Critical groups of McKay-Cartan matrices

26 Pages·2016·0.354 MB·English

by  Christian Gaetz

CRITICAL GROUPS OF MCKAY-CARTAN MATRICES
CHRISTIANGAETZ
Abstract. This thesis investigates the critical groups of McKay-Cartan matrices, a
certaintypeofavalanche-finitematrixassociatedtoafaithfulrepresentationγofafinite
group G. It computes the order of the critical group in terms of the character values
of γ, and gives some restrictions on its subgroup structure. In addition, the existence
of a certain Smith normal form over Z[t] is shown to imply a nice form for the critical
group. ThisisusedtocomputethecriticalgroupforthereflectionrepresentationofSn.
Inthecasewhereim(γ)⊂SL(n,C)itdiscussesforwhichpairs(G,γ)thecriticalgroup
is isomorphic to the abelianization of G, including explicit calculations demonstrating
theseisomorphismsforthefinitesubgroupsofSU(2)andSO(3,R). Inthiscaseitalso
identifiesasubsetofthesuperstableconfigurations,answeringaquestionposedin[3].
Submitted under the supervision of Professor Victor Reiner to the University Honors
Program at the University of Minnesota–Twin Cities in partial fulfilment of the
requirements for the degree of Bachelor of Science summa cum laude in Mathematics.
Date:June12,2016.
1
2 CHRISTIANGAETZ
Acknowledgements
I am very thankful to Vic Reiner for introducing me to the topic of critical groups and
for always knowing the right references to check and the right ideas to try, as well as for
his careful reading of earlier drafts of this thesis. I also thank Gregg Musiker and Pavlo
Pylyavskyy for agreeing to be readers for this thesis.
Contents
Acknowledgements 2
1. Introduction 3
1.1. Graph Laplacian matrices 4
1.2. Critical groups of graphs 5
1.3. Smith normal form and the cokernel of a linear map 5
2. Avalanche-finite matrices 5
2.1. McKay-Cartan matrices 6
3. The structure of critical groups for group representations 8
4. Critical groups for the reflection representation of S 11
n
5. An alternative expression for critical groups 12
5.1. Critical groups for special linear representations 12
5.2. Superstable configurations 14
Appendix A. The McKay correspondence 16
Appendix B. Calculation of K(γ) for finite subgroups of SU(2) 17
B.1. Type A(cid:101)n−1 17
B.2. Type D(cid:101)n+2 18
B.3. Types E(cid:101)6,E(cid:101)6 and E(cid:101)8 20
Appendix C. Calculation of K(γ) for finite subgroups of SO(3,R) 22
C.1. Type A(cid:101)n−1 22
C.2. Type D(cid:101)n+2 22
C.3. Types E(cid:101)6, E(cid:101)7, and E(cid:101)8 24
References 26
CRITICAL GROUPS OF MCKAY-CARTAN MATRICES 3
1. Introduction
This thesis will study the critical group, K(C) := coker(Ct : Z(cid:96) → Z(cid:96)), which is defined
for C an avalanche-finite matrix. Specifically it will investigate the structure of K(C) when
C istheMcKay-Cartan matrix associatedtoafaithfulrepresentation γ ofafinitegroupG.
This matrix has entries c =nδ −m for 1≤i,j ≤(cid:96) where the m are defined by
ij ij ij ij
(cid:96)
(cid:88)
χ ·χ = m χ
γ i ij j
j=0
andwhere{1 =χ ,χ ,...,χ }isthesetofirreduciblecomplexcharactersofG;inthiscase
G 0 1 (cid:96)
we define K(γ):=K(C). McKay-Cartan matrices were shown to be avalanche-finite in [3].
Later in Section 1 critical groups are introduced in the context of Laplacian matrices of
graphs, which are themselves avalanche-finite matrices. Section 2 defines avalanche-finite
matrices and McKay-Cartan matrices in general and gives certain basic results about the
critical groups of the latter. Section 3 applies some results of Lorenzini in [8] to obtain the
first main Theorem:
Theorem 1. Let G be a finite group with faithful complex representation γ and critical
group K(γ). Let e=c ,c ,...,c be a set of conjugacy class representatives for G, then:
0 1 (cid:96)
i.
