Frontier census properties and eigenvalue bounds in the general case

Now we extend the frontier census bounds of Section 4.3 to the general case. For eachk≥1 we extend two of thefrontier census properties:

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

have their colors adjacent toais finite andP

y>x−θκ(y),a ≤k.

(MnkSB): For every colora∈Γ: Ifxis minimal inPa, then the number of elements less than xthat

have their colors adjacent toais finite andP

only finitely many nonzero terms. We have just interpreted “P

” as follows: If there are infinitely many elementsygreater (or less) thanxfor whichθκ(y),a =0, ignore those terms. We call the sumPy>x−θκ(y),a

(respectivelyP

y<x−θκ(y),a) anupper(respectively alower)adjacency sumsince−θκ(y),a ,0 if and only if

κ(y)∼a. We note that these properties have the following equivalent forms:

(MxkSB): The property MxFGA holds and for everya∈Γ: Ifxis maximal inPa, thenPy>x−θκ(y),a≤k.

(MnkSB): The property MnFLA holds and for everya∈Γ: Ifxis minimal inPa, thenPy<x−θκ(y),a≤k.

The properties Mx1SB and Mn1SB continue to be the most important of these properties. The property Mn2SB also plays an important role in the classification in Chapter 8; see Proposition 8.2.2. In Proposition 7.4.5 we indicate how Mx1SB revamps axioms considered by Stembridge [Ste]. This property was retrospec- tively found to be implicitly present in Proposition 2.5 of [Ste]. That early statement in [Ste] was formulated in terms of decompositions of Weyl group elementsw, before the heap finite colored posets were introduced. Both properties Mx1SB and Mn1SB are satisfied vacuously by the full heaps of Green. So these updated frontier census bounds revamp, generalize, and unify the axioms considered by Stembridge and Green.

Continue to denoteEµ ={µa(F,I)|a ∈Γ,(F,I)∈ F I(P)}. All of the results from Section 4.3 can be extended to the general case. To update Proposition 4.3.1, just replace I2A, MxkGA, and MnkLA with I2∨1A, MxkSB, and MnkSB, respectively:

Proposition 4.6.1. Suppose P satisfies EC, AC, and I2∨1A. Then for every integer k≥2: (a) The number1−k is a lower bound forEµif and only if P satisfies MxkSB.

(b) The number k−1is an upper bound forEµif and only if P satisfies MnkSB.

We provide a new proof since the computations have changed with the updated properties.

Proof. Fix an integerk ≥ 2. Suppose there is someb ∈ Γand (F,I) ∈ F I(P) such thatµb(F,I) < 1−k. 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 MxkSB fails and we

are done. Otherwise we haveP

z∈Υb(F,I)−θκ(z),b=υb(F,I)> k≥2. Each element inΥb(F,I)⊆ I is greater

thanyand has color adjacent tob. SupposePb∩F,∅. Sinceyis maximal inPb∩I, we seePb∩Fmust have

a minimal elementxby local finiteness. Notey< xare consecutive occurrences of the colorb. By AC each element inΥb(F,I) is in (y,x). But this violates I2∨1A since it impliesPz∈(y,x)−θκ(z),b≥Pz∈Υb(F,I)−θκ(z),b>2.

ThusPb∩F=∅and soyis maximal inPb. But even if MxFGA holds, we see MxkSB still fails since then

P

z>y−θκ(z),b≥Pz∈Υb(F,I)−θκ(z),b>k.

Now suppose MxkSB does not hold for somek ≥ 2. Then there is a colorb ∈ Γand an element y maximal inPb such that either there are infinitely many elements greater thanywith colors adjacent tob,

or there are finitely many such elements andP

z>y−θκ(z),b >k. Either way, choose a finite setSof elements

greater thanywith colors adjacent tobsuch thatP

z∈S−θκ(z),b > k. LetI be the ideal generated bySand

letF := P−I. By local finiteness, the intervals betweenyand the elements ofSare all finite. Thus for allc ∼ bthere are finitely many elements greater thanyin Pc∩I. HenceS ⊆ Υb(F,I) and so we have

υb(F,I)=P

z∈Υb(F,I)−θκ(z),b ≥Pz∈S−θκ(z),b>k. Henceµb(F,I)<1−k.

Dualize to get (b).

To update Corollary 4.3.2, replace I2A, Mx2GA, and Mn2LA with I2∨1A, Mx2SB, and Mn2SB, respectively:

Corollary 4.6.2. Suppose P satisfies EC, AC, and I2∨1A.

(a) We haveEµ⊆ {−1,0,1}if and only if P satisfies Mx2SB and Mn2SB. (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

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

The proof of Proposition 4.3.2 from Section 4.3 applies once Equation (3.6) is referenced.

To extend Corollary 4.3.3, replace I2A, MxkGA, and MnkLA respectively with I2∨1A, MxkSB, and MnkSB:

Corollary 4.6.3. Suppose P satisfies EC, AC, and I2∨1A.

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

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