Download Dowling, T.A.; (1972)A class of geometric lattices based on finite groups."

1. This research was supported in part by the Air Force Office of Scientific
Research under Contract AFOSR-68-l4l5.
2. Present address: Department of Mathematics, Ohio State University,
Columbus, Ohio 43210.
A CLASS OF GEOMETRIC LATTICES
BASED ON FINITE GROUPS
by
T. A. Dowling l ,2
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 825
May, 1972
A CLASS OF GEOMETRIC LATTICES BASED ON FINITE GROUPS
T. A. Om'll i ng 1,2
ABSTRACT
For any finite group
tice
Let
Q (G)
of rank
n
P +
n,
n
a finite geometric 1at-
the lattice of partial G-partitions, is constructed.
be the lattice of partitions of an (n+l)-set.
n 1
jection
G and positive integer
~:
Qn(G)
P + ,
n l
+
and an injection
preserves order and rank.
to an isomorphism,
When
Q (G)
n
Qn+1(G) ,
G is the trivial group,
1f
Q (G)
is established, implying that
n
n
~
3
if and only if
subgroup of the multiplicative group of the field.
representable over no field iff
Q (G)
n
Q (G)
n
reduces
is super-
m
=1
or 2,
q
iff
q-l,
n
Consequently,
G is noncyc1ic, and if
m divides
Q (G)
is
G is isomorphic to a
is representable over (a) every field iff
finite field of order
fields iff
-1
It is further shown that nonisomorphic groups give nonisomorphic
representable over a field when
then
= \
The existence of a Boolean sub1attice of mod-
lattices, and the representation problem is solved completely:
ro,
each of which
are determined, and Stirling-like identities for
the Whitney numbers obtained.
solvable.
+
The interval structure, MBbius function, and character-
istic polynomial of
u1ar elements in
\: P +
n 1
There exists a sur-
Q
n
(G)
G is cyclic of order
m· 1,
(b) a
(c) the rational or real
and (d) the complex field for all
is
m.
1.
The set
P
n
INTRODUCTION
of all partitions of an n-element set, when ordered by re-
finement, is a well-known geometric lattice enjoying a number of structural
properties.
Every upper interval of a partition lattice is a partition lat-
tice, and in general, every interval is a direct product of partition lattices.
The Whitney numbers of the partition lattices are the familiar Stirling numbers, and the characteristic polynomial is simply a descending factorial,
hence all its roots are integers.
The set of partitions with a single non-
trivial block is a Boolean sublattice of modular elements, so the partition lattice is supersolvable in the sense of Stanley [10].
Because of these and other
structural properties, the partition lattices occupy a middle ground between
the highly-structured projective (connected modular) geometric lattices and
arbitrary geometric lattices, thereby exhibiting some of the consequences of
the departure from modularity while retaining sufficient structure to facilitate their study and test conjectures.
We describe in this paper for any finite group
G a class of finite
geometric lattices, here called the partial G-partition lattices, which share
a number of the properties of the partition lattices.
Following a review in
Section 2 of preliminary results on ordered sets and geometric lattices, the
lattice
rank
Q (G)
n
of partial G-partitions of an n-set, a geometric lattice of
is defined and its structure investigated in Section 3.
n,
existence of a surjective map
Qn(G),
Qn(G)
~
P +
n l
and an injective map
both of which preserve order and rank, is demonstrated.
tion embeds
Pn+
l
in
Qn (G)
P +
n l
~
The injec-
both as a sublattice and a subgeometry, and both
maps reduce to isomorphisms when
G is the trivial group of one element.
nature of covers and the interval structure in
Section 3.
There the
Q (G)
n
is also examined in
In Section 4, we prove the existence of a Boolean sublattice of
The
2
modular elements in
Qn(G) ,
implying its supersolvability, determine its
M6bius function and characteristic polynomial, and show that the \fuitney numbers of the partial G-partition lattices satisfy recursions and inverse relations analogous to those of the Stirling numbers.
primarily to the representation problem of
Qn(G) ,
Section 5 is devoted
following a description
of the structure of the rank three (planar) geometries
Q3(G)
and a proof that
nonisomorphic groups of the same order result in nonisomorphic lattices with
the same Whitney numbers.
over a field
F
group of
As a result,
F.
G is cyclic.
iff
We show that when
n
~
3,
Q (G)
n
is representable
G is isomorphic to a subgroup of the multiplicative
Q (G)
n
is not representable over any field unless
Thus simply by taking
G noncyclic, we obtain an infinite
class of moderately-structured geometric lattices which are not subgeometries
of any projective geometry.
The results of this paper generalize to arbitrary finite groups many of
the results in our earlier paper [5], which in the present context dealt with
the case where
G is the multiplicative group of a given finite field.
Theorem 6 and the specializations of Theorems 1-5, 7 and 10 to that case appear in [5]; Theorems 8, 9, and 11 have no counterpart there.
Although most of the extensions of the results in [5] to an arbitrary
finite group are straightforward, we include them here not only to make the
present paper self-contained, but also because of differences in notation,
terminologys and definitions required for the general case.
2.
PRELIMINARIES
We collect in this section a number of results and definitions required
later.
For further details the reader is referred to [2,7].
3
A preordered set
(PI, s)
transitive relation, written
simply
pI
(PI, s).
for
antisymmetric.
is a set
x
y.
S
together with a reflexive,
When the order is implicit, we write
is an (partially) ordered set if
pI
Every preordered set
pI
P of E-classes of
the equivalence relation
x
if
Ln (F)
P
x E y
is a finite set.
the set
P
An intel'Va l
{zlx s z S y}
P.
x S y
in
y > x
and
[x,y]
(u,v) s (x,y)
of an ordered set
P
u
An element
P
y
[x,y] ... {x,y}.
= {xo,xl' ..• ,x n }
in
An ordered set
S
x
in
P and
[x, y] ,
is a cover of
x
P
P.
P, Q is
v s y
C
[x,y] ...
= {xO,x l '···}
P: Xo < Xl < ••••
Suppose
If
of
C is
P has finite
P has finite height and
Then C is a maximal chain in
covers
in
defined whenever
C is one less than its cardinality.
and
is
(or covers x) iff
A finite or countable subset
is a chain in
iff
E is
is the ordered subset
P restricted to
height if all chains in P are finite.
for all
i
= 1, ... ,n.
[x,y]
An ordered
P satisfies the chain condition if it has finite height and all maximal
set
chains in any interval
o
iff
with the order of
finite, the length of
where
F an infinite field, all ordered sets
is a chain if it is totally ordered in
P
C
y, Y s x.
S
p' ,
The direct product of two ordered sets
Q with order
x
is also
With the exception of the projective lattices
in Section 5, in the case where
considered here are finite.
Q.
iff
s
is canonically associated with an
ordered set, namely the quotient set
finite
(0
~
x
for all
of an element
XEP
a unit element
1
Let
coatom of
[x,y]
If
is the length of all maximal chains in
(x s 1
for all
XEP
P have a 0 and 1.
P
have the same length.
P has a zero element
and satisfies the chain condition, the rank
x E P)
the corank of an element
~
pI
XEP)
[O,x].
If
P has
and satisfies the chain condition,
is the length of all maximal chains in
An atom of
is an element covered by
p(x)
P
1.
is an element
P
cove~ing
[x,l].
O.
is a lattice iff any two
A
4
e
elements
x, y
have a unique minimal upper bound
x v y,
supremum, and a unique maximal lower bound x /\ y,
subset
M of a lattice
the order of
If
when
~
P, L are ordered sets, a function
x
S
y
~
~(x)
implies
is rank-preserving if
~-l
and
A lattice
~
L,
~(y).
s
p(~(x»
If both
= p(x)
M is a lattice under
P
~
L
L.
is order-preserving
P, Q satisfy the chain condition,
x€P.
~:
L are iso-
P and
P
~
L such that both
is oompZete if every subset has a supremum and infimum,
L
L is semimodutar if
covers both
x /\ y.
x v y
is the supremum of the set of atoms in
x
and
y
whenever
x
and
[O,x].
y
A geometrio Zattioe is a complete, atomic, semimodular lat-
tice of finite height.
v p
~:
iff there is a bijection
x
=x
iff
for all
and atomio if every element
y
P
A
preserve or d er.
'i'
cover
called their infimum.
