Matrices with
Integer Eigenvalues
4 1

1 4
1 0

0 1
 −1 −1

1 0 −1

0 1 −1
3 0 0

0 3 0
0 0 3 
Ron Adin
Bar-Ilan University
radin@math.biu.ac.il
1
A Recent Monthly Paper
Almost All Integer Matrices Have No Integer
Eigenvalues, G. Martin and E. B. Wong,
Amer. Math. Monthly 116 (2009), 588-597.
We want to deal with (specific) exceptions.
2
Highlights
Certain matrices with integer entries have,
conjecturally, integer eigenvalues only.
Partial results are known.
The multiplicity of the zero eigenvalue has
an algebraic interpretation.
3
A Signed Graph: Vertices
[0, r ] := {0,1,… , r}
For k ≥ 0 and 0 ≤ a, b ≤ k + 1 let
Ω k ( a, b) := set of all pairs (U , V ) s.t.
U , V ⊆ [0, k ], U = a, V = b.
Weight: wt (U , V ) := ∑ u + ∑ v
u∈U
v∈V
Ω k ( a, b, w) := set of all (U , V ) ∈ Ω k ( a, b)
with weight w.
4
A Signed Graph: Vertices (cont.)
Example: a = b = 2, w = 4.
k = 0,1: Ω k (2, 2, 4) = ∅
k = 2 : Ω k (2, 2, 4) = {(12, 01), (02, 02), (01,12)}
k = 3 : Ω k (2, 2, 4) = {(03, 01), (12, 01), (02, 02),
(01,12), (01, 03)}
Note: wmin
a b
a b
=   +   , wmax = k (a + b) −   −  
 2 2
 2  2
5
A Signed Graph: Edges
For (U , V ), (U , V ) ∈ Ω k (a, b, w)
and u ∈ U , v ∈ V , z ∈ », write
(U , V ) ∼ (U , V ) if
( u ,v , z )
1. U = (U \ {u}) ∪ {u + z}
2. V = (V \ {v}) ∪ {v − z}
3. u + v ≤ k
6
A Signed Graph: Edges
For (U , V ), (U , V ) ∈ Ω k (a, b, w)
and u ∈ U , v ∈ V , z ∈ », write
(U , V ) ∼ (U , V ) if
( u ,v , z )
1. U = (U \ {u}) ∪ {u + z}
2. V = (V \ {v}) ∪ {v − z}
3. u + v ≤ k
U
u
v V
u+v
U
u+z
v − z V
7
A Signed Graph
Gk ( a, b, w) = graph with vertex set Ω k ( a, b, w)
and edges corresponding to the various
(U , V ) ∼ (U , V ).
( u ,v , z )
Loops and multiple edges may occur.
Attach signs to edges:
8
Signs
For u = (u1 ,… , ua ), u = (u1 ,… , ua ) define:
ε (u , u ) = 0 if u or u has repeated elements;
ε (u , u ) = sign(σ ) sign(τ ) if
uσ (1) < … < uσ ( a ) , uτ (1) < … < uτ ( a ) .
For U = {u1 , … , ua }, U = {u1 ,… , ui + z ,… , ua }
where u1 < … < ua , define
ε (U , U ) := ε ((u1 ,… , ua ), (u1 ,… , ui + z ,… , ua )).
ε ((U , V ), (U , V )) := ε (U , U ) ⋅ ε (V , V )
9
Signed Adjacency Matrix
Tk ( a, b, w) = the (signed) adjacency matrix
of the graph Gk (a, b, w).
Note:
Diagonal elements are nonnegative
integers.
Off-diagonal elements are 0, 1 or −1.
10
Signed Adjacency Matrix
Example: k = 3, a = 2, b = 3, w = 8
4 1

