arXiv:1210.1446v1 [math.MG] 2 Oct 2012The Sum of Squares Law
Julio Kovacs∗, Fang Fang, Garrett Sadler, and Klee Irwin†
Quantum Gravity Research, Topanga, CA, U.S.
27 September 2012
Abstract
We show that when projecting an edge-transitiveN -dimensional polytope onto anM -dimensional
subspace of RN , the sums of the squares of the original and projected edges are in the ratio N/M .
Statement
Let X ⊂ RN a set of points that determines an N -dimensional polytope. Let E denote the number of
its edges, and σ the sum of the squares of the edge lengths. Let S be an M -dimensional subspace of
R
N , and σ′ the sum of the squares of the lengths of the projections, onto S, of the edges of X.
Let G be the group of proper symmetries of the polytope X (that is, no reflections). If G acts
transitively on the set of edges of X, then:
σ′ = σ ·
M
N
.
The orthogonality relations
The basic result used in our proof is the so-called orthogonality relations in the context of representations
of groups. The form of these relations that we need is the following:
Theorem 1. Let Γ : G → V ×V be an irreducible unitary representation of a finite group G. Denoting
by Γ(R)nm the matrix elements of the linear map Γ(R) with respect to an orthonormal basis of V , we
have:
|G|
∑
R∈G
Γ(R)∗nmΓ(R)n′m′ = δnn′δmm′
|G|
dimV
,
(1)
where the ∗ denotes complex conjugation.
A proof of these relation can be found in standard books on representation theory, for instance [1, p.
79] or [2, p. 14]. See also theWikipedia article http://en.wikipedia.org/wiki/Schur_orthogonality_relations.
∗Corresponding author. Email: julio@quantumgravityresearch.org
†Group leader. Email: klee@quantumgravityresearch.org
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Proof of the sum of squares law
The idea is apply the orthogonality relations (1) to the group G of proper symmetries of the polytope
X, considering its standard representation on the space RN (i.e., R · x = R(x)). This representation
is clearly unitary, since the elements of the group are rotations and hence orthogonal transformations.
Also, the representation is irreducible, since G takes a given edge to all the other edges, which do not
lie on any proper subspace due to the assumption of X being an N -dimensional polytope.
We can assume that the edge lengths of X are all equal to 1. Let {v1, . . . , vN} be an orthonormal
basis for RN such that v1 coincides with the direction of one of the edges e of X. Then, for R ∈ G, let
Γ(R) be the matrix of R in that basis, that is:
R(vj) =
N
∑
i=1
Γ(R)ijvi.
Since this is an orthonormal basis, we have:
Γ(R)ij = 〈R(vj), vi〉,
where 〈, 〉 denotes the standard inner product in RN . In particular, for j = 1:
Γ(R)i1 = 〈R(e), vi〉
(i = 1, . . . , N).
(2)
Note that this is exactly the length of the projection of each edge onto the line spanned by vi. Now,
from equation (1), by putting n′ = n and m′ = m, we get:
∑
R∈G
|Γ(R)nm|
2 =
|G|
N
.
(3)
Using the Γs given by the previous equation:
∑
R∈G
〈R(e), vi〉
2 =
|G|
N
(i = 1, . . . , N).
(4)
Now let v be any unit vector. We’ll show that the above equality holds for v as it does for vi. To
see this, write v as a linear combination of the basis vectors vi: v = sumiaivi. Since ‖v‖ = 1, we have
∑
a2i = 1. Then:
∑
R
〈R(e), v〉2 =
∑
R
〈R(e),
∑
i
aivi〉
2 =
∑
R
(
∑
i
ai〈R(e), vi〉
)
2
=
∑
R
(
∑
i
a2i 〈R(e), vi〉
2 + 2
∑
iaiaj〈R(e), vi〉〈R(e), vj〉
)
=
∑
i
a2i
∑
R
〈R(e), vi〉
2 + 2
∑
iaiaj
∑
R
〈R(e), vi〉〈R(e), vj〉
=
|G|
N
+ 2
∑
iaiaj
∑
R
Γ(R)i1Γ(R)j1,
due to eqs. (4) and (2). Now it turns out that the second term is 0. This is an immediate consequence
of eq. (1) with n = i, n′ = j, m = m′ = 1. Therefore, the equality:
∑
R
〈R(e), v〉2 =
|G|
N
(5)
2
holds for any unit vector v.
Now let S be the projection subspace of dimension M > 1, and let’s denote by PS : R
N → S the
projection operator. Choose an orthonormal basis {u1, . . . , uM} of S. Then:
PS(R(e)) =
M
∑
i=1
biui,
with
bi = 〈PS(R(e)), ui〉 = 〈R(e), ui〉.
Therefore,
∑
R∈G
‖PS(R(e))‖
2 =
∑
R
∑
i
b2i =
∑
R
∑
i
〈R(e), ui〉
2 =
M
∑
i=1
(
∑
R
〈R(e), ui〉
2
)
= |G| ·
M
N
,
where the last equality is because of eq. (5).
To obtain the required result, we observe that G can be partitioned in E “cosets” of the same
cardinality k, where E is the number of edges of X. To see this, let H = {g ∈ G | g · e = e} be the
subgroup of G that leaves edge e invariant. Then the coset RH = {g ∈ G | g · e = R(e)} is the subset
of elements of G that send edge e to edge R(e). Denote the cardinality of H by k. Since there are E
edges and the action is edge-transitive, there are E cosets, each of cardinality k. Therefore, |G| = kE.
Denoting the edges by e1, . . . , eE , and the corresponding cosets by C1, . . . , CE (so that R(e) = el for
R ∈ Cl), we have:
∑
R∈G
‖PS(R(e))‖
2 =
∑
R∈∪E
l=1
Cl
‖PS(R(e))‖
2 =
E
∑
l=1
∑
R∈Cl
‖PS(R(e))‖
2
=
E
∑
l=1
∑
R∈Cl
‖PS(el)‖
2 =
E
∑
l=1
k‖PS(el)‖
2 = k
E
∑
l=1
‖PS(el)‖
2.
On the other hand, we saw that the left-hand side of this equation equals |G| · M/N , which is
kE ·M/N . Equating this to the above and canceling the factor k, we obtain:
σ′ =
E
∑
l=1
‖PS(el)‖
2 = E ·
M
N
= σ ·
M
N
,
which completes the proof.
References
[1] T. Bröcker and T. tom Dieck. Representations of Compact Lie Groups. Springer-Verlag, New York,
1985.
[2] J.-P. Serre. Linear Representations of Finite Groups. Springer-Verlag, New York, 1977.
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