P and suprema and infima in M agree with those of
marphio, written P
'i'
is a sublattioe of
P
called their
A finite lattice is geometric when
for some atom
p
~
x.
A geometric lattice
y
covers
x
iff
L satisfies the chain
condition and its rank function obeys the semimoduZar inequality:!
p(xvy) + p(x/\y) S p(x) + p(y)
for all
x,y € L.
Elements of rank I (atoms),
2, 3 are points, lines, planes, respectively, and elements of corank I (coatoms), 2, 3, are copoints, colines, coplanes, respectively.
Every interval
of a geometric lattice is geometric.
A oombinatorial geometry is a set
opera tor
A f+ A on subsets of
S satisfying (a) the exohange property:
p,q € S,
A £ S,
but
basis property:
AO = A,
subset
~
S of "points" together with a closure
and
if
q € Aup
q
~
A,
then
p € Auq,
A £ S there exists a finite subset
A of
otherwise.
S
is independent if
A-p f A for all
All maximal independent subsets of any set
(b) the finite
A of
O
and (c) the empty set and all singleton subsets of
pEA,
if
A such that
S are closed.
A
and dependent
A, called bases of
A,
5
have the same cardinality, the rank of
~
geometry on
subset of
S
T
is a subset
T
of
A subgeometry of a combinatorial
A.
S with closure operator
is independent in the subgeometry on
T
A
~A
n T.
A
iff it is independent
in the original geometry.
The set of closed sets of a combinatorial geometry, ordered by inclusion,
is a geometric lattice.
geometry on its set
geometry on
T
some subset
A of
S
Conversely, every geometric lattice
A= {pip
of points by
consists of all elements
T.
~
x€L
sup A}.
L defines a
The lattice of the sub-
such that
x = sup A for::-:
It is not in general a sublattice of
L.
We shall
A minor of
identify a geometry with its (geometric) lattice of closed sets.
L
is a subgeometry of some interval of
If
a: P
+
P, L
are geometric lattices, an injective strong map is an injection
L which takes points to points and preserves suprema:
a(x) v a(y).
In this case
a subgeometry of
F
L.
P
A projective geometry of dimension
A representation of a rank
a: P
+
L (F).
n
n
n.
geometry
Equivalently,
P
~(A)
image
P,
P
over
P
A of
is representable over
X be a finite set of
n
n
F .
~:
S
If
partitions of
~
are the blocks of
1T.
P
=
the latter
over a field
L (F).
n
is an injective strong
S
+
F
n
F,
iff there
where
is independent in
S
P
is the
iff its
is representable over
F,
F.
elements.
A partition of
of disjoint, nonempty subsets of
subsets
F
is representable over
is linearly independent in
every minor of
Let
such that a subset
n-l
L,
We denote its lattice by
exists an injection (called a coordinatization)
point set of
a(xvy)
is isomorphic to its a-image in
is a combinatorial geometry of rank
map
L.
X with
X is a set
The
There is an obvious correspondence between
X and equivalence relations defined on
partition being the equivalence classes.
X,
the blocks of the
6
The set
ment:
1T
S;
l
P
1T
of all partitions of
n
X
is (partially) ordered by refineSo ordered,
P
with zero element the partition of
X
iff every 1T -block is a union of 1T -blocks.
Z
l
Z
is a geometric lattice of rank
n-l,
n
into singleton subsets (the identity relation) and unit element the single
block partition
{X}
(the universal relation).
The supremum and infimum of two partitions
1T
Z
= {Bl, ••• ,B }
s
A
j
(j = 1, .•• ,r)
with vertices
{Aj,Bk }
A.
(l
J
all
A.
J
1T
Z
Bk(k = 1, ••• ,5).
versus the
A , ••• ,A , B , •.• , Bs'
1
1
r
J
B :F (/J.
k
for any edge
{Aj'BkL
A.
(l
This is the bipartite graph
and edges the set of all pairs
Then a block of
A block of
1T
1
1T
v 1T
1
Z
"
1T
Z is a subset
is a union
over
UA.
J
in a connected component of the graph.
The lattice
n-l.
= {A , •.• ,A },
r
1
1
is easily found by means of the intersection graph of the
such that
B ,
k
1T
P
n
of partitions of
The rank function is
can be obtained from
3.
Let
X is a geometric lattice of rank
p(1T) = n-I1TI.
1T
1T
Z covers
1T
iff
1
by replacing two 1T -b1ocks by their union.
l
1
THE LATTICE
X = {xl' .•. ,x }
n
A partition
Qn(G}
OF PARTIAL G-PARTITIONS
be a finite set of
n
elements.
By a partiaZ
partition of X we shall mean a set a = {A1 , ..• ,Ar } of disjoint, nonempty
subsets of
X,
i.e. a partition of a subset
are the bZocks of
a.
(partially) ordered by
The set
a s; a
a-blocks, i.e. iff for each
r
Uj=lA j
of
X.
The subsets
Qn of all partial partitions of
J
X is
iff every a-block is the union of a set of
BkEa
there exists a nonempty subset
a
k
of
B = U A.• So ordered, Q is isomorphic to the lattice P +
n
n l
k
ak J
of partitions of an (n+1)-set X u {x }' the isomorphism P + -+ Q is
n l
o
n
such that
e
A.
given simply by deleting from each partition
{A u{x },A ,···,A }
O O 1
r
of
a
7
xu
{x }
O
the distinguished block
A u {x }
O
O
the block of any partition of
of the partition.
(3.l)
4>
Formally, we define the inverse map
J=
of all completions of
has Q -rank
a.
n
=
p{a.)
The partition
J
to a partition of
a.
as the zero bZock
$: Q + P +
n
£
=
p{a.)
= {El, .•• ,En }
X u {x }.
O
(n+l)-{r+l)
= n-r,
Q.
n
Qn
Q.
Covers in
Q
n
are of two types.
tained by deleting some block
block of
a.
E
i
= {xi}
a.
is a partial
is
n
replacing two blocks
Aj ,
~
from
A.
J
of
A I-cover of
a.
a.,
a.
= {Al, ••• ,Ar }
is ob-
while a II-cover is obtained by
by their union
is obtained simply by combining some
A u ~.
j
A.
J
€
a.
The <j>-image of
with the zero
$<0.) •
Now let
Elements of
G be a finite (multiplicative) group, with unit element 1.
G will be denoted
K,
A,
~,
•••
with or without subscripts.
We
define a partiaZ G-partition of X as a set
(3.3)
e
of functions into
sets of
X.
Thus
$
and the partial
Every subset of a partial partition
in
a I-cover of
Note that
i.e.
X into its singleton subsets
of
a.
2:
by
10.1.
n -
the zero element of
partition
n l
is the supremum
</>{a.)
Thus the empty partial partition is the unit element of
partition
We refer to
X to an (r+l)-block partition of
takes an r-block partial partition of
(3.2)
o.
=
(a.)
O
P +
n l
x
'l1hich contains
r
A = X-U. lA ..
in
containing
G for which the domains
are disjoint, nonempty sub-
8
(3.4)
1T(a)
=
is a partial partition of
X.
partitions of
X.
The map
Let
denote the set of all partial G-
Q'(G)
n
1T: Q'(G)
-+
n
defined by (3.4), takes each par-
Q ,
n
X to its underlying partial partition of
tial G-partition of
X.
To simplify expressions encountered below, we adopt the convention that
the domains of functions
are always denoted by the capitals
A , B , C , etc. of the letters denoting the functions, with appropriate subj
k
t
scripts.
written simply
If
a
k
is a partial G-partition of
J
to
n
J
= {a.lj = l, •.. ,r}
be a function
b k : Bk
G,
-+
for all
Bk
where
A.a.
ing it; no addition operation in
b
k
= La
by
ei(x )
i
function
=1
for each
a : A
j
j
-+
i
~
union\!
The summation sign
of the functions follow-
Let
E
e : E
i
i
= {eili = l, ..•• n}.
-+
G
Then any
G may be written as a linear combination
The analogue in
a
A.a •
J j
and define the unit functions
= l, ... ,n.
of the unit functions, where
lowing:
k
G is assumed.