(cid:96)
(cid:89)
(n−χ (c ))=|K(γ)|·|G|
γ i
i=1
ii. If χ is real-valued, and χ (c) is an integer character value achieved by m different
γ γ
conjugacy classes, then K(γ) contains a subgroup isomorphic to
(Z/(n−χ (c))Z)m−1.
γ
Section 4 derives explicit formulas for critical groups in the case of the reflection repre-
sentation of S :
n
Theorem 2. Let γ be the reflection representation of S and let p(k) denote the number
n
of partitions of the integer k. Then
p(n)
K(γ)∼=(cid:77)Z/q Z
i
i=2
where
(cid:89)
q = k
i
1≤k≤n
p(k)−p(k−1)≥i
Section5givesanalternativedescriptionofthecriticalgroupofaMcKay-Cartanmatrix
intermsoftherepresentationringoftheassociatedfinitegroupanddiscussestwoquestions
raised in [3], answering one of them:
Theorem 3. Let γ : G (cid:44)→ SL(n,C) be a faithful representation of G with McKay-Cartan
matrixC. Thenthestandardbasisvectorse correspondingtothenon-trivialone-dimensional
i
characters χ of G are superstable configurations for C.
i
4 CHRISTIANGAETZ
1.1. Graph Laplacian matrices. Let Γ be a loopless undirected graph on n vertices
labelled {1,...,n}. Let c be the number of edges between vertices i and j, and let d
ij i
denote the degree of vertex i. The Laplacian matrix L(Γ) is an n×n integer matrix with
entries:
(cid:40)
d if i=j
L(Γ) = i
ij
−c if i(cid:54)=j
ij
ManyclassicalresultsinalgebraicgraphtheoryrelatetotheLaplacianmatrix. Mostfamous
is Kirchhoff’s Matrix Tree Theorem:
Theorem 4 (Kirchhoff). Let Γ be an undirected graph with Laplacian matrix L = L(Γ).
Let κ(Γ) denote the number of spanning trees of Γ. Then if Lij denotes the matrix L with
row i and column j removed, we have
detLij =κ(Γ)
for all 1≤i,j ≤n.
In fact, finer results may be obtained. Results (i) and (ii) below are very classical (a
proof can be found, for example, in [4, Corollary 6.5]). Part (iii) is clear from part (i) and
the existence of Smith normal form over Z (see Section 1.3).
Proposition 1. Let Γ be a connected graph on n vertices and L = L(Γ) be its Laplacian
matrix, then
i. L has rank n−1, with kernel generated by (1,...,1)t.
ii. Let λ ,...,λ be the nonzero eigenvalues of L, then
1 n−1
λ ···λ
κ(Γ)= 1 n−1
n
iii. coker(Zn −→L Zn) is isomorphic to Z⊕K(Γ) where K(Γ) is a finite abelian group
with |K(Γ)|=κ(Γ).
Example 1. Let Γ be the graph
1
3 4
2
Then
 
2 −1 −1 0
−1 2 −1 0 
L(Γ)= 
−1 −1 3 −1
0 0 −1 1
WefindthattheeigenvaluesofL(Γ)are0,1,3,4anditsSmithnormalform(seeSection1.3)
isdiag(1,1,3,0). Thereforeκ(Γ)= 1·3·4 =3andcoker(L)∼=Z⊕Z/3Z, soK(Γ)∼=Z/3Z. It
4
iseasytoseethatΓdoesinfacthave3spanningtreesandthat(1,1,1,1)t istheeigenvector
corresponding to the eigenvalue 0.
CRITICAL GROUPS OF MCKAY-CARTAN MATRICES 5
1.2. Critical groups of graphs. The finite abelian group K(Γ) from Proposition 1 is
called the critical group of Γ and it provides a finer invariant for Γ than κ(Γ), since there
are many abelian groups of a given order. The group K(Γ) encodes information about the
critical configurations in the abelian sandpile model, a certain dynamical system on Γ. An
introduction to this topic can be found in [7], and applications to areas such as economic
models and energy minimization can be found, for example, in [1, 5].
1.3. Smith normal form and the cokernel of a linear map. Smith normal forms will
be one of our primary tools for calculating critical groups. Several basic facts about Smith
normal form are summarised in this section.
Let R be a ring and A∈Rn×n be a matrix. A matrix S is called the Smith normal form
of A if:
• There exist invertible matrices P,Q∈Rn×n such that S =PAQ.
