Smith invariants and dual graded graphs

25 Pages·2007·0.288 MB·English

by  Alexander Miller

SMITH INVARIANTS AND DUAL GRADED GRAPHS
ALEXANDERMILLER
Abstract. The aim of this paper is to present evidence fora simpleconjec-
tural relationbetween eigenvalues and invariant factors of incidence matrices
associated withadjacent ranksindifferentialposets. Theconjectural relation
yields the Smith invariants immediately, as the eigenvalues are completely
understood [3, 15]. Furthermore, we consider more general structures: dual
gradedgraphs[3]. Inthissetting,theaforementionedrelationssometimeshold
andothertimesfail.
One particularly interesting example is the Young-Fibonacci lattice YF
studiedbyOkada[14],whichweshowpossessestheaforementionedconjectural
relation.
Contents
1. Background 1
1.1. Differential posets 1
1.2. Eigenvalues 3
1.3. Smith invariants 3
2. Conjecture for r-differential posets 4
3. Constructions 7
3.1. Properties of constructions 7
3.2. Proof for YF 12
4. Young’s lattice Y 12
5. Dual graded graphs 17
5.1. Duality of graded graphs 17
5.2. Examples 18
5.3. Eigenvalues 19
5.4. Constructions for dual graded graphs 20
5.5. Notes on Conjecture 2.6 and dual graded graphs 21
6. Remarks and questions 23
References 24
1. Background
1.1. Differential posets. Let P be a locally finite poset with ˆ0, having only
finitely many elements of each rank. In P we define up and down maps U,D :
Key words and phrases. Smith-eigenvalue relation, invariant factors, cokernel, differential
posets,dualgradedgraphs,Wagner’sconstruction, Hopfalgebras,fundamental construction.
IwishtothankVicReinerforbeingsuchanextraordinarymentorthroughoutmyundergrad-
uatecareer. IwouldalsoliketothankJ.NzeutchapandR.P.Stanleyforhelpfulcorrespondences.
1
2 ALEXANDERMILLER
ZP→ZP by
Ux= y Dy = x
y≻x y≻x
X X
for vertices x,y in P and extending linearly. We call P r-differential, for positive
integer r, if U and D satisfy the commutation relation
DU −UD =rI.
Wewilloftendropther whenitisarbitraryandthestatementathanddoesnot
depend on it, going against the tradition [3, 15] of its omission only when r =1.
1.1.1. Young’slattice. Theprototypicalexampleofa1-differentialposetisYoung’s
lattice, denoted by Y. This is the set of all partitions P, ordered by inclusion of
Young diagrams; see Figure 1.
∅
Figure 1. Young’s lattice.
Recall that Y describes the branching from S to S in the representation
n n−1
theory of S , a fact we will later use.
n
1.1.2. Fibonacci posets. As aset,theFibonacci r-differential poset Z(r)consistsof
all finite words using alphabet {1 ,1 ,...,1 ,2}. For two words w,w′ ∈ Z(r), we
1 2 r
define w to cover w′ if either
(1) w′ is obtainedfromw by changinga2 to some 1 , aslongas only 2’soccur
i
to its left, or
(2) w′ is obtained from w by deleting its first letter of the form 1 .
i
112 22 121 211 1111
12 21 111
2 11
1
∅
Figure 2. The Young-Fibonacci lattice YF.
It is an easy exercise to show Z(r) is an r-differential poset [15]. It is also easy
to see why it has such a name: the jth rank of Z(1) = YF (the Young-Fibonacci
lattice) has size f , the jth Fibonacci number; see Figure 2.
j
SMITH INVARIANTS AND DUAL GRADED GRAPHS 3
1.2. Eigenvalues. A particularly nice feature of differential posets is that the
eigenvalues of DU and UD are simple to write down. To be more precise, we
have the following proposition [15]:
Proposition 1.1. Let P be an r-differential poset. Then
n
Ch(DUn,λ)= (λ−r(i+1))∆pn−i
i=0
Y
and
n
Ch(UD ,λ)= (λ−ri)∆pn−i.
n
i=0
Y
1.3. Smith invariants. Henceforth,wetakeRto be aUFD.Furthermore,except
when explicitly noted otherwise, matrices in this section are over R.
Definition 1.2. The units of Rn×n are called unimodular matrices.
It is a standard exercise to show that a matrix M is unimodular if and only if
detM is a unit of R.