= l, ••. ,n,
i
will
k
of
J J
is to be interpreted as the "domain-disjoint
= {xi}'
is
such that the restriction of
UakA j ,
=
In this case we write
Ei
and
X.
a (left-) linear combination (over G) of a
a,
is a (left-) G-multiple
Let
given by (3.3) may be
Q'(G)
E
a = {a.lj = l, •.• ,r}.
any non-empty subset of
b
a
Thus for example the element
= {e.IE.
cA.}
1.
1.J
E.
J
and
K
i
of the order relation of
Q' (G)
n
Qn
=
is then the fol-
B iff every S-function is a linear combination of a set of a-
functions. i. e. , iff for each
b
k
e: 13
there exists a subset
a
k
of
a
and
= La
A.a j • The relation :;; is clearly rek J
flexive and transitive, hence is a preorder on Q' (G) • Suppose a :;; B. Then
n
elements
A. e: G such that
for each
bk e: 13,
1T(f3)
in
J
Q.
n
bk
Thus if
=L
ak
bk
Aja j ,
a:;; a',
so
a'
~
Bk
a,
= ~a
then
A.•
k
J
1T(a)
It follows that
= 1T(a')
1T(a):;;
and there exists
9
a bijection
aj
+-l-
aj
equivalence relation:
E-class containing
such that
aj
aEa'
0.:50.'.0.' :5
a.
iff
Any member
= Ajar
a'
a.
J
= A-1
j aj'
a.
and let
of an E-class
mined up to scalar multiples of its elements.
Let
(a)
E denote this
(a)
denote the
is uniquely deter-
The situation is analogous to
that of a set of homogeneous coordinate vectors for some set of points in a
projective geometry.
also denoted
The preorder
on
:5
on the quotient set
:5,
Q'(G)
n
induces a partial order,
= Q~(G)/E
Qn(G)
of E-c1asses in the
usual way:
iff
We will be concerned primarily with the ordered set
proofs will often be given in terms of
Q (G)
n
henceforth, but
and its preorder, with
Q' (G)
n
E-equivalence replacing equality.
Any func tion
as a function on
the f-image of
preserving:
(a)
f
on
Q' (G)
To avoid double parentheses, we write
Q (G).
n
(a)
€
:5
(8)
which is constant on E-classes will be taken
n
Then from the above,
Q (G).
n
n(a)
implies
Given any nonempty subset
define as the function
j
set of indicator functions
x.
G with
~
J
is order-
the indicator function of
lA. (xi)
= l, ••. ,r}
{lA \j
Q
n
~
J
=1
for all
xi
= l, •.. ,r}
of
X will be
we
The
€
of a partion partition
j
{Ajlj
for
n (8) •
:5
of
lA: A.
n: Q (G)
n
f (a)
called the indicator set of
{A Ij
j
= 1, ... ,r}.
The E-class of the indicator set thus consists of all partial G-partitions of
constant functions with n-image
{Ajlj
= l, .•. ,r}.
The map
clearly injective and order-preserving, as disjoint unions
I lA .•
J
surjective.
The composite
If
G
=1
'11'°1:
Qn
~
Q
n
~
Q
n
(G)
X the E-class of its indicator set is
assigning to each partial partition of
sums
1:
Qn
is the identity on
UA j
Qn'
correspond to
so
is the trivial group, the l-image of a partial
'11'
is
10
partition is its only n-preimage, so
~·n
is an isomorphism
As in
Qn'
~
Qn(l)
covers in
obtained by deleting some
are of two types.
from
j
tained by replacing two functions
a j + Aa
k
is an isomorphism
a,
a , a
j
k
either case, the covering element
of
a
(8)
a.
lsi =
has
(a)
(a)
is
is ob-
may be taken as 1).
J
la 1-1.
Qn (G)
Thus
all maximal chains in any interval
have the same length
The zero element of
where
e:
= {eili = l, •.• ,n}
is the set of unit functions.
Q (G)
Qn' hence
by a linear combination
Qn (G)
of
~
A I-cover of
while a II-cover of
(by E-equivalence, the coefficient of
isfies the chain condition:
Qn(l)
Pn .
Qn(G)
a
n
In
sat-
[(a),(8)]
Q (G)
is
n
of
(e:) ,
The rank function
is therefore
n
(3.5)
Note that (3.5) is also the rank (3.2) of
rank.
Clearly,
does also.
1
n(a)
in
Qn'
so
preserves
We summarize these results in
THEOREM 1: Let X = {xl""'xn } be a finite set of n elements, G a
P +
n l
finite multiplicative group,
Qn
~
P +
n l
the lattice of partitions of
the lattice of partial partitions of
set of E-classes of partial G-partitions of
(a)
13
An element
(8')
in
Q (G)
n
X.
X,
S
=
a - {a },
j
covers
l3
=
a - {aj,ak } U {aj+Aak },
(I-cover)
or
(3.7)
e
where
aj,a k
€
a,
A
€
G.
Qn (G) the ordered
Then
of the form
(3.6)
and
X u {x },
O
(II-cover)
(8')
=
(8)
for some
11
(b)
Q
n
satisfies the chain condition, with rank function
(G)
= n - lal.
p(a)
(c)
~.
The map
. Qn (G)
+
Qn'
which assigns to each E-class of partial
G-partitions its underlying partial partition, is surjective and preserves
order and rank.
(d)
The map
Q
1:
n
Q (G),
n
+
which assigns to each partial partition the
E-class of its indicator set, is injective and preserves order and rank.
(e)
~
Qn(l)
If
Qn'
G
so
=1
is the trivial group,
~.~
=
~
Qn(l)
is an isomorphism
1
~
-1
is an isomorphism
Pn+ l ,
where
~:
Qn + Pn+l
the isomorphism (3.1).
COROLLARY 1.1:
Each element of rank
( r)
1
elements of rank
n-r+l,
COROLLARY 1.2:
where
m
Qn (G)
in
is covered by
(r)
2
m is the order of
The atoms of
=
+
n-r
Q (G)
n
are
G.
(a(i», (a(ii')(A»,
where
{e.},
E: -
l.
=
defined for all
(a(i'i)(A- l
1
~ i,
i'
~ n,
i
~ iV,
A
E
G.
Note that
(a(ii')(A»
=
».
Our next theorem describes the structure of upper and lower intervals of
Q (G).
n
From these the structure of an arbitrary interval can be obtained.
THEOREM 2:
(a)
If
(a)
E
Q (G)
n
[(a),l]
is of corank
Qr (G).
r,
then
is
12
(~) €
Let
(b)
Qn(G),
where
~
[0, (8»)
Proof:
each
b
k
S=
in
bk = La.kAjaj
{b
k
J
: B
k
Glk
+
of a subset
= l, ••• ,s}
s
Then
iff
is a linear combination
b~a.): {AjlAj S Bk }
namely
b
(a. )
(A )
k
j
= Aj
+
.
b
k
corresponds
G on a nonempty subIn particular, the
are the unit functions of
j
{Al, .•. ,A }. This correspondence preserves
r
(\A b )(a) = \A b(a.)
so the map (~) + (S(a», where
l.kk
l.kk'
linear combinations:
= {b (a.) Ib
k
(b)
k
€8}
is an isomorphism
For any
~ ~
tition
TI(S)
(a.)
If
(a.)
we have
[0, (S») ,
€
~
[(a.),l]
TI(a.)
[O,(~)]
€
and
TI(a.)
= ~,
Qr(G).
Every partial par-
TI(8).
~
~k = {~j Ij =l, ... ,r k } of
consists of partitions
together with a partial partition
k = 1, ... ,8,
BO'
n
Every such function
a..
a. k of
set of the r-set
(a.)
••• x p
x
nl
J
in a one-one manner to a function
a
x P
nO
a. = {a.: A. + GIj = l, ... ,r}.
Let
(a)
Q (G)
then
a.
~O
B ,
k
= {AOjlj = 1, ••• ,r O} of
must be of the following
(j = 1, .•. ,r ) is arbitrary, while a kj (k = l, .•• ,s;
O
Oj
j = l, ••. ,r ) is uniquely determined up to scalar multiples as the restrick
form:
a
~j'
tion of
b
between
[O,(S)]
k
to
Hence by E-equivalence there is a one-one correspondence
x
Q (G) x P
nO
nl
the product of the orders in B and
O
is that of
~
and
(G),
°
= {bklk
= l, .•• ,s}.
If
b
k
~
(8)
n
The order in
B
k
in
(k
[0,(8)]
is
Clearly the order in
B
O
s
B , ••• , Bs .
l
and the order in
COROLLARY 2.1: Let (a.)