• S is a diagonal matrix S =diag(s ,...,s ) with s |s for i=1,...,n−1.
1 n i i+1
The Smith normal form of A, if it exists, is unique up to multiplication of the s by units
i
in R.
Proposition 2. Let A ∈ Rn×n be a matrix and suppose A has Smith normal form S =
diag(s ,...,s ). Then
1 n
n
coker(A:Rn →Rn)∼=(cid:77)R/(s )
i
i=1
Proof. Clearly coker(S) = (cid:76)n R/(s ). The invertible maps P,Q provide the desired
i=1 i
isomorphism. (cid:3)
Proposition 3. Let A ∈ Rn×n be a matrix. If R is a PID then A has a Smith normal
form.
Proof. BytheclassificationoffinitelygeneratedmodulesoveraPID,coker(A)isisomorphic
to (cid:76)n R/(s ) for some choice of the s ∈ R with (s ) ⊃ ··· ⊃ (s ). It is easy to see that
i=1 i i 1 n
S =diag(s ,...,s ) is the Smith normal form of A. (cid:3)
1 n
2. Avalanche-finite matrices
The abelian sandpile model, which motivated our interest in the critical group K(Γ) of
a graph has recently been found to have a natural generalization to integer matrices other
than graph Laplacians. These more general matrices are called avalanche-finite matrices.
As before, we define the critical group K(C) of such a matrix C as
K(C):=coker(Ct :Z(cid:96) →Z(cid:96))
And, as before, several important classes of configurations (the superstable and the critical
configurations) form distinguished sets of coset representatives in this group.
Definition 1. A matrix C =(c ) in Z(cid:96)×(cid:96) with c ≤0 for all i(cid:54)=j is called a Z-matrix.
ij ij
Given a Z-matrix C, we call the elements v = (v ,...,v )t ∈ N(cid:96) chip configurations, and
1 (cid:96)
we define a dynamical system on the set of such configurations as follows:
• A configuration v is stable if v <c for i=1,...,(cid:96).
i ii
6 CHRISTIANGAETZ
• If v is unstable, then choose some i so that v ≥ c and form a new configuration
i ii
v(cid:48) = (v(cid:48),...,v(cid:48))t where v(cid:48) = v −c for j = 1,...,(cid:96). The result v(cid:48) is called the
1 (cid:96) j j ij
C-toppling of v at position i.
Definition 2. A Z-matrix is called an avalanche-finite matrix if every chip configuration
can be brought to a stable one by a sequence of such topplings.
The superstable configurations of an avalanche-finite matrix are those which are not only
stable, but cannot even be toppled at several positions at once, leaving the accounting to
the end. It is known that the superstable configurations form a set of coset representatives
for K(C) (see [11, Theorem 13.4]).
Definition 3. Let u∈N(cid:96) be a chip configuration for an avalanche-finite matrix C. Then u
is called a superstable configuration if z ∈N(cid:96) and u−Ctz ∈N(cid:96) together imply that z =0.
In this thesis we will be interested in understanding the critical groups and superstable
representations of a subset of avalanche-finite matrices called McKay-Cartan matrices.
2.1. McKay-Cartan matrices. Let G be a finite group with irreducible complex charac-
tersχ ,...,χ . Wewillalwaysletχ denotethecharacterofthetrivialrepresentation. Letγ
0 (cid:96) 0
be a faithful (not necessarily irreducible) n-dimensional representation of G with character
χ . Let M ∈Z((cid:96)+1)×((cid:96)+1) be the matrix whose entries are defined by the equations
γ
(cid:96)
(cid:88)
χ ·χ = m χ
γ i ij j
j=0
Definition 4. Let γ be an n-dimensional faithful complex representation of a finite group
G, and let M be defined as above. The extended McKay-Cartan matrix C(cid:101) := nI −M
and the McKay-Cartan matrix C is the submatrix of C(cid:101) obtained by removing the row and
column corresponding to the trivial character χ .