Definition 1.3. A(possiblyrectangular)diagonalmatrixD is adiagonalformfor
a matrix A if there exist unimodular matrices P and Q such that D =PAQ. It is
called the (up to units) Smith normal form of A if the diagonal entries d ,d ,...
11 22
of D are such that d |d for all i≤j; in this case, we say the Smith entries of A
ii jj
are s =d .
i ii
It should be noted that a matrix A need not have a Smith normal form, as R is
only a UFD, not a PID. However, if it does have a Smith form, then it is unique
up to units.
For anintegralmatrixA, let d (A) be the greatestcommondivisor ofthe deter-
i
minants of all the i×i minors of A, where d (A)=0 if all such i×i determinants
i
are zero. The number d (A) is called the kth determinantal divisor of A. The
k
following is quite useful when studying Smith invariants.
Theorem 1.4. The Smith normal form entries (s ,s ,...) of a matrix A∈Zn×n
1 2
are given by the equation
d (A)
j
s (A)= ,
j
d (A)
j−1
where d (A) is taken to be 1.
0
From this, when A is integral and invertible we get a useful description for the
largest Smith invariant.
Proposition 1.5. Let A∈Zn×n be nonsingular. Then s is the smallest positive
n
integer for which
s A−1 ∈Zn×n.
n
Proof. From basic linear algebra we know that
M
A−1 =(−1)j+i ji ,
ij detA
4 ALEXANDERMILLER
whereM isdefinedtobe thedeterminantofthe(n−1)×(n−1)matrixresulting
ij
from the deletion of row i and column j in A. Hence
M
sA−1 ∈Zn×n ⇔ s ij ∈Z ∀i,j
detA
⇔ detA|sM ∀i,j
ij
⇔ detA|gcd(sM )=sgcd(M )
ij ij
i,j i,j
gcd (M )
⇔ s i,j ij ∈Z
detA
s
⇔ ∈Z,
s
n
since Theorem 1.4 implies
d (A) detA detA
n
s = = = .
n
d (A) d (A) gcdM
n−1 n−1 ij
(cid:3)
2. Conjecture for r-differential posets
The following definitions are central to our conjectures.
Definition 2.1. Let ϕ be an endomorphism of a rank n free R-module, with all
eigenvalues in R. To ϕ we associate a partition E of its eigenvalues, defined to be
the multiset of sets
E ={eigenvalues with multiplicity at least i},
i
where 1≤i≤n.
Definition 2.2. Let ϕ ∈ Rn×n. Then ϕ is said to possess the Smith-eigenvalue
relation if it has all its eigenvalues in R and it has a Smith form over R, with
s = λ,
n+1−i
λY∈Ei
taking the empty product to be 1.
Notethatϕ+tI possessingtheSmith-eigenvaluerelationoverR[t]impliesϕ+kI
has the Smith-eigenvalue relation for all k ∈R.
2 1
Example 2.3. Let ϕ = . Then we have ϕ+tI has eigenvalues t+1 and
1 2
t+3. Thus, E ={t+1,(cid:18)t+3}(cid:19)and E =∅. Computing, we have
1 2
2+t 1 0 1−(2+t)2 0 1−(2+t)2 1 0
∼ ∼ ∼ ,
1 2+t 1 2+t 1 0 0 (t+1)(t+3)
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
where the last step simply involves swapping rows and factoring. Looking back at
Definition 2.2, one can now see that ϕ+tI possesses the Smith-eigenvalue relation
over Z[t].
The followingpropositionshowsthatthe Smith-eigenvaluerelationis specialfor
even diagonal matrices.
Proposition 2.4. For diagonal matrices ϕ∈Zn×n over Z
SMITH INVARIANTS AND DUAL GRADED GRAPHS 5
(i) ϕ has the Smith-eigenvalue relation if and only if every pair of distinct eigen-
values λ 6= λ are relatively prime. (In particular, a singular ϕ has the
i j
Smith-eigenvalue relation if and only if ϕ∈±{0,1}n×n.)
(ii) ϕ+tI has the Smith-eigenvalue relation over Z[t] if and only if
ϕ∈{0,1}n×n+ZI.
