Conditional Marked Order Polyhedra

Definition 6.5.1. Given a marked poset (P,λ), a linear map s: R^P → R^k and b ∈ R^k, we define the conditional marked order polyhedron O(P,λ,s,b) as the intersection O(P,λ) ∩s⁻¹(b).

The faces ofO(P,λ,s,b) correspond to the faces of O(P,λ) whose relative interior meetss⁻¹(b). Hence, they are also given by face partitions. However, given a face partitionπ ofO(P,λ), deciding whether it is a face partition ofO(P,λ,s,b)can not be done combinatorially in general. The problem is in determining whether the linear systems(x) = b admits a solution in the relative interior ofF_π. We come back to this issue later in the section. Still, given a pointx ∈ O(P,λ,s,b), we obtain a face partition π_x and we can find the dimension ofF_x ⊆ O(P,λ,s,b)by calculating a kernel of a linear map associated toπ_x.

Given a partitionπ ofP, we define the linear injectionr_π: R^π˜ →R^P by r_π(z)_p =

{z_B ifpis an element of the free blockB ∈π˜, 0 otherwise.

We can describer_π as taking a labelingzof the free blocks ofπ with real numbers and making it into a labeling ofP with real numbers, by putting the values given byzon elements in free blocks, while labeling elements in non-free blocks with zero. Ifπ is a face partition ofO(P,λ), we have seen in the proof of Proposition6.2.2that the affine hull ofF_π ⊆ O(P,λ)is a translation of im(r_π). The following proposition is a generalization of this observation to conditional marked order polyhedra.

Proposition 6.5.2. Letxbe a point of O(P,λ,s,b)with associated face partitionπ =π_x. LetU be the linear subspace of R^P parallel to the affine hull of the face F_x ⊆ O(P,λ,s,b). The mapr_π restricts to an isomorphismker(s◦r_π) ∼−→U. In particular, the dimension of F_x is the same as the dimension of ker(s ◦r_π).

Proof. LetF_x^′ be the minimal face ofO(P,λ)containingx, so thatF_x = F_x^′ ∩s⁻¹(b). For the affine hulls we also have aff(F_x)=aff(F_x^′) ∩s⁻¹(b). LettingU^′be the linear subspace parallel to aff(F_x^′), just asU is the linear subspace parallel to aff(F_x), we obtain

U =U^′∩ker(s)=im(r_π) ∩ker(s),

since ker(s) is the linear subspace parallel tos⁻¹(b). This description implies thatr_π restricts to an isomorphism ker(s ◦r_π) ∼−→U.

Remark 6.5.3. In the special case of Gelfand–Tsetlin polytopes with linear conditions given by a weightµ, this result appeared in [DM04] in terms of tiling matrices associated to points in the polytope. The tiling matrix is exactly the matrix associated to the linear maps ◦r_π.

Example 6.5.4. Let(P,λ)be the linear marked poset 0 ≺p ≺q≺r ≺s ≺ 5

6.5. Conditional Marked Order Polyhedra

x_p x_q

1 2 1

2 3

u v w

Figure 6.5.: The conditional marked order polytope O(P,λ,s,b)from Example 6.5.4 to-gether with three points on faces of different dimensions.

and impose the linear conditionsx_p +x_r = 4 andx_q +x_s = 6 onO(P,λ). We describe these conditions by intersecting with s⁻¹(b)for the linear maps: R^P → R² given by s(x)=(x_p +x_r,x_q+x_s)andb =(4, 6). Any point inO(P,λ,s,b)is determined byx_p and x_q, so we can picture the polytope inR². Expressing the five inequalities in terms ofx_p, x_q using the linear conditions, we obtain

0 ≤x_p, x_p ≤ x_q, x_q ≤ 4−x_p, x_q ≤ 2+x_p, 1 ≤x_q. The resulting polytope inR^{p,q} R² is illustrated in Figure6.5.

