lattice representation of the matroid type defined by their constraints.

We see that M1 is loopfree, thus it is the matroid type of a face of the Bergman complex, and that 8 is the only singleton which is a vertex of this face. On the other hand, the p-bayp3, 3q and the q-bay p4, 3q do not define a face of the Bergman com-plex, since every path that goes through both will go East on the 7th step, thus 7 will be a loop of the corresponding matroid type. Analogously, the p-bayp3, 3q and the two singletons 5 and 6 do not define a face of the Bergman complex because every path that goes throughp3, 3q and goes North at the 5th and 6th step must go East (along q) on the 4th step, thus 4 will be a loop of the corresponding matroid type.

(42)

3.2 A Simpliciality Criterion

The goal of this section is to give a complete characterization of which lattice path matroids possess a simplicial Bergman complex.

Theorem 3.7. Let Mω be a face of the Bergman complex of a connected lattice

path matroid Mpp, qq. Then the face Mω is simplicial unless its vertices include

(1) a fundamental flat F corresponding to a baypx, zq of p and (2) a fundamental flat G corresponding to a baypx, yq of q with z y ¡ 1.

Proof. Recall a face Mωof the Bergman complex is simplicial if every proper subset

of its vertices determines a proper subface. Conversely, Mω fails to be simplicial

if and only if

Mω tB P M : |B X F | rank F for F P Uu

for a proper subset U V .

Let the vertices of Mω include F and G as in (1) and (2) above. Then, for

all B P Mω, the lattice path ppBq passes through px, yq and px, zq. So it satisfies

ppBqj N for j x y 1, . . . , x z. The converse is also true: if all lattice paths

ppBq with B P Mωsatisfy ppBqj N for j x y 1, . . . , x z, then every ppBq

must pass through bothpx, yq and px, zq. We conclude that F and G correspond to vertices of V if and only if the flacets tx y 1u, . . . , tx zu correspond to vertices of V . So a subset of constraints already impose the constraints of other vertices.

For the reverse implication, suppose the face Mω is not simplicial. So there is

F0P V such that

(43)

Case 1: F0 is a fundamental flat. Without loss of generality F0 corresponds to a

p-baypx, zq.

Claim: Let u be minimal such thatpx, uq is a node of Mω. Thenpx, uq lies on q.

Let v be maximal such thatpx, vq is a node of Mω (so in particular v¥ z).

There is an Mω-path

s s1. . . sx u1N . . . N Esx v 2. . . sr m.

Now consider the path

s1 s1. . . sx u₁EN . . . N sx v 2. . . sr m.

If s1 ppBq for a basis B P M, then it would satisfy the constraints given by F for F P V ztF0u but not the one given by F0(it does not pass through

the p-bay px, zq). Thus, it would be an element in the right hand side but not in the left hand side of Equation (3.1): a contradiction.

We conclude that s1 R rp, qu, meaning that the point px 1, u 1q is not a lattice point between q and p, i.e. px, uq lies on q.

4 The claim shows that there is a node px, uq of Mω which lies on q. If there

is no q-bay px, yq, all paths of Mω go east in the px uq-th step. But this is

a contradiction to Mω being loopfree. So there has to be a q-bay px, yq with

u¤ y ¤ z which is also a node of Mω, q.e.d.

Case 2: F0 is a singletonti1u. Then every path s of Mω has si1 N.

(44)

To see this, consider two paths s, s1of Mωsuch that s1. . . si1 ends atpy1, y2q

and s₁1 . . . s1_i1 ends atpy11, y21q with y21 ¥ y2. We want to prove that y21 y2.

By way of contradiction suppose y1₂ y2 ¡ 0 and let pa1, a2q, pb1, b2q with

a1 a2 i1 b1 b2be on both s, s1, such that the path sa1 a2 1. . . sb1 b2

is always south of s1_a₁ _a₂ ₁. . . s1_b

1 b2.

In particular, s1_a₁ _a₂ N and s1_b

1 b21 E and there are no nodes of Mω

betweenpa1, a2q and pb1, b2q. The path

s11. . . s1a1 a2N sa1 a2 2. . . sb1 b21Esb1 b2 1. . . sr m

is thus a path of Mω. After i1steps ends at a pointpy12, y22q with y22 y2 1.