Proof. Let {λ } be distinct representatives for the diagonal entries of ϕ. Suppose
i
all λ are nonzero. Then the claim follows by induction and
i
s =lcm λ = λ ⇔gcdλ =1.
n i i i i
i
i
Y
Suppose now that ϕ is singular. If ϕ∈{0,1}n×n, it is already in Smith form, and
it is easy to see that
s = λ.
n+1−i
λY∈Ei
Onthe other hand, if ϕ contains an entry strictly greaterthan 1, then the product
of the nonzero products
λ
λY∈Ei
is strictly less than the product of ϕ’s nonzero Smith entries.
By the first part, setting t to the negative of a diagonal entry of ϕ gives the
necessity in the second assertion. Sufficiency follows from
t+j+1 1
∼
t+j (t+j+1)(t+j)
(cid:18) (cid:19) (cid:18) (cid:19)
over Z[t] for j ∈ Z, and that scalar matrices clearly possess the Smith-eigenvalue
relation. (cid:3)
1
Example 2.5. Consider ϕ= . Then
3
(cid:18) (cid:19)
(a) ϕ is nonsingular and has the Smith-eigenvalue relation.
(b) ϕ+I is nonsingular and does not have the Smith-eigenvalue relation.
(c) ϕ−3I is singular and has the Smith-eigenvalue relation.
(d) ϕ−I is singular and does not have the Smith-eigenvalue relation.
Because ϕ+kI does not have the Smith-eigenvalue relation for some k ∈ Z, we
have that ϕ+tI does not have the relation over Z[t].
With the above terminology, we are now in a position to state our main conjec-
ture:
Conjecture 2.6. Let P be an r-differential poset, and set n≥0. Then
(i) U has all Smith entries equal to 1;
n
(ii) ∆p ≥∆p ;
n+1 n
(iii) DU +tI has the Smith-eigenvalue relation over Z[t].
n
It should be remarked that Conjecture 2.6 is invariant under interchanging U’s
and D’s; this follows from D = Ut and the relation DU = DU +rI. The
n n−1 n n
following is also an important observation:
6 ALEXANDERMILLER
Observation 2.7. Let P be an r-differential poset. Assume p <p <p <···1,
1 2 3
andparts(ii)and(iii)ofConjecture2.6holdinP. ThenwehavetheSmithentries
of DU +tI over Z[t] are given by
n
entry multiplicity
(n+t)−δr,1(n+1)!r,t ∆2p0 =1
(i+1)! ∆2p
r,t n−i
1 p
n−1
where 0≤i≤n−1, we take p =0 for k negative, and
k
ℓ! =(r·ℓ+t)(r·(ℓ−1)+t)···(r·1+t).
r,t
Another observation one should make is that we have the following proposition
relating the parts of Conjecture 2.6:
Proposition 2.8. Let P be an r-differential poset in which part (i) of Conjec-
ture 2.6 holds. Then DU −rI has the Smith-eigenvalue relation if and only if
n
∆p ≥∆p ,∆p ,...
n n−1−δr,1 n−2−δr,1
for all n. In particular, (i) and (ii) of Conjecture 2.6 together imply the aforemen-
tioned special case of part (iii).
Proof. We begin by making a small observation: the assumption that (i) holds in
P provides a unimodular matrix M˜ so that
I
M˜U = .
0
(cid:18) (cid:19)
To see this, we start with unimodular matrices M and N such that
I
MUN = .
0
(cid:18) (cid:19)
From this, we have
I N−1 0 I
MU = N−1 = .
0 0 I 0
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
As
N−1 0
0 I
(cid:18) (cid:19)
is surely unimodular, the assertion follows.
Now that Ut =D and DU +UD =rI, the above observation shows
n n
I 0
M˜(DU −rI)M˜t =M˜UD M˜t = ,
n n 0 0
(cid:18) (cid:19)
where the right hand side has rank p by the injectivity of U . That is,
n−1 n−1
coker (DUn−rI)∼=Z∆pn.
The result now follows from the fact that
dimkerUD =∆p ,
n n
i.e. 0∈E ,...,E ,andthateachofthe remainingp setsE hasproduct
1 pn−pn−1 n−1 i
1 if and only if ∆p ≥∆p ,∆p ,.... (cid:3)
n n−1−δr,1 n−2−δr,1
1Itisanopenproblemtoshowthisholdsforalldifferentialposets;itwasoriginallyposedin
in[15].
