Frontier census properties and eigenvalue bounds

Here we introduce our last coloring properties; they limit the eigenvalues of the actions of coroots. For eachk≥1 we define twofrontier census properties:

(MxkGA): For every colora∈Γ: If xis maximal inPa, then there are at mostkelements greater than

xthat have their colors adjacent toa,

(MnkLA): For every colora∈Γ: Ifxis minimal inPa, then there are at mostkelements less thanx

that have their colors adjacent toa.

We also introduce two more general such properties:

(MxFGA): For every colora∈Γ: Ifxis maximal inPa, then the number of elements greater thanx

that have their colors adjacent toais finite,

(MnFLA): For every colora∈Γ: Ifxis minimal inPa, then the number of elements less thanxthat

have their colors adjacent toais finite.

The properties Mx1GA and Mn1LA are the most important of these properties. They will be used to finish the characterizations of (upper)P-minuscule representations ofg0(respectivelyb0₊) in Chapter 5. However, their primary importance comes from revamping, generalizing, and unifying axioms considered by Stembridge and Green. See the paragraph following the extended definitions of the frontier census properties in Section 4.6 for more on this.

The three results below describe the interactions between the frontier census properties and the component weight function{µa}a∈Γconstructed in Section 4.1. DefineEµ:={µa(F,I)|a∈Γ,(F,I)∈ F I(P)}.

Proposition 4.3.1. Suppose P satisfies EC, AC, and I2A. Then for every integer k≥2: (a) The number1−k is a lower bound forE_µif and only if P satisfies MxkGA. (b) The number k−1is an upper bound forE_µif and only if P satisfies MnkLA.

Sincek ≥2, we getµb(F,I)<−1. We havePb∩I , ∅sinceµb(F,I)≥ −1 wheneverPb∩I =∅. Hence µb(F,I)=1−υb(F,I), soυb(F,I)>k. This showsPb∩I has a maximal elementysinceυb(F,I)=1 when

Pb∩Ihas no maximal element. Letc∈Γbe such thatc∼band suppose there are infinitely many elements

greater thanyinPc∩I. Thenymust be maximal inPbby local finiteness and AC. Thus MxkGA fails and

we are done. Otherwise we have|Υb(F,I)| =υb(F,I)> k. Each element in Υb(F,I) is greater thanyand

has color adjacent tob. So by AC each element inΥb(F,I) is less than any element in Pb∩F. But since

|Υb(F,I)|>2, by I2A we seePb∩F =∅. Thusyis maximal inPb. Thus MxkGA fails again sinceΥb(F,I)

contains more thankelements.

Now suppose MxkGA does not hold for somek≥2. Then there is someb∈Γfor whichPbcontains a

maximal elementysuch that there are more thankelements greater thanywith colors adjacent tob. LetI be an ideal generated byk+1 of these elements and letF:=P−I. By local finiteness, the intervals between yand thesek+1 elements are all finite. Thus for allc ∼ bthere are finitely many elements greater than yin Pc∩I. So each of thesek+1 elements is inΥb(F,I) and we haveυb(F,I)= |Υb(F,I)| > k. Hence µb(F,I)<1−k.

Dualize to get (b).

We use this result to get:

Corollary 4.3.2. Suppose P satisfies EC, AC, and I2A.

(a) We haveEµ⊆ {−1,0,1}if and only if P satisfies Mx2GA and Mn2LA. (b) The setEµis finite if and only if P satisfies MxFGA and MnFLA.

(c) Let{ηa}a∈Γbe any component weight function and letEη:={ηa(F,I)|a∈Γ,(F,I)∈ F I(P)}. Then the

setEηis finite if and only if P satisfies MxFGA and MnFLA.

Proof. Part (a) follows immediately from Proposition 4.3.1 sinceEµ ⊆ _Z. For any k ≥ 1, the property MxkGA (respectively MnkLA) implies MxFGA (respectively MnFLA). Conversely, the property MxFGA (respectively MnFLA) implies the existence of somel≥1 such that MxkGA (respectively MnkLA) holds for allk ≥l. Combined with both parts of Proposition 4.3.1, this gives (b). For (c), letCbe a component ofF I(P) and fix (F0,I0)∈ C. Letb∈Γand let (F,I)∈ C. Since both{µa}a∈Γand{ηa}a∈Γare component

weight functions, applying Equation (3.1) twice gives

µb(F,I)−µb(F0,I0)=2∆b[(F,I),(F0,I0)]−

X

c∼b

∆c[(F,I),(F0,I0)]=ηb(F,I)−ηb(F0,I0).

Thusµb(F,I)−ηb(F,I)=µb(F0,I0)−ηb(F0,I0), so the differenceµb−ηbis constant onC. Since this holds

for all components, we see thatEµis finite if and only ifEηis finite. Thus (c) follows from (b). We close by specifying when the bounds forE_µ are attained:

Corollary 4.3.3. Suppose P satisfies EC, AC, and I2A.

(a) Suppose P additionally satisfies MxFGA. Let k be the smallest positive integer for which P satisfies MxkGA. If k=1, thenminEµ=−1. If k>1, thenminEµ =1−k.

(b) Suppose P additionally satisfies MnFLA. Let k be the smallest positive integer for which P satisfies MnkLA. If k=1, thenmaxE_µ=1. If k>1, thenmaxE_µ=k−1.

Proof. To prove (a), first supposek=1. Since Mx1GA implies Mx2GA, it follows from Proposition 4.3.1 that−1 is a lower bound forE_µ. Lety∈Pand setb:=κ(y). LetF be the principal filter generated byyand setI :=P−F. By the last statement of Proposition 4.2.1 we know thatµb(F,I)=−1. Hence−1∈ E_µ, so minE_µ =−1. Now supposek>1. The same argument as above shows that−1∈ E_µ. By Proposition 4.3.1 we know that 1−kis a lower bound forEµ. By assumptionPdoes not satisfy MxlGA for any 1≤l<k. So Proposition 4.3.1 implies 1−lis not a lower bound forEµfor any 1<l<k. Since−1∈ Eµ, we see 1−lis not a lower bound forEµfor any 1≤l<k. SinceEµ ⊆_Z, this implies that 1−k∈ Eµ. Thus minEµ =1−k.

Dualize to get (b).