By repeating this operation we can assume without loss of generality that y1₂ y2 1. Now, in this case the path

s1₁. . . s1_i1₁Esi1 1. . . sr m

is an element of the right-hand side but not on the left-hand side of

Equa-tion (3.1): a contradicEqua-tion. 4

Thus, there is an x such that for all B P Mω, the path ppBq1. . . ppBqi1 ends at

px, i1_{xq. Since F}

0 ti1u, we have that both px, i1 x 1q and px, i1 xq are

nodes of Mω.

To prove the theorem it now suffices to prove the following claim. Claim: There is a p-baypx, zq north and a q-bay px, yq south of px, i1 xq.

Let px, uq be the lowest node of Mω south of px, i1 x 1q. Then there is

(45)

node of Mω and let t be a path of Mω with tx v 1 E. Consider the paths

s1: s1. . . sx u1N . . . N Esi1 1. . . sr m,

t1: t1. . . ti11EN . . . N tx v 2. . . tr m.

A schematic view of the situation is illustrated in Figure 3.3. On the left we see s (red) and s1 (blue), on the right t (red) and t1 (blue).

px, uq px, i1_xq px 1, i1_xq px, vq px, i1_{x 1q}

s

1 px, vq px, uq px 1, i1_{x 1q}

t

1

t

px, i1_{x 1q} px, i1_xq

Figure 3.3: The paths s, s1, t and s in the proof of the claim.

Neither s1 nor t1can be a path of Mω because they violate the constraint of

F0 by s1i1 t1i1 E. Therefore, in order for Equation (3.1) to hold, each of

the paths s1 and t1, if at all an element ofrp, qu (if not - case si), must violate the constraint given by some F with F P V ztF0u. Since the step where s1

goes east but s does not is F0 the vertex F can not be a singleton. Thus it

must correspond to a bay - case sii.

(si) the pointpx, i1 x 1q lies on p (since s1passes throughpx 1, i1 xq), or

(sii) there is F P V such that F corresponds to a q-bay px, yq between px, i1_{x 1q and px, uq.}

(46)

Similarly, for t1 not to be a path of Mω one of the following must enter:

(ti) the pointpx, i1 xq lies on q (since t1passes throughpx 1, i1 x 1q), or

(tii) there is F P V such that F corresponds to a p-bay px, zq between px, i1_{xq and px, vq.}

Now, if both (si) and (ti) were true, Mpp, qq would not be connected; on the other hand, (si) and (tii) together imply that there is a point on p south of a p-bay, and similarly from (ti) and (sii) follows the existence of a point on q north of a q-bay. Thus, no combination of the above cases can occur. We conclude that both (sii) and (tii) must hold, proving the claim.

Finally, z y 1 belongs to the case where px, i1 xq and px, i1 x 1q are those bays themselves. Then i1 would be a land neck and hence is no flacet

at all. 4

Example 3.8. Consider the lattice path matroid given in Figure 3.4. The face Mω whose vertices correspond to the baysp4, 3q, p4, 6q and the singletons 8, 9, 10 is

a minimal non-simplicial face.

8 8 9 9 10 10 (4,3) (4,6) 8 9 10

Figure 3.4: Illustration of flacets of the Bergman complex that define a non-simplicial face as well as a lattice path representation of the matroid type of this face

On Complexes of Products and Products of Complexes

Thesis

On Products of Complexes and

Complexes of Products

Vom Fachbereich Mathematik und Informatik

der Universit¨at Bremen

zur Verleihung des akademischen Grades

Doktor der Naturwissenschaften

(Doctor rerum naturalium, Dr. rer. nat.)

genehmigte Dissertation.

Author

Dipl.-Math. Martin Dlugosch

Advisor

Prof. Dr. Eva Maria Feichtner

Referees

Prof. Dr. Eva Maria Feichtner

Prof. Dr. Laura Anderson

Abgabe am 22.08.2013

Kolloquium am 24.09.2013

Contents

Acknowledgments

1 Introduction

1

1

1

1

2

3

A

B

C

D

A

B

C

D

2

3

3

2

2

2

A

B

C

D

2

3

5

2

5

1

1

1

1

2

A

B

C

D

2

1

1

1

1

3

A

B

C

D

A

B

C

D

6

6

6

4

4

2

₀

_{A|B|CD AB|CD ˆ1}

₀

_{AB|CD ˆ1}

₀

_{AB|C|D AB|CD ˆ1}