We want to calculate the dimensions of the minimal faces ofO(P,λ,s,b)containing the pointsu = (1, 2),v = (1.5, 2.5) andw = (2, 2) inR². InR^P these points and their associated partitions ofP are

0 |1 |2 | 3 |4 |5, 0 |1.5 | 2.5 2.5 |3.5 | 5, and 0 |2 2 2 |4 | 5.

Hence, we have 4, 3 and 2 free blocks, respectively. The associated linear mapss◦r_π can be represented by the matrices

(

1 0 1 0 0 1 0 1 )

, (

1 1 0 0 1 1 )

, and (

2 0 1 1 )

,

respectively. The kernels of these maps have dimension 2, 1 and 0 corresponding to the dimensions of the minimal faces containingu,vandw as one can see in Figure6.5. ♢ Given any(P,λ)-compatible partition ofP, we obtained a polyhedronF_π^′ contained in O(P,λ)in the previous section. Hence, we have a polyhedronF_π contained inO(P,λ,s,b) given byF_π =F_π^′ ∩s⁻¹(b). As in the unconditional case, these polyhedra are canonically affine isomorphic to conditional marked order polyhedra given by the quotient(P/π,λ/π). Proposition 6.5.5. Let(P,λ)be a marked poset,π a(P,λ)-compatible partition,s: R^P → R^k a linear map and b ∈ R^k. Define s/π to be the composition s ◦ q^∗, where q^∗ is the inclusion R^P/π ↪→ R^P induced by the quotient map of marked posets. The polyhe-dron F_π ⊆ O(P,λ,s,b)is affinely isomorphic to the conditional marked order polyhedron O(P/π,λ/π,s/π,b)via the canonical isomorphism obtained by restrictingq^∗.

Proof. By definition,F_π is the intersection of the faceF_π^′ ofO(P,λ)withs⁻¹(b). We know thatq^∗ restricts to an affine isomorphismO(P/π,λ/π) ∼−→F_π^′. Hence,F_π is contained in the image ofq^∗as well and we have

F_π =F_π^′ ∩s⁻¹(b)= F_π^′ ∩imq^∗∩s⁻¹(b)= F_π^′ ∩q^∗((s ◦q^∗)⁻¹(b)).

We may writeF_π^′ asq^∗(O(P/π,λ/π))and use injectivity ofq^∗to obtain

F_π =q^∗(O(P/π,λ/π)) ∩q^∗((s/π)⁻¹(b))=q^∗(O(P/π,λ/π) ∩ (s/π)⁻¹(b)).

By definition of conditional marked order polyhedra, this is just the injective image of O(P/π,λ/π,s/π,b)underq^∗, which finishes the proof.

WhenF is a non-empty face of O(P/π,λ/π,s/π,b)we have an associated partition π = π_F, so thatF = F_π. Thus, we obtain the same corollary on faces of conditional marked order polyhedra as in the unconditional case.

Corollary 6.5.6. For every non-empty face F of a conditional marked order polyhedron O(P,λ,s,b)we have a canonical affine isomorphism

O(P/π_F,λ/π_F,s/π_F,b) F.

The next proposition will allow us to consideranypolyhedron as a conditional marked order polyhedron up to affine isomorphism. Thus, there is little hope to understand general conditional marked order polyhedra any better than we understand polyhedra in general.

Proposition 6.5.7. Every polyhedron is affinely isomorphic to a conditional marked order polyhedron.

Proof. LetQ ⊆ Rⁿbe a polyhedron given by linear equations and inequalities

∑n i=1

a_kix_i =c_k fork =1,. . .,s,

∑n i=1

b_lix_i ≤d_l forl =1,. . .,t.

Define a posetP with ground set{p¹,. . .,p_n,q¹,. . .,q_t,r}and covering relationsq_l ≺r

forl =1,. . .,t. Define a marking onP^∗ = {r}byλ(r)= 0. The marked poset obtained

this way is depicted in Figure6.6. Let the linear systems(x)=bforx ∈R^P be given by

∑n i=1

a_kix_pi =c_k fork = 1,. . .,s,

∑n

i=1

b_lix_pi −x_ql =d_l forl = 1,. . .,t.