8
••• x p
= l, ..• , s)
Q (G),
n
where
is that of
a.
is a linear combination of
P
n
k
.
= {a j /j = l, ... ,r},
a.
k
Sa.,
let
0
13
k=l, .•. ,s,
where
~
[(a),«(3)]
COROLLARY 2.2: Let
If
B = X,
«(3)
Q
r
(G) x P
°
r
x ••• x P
be a copoint of
B
= {xi}'
where
S = {b: B + G}.
COROLLARY 2.3: Let (S)
=
S
s
X-Uk=lB k •
(a
(ii')
(K
P,
n
then
[0,«(3)]
k
Qn(G),
then
,...,
Let
.
rk
1
=
while if
Then
{b
k
Q (G),
EO
=
where
n
I
E:
"" Qn-l (G).
=
Kie i : Bk
i
K ,»,
i
Glk = l, .•. ,s}.
k
Then the atoms of
-1
+
for all
[0,«(3)]
i, i'
are
such that
= l, ••• ,s;
(b)
(a(i»,
for all
(c)
(a(ii')(A»,
i
such that
for all
i, i'
xi
EO
B .
0'
such that
xi,x '
i
EO
B '
O
and all
A
€
G.
THEOREM 3: Qn (G) is a geometric lattice.
Proof: We prove first that Qn(G) is a lattice. Let (a),(S)
Since
n
preserves order,
(a), «(3).
where
tion
al s
u
Let
~
t,
c : C
t
l
(3,
n(y)
n(a) v ~«(3) = {clll
are the blocks of
+
where
n(a) v n(S)
~
= l, ... ,t},
n(a) v n(S)
for any upper bound
and suppose
l
Sl are defined by
=
Qn(G).
(y)
of
{clll = l, ••. ,u},
such that there exists a func-
G which is simultaneously a linear combination of
a ,
EO
al S a and
14
Then if
of
Y
= {cl
in
(a), «(3)
the
c
l
Gil = l, ••• ,u},
Q (G).
n
= 1, ... ,u,
l
l'
: C ~
is clearly a minimal upper bound
(y)
To show that
(a) v (f3)
exists, we must prove that
are uniquely defined up to scalar multiples.
Suppose
then that
cR..
= I
K.a.
J J
a,e.
= I
c'
l
Cl
are two such functions
K~a.
J J
al
~
G.
Let
= I
Akb k ,
= L
Akb k
(3l
S,e
xi E C,e'
and define
Aj,Bk
by
Then
=
K.a.(x )
=
Akbk(x i ),
=
K~a,(xi)
=
Akbk(x i ),
-t
J J
J J
hence
,-1, ,
=
-1
j j
or
K'K
empty.
the
=
J
C
l
(j: a.Ea o )
J.{~
that the elements
Kj
= ~Kj'
Ak
is a block of
versus the
,-1
",-1
k
so
ck
= {bklk = l, ••• ,s},
preserves order,
(13).
TI(Y)
n(a)
We obtain the blocks of
A n B
j
k
is non-
the intersection graph of
is connected.
~,
for all
It follows then
j, k.
Thus
= ~cl'
and define
~
(k: bkES )
l
are equal, say to
KjK j , I\ l\k
= ~Ak'
TI(a) v TI(S),
B
k
Consider next the infimum of
13
I\k'
This latter equality must hold whenever
But since
A.
Kj
A
A
O
empty intersections of the blocks of
=
r
J=
A
= l, ... ,r},
X-u. lA.,
TI(S)
As
J
for any lower bound
TI(S)
n(a)
«3).
and
(a)
(y)
of
TI
(a)
by deleting the zero block of
and
15
=
with the blocks of
=
C be a block of
Let
= l, .•• ,r,
j
C '
jO
C
define
Similarly, if
Ok : COk ~ G by
= l, ••. ,r;
j
k
c Ok
If
n(a) A n(S).
C
= CjO = Aj
c jO : C ~ G as the restriction
jO
C
= COk = AO n
= bklcOk'
= l, •.• ,s,
B
k
for some
Finally if
C
k
n B '
O
ajlcjo
= l, ••• ,s,
= Cjk = Aj
n B
k
define an equivalence relation
for some
of
aj
to
define
for some
R
jk
on
C
jk
by
iff
Kie i , bklc ' k = I
A.e .• Then if ~J'1-0 is the value of
€jk ~ 1
~
J
jk
on an Rjk-block (equivalence class) Cjkt of Cjk ' Ai = ~jktKi for
where
a.lc' k
J
-1
AiK i
J
= IE
Cjkt ·
all
xi
c jkt
= ajlcjk!.
C
jkt
€
Thus
bklc jkt
= ~jkt
(ajlc jkt )·
It is clear that the partition of
is the maximal partition of
C
jk
c jkt : Cjkt ~ G by
Define
Cjk
into its Rjk-blocks
for which functions may be defined on
the blocks which are simultaneously G-multiples of the restrictions of both
a
j
and
bk ·
above, then
(y)
=
(a)
A
Thus if
(y)
(8).
y
is the set of functions
is the unique maximal lower bound of
It follows that
(a(i», (a(ii')(A»
easily verified that
(13)
i
with
n
while
=
and
(a),
(13) ,
(a)
Then it is
iff
(a) v (a (i»
(13)
i. e. ,
and the definition
in Corollary 1.2.
is a I-cover (3.6) of
(13)
for any
Q (G)
of
(y)
defined
is a lattice.
Q (G)
n
Recall now the form (3.6), (3.7) of covers of
of the atoms
c jO ' c Ok ' c jkt
is a II-cover (3.7) of
(a)
iff
16
for any
such that
i, i'
where
Q (G)
Thus
COROLLARY 3.1:
subgeometry of
Proof:
t:
Qn
+
n
o
is geometric.
The partition lattice
Pn+l
is both a sublattice and a
Q (G).
n
It is clear from the proof of Theorem 3 that the injective map
Q (G)
preserves suprema and infima, so the t-image of
n
lattice of
Q (G).
points, so
l.(Q )
n
Since
n
t
is a sub-
Q
n
is also rank-preserving, it takes points to
is also a subgeometry of
But then
Q (G).
n
o
Q (G)
We refer to
as the lattice of partial G-partitions.
n
The ele-
ments are, of course, E-classes of partial G-partitions.
THE r~OBIUS FUNCTION, CHARACTERISTIC POLYNOMIAL
4.
Qn(G)
AND WHITNEY NUMBERS OF
A
modular eZement [ 9] of a geometric lattice L with rank function
is an element
XEL
such that the modular identity
p(XVy) + p(XAy)
holds for
morphism
yEL.
[XAy,y]
If
~
p
x
= p(x) + p(y)
is a modular element, the map
[x,xVy]
with inverse
w 1+ wAy,
z
~
x v z
for any
is an isoyEL.
Every
point of a geometric lattice is a modular element.
THEOREM 4:
tions of
X,
Let
i.e.,
E =
E
i
{e : E
i
i
= {Xi}'
+
Gli = l, ••. ,n}
ei(x )
i
= 1,
i
be the set of unit func-
= l, ••• ,n.
Then the subset
17
is a (Boolean) sublattice of
Q (G).
Every element of
n
M is modular in
Q (G).
n
Proof:
Let
(a)
E
M,
say
= {eili = 1, ... ,r},
a
and let
(13)
Q (G),
E
n
where
Then the blocks of
= l, •.. ,s
k
all
The blocks of
are
are all
also
p(y)
(a)
E
and that
= n-Iyl
M,
(ana)
M
k
Thus
= n(a)
n«a)v(a»
then each
E ,
M.
B ,
k
k
,r,
AO = {Xr+l' •.. ,X n }.
where
= l, ..• ,s,
r+s
=
v n(a).
such that
b
k
so
(a) A (13)
=
=
n«a)A(S»
laval+laAal
Qn(G),
is a unit function
and we have
~
B
k
The total number of blocks in
lal+ISI,
is the rank function of
singleton subset
E
].
It is clear from the proof of Theorem 3 that
these two partial partitions is
Since
i=l, •.•
E.,
Bk S {xl, .•• ,x r },
such that
n(a) v n(S)
{Xl' ... ,X }.
r
n(a) A n(S)
n(a) A n(S)
(a)
e ,
k
(aua)
is a sublattice, and the map
(a)
lal+ISI.
is modular.
so each
E
=
M,
ilk
(a) v (a)
1+ {e.le.
