Proposition 1.1 W(P) is independent of
k1, k2,..., km,
and invariant under the action of the symmetric group Sm on the
set {a1, a2,..., am}, given by
pai = api,
p Î Sm.
The vertex
given by l=0 in (1.1) is called the anchor of P. It is
obviously invariant under the action of Sm. Let us now assume that
the integers a1, a2,..., am
are all distinct, and let
p Î Sm
act on {a1, a2,..., am}
without fixed point. Then no other vertex of P is invariant under
p.
The permutation p.
maps P to another m-hppd P*
which shares with P just their common anchor. Thus we obtain, by
equating W(P) with W(P*),
a rule declaring that, omitting the
anchor,
( |
Õ |
even vertices
of P)( |
Õ |
odd vertices of
P*) = |
|
( |
Õ |
odd vertices
of P)( |
Õ |
odd vertices of
P*) (1.4) |
|
Notice that there are 2m-1 vertices on each side of (1.4). These
may be described as the 2m+1-2 vertices of an m-hyperstar (of
David), so (1.4) enunciates an m-Hyperstar of David
Theorem.
Suppose instead that the permutation p
leaves j elements of the set
{a1, a2,..., am}
fixed, j ³ 1. Then P and P*
share 2j common vertices (including the common anchor). Of these
vertices 2j-1 are even and 2j-1 are odd. Thus, when we
cancel out the common vertices, we get a statement like (1.4), but with
2m-2j vertices on each side.
In fact, this statement is compounded of
2j cases of (1.4), each exhibiting an instance of the
(m-j)-Hyperstar of David Theorem.
In fact, we obtain a non-degenerate Hyperstar of David identity precisely
when p
is an m-cycle. When
p
is a composition of cycles on disjoint subsets of
{a1, a2,..., am}
the resulting identity will always
break down into a set of identities in lower dimensions. As an example,
consider the case m=4 and the permutation
p = (12)(34).
Then the Hyperstar of David identity asserts the equality of two products,
each involving 15 multinomial coefficients. However this identity is
really composed of two genuine Star of David identities, each asserting the
equality of two products of 3 coefficients, and a new identity
asserting the equality of two products of 9 coefficients. It is
natural to ask: "What is the geometric, and algebraic, meaning of this last
equality?" We propose to study this type of question in a later article.
So far we have treated n as a distinguished variable in our anchor
formula. It is, of course, evidently possible to treat any of
k1, k2,..., km
as the distinguished variable. To
simplify notation (as much as possible) we will deal explicitly with the case
in which km is the distinguished variable.
We choose positive integers
(a1, a2,..., am-1)
with
a £ n.
Then, for each l,
0£l£m-1
let 0 < il < m, ir < is
if r < s, and
V0(i1,...,il)
= |
æ
ç
ç
ç
ç
è |
|
|
|
ö
÷
÷
÷
÷
ø |
(1.5) |
|
where
so that
|
km |
|
= n- |
m-1
å
i = 1 |
ki¢ = n- |
m-1
å
i = 1 |
ki- |
l
å
j = 1 |
aij = km- |
l
å
j = 1 |
aij (1.7) |
|
We say that V0, is an even vertex if l is even. This, of
course, includes the case where l=0, when V0 is the anchor.
If l is odd, V0 is an odd vertex.
Next let
Va(i1,...,il)
= |
æ
ç
ç
ç
ç
è |
|
|
|
ö
÷
÷
÷
÷
ø |
(1.8) |
|
so that
|
km |
|
= n-a- |
m-1
å
i = 1 |
ki¢ = n-a- |
m-1
å
i = 1 |
ki- |
l
å
j = 1 |
aij = km-a- |
l
å
j= 1 |
aij (1.10) |
|
We say that Va is an even vertex
if l is odd; and Va
is an odd vertex if l is even.
The vertices
V0(i1,...,il),
Va(i1,...,il)
constitute the 2m vertices of an m-hppd P and we
define, as before, the weight of P by the formula
Just as in