Homotopy of Non-Modular Partitions and the Whitehouse Module.
We present a class of subposets of the partition lattice n with the following property: The order complex is homotopy equivalent to the order complex of n  1, and the Sn-module structure of the homology coincides with a recently discovered lifting of the Sn 1-action on the homology of n  1. This is the Whitehouse representation on Robinsons space of fully-grown trees, and has also appeared in work of Getzler and Kapranov, Mathieu, Hanlon and Stanley, and Babson et al.One example is the subposet Pnn 1 of the lattice of set partitions n, obtained by removing all elements with a unique nontrivial block. More generally, for 2  k  n the subposet of the partition lattice obtained by removing all elements with a unique nontrivial block of size equal to k, and let Pnk &equals; \bigcapi &equals; 2k Qni. We show that Pnk is Cohen-Macaulay, and that Pnk and Qnk are both homotopy equivalent to a wedge of spheres of dimension (n  4), with Betti number (n  1) \!{n - k\over k}. The posets Qnk are neither shellable nor Cohen-Macaulay. We show that the Sn-module structure of the homology generalises the Whitehouse module in a simple way.We also present a short proof of the well-known result that rank-selection in a poset preserves the Cohen-Macaulay property.
Introduction
In this paper we consider subposets of the partition lattice \Pi n obtained by
removing various modular elements. Recall that \Pi n is the lattice of set partitions of
an n-element set, ordered by refinement. We say a block of a partition is nontrivial
if it consists of more than one element. The modular elements of \Pi n are precisely
those partitions with a unique nontrivial block (for this and other basic definitions
see [St3]). For a bounded poset P we denote by -
P the proper part of
the poset P with the greatest element - 1 and the least element - 0 removed. We
for the order complex of P ; the simplices of \Delta(P ) are the chains of -
By the ith (reduced) homology ~
of P we mean the ith (reduced) simplicial
homology of its order complex \Delta(P homology in this paper is taken with
integer coefficients except for representation theoretic discussions, in which case we
take coefficients over the complex field. All posets are bounded unless explicitly
stated otherwise.
1991 Mathematics Subject Classification. Primary 05E25, 06A08, 06A09; Secondary 20C30.
Key words and phrases. poset, homology, homotopy, set partitions, group representation.
Research supported by National Science Foundation Grant No. DMS9400875.
Typeset by A M S-T E X
For
n to be the subposet of \Pi n obtained by removing all
modular elements whose unique nontrivial block has size 2 - i - k; and define Q
to be the subposet of \Pi n obtained by removing all modular elements whose unique
nontrivial block has size k: In particular, P
n consists of all partitions in \Pi n with
at least two nontrivial blocks, together with the greatest and least elements. It is
not hard to see that the posets P k
are ranked, of rank (n \Gamma 2); one less than the
rank of \Pi n . On the other hand the subposets Q k
n have full rank (n \Gamma 1):
The figures below show the (order complexes of) the posets P 3
4 and Q 3
4 is not Cohen-Macaulay. Note that the 0-dimensional order complex of P 3
4 and
the 1-dimensional order complex of Q 3
4 both have the same homotopy type, and
hence have the same homology.