SMITH INVARIANTS AND DUAL GRADED GRAPHS 7
3. Constructions
From existing differential posets, there are two natural ways one can construct
new differential posets. The first way is by employing the Cartesian product; the
second is known as Wagner’s construction.
Definition 3.1. If P and Q are posets, we define their Cartesian product to be
the poset P×Q on the set
{(p,q) : p∈P and q ∈Q}
such that (p,q)≤(p′,q′) if p≤p′ in P and q ≤q′ in Q.
Example 3.2. The following is a small example for the Cartesian product of two
posets.
× ∼=
One then observes the following
Lemma 3.3. [15] Assume that P and Q are r- and s-differential posets. Then
P×Q is an (r+s)-differential poset.
WenowdescribeWagner’s construction,amethodusedtoproducer-differential
posets from partial r-differential posets of some finite rank (see [15, §6]).
Let P be a finite graded poset of rank n, with ˆ0. Furthermore, assume that
DU −UD =rI
as operators on P ,P ,...,P . We call P a partial r-differential poset of rank
0 1 n−1
n. Let P+ be the poset of rank n+1 obtained from P in the following way: for
eachv ∈P , addavertexv∗ ofrankn+1 to Pthat coversexactlythose x∈P
n−1 n
coveringv. Finally, above each x∈P we adjoin r new vertices in P+. We denote
n
the resultingposetofrankn+1by E (P). Iteratingthis constructionproducesan
r
r-differential poset:
Proposition 3.4. Let P be a partial r-differential poset of rank n. Let
W(P)= lim Eℓ(P).
r
ℓ→∞
Then W(P) is r-differential. Moreover, W(P) =P.
[0,n]
An important example of the construction is Z(r), obtained by applying the
Wagner construction to
1 r
···
As an example, see Figure 3 for Z(2).
3.1. Properties of constructions.
8 ALEXANDERMILLER
Figure 3. Z(2).
3.1.1. Conjecture 2.6(i).
Proposition 3.5. Let P and Q be differential posets in which Conjecture 2.6(i)
holds. Then Conjecture 2.6(i) holds in P ×Q.
Proof. We begin by noting that
U(P×Q) :Z(P ×Q) →Z(P ×Q)
n−1 n−1 n
is given by
I ⊗U(P)
q0 n−1
U(Q)⊗I I ⊗U(P)
 0 pn−1 q1 n−2 
 U1(Q)⊗Ipn−1 ...... ... ,
 
 
 ... Iqn−1 ⊗U0(P)
 U(Q) ⊗I 
 n−1 p0 
 
for a proper indexing. By our assumption on U(P) and U(Q), one can perform
elementary row and column operations to arrive at a lower triangular matrix with
factors
I
I ⊗ pn−1−i ,
qi 0
(cid:18) (cid:19)
0≤i≤n−1, along the diagonal. The assertion follows. (cid:3)
The analogous proposition for Wagner’s construction is clear by construction:
Proposition 3.6. Let P˜ be a partial differential poset of rank n in which coker U
i
is free for all 0 ≤ i≤ n−1. Then Conjecture 2.6(i) holds in the differential poset
P obtained from P˜ by Wagner’s construction.
Corollary 3.7. Let P be a differential poset constructed using Wagner’s construc-
tion and Cartesian products on differential posets and partial differential posets in
which coker U is free for all applicable i. Then Conjecture 2.6(i) holds in P.
i
3.1.2. Conjecture 2.6(ii).
Proposition 3.8. If P and Q are arbitrary differential posets, Conjecture 2.6(ii)
holds in P×Q. That is, Conjecture 2.6(ii) holds in all decomposable differential
posets.
SMITH INVARIANTS AND DUAL GRADED GRAPHS 9
Proof. Let P(t) := p ti, Q(t) := q ti, and (P ×Q)(t) := r ti = P(t)Q(t).
i i i
Then
P (1−t)P(t)P, (1−t)Q(t)∈N[[t]] P
because U is injective. Thus,
(1−t)2(P ×Q)(t)=(1−t)2P(t)Q(t)∈N[[t]].