].
].
~ a}
If
is a
=
is an
anti-isomorphism from M to the Boolean lattice of subsets of
The MBbius function [7]
is defined recursively by
~
~:
LxL
~~
~(x,x)
= I,
(x,y)
=
of a finite partially order set
~(x,y)
L
=0
if
x
~
y,
L
and
~ (x, z)
z:x~z<y
if
x
~
y.
If
L
is a geometric lattice of rank
the characteristic polynomial [4] of
L
is
n
with rank function
p,
18
~(O,x)vn-p(x).
L
=
xEL
The characteristic polynomial extends to geometric lattices the notion of the
chromatic polynomial of a graph.
In particular, if
r
contractions [7] of a linear graph
r
polynomial of
with
is
k
L
is the lattice of
components, then the chromatic
is
The lattice of partitions
(isomorphic to) the lattice of contractions of the complete graph
chromatic polynomial
Pn+l
is
v(v-l) .•• (v-n) ,
(v-l)(v-2) ... (v-n),
finite group
G of order
p (v,m)
n
P
m (recall
with
so the characteristic polynomial of
i.e., the falling factorial
obtain the characteristic polynomial
K +l
n
~
of
Q (G)
Q (1»
We may
for an arbitrary
n
~=n
(v-l)(n).
with the aid of the
following special case of a result due to Crapo ([3], Th. 6, Cor. 5):
If
L is a finite geometric lattice of rank
a copoint of
=
val
L
x:xAc=O
P[x,l](v),
If
G is of order m,
the characteristic polynomial of
is
(4.2)
n-l
ilr (v-l-mi)
O
p (v;m)
n
where
(x)(n)
Proof:
(S)
[O,(S)].
A (a)
=
n(
)
m v~l (n)'
is the falling factorial
We take as our
By Corollary 2.2,
4,
is
P[a,b](v) is the characteristic polynomial of the inter[a,b] of L.
THEOREM 5:
Qn (G)
c
L, then
(4.1)
where
nand
=
[O,(S)]
° iff
(a)
c
~
in (4.1) the copoint
Qn-l (G).
=
x(x-l) ••• (x-n+l).
° or
The number of such atoms is
Since
(a)
(S)
E
(S),
where
S
= {ell.
M is modular, by Theorem
is an atom of
Qn (G)
not in
19
(n- 1) _ m(n- 1),
(1n) + m(n)
2 1
2
i.e.,
l+m(n-l).
By Theorem 2,
~
[(a),l]
Qn-l (G)
for every atom
(a).
Thus (4.1) gives
(4.3)
p (v;m)
n
Since
Pl(v;m) = v-l,
=
(v-l-m(n-l»p
n- l(v;m).
o
we obtain (4.2) by iteration of (4.3).
Stanley [10] has recently investigated the class of finite geometric lattices containing a maximal chain
••• < x
=1
n
of modular ele-
Such lattices, called supersoZvabZe, have the property that all zeros
ments.
of the characteristic polynomial are positive integers, namely,
where
a
is the number of atoms in
i
Q
Theorem 4,
n
COROLLARY 5.1:
u
m
n
= ~(O,l),
where
Let
~
m is the order of
=
(x)(n)
Proof:
When
value of
~
(_l)n
m = 1,
~(O,l)
v
=0
i.e.,
G.
[O,x _ ).
i l
By
= l+(i-l)IGI
Qn(G),
and let
Then
n-l
IT
(l+mi)
i=O
(-m)
is the rising factorial
Set
a.1.
be the Mobius function of
m
u
=
n
where
1.
is supersolvable, with
(G)
but not in
[O,x.)
n (~) (n) ,
(x)(n) = x(x+l) ..• (x+n-l).
o
in (4.2).
G is the trivial group
for the partition lattice
P +
n l
1,
~
u
1
n
Qn(l).
=
n
(-1) nl
is the
Since the Mobius
function is multiplicative over direct products, we obtain from Corollaries
2.1 and 5.1,
20
COROLLARY 5.2: Let
Corollary 2.1.
(a) :s; (S)
in
Q (G),
n
where
a, B are as in
Then
~«a),(B»
m 1
u
u
r
r
O l
=
u
1
r
s
r
(r )
1
(_l)r-s m O (-) O
m
=
s
n
k=l
The Whitney numbers of a finite geometric lattice
(r -l)!
k
L
of rank
n
are
defined by
w(n,r)
(4.4)
L
=
~(O,x),
(First kind),
x:p(x)=n-r
the coefficient of
W(n,r)
(4.5)
v
r
in the characteristic polynomial, and
L
=
(Second kind),
1,
x:p(x)=n-r
the number of elements of corank
r.
The most well-known examples are the
following:
(1)
If
L
= Bn'
the lattice of subsets of an n-set,
W(n,r)
(2)
If
L
= Ln (F),
=
the lattice of subspaces of an n-dimensional vector
space (or (n-l)-dimensional projective space) over a finite field
q,
)
w(
n,r
where
(~)q
=
(_l) n-r q
(n-r)
2
(n)
r q'
w(n,r)
is the Gaussian coefficient [6],
=
(gn_l) ... (gn-r+l_ l )
r
(q -1) .•. (q-l)
=
n
(r) q'
F
of order
21
(3)
If
L =P + ,
n l
the lattice of partitions of an (n+l)-set,
=
w(n,r)
s(n+l,r+l),
=
W(n,r)
S(n+l,r+l),
the Stirling numbers of the first and second kind, respectively.
Q (G)
Each of these examples, as well as the lattices
considered here,
n
are classes of geometric lattices which satisfy the hypotheses of
THEOREM 6:
L
n
that
Let
{Ln In
is of rank
the interval
[x, l]
Whitney numbers of
= 1,2, ... }
and for all
n,
be a class of geometric lattices such
x€L
is isomorphic to
L
r
of corank
n
.
r
(0
~
v1(n, r), W(n, r)
Let
r
~
n < 00),
be the
Then
L .
n
L W(n,r)
w(r,s)
=
o(n,s),
W(r,s)
=
o(n,s),
r
L w(n,r)
r
where
o(a,b)
W(n,r)
(4.6)
=1
if
a
= b,
and
0
otherwise,
and the numbers
satisfy the inverse relations
an
Proof:
= L\
r
W(n,r)b r ,
b
n
L w(n,r)a r .
=
r
We use the identities
o(O,y)
L
=
~(x,y)
L
=
x:x~y
~(O,x).
x:x~y
Then
L W(n,r)
w(r,s)
=
=
)1
(x, y) o(s,n-p(y»
y~x
o (s,n-p (y»
L
)1
(x, y)
x~y
n
I
yEL
=
n
L
y€L
=
L
L
x€L
r
o(s,n-p(y»
n
o(n,s).
O(O,y)
w(n,r),
22
Similarly,
~ w(n,r) W(r,s)
L
=
o(s,n-p(y»
yEL
an
= l..r
\
]J(O,x)
x:::y
n
= L
Then if
~
O(s,n-p(y»
yEL
=
O(s, n-p (y»
y2::x
n
L
=
L
]J(O,x)
XEL
r
o(O,y)
n
o(s,n).
W(n,r)b,
r
L w(n,r)a r = L w(n,r) L W(r,s)b s
r
r
s
= L b s L w(n,r)
s
W(r,s)
r
= L bs
Hn,s)
s
=
b •
n
o
The converse is proved analogously.
COROLLARY 6.1:
Let
for any fixed group
t (n,r),
T (n,r)
m
G of order
m.
n v--l
m (---;-) (n)
v
be the Whitney numbers of
m
n
Then
=
L t m(n,r)v
nomial
Set
p (v;m),
n
a
r
= vr
in (4.6).
r
,
r
=
L Tm(n,r)
r
Proof:
Qn(G) ,
Then
b
n
r v-I
m (~) (r)'
is the characteristic poly-
o
given by (4.2).
Observe that the inverse relations in Corollary 6.1 are the analogs of
the defining relations of the Stirling numbers, which are obtained from these
by setting
m
=I
and multiplying both equations by
v.