Figure

1: The poset b

Figure

2: The poset b
describe briefly the motivation for this work. In [Su] some general techniques
were developed for computing the homology representation of a poset for a finite
group of automorphisms, and applied to Cohen-Macaulay subposets of the partition
lattice. Note that the subposets P k
are invariant under the action of the
In particular the Lefschetz module (i.e, the alternating sum
(by degree) of the reduced homology modules ), Alt(P k
n ); is a virtual Sn-module.
By applying [Su, Theorem 1.10 and Remark 1.10.1] to the subposets P k
; we can
show that as (virtual) Sn-modules, (\Gamma1) n\Gamma4 Alt(P k
are both
isomorphic to
(Here the up arrow indicates induction.) For the representation given by
(0.1) is the complement of ~
in the induction of ~
This is precisely the representation of Sn on Robinson's space of fully grown trees,
as computed by Sarah Whitehouse (see [R], [RW], [W], [Ha2]). The restriction of
this representation to Sn\Gamma1 is ~
to tensoring with the sign, this is also
the lifting of the Sn\Gamma1 action on the multilinear component of the free Lie algebra
generators up to Sn ; described in [GK].
For arbitrary k it is not hard to see that (0.1) is in fact a true representation
of Sn : In view of these facts, one is naturally led to ask whether the homology of
the subposets P k
n is concentrated in a unique dimension. We answer this
question in the affirmative, showing that both posets have nonvanishing (reduced)
homology only in dimension
We show that the order complexes of the posets P k
are homotopy equiva-
lent, with free integral homology, and that P k
n is Cohen-Macaulay over the integers.
(It follows that the pure posets Q k
n are not Cohen-Macaulay.) We conjecture that
the order complex of P k
(and hence of Q k
n ) is homotopy equivalent to a wedge of
4)-spheres, and verify this conjecture for
Our main tool is Quillen's fibre lemma (see [Q], [Bj3])). In Section 1 we investigate
the effect on homology of deleting an antichain from a poset (Theorem
1.1) and generalise this to an analogue for simplicial complexes (Theorem 1.5).
As a consequence we obtain a simple proof, using only the exact homology sequence
of a pair, of the well-known result that rank-selection in a poset preserves
the Cohen-Macaulay property. In Section 2 we show that the subposets c
n and
c
are homotopy equivalent (in fact Sn-homotopy equivalent). The representation
theoretic aspects are addressed in Section 3, where we give a simple formula (equa-
tion (3.1)) describing the Sn-module stucture of the homology of Q k
(and hence of
terms of the homology of the partition lattices \Pi k and \Pi n : We conclude in
Section 4 with a brief discussion of possible generalisations of this work.
1. Deleting an antichain from a Cohen-Macaulay poset
Let P be any poset, and let A be an antichain in P: For our first result we
use the exact sequence of a pair to obtain information on the homology of the
subposet PnA of P obtained by removing all elements of A; in the case when P is
Cohen-Macaulay.
The hypotheses in the theorem below may be relaxed somewhat by considering
the more general case of simplicial complexes; see Theorem 1.5 at the end of this
section.
Theorem 1.1. Let P be a Cohen-Macaulay poset of rank r over the integers. Let
A be an antichain in -
P such that for every a 2 A; the homology of at least one
of the intervals ( - 0; a); (a; - 1) is free. Let PnA denote the subposet of P obtained by
deleting the elements of A: Then the reduced integral homology of PnA vanishes in
all dimensions except possibly r \Gamma 2 and r \Gamma 3: If in addition the homology of P and
all intervals ( - 0; a); (a; - 1); a 2 A; is free, then so is the homology of PnA:
Proof Consider the long exact homology sequence of the pair (\Delta(P ); \Delta(P nA))
(see [Mu1]). Since P is Cohen-Macaulay, the reduced homology of P vanishes for
degrees not equal to r \Gamma 2; and the long exact sequence reduces to the following two
sequences:
and, for
We must first compute the relative homology groups H i (\Delta(P ); \Delta(P nA)): Clearly
the ith quotient chain group C i (\Delta(P ))=C i (\Delta(P nA)) consists of classes of chains
going through at least one element of A: Since A is an antichain, each such chain
must go through exactly one element of A: Now consider the boundary ~
@ map of
this relative complex. By the preceding remarks it is clear that if
(representative of) a nonzero relative i-chain, where x
is the unique element of A in the chain, then
~
where as usual the hat denotes suppression of an element.
Hence the complex of relative chains is isomorphic to the direct sum of tensor
products (over the integers) of chain complexes
a2A
~
)\Omega ~
By hypothesis, in each summand of (1.3) at least one of the intervals has free
homology. Consequently, by the K-unneth theorem, the relative homology is given
by
a2A
~
P\Omega ~
Now use the fact that for the intervals ( - 0; a) and (a; - 1) in P; the reduced homology
vanishes except in the top dimension. Hence in the above sum, the right-hand side
vanishes unless 2: The
first conclusion now follows from (1:2):
For the second statement, observe that if the top homology of P is free, then
from the exact sequence (1.1) it is clear that the homology of PnA in degree r \Gamma 2
(being a subgroup of a free abelian group). The relative homology is free as
well, and hence PnA has free homology in both dimensions.
As a by-product of this general result, we obtain a simple proof of the fact
that rank-selection preserves the Cohen-Macaulay property, a theorem due inde-
pendently, and with different proofs, to Baclawski, Stanley and Munkres.
Corollary 1.2. ([Ba1, Theorem 6.4], [St1, Theorem 4.3], [Mu2, Corollary 6.6])
Let P be a Cohen-Macaulay poset over the integers, and let Q be a rank-selected
subposet of P: Then Q is Cohen-Macaulay over the integers.
Proof Let some subset of -
It suffices to consider the case
of removing one rank, so that A is an antichain. Then Q is ranked of rank
where r is the rank of P: Hence ~
H 0: Now use the preceding result.
The same argument applies to an open interval in Q; which either coincides with