The conditional marked order polyhedronO(P,λ,s,b)is affinely isomorphic toQ by the mapO(P,λ,s,b) →Q sendingx ∈R^P to(x_p1,. . .,x_pn) ∈ Rⁿ.

6.5. Conditional Marked Order Polyhedra

p1 p2

· · ·

p_n q1 q2

· · · q_t 0

Figure 6.6.: The marked poset constructed in the proof of Proposition6.5.7.

We may now come back to the question of when a face partition π of (P,λ) still corresponds to a face of O(P,λ,s,b). As discussed at the beginning of this section, we have to decide whethers(x)=badmits a solution in the relative interior of the faceF_π^′ ofO(P,λ), that is, relint(F_π^′) ∩s⁻¹(b),œ. Using the affine isomorphism induced by the quotient map this is equivalent to

relint(O(P/π,λ/π)) ∩ (s/π)⁻¹(b),œ.

Hence, we reduced the problem to deciding whether a linear systems(x)=badmits a solution in the relative interior of a marked order polyhedron O(P,λ). However, even deciding whether s(x) = b admits any solution in O(P,λ) is equivalent to deciding whetherO(P,λ,s,b)is non-empty, which is in general just as hard as determining whether an arbitrary system of linear equations and linear inequalities admits a solution by Proposition6.5.7.

We conclude that the concept of conditional marked order polyhedra is too general to obtain meaningful results. Still, in special cases the structure of an underlying poset and faces still corresponding to a subset of face partitions might be useful. An interesting class of conditional marked order polyhedra might consist of those, whereP is connected and conditions are given by fixing sums along disjoint subsets ofP, as is the case for Gelfand–Tsetlin polytopes with weight conditions.

In Chapter10we study a special case of such Gelfand–Tsetlin polytopes with weight conditions in the context of frame theory, namely polytopes of eigensteps as in Section4.3 for the case of equal norm tight frames. We see that even determining the dimension and facets of such a conditional marked order polyhedron is a non-trivial task.

Polyhedra

Having studied marked order polyhedra in Chapter6, we want to move towards marked chain polytopes in this chapter. As already mentioned in Chapter1, little is known about the combinatorics even of ordinary unmarked chain polytopes. In the marked setting, the situation is even worse, since the transfer mapO(P,λ) → C(P,λ)from the marked order polytope to the marked chain polytope does not even preserve vertices. In this chapter, we take the following approach to gain knowledge about marked chain polytopes: given a marked poset(P,λ) introduce a parametert ∈ [0, 1]to the transfer map to obtain a homotopyφ_t between the identity onO(P,λ)and the transfer map toC(P,λ). It turns out thatφ_t(O(P,λ))is a polytope for allt ∈ [0, 1]and the combinatorial type does not change whent varies in(0, 1). Thus, we may think of the marked order polytopeO(P,λ) att =0 and the marked chain polytopeC(P,λ)att =1 as continuous degenerations of ageneric marked poset polytopeobtained fort ∈ (0, 1). We will give a precise definition of what we mean bycontinuous degenerationand indicate how this point of view might help understanding the face structure of marked chain polytopes.

Inspired by previous representation theoretically motivated work of Fang and Fourier onmarked chain-order polytopesin [FF16], we allow the parametert to be different in each coordinate and obtain a family of polytopesO_t(P,λ)fort ∈ [0, 1]^P˜ that we refer to as thecontinuous family of marked poset polytopes. Analogous to the case of just one parameter, the combinatorial type stays constant along the relative interiors of the faces of the parametrizing hypercube. We recover the marked chain-order polytopes of Fang and Fourier at some of the cube’s vertices, hence putting them in an elegant unified framework.

The results of this chapter are joint work with Xin Fang, Ghislain Fourier and Jan-Philipp Litza and have also appeared in [FFLP17].

Im Dokument The Polyhedral Geometry of Partially Ordered Sets (Seite 75-80)