0
Proposition 4 (A special case of Proposition 5.3 in [3]). Let γ be a faithful complex repre-
sentation of a finite group G.
i. A full set of orthogonal eigenvectors for C(cid:101) is given by the set of columns of the
character table of G:
δ(g) =(χ (g),...,χ (g))t
0 (cid:96)
as g ranges over a collection of conjugacy class representatives for G.
ii. The corresponding eigenvalues are given by
C(cid:101)δ(g) =(n−χγ(g))·δ(g)
iii. The vector δ(e) = (χ0(e),...,χ(cid:96)(e))t spans the nullspace of both C(cid:101) and C(cid:101)t. In par-
ticular, this implies that these matrices have rank (cid:96).
Proof. The equation Mδ(g) = χ (g)δ(g) follows immediately from evaluating both sides of
γ
the defining equations of M at g. Then
C(cid:101)δ(g) =(nI−M)δ(g) =(n−χγ(g))δ(g)
This proves (ii). The orthogonality in (i) follows from the second orthogonality relation for
irreducible characters.
CRITICAL GROUPS OF MCKAY-CARTAN MATRICES 7
For (iii), note that since γ is faithful, χ (g) (cid:54)= n for n (cid:54)= e. Therefore (ii) gives that the
γ
nullspace of C(cid:101) is 1-dimensional, spanned by δ(e). This shows that C(cid:101), and therefore C(cid:101)t, has
rank (cid:96). Now, the i-th entry in Mtδ(e) is
(cid:96)
(cid:88)m χ (e)=(cid:88)(dimχ ·(cid:104)χ χ ,χ (cid:105))=(cid:104)((cid:77)χ⊕dimχj)·χ ,χ (cid:105)
ji j j j γ i j γ i
j=0 j j
1 (cid:88) 1
=(cid:104)χ χ ,χ (cid:105)= χ (g)χ (g)χ (g)= (|G|χ (e)χ (e))=χ (e)χ (e)
reg γ i |G| reg γ i |G| γ i γ i
g∈G
Therefore C(cid:101)tδ(e) =(n−χγ(e))δ(e) =0. (cid:3)
Finally,wenotethatMcKay-Cartanmatricesareindeedavalanche-finite,motivatingour
study of their cokernels.
Theorem 5 ([3], Theorem 1.2). The McKay-Cartan matrix C associated to a faithful rep-
resentation γ of a finite group G is an avalanche-finite matrix.
Definition 5 ([3], Definition 5.11). Given a faithful complex representation γ of a finite
group G with McKay-Cartan and extended McKay-Cartan matrices C,C(cid:101), we define its
critical group in either of the following equivalent ways:
K(γ):=coker(Ct)=K(C)
Z⊕K(γ):=coker(C(cid:101)t)
Proposition 5 ([3], Proposition 6.12). The critical group K(γ) is unchanged by
a. adding copies of the trivial representation to γ.
b. precomposing with a group automorphism σ :G→G.
Proof. For (a), replacing χ with χ +d·χ replaces M with M(cid:48) =M +dI and therefore
γ γ 0
replaces C(cid:101) with (n+d)I−(M +dI)=nI−M =C(cid:101).
For (b), precomposition by σ permutes {χ ,...,χ }, this corresponds to a permutation of
0 (cid:96)
the basis for the map C(cid:101), which does not affect the cokernel up to isomorphism. (cid:3)
Example 2. Let G = S and consider the reflection representation γ of G (this is the
4
action by permutation matrices on C4 with the copy of the trivial representation removed;
the associated partition is (3,1)). The character table is
e (12) (123) (1234) (12)(34)
χ 1 1 1 1 1
0
χ 1 -1 1 -1 1
1
χ 3 1 0 -1 -1
γ
χ 3 -1 0 1 -1
3
χ 2 0 -1 0 2
4
The matrices M,C(cid:101) and C associated to γ are
   
0 1 0 0 0 3 −1 0 0 0  
2 −1 −1 0
1 1 1 1 0 −1 2 −1 −1 0 
    −1 3 −1 0 
M =0 1 0 1 0C(cid:101) = 0 −1 3 −1 0 C = 
    −1 −1 2 −1
0 1 1 1 1  0 −1 −1 2 −1
0 0 −1 3
0 0 0 1 0 0 0 0 −1 3
8 CHRISTIANGAETZ
TocalculateK(γ),wecalculatethatC hasSmithnormalformdiag(1,1,1,4),orequivalently
that C(cid:101) has Smith form diag(1,1,1,4,0). Thus K(γ) ∼= Z/4Z. Theorem 2 will give the
structure of the critical group for the reflection representations for all S .
n
3. The structure of critical groups for group representations
The following general results of Lorenzini about ((cid:96)+1)×((cid:96)+1)-integer matrices of rank
(cid:96) will be useful.