The analogs of the
23
Stirling recurrences are given in
THEOREM 7:
fixed group
The Whitney numbers
G of order
(4.7)
T (n,r)
(4.8)
t
m
m
Proof:
(n,r)
m,
Tm(n-l,r-l) + (l+m(r-l))T (n-l,r),
X-{x }
n
J
=
t (n-l,r-l) - (l+m(n-l))t (n-l,r).
m
of size
r.
for any
m
(AEG),
n
Qn(G) ,
m
A partial G-partition of
a.+Ae
of
m
=
X
X-{x }
n
a unique partial G-partition
by
m
satisfy the recursions
unique partial G-partition of
a.Ea
J
t (n,r), T (n,r)
of
a
of size
of size
X-{x}
n
r
r-l
is obtainable from a
by adding
r
of size
or from
by replacing some
or else is equal to a partial G-partition of
This proves (4.7), while (4.8) follows from a comparison
of the coefficients of
\!
r
o
in (4.3).
5.
REPRESENTATION OF Qn(G)
In this section we solve the representation problem (Theorems 9, 10,11)
of
Q (G)
n
after first considering whether nonisomorphic groups can result in
isomorphic lattices (Theorem 8).
Q3(G)
The structure of the rank three geometries
will be required in the proofs of Theorems 8 and 9, and we begin with
a description of their structure.
resentative
a
Let
G be of order
of each E-class is fixed.
by its chosen representative
a
The element
throughout.
m,
(a)
and assume a repwill be denoted
The particular representatives
chosen will be those given below.
Recall the Boolean sublattice
Mare
Q3(G)
{e }, {e }
3
Z
meets each
and
{e }.
i
{ell.
M = {(a)!a ~ s}
of
Q3(G).
The lines of
These are modular lines, so every line of
The three M-lines
{ei}
intersect in pairs; we
24
regard them as the sides of a triangle.
section of
{e } and
j
In what follows,
The vertices of the triangle are the
{e }.
k
(i,j,k)
will denote an arbitrary element of
{(1,Z,3), (2,3,1), (3,1,2)}.
In addition to its two vertices
the triangle contains
form
{ei ' ej +Ai e k } ,
{e } will be denoted
i
Ai
E
G,
3m
and
{ei,e },
each side
k
{e i }
of
m other points, called its interior points, of the
one for each
A.
1-
E
The set of interior points of
G.
Thus
8i •
8
1
=
{{ e , e +A e } IA
l
2 l 3
l
E
G}'
8
=
{{e 2 , e 3+A 2e l }IA 2
E
G}'
=
{{e 3 , e l +A 3 e A}IA 3
E
G}.
8
There are
{ei,e }
j
2
3
trivial (2-point) lines in
join the vertex
{ej,e }
k
Q3(G).
The
m lines
to the interior points
{e.+A.e },
J 1- k
{e i , ej+Aiek }
of
the opposite side (1. e. , to
Si)'
2
The remaining lines are m in number.
These are the lines, to be called
transversal lines.. of the set
contains three points, one interior
Each transversal line
point on each side of the triangle, namely
e
(5.1)
{e , e + A e }
1 3
l
Z
E
8 ,
1
{e , e + A e }
2 l
2
3
E
8 ,
2
{e , e + A e }
3 2
3
3
E
8 ,
3
25
where
of
Then the subset
GxGxG
is the image of
-1
(A l ,A 2 ,(A 1 A2 )
incident with
).
-1
T under the injection
Each interior point
{A e +A e +e }
2 l l 2 3
~
on the side
{e }
i
{ei,ej+Aie }
k
is
m transversal lines, joining it to the interior points of the
2
Combinatorially, the m transversal lines of Q3(G) rep-
remaining two sides.
resent the triples of a latin square of order
m,
of rows, columns and symbols (in any order).
with
Sl' S2' S3
the sets
Clearly any latin square (alge-
braically, a quasi-group) can be used to construct a planar geometry with the
incidence properties described above for
Q3(G).
However, the nonassocia-
tivity of quasi-groups prevents generalizing the construction of
letting
G be a quasigroup, for dimensions
Q
n
(G)
by
n > 3.
It is evident from the results of Section 4 that the Whitney numbers,
hence the characteristic polynomial, depend on
Further, if
G, G'
G only through its order
are two groups of the same order
images of any partial partition of
m,
X is the same in
m.
the number of n-pre-
Q (G)
n
and
and the number of elements covering and covered by any element depends only
on
m and its n-image.
These similarities naturally suggest the question as
to whether two groups of the same order give rise to isomorphic lattices.
The answer is given by
THEOREM 8:
If
n
~
3
and
Q (G) ~ Q (G')
n
- n
for two groups
G, G',
then
G ~ G'
Proof:
Q (G')
4It
n
[(a),l],
Clearly,
G, G'
be an isomorphism.
where
(a)
must be of the same order
The restriction of
0
m.
Let
0:
Qn(G)
to an upper interval
is of corank three, is an isomorphism
[(a),l]
+
26
[a(a),a(l»).
By Theorem 2(a),
[(a),l)
~
Q3(G)
Thus it is sufficient to prove the theorem for
the sub1attice
M of modular elements in
Q3(G)
{e ,e ,e }.
1 2 3
Then for each
{ei,ej+Aie }
k
a-image of a point
where
A.
1.
€
thus regard as a bijection
transversal lines of
Q3(G)
G,
(i,j,k)
Si
in
A~€G'.
1.
= 3.
Clearly
onto
M in
aiM
we may assume without loss of generality that
sets of
n
€
~
[a(a),a(l»)
and
a
Q3(G').
must take
Q3(G'),
and
is the identity on sub-
{(1,2,3),(2,3,1),(3.l.2)},
is a point of the form
The restriction of
a
to
S.
1.
we may
G+G'.
Clearly
°
to transversal lines of
Q3(G'),
so that for
0i:Ail+AI
the
of
must take
all
=
=
iff
i.e.
=
Let
A ,A
1 2
€
Then for all
be the bijection
'3: G+G'
G,
-1
») -1
=
[a «A A )
l 2
3
=
a (A 2 )ol(A 1 )·
2
The proof is then completed by application of the following
LEMMA:
bijections
If
4>.:
1.
i
G ~ G'.
are two (multiplicative) groups, and there exist three
= 1,2,3,
=
(5.2)
then
G, G'
such that for all
K,A
€
G,
27
Proof of Lemma:
<P
(5.3)
<P
Let
l
2
Taking
= 1,
K
(A)
=
<P 3 (A)[ <P 2 (1)
(K)
=
[<Pl(l)]
-1
= 1,
respectively in (5.2), we have
]-1
<P (K).
3
= [<Pl(l)] -1 <P 3 [<P 2 (1)] -1 •
<P
A
Then
<p: G -)- G'
is bijective, and by (5.2)
and (5.3),
so
o
is an isomorphism.
<P
We turn now to the representation problem for
field, denote by
THEOREM 9:
then
If
Q (G)
is representable over a field
n
any (0)
Since
Q3(G)
F
F.
F
is any
We then have
and
n
~
3,
F*.
[(0),1]
is isomorphic to the minor
of
Q (G)
n
for
of corank three, it is again sufficient to prove the theorem for
= 3. We assume a coordinatization of Q3(G)
a fixed coordinate vector in
point of
Q3(G)
injection
a
F
3
over
F
is given and choose
from the homogeneous set representing each
by taking one of the coordinates unity, according to the con-
ventions described below.
Q3(G)
If
the multiplicative subgroup of
G is isomorphic to a subgroup of
Proof:
n
F* = F-{O}
Qn (G).
into
F3
The
~epresentation
may then be regarded as an
such that any subset of three or fewer points of
is independent iff its a-image is linearly independent over
Again let
(i,j,k)
E
{(1,2,3),(2.3.l),(3,1.2)}.
three (independent) M-points of
of
3
F ,
k.
Then since an interior point
with
a({ere })
k
Q3(G)
F.
We may assume that the
are represented by the unit vectors
having 1 in position
{e. ,e.+A .e }
J.
