Lecture 3
Lecturer: Doron Puder
Preliminary Scriber: Karthik C. S.
Scribe Upgrader:
Lemma 1. If f ∈ R[x] of degree n and g ∈ R [x] of degree n − 1 then f and g interlace if and
only if f + tg is real rooted for all t ∈ R..
We will see the proof of the reverse direction today. The proof is by induction on t. The
case when t = 0 is trivial. If f is real rooted then ri (f + tg) moves monotonically with t. For
all t1 , t2 we can choose T such that t ∈ [−T, T ] and f − T g, f + T g have a common interlacing.
This implies that {f + tg} have a common interlacing. Without loss of generality assume f, g
has positive leading coefficient.
Note that as t tends to −∞, the roots tend to infinity, and in the other direction if t tends
to ∞, the roots tend to β1 .
An important corollary of the above lemma is the following:
Corollary 2. The roots of (f + tg) move monotonically with t (downwards if f, g have positive
leading coefficients)
Lemma 3. If A ∈ Z(n) and v ∈ Cn then Φ(A) and Φ(Atvv ? ) interlace.
Proof. Let roots of Φ(A) be α1 , . . . , αn and roots of Φ(A1,1 ) be β1 , . . . , βn−1 . Without loss of
generality we can assume v = (η, 0, . . . , 0). Note that Φ(Atvv ? ) = Φ(A) + t|η|2 Φ(A1,1 ), and
by Lemma 1, we have Φ(A) and Φ(A1,1 ) interlace, and applying Corollary 2 for t=1 we have
β1 ≥ α1 ≥ β2 ≥ · · · .
Lemma 4. Assume B is a random matrix in H(n) such that EB [Φ(B + A)] is real rooted for
all A ∈ H(n). Then for all A ∈ H(n) and v ∈ Cn we have EB [Φ(B + A)] and EB [Φ(B + tvv)]
interlace.
Proof. Without loss of generality, we can assume that v = (η, 0, . . . , 0). If we look at
?
EB [Φ(B + A0 + tvv
that it is real
)] then we note
rooted for all real t, and by assumption
2
we have that EB Φ(B + A0 ) − t|η| Φ((B + A0 )1,1 ) . Now by linearity of expectation we have:
EB Φ(B + A0 ) − t|η|2 Φ((B + A0 )1,1 ) = EB [Φ(B + A0 )] − EB t|η|2 Φ((B + A0 )1,1 )
and the proof follows by applying Lemma 1.
Finally, we can go to prove the theorem of MSS:
Theorem 5 (MSS15). If q ∈ ∆2 (G) where the distribution of every edge is independent of all
the other edges then Φq is real rooted.
1
(
1 with probability pr
Proof. Let m = |E(G)|. Let j ∈ [m]. We consider the following signing, er =
−1 with probability 1 − pr
Let Ar be the corresponding matrix with entries in the (i, j) and (j, i) coordinates only
where r = (i, j) or (j, i). Note that AG = summ
r=1 Ar .
Consider the following random matrix:
(
In with probability pr
Wr =
In − (~0, ~0, . . . , 2er , ~0 . . . , vec0) with probability 1 − pr
.
So we have that Wr Ar Wr? is Ar with probability pr and −Ar with probability 1 − pr . We
define Φq as follows:
" m
#
X
?
Φq = EW1 ,...Wm Φ(
W r Ar W r ) .
i=1
We will prove that the following is real rooted for all H ∈ H(n) and r ∈ 0 ∪ [m]:
"
#
r
X
?
EW1 ,...Wr Φ(H +
W r Ar W r ) .
i=1
We will prove by induction on r. The case when r = 0 is trivial. Thus, we make the
induction hypothesis and proceed as follows:
"
#
"
#
r
r−1
X
X
EW1 ,...Wm Φ(H +
Wr Ar Wr? ) = pr EW1 ,...Wr−1 Φ(H + Ar +
Wr Ar Wr? )
"
+(1 − pr )EW1 ,...Wr−1 Φ(H − Ar +
i=1
r−1
X
i=1
#
Wr Ar Wr? ) ,
i=1
We observe that Ar + uu? = −Ar + vv ? , where v = (1 1) and u = (1 − 1).
i
h
P
? ) and
Now let α1 ≥ . . . ≥ αn be the roots of EW1 ,...Wr−1 Φ(H + Ar + r−1
W
A
W
r
r
r
i=1
h
i
Pr−1
?
δ1 ≥ . . . ≥ δn be the roots of EW1 ,...Wr−1 Φ(H + Ar + uu + i=1 Wr Ar Wr? ) . By Lemma 4,
we have that δ1 ≥ α1 ≥ hδ2 ≥ · · · . But we know Ar +uui? = −Ar +vv ? . Thus, if β1 ≥ . . . ≥ βn be
P
?
the roots of EW1 ,...Wr−1 Φ(H − Ar + r−1