Proposition 6 ([8],Propositions2.1and2.3). Let M be any ((cid:96)+1)×((cid:96)+1)-integer matrix
of rank (cid:96) with characteristic polynomial char (x) = x(cid:81)(cid:96) (x−λ ). Let R be an integer
M i=1 i
vector in lowest terms generating the kernel of M, and let R(cid:48) be the corresponding vector
for Mt. Let H be the torsion subgroup of the cokernel of Z(cid:96)+1 −M→Z(cid:96)+1.
i.
(cid:96)
(cid:89)
λ =±|H|(R·R(cid:48))
i
i=1
ii. Let λ (cid:54)= ±1,0 be an integer eigenvalue of M, and µ(λ) be the maximal number
of linearly independent eigenvectors for the eigenvalue λ. If M is symmetric and
R has at least one entry with value ±1 then C contains a subgroup isomorphic to
(Z/λZ)µ(λ)−1.
Translating this proposition into the context of critical groups of group representations
allows us to prove Theorem 1.
Theorem 1. Let G be a finite group with faithful complex representation γ and critical
group K(γ). Let e=c ,c ,...,c be a set of conjugacy class representatives for G, then:
0 1 (cid:96)
i.
(cid:96)
(cid:89)
(n−χ (c ))=|K(γ)|·|G|
γ i
i=1
ii. If χ is real-valued, and χ (c) is an integer character value achieved by m different
γ γ
conjugacy classes, then K(γ) contains a subgroup isomorphic to
(Z/(n−χ (c))Z)m−1.
γ
Proof. Part(i)followsfromProposition6(i)withM =C(cid:101): theeigenvaluesofC(cid:101) aregivenin
Proposition 4 (ii), and we have that in this case R=R(cid:48) =(χ (e),...,χ (e))t by Proposition
0 (cid:96)
4 (iii). Therefore
(cid:96)
(cid:88)
R·R(cid:48) = (dimχ )2 =|G|
i
i=0
Theproductonthelefthandsideispositivesincethecomplexconjugateofatermn−χ (c)
γ
also appears in the product as n−χ (c(cid:48)) where c(cid:48) is conjugate to c−1.
γ
CRITICAL GROUPS OF MCKAY-CARTAN MATRICES 9
For part (ii), notice that µ(n−χγ(c))=m since the eigenvectors of C(cid:101) are orthogonal by
Proposition 4 (i). If χ is real valued then
γ
1 (cid:88)
(cid:104)χ ,χ χ (cid:105)= χ (g)χ (g)χ (g)
j γ i |G| j γ i
g∈G
1 (cid:88)
= χ (g)χ (g)χ (g)=(cid:104)χ χ ,χ (cid:105)
|G| j γ i j γ i
g∈G
so C(cid:101) is symmetric in this case. Finally, χγ(c)<n, so we can rule out the cases λ=0,−1 in
Proposition 6 (ii). Applying this proposition then gives the desired result. (cid:3)
Example 3. Let γ be the defining permutation representation of the alternating group A
n
for n ≥ 5, which clearly has a real-valued character. This group always has a split class, a
conjugacy class of S which splits into two classes in A . This means that χ has the same
n n γ
value on representatives c1,c2 of two different conjugacy classes, with c1,c2 (cid:54)= e. Thus C(cid:101)
has a repeated eigenvalue λ = n−χ (c ) ≥ 2. By Theorem 1 (ii), this implies that K(γ)
γ 1
has a subgroup isomorphic to Z/λZ.
Corollary 1. In the context of Theorem 1, if χ is Q-valued, then Syl (K(γ)) = 0 unless
γ p
p≤2n.
Proof. The fact that χ is Q-valued implies that it is Z-valued since a sum of roots of unity
γ
is rational if and only if it is integral. Thus −n ≤ χ (c ) < n for all i, and so the product
γ i
on the left hand side of part (i) of the theorem is a product of integers which are at most
2n. (cid:3)
Example 4. The requirement that χ is Q-valued in Corollary 1 is necessary. If G =
γ
(cid:104)g|gm =1(cid:105) is a cyclic group and γ is given by
(cid:18)e2πi/m 0 (cid:19)
g (cid:55)→
0 e−2πi/m
then K(γ)∼=Z/mZ (see Appendix B.1). In this case |K(γ)| may have prime divisors up to
m which are larger than 2n=4.