J J. k
i
E
Si
and
o in
positions
is collinear with
j
and
28
{ei,e j }
{ei,e },
k
and
°
it is represented by a vector with
k.
and nonzero elements in positions
j
coordinate in position
Denote nonzero elements of
c, ... ,
Hi
= 1,Z,3,
J
~
~i(Ai)
by
~
By convention we assume the
F by
a,b,
=
{all (0,1,a1 )
EO
o(Sl)}'
HZ
=
{azl (a Z,O,l)
EO
o(SZ)},
H
3
=
{a
(1,a ,0)
3
EO
a(S3)}·
vectors
= ai'
-1
{A Ze 1+A 1 e Z+e }
3
respectively,
3
1
F*.
where
A
3
so for all
= 1,
G ~ Hi'
~i:
Define bijections
the coordinate in position
are
(0,1'~1 (A
of
k
1
»,
(~Z O'Z)
,0,1),
(1'~3 (A
3 ) ,0),
= (A 1AZ) -1 • It is easily verified that the three
(0,1,a ), (a ,O,l), (1,a ,0)
1
Z
3
a 1 a Za 3 = -1,
A1AZA 3
H
1
Then the a-images of the three points (5.1) of the trans-
k
versa1 line
A1 ,A Z,A 3
EO
of
G,
F
3
are linearly dependent iff
~1(A1)~Z(AZ)~3(A3)
= -1
iff
i. e. ,
(5.4)
=
Interchanging
*
1.
is thus an m-subset of
a({e.,e.+A.e }}.
F
is
i
and let
Each
i
j
and
in position
is abelian.
~3: G
~
H
3
A
1
and
Thus
A does not affect the right side of (5.4), since
2
-1 -1
-1 -1 have the same
A A
and A A
1jJ3-image. Since
1 Z
2 1
is bijective, i t follows that
We next prove that each
Note that
the subset
Hi
G is abelian.
is a coset of a subgroup of
U is the image under the bijection
*
F.
Define the
~lxIjJZxIjJ3: G3 ~ H1xHZxH3 of
29
corresponding to the transversal lines of
(i,j,k)
ai
of
(1,2,3),
= -l/ajak •
ments of
Let
H..
E
~
a.
E
J
H.
iff there exist
~
a.
imply
H.,
J
a.
-l/aja i
-l/aja i
€
Hk ,
ci
E
Hi
imply
ajai/c i
(iii)
ajai/c i
E
Hj ,
bi
E
Hi
imply
-ci/ajaib i
-ci/ajaib i
E
Hk ,
H.,
J
a
E
k
H
k
such that
Hk ,
E
(ii)
(iv)
E
J
Then
H••
J
E
J
Thus for any permutation
be any three (not necessarily distinct) ele-
ai' b i , c i
Choose any
~
(i)
a.
Q3(G).
aj
E
Hj
imply
Hj ,
E
E
aibi/c i
E H .•
~
Thus
(5.5)
Choose fixed elements
c
Hi'
E
i
~
b. /c.
~
c/a i
H.,
E
~
1
next prove
Let
so
G
l
a.
1
= K.C.
1 1
we have
E
E
G .
k
J
E
K.
E
G
i
Thus
Thus
implies
H ,H ,H
1 2 3
quotient group
-1
1
Hi}.
E
~
= ci/a i
1
1
m and
H.
1
a.lc.
1
~
ai/c i ,
(a/c i ) (bi/c i )
= Gic i
(5.5)
G"
1
€
Finally, i f
G.•
E
so
E
and i f
G.,
€
Hi'
aibi/c i
of order
a. = K.C.
Hi'
1.
k
O
l/K i K/,O
E
K
E
G. ,
K.
K
i
E
G. ,
so
i
1
J
J
0
=
H .
j
E
G •
i
Thus
is a coset of
G.•
~
G.
J
G
l
Then for any
K.
1
= K.J = 1,
imply
Gk ,
E
imply
KiK j
= Gz = G3 = H,
are three cosets of the subgroup
F* /H, HIH H
2 3
= -H,
-l/aia j = -l/K i Kj c i Cj
Then
-c l c Zc ·
3
Putting
1/K K A EG •
i j
E
J J
J
Define
Gk ·
But then
1
~
We
= G2 = G3 ·
-1/KiKjClC2C3
K.K.
~
Clearly
(ai/c i )
F*
is a subgroup of
G.
1
so
~
then by (5.5),
G. ,
E
1
1
2
implies
is a group.
G.
and let
{a. / c. Ia.
=
G.
We claim each
= 1,2,3,
i
the coset of
E
1/1.
K
0
i
l/K.K. E G ,
k
1 J
Gk ·
Putting
H ,
k
G ,
i
E
so
G ,
k
E
E
so
= 1,
K.
J
-c c c
l 2 3
say,
and
1.
H of
F* ,
such that in the
H containing
0
=
-1.
E
H.
30
Choose any three elements
define bijections
f i : Hi + -H
~1'~Z'~3 E
H such that
by
=
fi(a i )
-~i(ai/ci).
~1~Z~3
= AO'
and
Then
=
so the image under
U
flxfZXf3: HlxHZXH3
= {(al,aZ,a3)lalaZa3 = -I}
(-H)
+
is a set in
3
(-H)
of the subset
3
with the same property.
It
follows that the maps
(O,l,a )
l
1+ (O,l,f (a l ))
l
(aZ,O,l)
1+ (fZ(aZ),O,l)
(1,a ,0)
3
~
(1,f (a ),0)
3 3
give a representation of
H
1
= HZ = H3 = -H.
We may assume,
therefore, that the given representation is of this form.
Recall now the bijections
(5.6)
Let
~.:
G
1.
+
-H
satisfying (5.4).
Let
T
i
=
=
T: G + H be defined by
Then from (5.6),
=
so by the lemma,
G ~ H.
The converse of Theorem 9 is
o
31
THEOREM 10:
Qn (G)
then
If
F
is a field and
G is isomorphic to a subgroup of
F*,
F.
is representable over
Proof: Let Ln (F) denote the lattice of subspaces of the projective
geometry of rank
points of
L (F),
n
n
(projective dimension
and assume that
xi
E
X.
Then if
L (F)
n
(K , ..• ,K )
1
n
is coordinatized over
X,
f: X + F,
representing
(K , .•• ,K )
1
n
E
n
F ,
not identically zero, represents a copoint of
=
is the point set of
V{plp
iff
The set
f' = Af
S(f)},
{p = LK.x.
ILKif(x.)
=
1. 1.
1.
=
for some
{e , ..• ,e }
n
1
E
f:
As in the case of points, two functions
o
n
F
with the unit vectors of
Since the coordinates are homogeneous, the vector
S (f)
n
the set of
namely
S(f)
L (F)
S
F with respect to any
is a point with coordinate vector
f
where
and
is determined only up to a constant nonzero scalar multiple.
Any function
L (F),
n
p
p = L~=lKiXi'
we write
F,
we take as a fixed basis of
L (F).
n
system of reference containing
the
n-1) over
A
E
f, f'
represent the same copoint of
F*.
of unit functions, with
otherwise, is the copoint basis dual to
e.
1.
a}.
ei(x ,) = 1
i
if
i = it,
x:
=
V{x.,li' 1: i},
=
!dei' Ii'
1.
1: i}.
A copoint f: X + G is thus a linear combination f = LAie , where Ai =
i
f(x i ).
Then f is minimally dependent on the subset {eilAi 1: O} of {ei, •. ,e }.
n
That is, f ~ A{e.IA. 1: O} but f ~ the infimum of any proper subset of
1.
1.
32
{eilAi of OJ,
since a point
is
{ilK.1
= OJ,
so IK.A. = O.
1 1
Conversely,
Ai of O.
More generally, if
then
is dependent on
g
J J
dent on
~
Xi
{ f l ' ...
,f }
r
iff
a
E
Aj
J
for any
i
such that
is a linear combination
and in this case,
g
is minimally depen-
(a) E Qn(G),
= {a j :
a
X + G u {OJ
= l, .•. ,r},
A + Glj
j
F,
=0
aj(x )
i
X and defining
~
G is a subgroup of
F*.
For
we now regard the
simply by extending the domain from
for all
xi
X-A .
j
E
Note that any two
Ln (F).
E-equivalent partial G-partitions represent the same set of copoints in
Further since the subsets
= {xilaj(xi)
A
j
{eilaj(xi) of OJ,
the subsets
a
pendent set of copoints in
of OJ,
= l, ... ,r,
j
minimally depend are disjoint.
j
=
g
J
as functions
to
I
{i A. of O} ~
{f.IA. of OJ.
any element
J
f
g
Assume without loss of generality that
a.
if f
1 1 1
is any independent set of copoints.