That is, ∆2r ≥0. (cid:3)
n
Proposition 3.9. Let T be a partial r-differential poset of rank n in which Con-
jecture 2.6(ii) holds for available ranks, and let T be the differential poset obtained
via Wagner’s construction. If r > 1, Conjecture 2.6(ii) holds in T. If r = 1,
Conjecture 2.6(ii) holds in T if and only if p ≤2p .
n n−1
Proof. First note
p =p +rp .
n+1 n−1 n
Suppose now r >1. In this case we have
∆p ≥p ≥∆p .
n+1 n n
For r=1 we have
∆p =p ≥∆p ⇔2p ≥p .
n+1 n−1 n n−1 n
The assertion now follows from the fact that
p =p +p ≤2p
n+1 n−1 n n
by the injectivity of U. (cid:3)
3.1.3. Conjecture 2.6(iii). We begin with a special case of Conjecture 2.6(iii) that
follows from the previous two sections.
Proposition 3.10. Let P be an r-differential poset constructed using Wagner’s
construction and Cartesian products on differential posets and partial differential
posets in which cokerU is free forall applicable i. Ifr >1, thenDU−rI possesses
i
theSmith-eigenvaluerelation. Ifr =1,thenDU−rI possessestheSmith-eigenvalue
relation if and only if
∆p ≥∆p ,∆p ,...
j j−2 j−3
holds for all applicable j in the poset used to obtain P.
Proof. Ther >1casefollowsimmediatelyfromCorollary3.7andPropositions3.8,3.9,
and 2.8.
Suppose r = 1. By Corollary 3.7, we know that coker U is free in P. Thus, by
Proposition 2.8 we have that DU −rI possesses the Smith-eigenvalue relation if
and only if
∆p ≥∆p ,∆p ,...
j j−2 j−3
in P for all j.
Becauser =1,itis clearthatPwasobtainedfromasingle posetP′. Moreover,
it is clear that either P = P′ or P = W(P′). If P = P′, then we are done by
our hypothesis. Assume that P = W(P′), where P′ is a partial differential poset
of rank n−1. We need to show that the above inequalities hold in P if and only
if they hold in P′. The necessity is clear, as W(P) = P′. To see the other
[0,n−1]
direction, recall
p =p +rp
n+i n+i−2 n+i−1
10 ALEXANDERMILLER
for all i≥0. Thus, with r =1 we have
∆p =p ≥∆p
n+i n+i−2 n+i−2−j
for all i,j ≥0, by the injectivity of U. This finishes the proof. (cid:3)
Remark 3.11. Here we note that
DU(P×Q) =D(P)⊗U(Q)⊕(I⊗D(Q)U(Q)+D(P)U(Q)⊗I)⊕U(P)⊗D(Q).
But it is not clear how to use this.
We now develop a tremendously useful lemma, though unattractively technical,
for detecting the Smith-eigenvalue relation for DU + tI over Z[t] in differential
posets obtainedthroughWagner’s construction. It will follow as a simple corollary
that Conjecture 2.6(iii) holds for YF. For the remainder of this section, we take
r=1.
From the description of Wagner’s construction, we know that
D
(3.1) U = n and D = U I
n I n+1 n−1
(cid:18) (cid:19)
(cid:0) (cid:1)
for n≥ℓ. This gives a useful recursive description of DU , for n≥ℓ+2:
n
DU +I U I
n−2 pn−2 n−3 pn−2
(3.2) DU = D 2I .
n  n−2 pn−3 
I 2I
pn−2 pn−2
 
Next, we note a pivotal lemma.
Lemma 3.12. LetPbe a(1-)differentialposetobtainedfromapartialdifferential
poset of rank ℓ. We have
I
aI −(a−1)D pn
pn n+1 ∼ bI −a(a−1)D
(cid:18)−aUn −bIpn+1 (cid:19)  −bUpnn−−11 −a(a+b−1)Inpn
 
for n≥ℓ.
Proof. We simply use row and column operations together with (3.1). We have
aI −(a−1)U −(a−1)I
aI −(a−1)D pn n−1 pn
pn n+1 ∼ −aD −bI
(cid:18)−aUn −bIpn+1 (cid:19) −aIpnn pn−1 −bIpn 
I −(a−1)U −(a−1)I 
pn n−1 pn
∼ −aD −bI
 n fn−1 
−(a+b)I −bI
pn pn
I 
pn
∼ −a(a−1)DU −bI −a(a−1)D
 n−1 pn−1 n 
−(a+b)(a−1)U −(a+b)(a−1)I −bI
n−1 pn pn
I 
pn
∼ bI −a(a−1)D .
 pn−1 n 
−bU −a(a+b−1)I
n−1 pn
  (cid:3)