Corollary 2. For any faithful n-dimensional representation γ of G
(cid:81)(cid:96) (n−χ (c ))
i=1 γ i
|G|
lies in Z.
Proof. By Theorem 1 (i), this is, up to sign, the order of K(γ). (cid:3)
Theorem 1 (i) gives us the order of the abelian group K(γ) and part (ii) gives some
restrictions on the structure of this group, however this is not enough to uniquely specify
K(γ) in general. Below we analyse the Smith normal form of tI −C(cid:101) over Q[t] which will
allowustoexplicitlydeterminethecriticalgroupcorrespondingtotheirreduciblereflection
representation of S in Section 4.
n
10 CHRISTIANGAETZ
Proposition 7. Let G be a nontrivial finite group whose irreducible characters are Z-valued
(for example, a Weyl group) and let γ be a faithful complex representation of G. Let m(λ)
denote the multiplicity of an eigenvalue λ of C(cid:101). Then, up to multiplication of the rows by
nonzero rational numbers, the Smith normal form over Q[t] of tI −C(cid:101) is diag(α(cid:96)+1,...,α1)
where
(cid:89)
α = (t−λ)
i
λ
m(λ)≥i
In addition, α =1 and t|α if and only if i=1.
(cid:96)+1 i
Proof. Recall that two matrices A,B with entries in a common field F are similar (over F)
if and only if tI−A and tI−B have the same Smith normal form over F[t]. Let T be the
character table of G, the ((cid:96)+1)×((cid:96)+1)-matrix whose rows are the irreducible character
values. By Proposition 4, T is the matrix of eigenvectors for C(cid:101), therefore D := T−1C(cid:101)T is
the diagonalization of C(cid:101). Since T has integer entries we know T−1 has rational entries; this
implies that tI−C(cid:101) and tI−D have the same Smith normal form over Q[t].
Now, tI −D =diag(t−λ ,...,t−λ ). Since Q[t] is a Euclidean domain, 2×2 principal
0 (cid:96)
minors can be transformed
(cid:18) (cid:19) (cid:18) (cid:19)
f(t) 0 gcd(f,g) 0
→
0 g(t) 0 lcm(f,g)
usingelementaryrowoperationswithoutaffectingtherestofthematrix. Thisisessentially
the same algorithm used for computing integer Smith normal forms. Repeated applications
of these local transformations gives the desired Smith normal form.
Finally,sinceγ isfaithful,χγ takesonatleasttwodistinctvalues,andthereforeC(cid:101) hasat
leasttwodistincteigenvalues. Thus{λ|m(λ)=(cid:96)+1}isempty,soα =1. ByProposition
(cid:96)+1
4, m(0)=1 so t|α if and only if i=1. (cid:3)
i
Corollary 3. In the context of Proposition 7, if in addition tI −C(cid:101) has a Smith normal
form 1 over Z[t] then
i. The Smith normal form over Z[t] is diag(α ,...,α ) up to multiplication of each
(cid:96)+1 1
α by ±1.
i
ii. The critical group is
(cid:96)+1
K(γ)∼=(cid:77)(Z/|α (0)|Z)
i
i=2
Proof. Let S =diag(s(cid:96)+1,...,s1) be the Smith normal form of tI−C(cid:101) over Z[t]. It is a basic
factaboutSmithnormalformsthatdetS isequaltodet(tI−C(cid:101))uptoaninvertiblemultiple
in Z[t], that is, up to ±1. Since det(tI−C(cid:101)) is monic, so must be det(S), and therefore each
of the si is monic. Now, S is also a Smith normal form for tI −C(cid:101) over Q[t], therefore the
s agree with the α up to nonzero rational multiple. Since both are monic, we see that in
i i
fact s =±α .
i i
For (ii), evaluating at t=0 gives that diag(α (0),...,α (0)) is the Smith normal form
(cid:96)+1 1
for C(cid:101). By the proposition, αi(0)=0 if and only if i=1, which gives the desired result. (cid:3)
1Thisisnotguaranteed,sinceZ[t]isnotaPID.