1 4
Tk ( a, b, w) =  1 0

0 1
 −1 −1

1 0 −1 

0 1 −1 
3 0 0

0 3 0
0 0 3 
11
Conjectures
Conjecture: [Hanlon, ’92]
All the eigenvalues of Tk (a, b, w) are
nonnegative integers.
12
Conjectures (cont.)
Let M k ( x, y, λ ) :=
∑ m ( a , b, r ) x
k
a
yλ
b
r
a ,b , r
where mk (a, b, r ) = multiplicity of r as an e.v.
of Tk ( a, b) = ⊕ Tk ( a, b, w).
w
Conjecture: [Hanlon, ’92]
M k ( x, y , λ ) = ∏ (1 + x + y + λ i +1 xy )
k
Still open (in general)!
i =0
13
Background
Macdonald’s root system conjecture
14
Background
Macdonald’s root system conjecture
Hanlon’s property M conjecture:
H * ( L ⊗ »[t ] /(t k +1 )) ≅ H * ( L ) ⊗ ( k +1)
where the Lie algebra L is either
* semisimple, or
* upper triangular (nilpotent), or
* Heisenberg (nilpotent)
15
Background
Macdonald’s root system conjecture
Hanlon’s property M conjecture:
H * ( L ⊗ »[t ] /(t
)) ≅ H * ( L )
where the Lie algebra L is either
k +1
⊗ ( k +1)
* semisimple, or
* upper triangular (nilpotent), or
* Heisenberg (nilpotent)
16
Background
Macdonald’s root system conjecture
TRUE [Cherednik, ’95]
Hanlon’s property M conjecture:
H * ( L ⊗ »[t ] /(t
)) ≅ H * ( L )
where the Lie algebra L is either
k +1
⊗ ( k +1)
* semisimple, or
* upper triangular (nilpotent), or
* Heisenberg (nilpotent)
17
Background
Macdonald’s root system conjecture
TRUE [Cherednik, ’95]
Hanlon’s property M conjecture:
H * ( L ⊗ »[t ] /(t
)) ≅ H * ( L )
where the Lie algebra L is either
k +1
* semisimple, or
* upper triangular (nilpotent), or
* Heisenberg (nilpotent)
⊗ ( k +1)
TRUE [FGT, ’08]
18
Background
Macdonald’s root system conjecture
TRUE [Cherednik, ’95]
Hanlon’s property M conjecture:
H * ( L ⊗ »[t ] /(t
)) ≅ H * ( L )
where the Lie algebra L is either
k +1
* semisimple, or
* upper triangular (nilpotent), or
* Heisenberg (nilpotent)
⊗ ( k +1)
TRUE [FGT, ’08]
FALSE [Kumar, ’99]
19
Background
Macdonald’s root system conjecture
TRUE [Cherednik, ’95]
Hanlon’s property M conjecture:
H * ( L ⊗ »[t ] /(t
)) ≅ H * ( L )
where the Lie algebra L is either
k +1
* semisimple, or
* upper triangular (nilpotent), or
* Heisenberg (nilpotent)
⊗ ( k +1)
TRUE [FGT, ’08]
FALSE [Kumar, ’99]
?
20
Background (cont.)
 0 * *  



For the 3-dim Heisenberg L3 =  0 0 *  ,


with basis{e, f , x},
 0 0 0  


the Laplacian ∂∂* + ∂ *∂
for L3 ⊗ »[t ] /(t k +1 )
is ⊕ Tk ( a, b, w)
(in a suitable basis).
a ,b , w
The property M conjecture is then equivalent to
(an extension of)
M k ( x, y, 0) = (1 + x + y ) k +1
21
Partial Results
[Hanlon, ’92]
Explicit eigenvalues in the stable case:
* a≤b
a b
* w ≤  2 +  2
   
a b
* k ≥ (a − 1) + (b − 1) + w −   −  
 2  2
22
Partial Results (cont.)
Theorem: [Hanlon, ’92]
1:1
In the stable case, Ω k (a, b) ↔ pairs (λ , µ )
of partitions with λ + µ = w − wmin .
The eigenvalues of Tk (a, b) are λ − µ + ab.
The proof uses Schur functions in two sets
of variables.
23
Partial Results (cont.)
[A.-Athanasiadis, ’96]
1. a = 1, b = 2, ∀k : The nonzero eigenvalues
of ⊕ w Tk (a, b, w) are λ = 1,… , k + 1,
each with multiplicity k (and explicit
distribution over w).
2. a = 1, ∀b, k : The multiplicity of the zero e.v.:
k +1 

k 
mk (1, b, 0) = 
 = ( k + 1)  
1 b k − b 
b
24
Partial Results (cont.)
[Hanlon-Wachs, ’02]
Extend result 2 above to ∀a, b, k :
k +1


mk ( a, b, 0) = 

 a b k +1− a − b 
25
Partial Results (cont.)
[Kuflik, ’06]
Distribution over w (∀a, b, k ) :
mk ( a, b, w, 0) = coefficient of q w− wmin in
where
k +1


a b k + 1 − a − b

q
[ m]!q := [1]q ⋅ [2]q [ m]q ,
[ m]q := 1 + q + … + q m −1.
26
‫תודה על ההקשבה!‬
‫‪27‬‬