i=1 Wr Ar Wr ) , by Lemma 4, we have that δ1 ≥ β1 ≥
δ2 ≥ · · · . Also we get that: β1 ≥ α1 ≥ β2 ≥ · · · . Therefore from the interlacing lemma, we
have that the following is a convex combination and thus real-rooted:
"
#
r
X
?
EW1 ,...Wr Φ(H +
W r Ar W r ) .
i=1
Finally, the proof follows, by setting r = m.
1
Introduction to Representation Theory
For a detailed reference to representation theory please see a book by Alperin-Bell or Fulton or
Sagan or the lecture notes by Wigderson on representation theory.
2
.
Moreover, the first explicit expander family due to Murgulis in 1973 used representation
theory properties.
Definition 6. G is a finite group and X is a finite set. An action of G on X, denoted by
G → X is a group homomorphism ρ : G → Sym(X).
We can look at some examples. Let C3 denote the cyclic group on 3 elements. We have
C3 → {0, 1, 2} by addition. This is a group action.
[n]
Another example is Sn →
(all unordered pairs in [n]).
2
Definition 7. A representation of G is a group homomorphism ρ : G → GLn (C).
Note that every action is a representation, as we can map Sym(X) can be mapped to
GLn (C) by just multiplying with a permutation matrix.
Also, sgn is also a representation where sgn : Sn → GL1 (C).
Finally, we can have a representation from S3 → GL2 (C) explicitly given as follows: Let
ω = e2πi/3 .
1
identity →
1
2
ω
(1, 2, 3) →
ω
ω
(1, 3, 2) →
ω2
ω2
(1, 2) →
ω
ω
(1, 3) →
ω2
1
(2, 3) →
1
Definition 8. ρ1 , ρ2 : G → GLn (C) are equivalent if there exists B ∈ GLn (C) such that
ρ2 (g) = Bρ1 (g)B −1 .
Claim 9. Every ρ : G → GLn (C) is equivalent to a unitary representation (ρ : G → U (n)).
Proof. We have that A ∈ U (n) if and only if A? A = I if and only if hAu, Avi = hu, vi, for all
u, v ∈ Cn . Next we define a new inner product on G, denoted by h·, ·iG as follows:
1 X
hu, viG =
hgu, gvi.
|G|
g∈G
However, note that for every g0 ∈ G we have:
1 X
hg0 u, g0 viG =
hgg0 u, gg0 vi
|G|
g∈G
1 X
=
hgu, gvi
|G|
g∈G
= hu, viG
3
Definition 10. Let ρ1 : G → GLm (C) and ρ : G → GLn (C) , then we define ρ1 ⊕ ρ2 : G →
GLm+n (C) as follows:
ρ1 (g)
0
0
ρ2 (g)
Definition 11. ρ : G → GLn (C) is called irreducible if ρ is not equivalent to a direct sum.
Equivalently, ρ is irreducible if there is no non-trivial invertible subspace.
Let us now look at some irreps (irreducible representations). We have that all 1-dimensional
representations are irreducible. The example we saw earlier of the representation S3 → GL2 (C)
is also irreducible as it has ωn on the right. Finally, consider the representation C3 → {0, 1, 2},
where:


1
0→ 1 
1


1
1
1→
1


1

2 → 1
1
Note that this representation is reducible, as we have ρ : C3 → GL2 (C) ∼
= triv ⊕ W ⊕ W 2 .
We conclude by noting three important results in representation theory which we will use
in the coming lectures.
Theorem 12 (Unique decomposition theorem). The decomposition of every representation to
irreducible representation is unique in the sense that the same irreps appear with the same
multiplicity.
Theorem 13. For all finite groups G, there are only finitely many irreps, and moreover, we
have that the number of irreps is equal to the number of conjugacy classes of G.
Theorem 14. If ρ1 , . . . , ρr are the irreps of G such that they are all unitary then
the matrix coef1 P
ficients are an orthonormal basis for {f : G → C}, with respect to hf1 , f2 i = |G|
g∈G f1 (g)f2 (g).
Moreover,
(
1
if i = I, j = J, k = K
h[ρi ]j,k , [ρI ]J,K i = dim (ρi )
.
0 otherwise
As a corollary, we have:
Corollary 15.
X
(dim ρi )2 = |G|.
i
4