A Z-grading on a matroidM is a functionHthat assigns a flat ofM to each integerp,
subject to the condition that
cl [ p Hp ! =M and X p ρ(Hp) =ρ(M).
Recalling that cl(0) denotes the possibly empty family of loops inM, we write Γ for the integer-valued function on ∪pHp−cl(0) such that
Γ(e) =p e∈ Hp−cl(0).
Example 5.3.1. IfM= (V,I), whereV is ak-linear vector space andI is the family of
k-linearly independent subsets of V, then theZ-gradings of Mare theZ-indexed families
of subspaces Hsuch that V is the internal direct sum of {Hp :p∈Z}.
A mapT :M → M isgraded of degree k if
THp ⊆ Hp+k
for all p. Unless otherwise indicated, we will writegraded for graded of degree one. Here and throughout the remainder of this discussion we will assume that Mhas finite rank, so that all graded graded maps on Mare nilpotent.
corresponds an integer interval,
Supp(Jq) ={p:Jq∩ Hp 6=∅}.
The associated multiset
{Supp(Jq) :q = 1, . . . , m},
whereJ1, . . . , Jm are the orbits that compose J, is the barcode ofJ.
Proposition 5.3.2 states that the barcode of a graded Jordan basis forT is uniquely determined by T.
Proposition 5.3.2. IfI1, . . . , Im and J1, . . . , Jn are the orbits that compose two graded
Jordan bases, then there exists a bijection ϕ:{1, . . . , m} → {1, . . . , n} such that
Supp(Ip) = Supp(Jϕ(p))
for each p in {1, . . . , m}.
Proof. Every Jordan basis is the orbit of aK-minimal basis ofM/I(T). A graded Jordan basis, therefore, is the orbit of aK-minimal basis in
(∪pHp)/I(T) =∪p(Hp/I(T)).
The orbits of any two such bases will determine identical barcodes.
The remainder of this section will be devoted to a class of graded nilpotent maps with a particularly simple combinatorial structure. Fix filtrationsZ, Bon a finite-rank matroid
M. Assume that the elements of these filtrations are closed, and that Bp ⊆ Zp for p∈Z.
Define
Hp =Zp//Bp,
and let N be the matroid union ∪pHp. For eachein the ground set ofM, writee(p) for the copy of ein Hp⊆ N. Finally, letT :N → N be the function sendinge(p) toe(p+1).
Given this data, it is natural to ask when doesT engenders a Jordan basis, or equivalently, when the powers of T are complementary.
Lemma 5.3.3. Ifm is a nonnegative integer, then Tm is complementary if and only if
(Zp,Bp+m) is modular, for every p.
Proof. Let us write Z(p) and B(p) for the filtrations on H
p engendered by Z and B. Since
the sublevel sets of Z and Bare closed, one has
K(Tm) =[ p Z(p) p ∩ B (p) p+m I(Tm) =[ p Z(p+m) p .
The ranks of K(Tm) and I(Tm) in N are therefore given by the left and right-hand sums below. X p ρ((Zp∩ Bp+m)/Bp) X p ρ(Zp/Bp+m).
The identity below follows from the inclusion ofBp into the intersection of Zp and Bp+m.
if and only if (Zp,Bp+m) is modular.
ρ((Zp∩ Bp+m)/Bp) =ρ(Zp∩ Bp+m)−ρ(Bp)
ρ(Zp/Bp+m)≤ρ(Zp)−ρ(Zp∩ Bp+m).
Since the rank of N is P
p(ρ(Zp)−ρ(Bp)), complementarity holds if and only if strict
equality holds in both estimates, for all p.
Since (Zp+m,Bp) is trivially modular for every nonnegative m, we have shown the
following.
Proposition 5.3.4. OperatorT is Jordan if and only ifZ ∪ B is modular.
In light of the preceding observation, it is reasonable to suppose that a basis that generatesZandBmay bear some special relation to the Jordan bases ofT. For convenience, define the orbit of a subset S⊆E to be the orbit ofψ(S), whereψ is the map that sends each e∈ M toe(Z(e)) inN.
Proposition 5.3.5. The Jordan bases of T are the orbits in N of theZ-B-minimal bases in M.
Proof. IfBfreely generatesZandBinM, and ifJis the orbit ofB, thenJ∩Hp=BZ≤(p)p<B
freely generates Hp=Zp//Bp for all p. ThereforeJ is a Jordan basis. Conversely, suppose thatJ is a Jordan basis, and let
B =[
p
SinceN is the matroid union of theHp,J freely generates both I(Tm)∩Hp and K(Tm)∩Hp
for all m and p. Thus B freely generates Z and Bon the minor Zp/Bp. Consequently, if
Z0 andB0 are the restrictions ofZ andB, respectively, toZ
p− Bp, thenB freely generates
(Zq0+1∩ Bp0+1)/Bp andZq0/Bpfor allq. By modularity, it generates (Zq0+1∩ Bp0+1)/(Zq0∪ Bp).
Whenq < p one hasZq0+1∩ B0p+1 =Zq+1∩ Bp+1 andZq0 =Zq, so this minor agrees with
(Zq+1∩ Bp+1)/(Zq∪ Bp). (5.3.1)
Whenp≤q the ground set of (5.3.1) contains only elementsefor which B(e) =p+ 1≤
q+ 1 =Z(e). Sincee(p) is a loop for every such e, it follows that B intersects the ground set of (5.3.1) trivially when p≤q.
In conclusion,Bis the disjoint union of some independent sets in the matroidM/(Z∪B), and may thus be extended to a basis that generates Z andB. The orbit of this set will be a Jordan basis containing J.
In closing, let us return to the linear regime. Suppose that Zp and Bp are linear
filtrations on a vector space W, and letVp denote the linear quotient space Zp/Bp. The
inclusion maps Zp→ Zp+1 induce linear mapsVp→Vp+1, which collectively determine a
graded linear operator on ⊕pVp. Let us denote this map by Q. There is a canonical set
functionφ:∪pHp→ ∪pVp, which may be described as the rule that assignse∈ Hp to the
Hp T // Q // Hp+1 Vp Vp+1
whose vertical maps are given by φ. Since S ⊆ Hp is independent inN iffϕ(S) is linearly
independent in Vp, it follows thatJ ⊆ N is a matroid theoretic Jordan basis ofT if and
only if ϕ(J) is a linear Jordan basis ofQ. Thus the following.
Proposition 5.3.6. The graded Jordan bases of Q are the orbits of the Z-B minimal bases in W.