= l, ... ,r},
g ;::: A{fjl j
iff
ILf.
A{e. IA. of O}
$
Hence
= l, ... ?r,
j
{el, •.. ,e }
n
of
n
on which the
= {a.lj
= l, .•. ,r}
J
a
Ti follows that the map
L (F).
are disjoint,
a:
is an inde-
Q (G)
n
+ L (F),
n
where
a(a)
=
is well-defined and preserves rank,
=
(5.7)
(S) ;::: (a)
If
b
=
k
(A.
IA.a.
J J
J
G),
E
a (S) ;::: a (a),
Thus
in
=
lal
n -
then each
Qn(G),
i. e.,
a
e
Q (G)
1
$
$
i,
i'
n
n,
are
a
A
S
is a linear combination
takes points to points.
(a(i», (a(ii')(A»,
i of i',
E
for all
b
k
E
is order-preserving.
It follows from (5.7) that
points of
k
bIt ;::: A{ajl j = l, ... ,r}
L (F),
n
so in
b
E
G,
and
Recall that the
defined in Corollary 1.2, where
(a(ii')(A»
=
(a(i'i)(A-l».
It
is
S.
33
easily verified that
(5.8)
Suppose
(0),(6)
xi ~ X-U~=lAj'
~(8)
Qn(G),
A1
~ ~(o),
of the other, say
Then
a l (x i ,)/a1 (x i )·
But
b1 (xi)-A
-1
(8),
~
xi'
-1
xi
~
a(o) = a(6).
and
X '
O
xi' xi'
so
a: Q (G)
n
L (F)
+
n
xi-A
~(o),
a
xi-A
(0) = (8),
so
a
-1
xi'
a
$
a(o),
Let
{AjlKj # O}
a set of
~(o),
Thus
(A»
xi' ~bl'
= xi-A
=
say,
-1
-1
xi' ~ a(o).
It follows that
~(o)
xi-A
A=
Let
xi'
~(8).
(0) #
If
containing two e1eBut then if
~
a(S),
a con-
preserves suprema.
It will then follow that
Q (G)
n
is isomorphic
~
Qn(G).
is a copoint of
L (F)
Let
L (F).
n
(o),(S)
is order-preserving,
c ~ a(o) va(S),
LkAkb k .
iff
is an injection.
$
a«o)v(S».
To prove the reverse inequality, suppose
and
a(o)
in different blocks
~(8).
1
is an injective strong map, so that
a(o) v a(S)
(5.9)
-1
~
(i")
Al = B ,
l
to its a-image, the latter a subgeometry of
Since
xi' xi'
a(o
~
say.
a 1 (x ,)/a1 (x ) # b (x ,)/b (xi)'
1
i
i
1 i
we have
We now prove that
xi
then there exists a block
B ,B
l 2
B1 , xi' ~ B2 , where
(o(ii')(A» ~ (0), so
such that
Thus
~(8),
a contradiction, and we conclude
A = a l (xi,)/a l (xi)'
tradiction.
#
containing two elements
then there exists a block of
ments
~(o)
If
b1 (xi') = b1 (xi) # 0,
a(S),
Since
X-U~=lAj = X-U~=lBk = XO'
it follows that
are each partitions of
of one, say
xi-A
~
i.e., there exist
C = {xilc(xi) #
a}.
Then
c: X + F
Kj,A k ~ F*
C is the union of the
and also of the ~(8)-blocks {BkiAk # O},
(~(0)v~(8»-blocks
such that
{Clil = 1, ... ,v},
say.
n
c = LjKja
j
=
~(o)-blocks
hence is the union of
Assume first that
34
v
= 1,
i.e., that
C is a single block
=
rr(a) v rr(S).
of
Then
t
c
L
=
K.a.
where
Let
Kja j (xi)
=
GK , GA
j
k
of
a block of
so
rr(a) v rr(S),
(k
in a common coset
of
n
Kj/A
xi
E
Aj
= bk(xi)/aj(x i )
E
G.
and suppose
k
GA
Thus the function
G.
c'
into
G U {OJ,
y'
In general, if
where
C= C
o
where
partial G-partition
~
c
o«a)v(S».
(5.9), that
a
such that
y'
It follows that
(y')
If
C'
= 1, ... t)
is an F*-multiple
E
l
c'
E
y'
Thus
c
~
o(y') =
v
2,
~
y',
c =
By the preceding,
l = l , ... ,v,
for some
Thus
so
o«a)V(S»,
~
then
for some
and hence by
This completes the proof.
G is a subgroup of
(a (i»
is
are contained
otherwise.
= (a)v(S).
o(a) v o(S)
is supremum-preserving.
COROLLARY 10.1:
where
v
if
each
(j
J
C
hence represents the same copoint
C ,
U ... U
l
cl(x i ) = c(x i )
But since
Hence all
(y') = (a)V(S).
such that
Then
A.
the intersection graph of the
of
Bk .
Thus the cosets
It is clear from the proof of Theorem 3 that
o«a)V(S».
c
U
is nonempty.
is connected.
partial G-partition
Ll=lcl ,
C,
= 1, ••• , t)
of a function
L (F).
E
are equal whenever
G
versus the
c = AC'
Xi
=
J J
j=l
F*,
then
1-+ x. ,
1.
where
1
~
i,
i'
~
n,
iii',
A
E
G,
and
is a coordinatization of the points of
X
= {xl' .•. ,xn }
Qn (G)
over
F.
is a basis of
o
35
THEOREM 11: Let G be of order m, and let n
Q (G)
noncyclic,
Qn (G)
is representable over no field.
n
If
3.
Then if
G is
G is cyclic, then
is representable over
every field iff
(b)
a finite field of order
(c)
the rational or real field iff
(d)
the complex field for all
If
F
m
=I
(a)
Proof:
(i.e., G is trivial),
q
iff
m divides
m
=1
m.
is any field, and
is representable over no field if
cyclic of order
m,
every field iff
m
then
G is a finite subgroup of
G is noncyc1ic.
F*,
then
Thus by Theorem 9, Q (G)
n
Conversely, if
G is
G is a subgroup of the multiplicative group of
= 1, of a finite field of order q iff m divides q-l,
of the rational or real field iff
m.
q-1,
or 2,
G is necessarily cyclic (see, e.g. [1], Thm. 17).
every
~
m
=1
or 2,
Thus (a)-(d) follow form Theorem 10.
and of the complex field for
o
36
REFERENCES
[1]
Artin, E., caZois Theory. Notre Dame Mathematical Lectures, Number
2, Notre Dame, Indiana, 1959.
[2]
Birkhoff, G., Lattiae Theory. knerican ~~thematical Society Colloquium
Series, Volume 25, Providence, R.I., 1967.
[3]
Crapo, H.,
595-607.
[4]
Mobius Inversion in Lattices.
Crapo, H. and Rota, G.-C.,
CombinatoriaZ Geometries.
On
Arah. Math. XIX (1968),
the Foundations of CombinatoriaZ Theory:
MIT Press, Cambridge, Mass., 1970.
[5]
Dowling, T.A., A q-Analog of the Partition Lattice. To appear in the
Proceedings of the International Symposium on Combinatorial Mathematics
and Its Applications, Colorado State University, Fort Collins,
Colorado, September 9-11, 1971.
[6]
Goldman, J. and Rota, G.-C., On the Foundations of Combinatorial
Theory IV: Finite Vector Spaces and Eulerian Generating Functions.
Studies in AppZ. Math. XLIX (1970), 239-258.
[7]
Rota, G.-C., On the Foundations of Combinatorial Theory I: Theory of
Mobius Functions. Z. WarsaheinZiahkeitstheorie und Verw. Gebiete
2 (1964), 340-368.
[8]
Rota, G.-C., The Number of Partitions of a Set.
71 (1964), 498-504.
[9]
Stanley, R.,
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[10]
Stanley, R.,
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Preprint.
Preprint.
FOOTNOTES
Primary Classification Number: 0535
Secondary Classification Numbers: 0505, 0527
Key Words and Phrases: partition, partition lattice, finite group, partial
G-partition, geometric lattice, combinatorial geometry, Stirling numbers. representation over a field.
1. This research was supported in part by the Air Force Office of Scientific
Research under Contract AFOSR-68-l4l5.
2. Present address: Department of lfuthematics, Ohio State University,
Columbus, Ohio 43210.