the corresponding open interval of or else is obtained from it by deleting one
rank. Hence if Q is P minus one rank, then Q is Cohen-Macaulay.
If P is an arbitrary poset and A is an antichain of -
special case of a
well-known formula for the M-obius number -(P ) of P (see [Ba2, Lemma 4.6]) says
that
Noting that -(P ) is simply the reduced Euler characteristic of the order complex of
we have the following formula (which also
follows from the proof of Theorem 1.1):
Corollary 1.3. Let P and A be as in Theorem 1.1 (assume all homology is free).
Then
dim ~
dim ~
H r\Gamma2 (P
We return now to the partition lattice \Pi n : Recall that if - is an integer partition
of n; then a set partition x in \Pi n is said to be of type - if x has block sizes
n be the subposet obtained by deleting the
antichain consisting of all elements of type (k; 1 n\Gammak
For k - 3; the poset Q k
n is ranked of rank (n \Gamma 1): For let a 2 \Pi n have a unique
nontrivial block of size k; and suppose a covers x and is covered by y: Then all
blocks of x are singletons except possibly for two blocks whose union is the
k-block A of a: Assume first that B 1 has size less than or equal to k \Gamma 2: Since y
covers a; either y is a modular element with unique nontrivial block A[ fpg or else
y has two nontrivial blocks A and fp 1 here the p's are singletons of a: In either
case there is a non-modular element z in \Pi n in the interval (x; y) : in the first case
merge the block B 1 of x with the singleton fpg to form z: In the second case merge
the singletons p 1 and
Now suppose x is obtained from a by splitting the unique nontrivial block A into
the block B 1 and a singleton (Thus x is itself modular.) If y is modular with
nontrivial block A[ fpg; merge the singletons p and If y has a second nontrivial
block then merge the singletons p 1 and In each case this produces a
non-modular partition z in the interval (x; y):
Note that Q 2
n is the rank-selected subposet obtained by deleting the atoms.
For
n is not a lattice. The smallest interesting example is Q 3
complex is disconnected and one-dimensional, and is homotopy equivalent to a
wedge of two 0-spheres (see Figure 2 of the Introduction). In particular Q 3is not
Cohen-Macaulay. In the next section we shall see that this is true in general.
Finally we note that Theorem 1.1 gives the following fact, which will play a
crucial role in the next section.
Proposition 1.4. The reduced integral homology of Q k
n vanishes in all dimensions
different from
In the next section we shall show that the homology of Q k
n is concentrated in a
unique degree. It is not difficult to construct examples of a Cohen-Macaulay poset
P and an antichain A which show that PnA can have homology in both degrees.
We can relax the hypotheses of Theorem 1.1 by considering the appropriate
analogue for simplicial complexes. Recall that the link 'k(v) of a vertex v of a
simplicial complex \Delta is the subcomplex whose simplices are the faces F of \Delta such
that
F and F [ fvg is (a simplex) in \Delta:
Theorem 1.5. Let \Delta be a finite simplicial complex, and let A be a subset of the
vertices of \Delta such that every facet (i.e., maximal face) of \Delta has at most one vertex
in A. Assume that there is an integer d such that
(i) the ith reduced homology of \Delta vanishes for all degrees i 6= d; and
(ii) For every vertex a 2 A; the ith reduced homology of the link of a in \Delta vanishes
for all degrees i
Let \Delta 0 be the subcomplex of \Delta obtained by removing all faces having a vertex in
the set A: Then ~
addition \Delta and the
links 'k(a) have free integral homology for all a 2 A; then \Delta 0 also has free integral
homology.
Proof The following observations are sufficient, since the essential ideas are as in
the proof of Theorem 1.1. The key point now is that the relative chain complex
isomorphic to the direct sum, over a 2 A; of the chain complex of
the suspension of the link 'k(a) of a in \Delta:
Hence the relative homology is given by the formula
~
a2A
~
But by hypothesis, the link 'k(a) has zero homology in degrees j 6= That
is, the relative homology is zero for degrees 6= d: Now the conclusion follows exactly
as in Theorem 1.1.
In the particular case when \Delta is a pure d-dimensional Cohen-Macaulay simplicial
conditions (i) and (ii) of the above theorem are automatically satisfied.
The conclusion of Theorem 1.1 may thus be obtained by taking \Delta to be the order
complex of a Cohen-Macaulay poset of rank d 2: Note in particular that the
hypothesis, in Theorem 1.1, that for each a 2 A at least one of the intervals ( - 0; a)
or (a; - 1) have free homology, is not necessary for the first conclusion.
The full result of [Mu, Corollary 6.6] also follows from the above. In addition,
just as we obtained Corollary 1.2, we recover Stanley's result on subcomplexes
of completely balanced Cohen-Macaulay complexes (see [St1, Theorem 4.3]) from
Theorem 1.5. The details are identical to the above proof and the proof of Corollary
1.
2. A homotopy equivalence
We begin by stating a powerful theorem of Quillen, which we shall use repeatedly
throughout this paper. For a survey of the variations on this useful principle see
[Bj3].
Theorem 2.0. (Quillen's fibre lemma) [Q, Proposition 1.6] Let P and Q be bounded
posets and let
Q be an order-preserving map of posets. Assume that for
all a 2 -
Q; the fibre F a
ag is contractible. Then f induces a homotopy
equivalence of the order complexes \Delta(P ) and \Delta(Q): (The same conclusion
holds if the fibre F 0
ag is contractible for all a 2 -
Recall that P k
n is the subposet of \Pi n obtained by deleting all modular elements
of type
It follows from the remarks
about
that P k
n is also ranked, but of rank the atoms have been
deleted). The aim of this section is to show that the (n \Gamma 4)-dimensional complex
n ) and the (n \Gamma 3)-dimensional complex \Delta(Q k
have the same homology. In
fact the following stronger result holds.
Theorem 2.1. The order complexes of P k
are homotopy equivalent. More
generally, for any subset I of the integers the inclusion c
homotopy equivalence of the corresponding order complexes

Proof We shall only prove the first statement, since the second follows by the
identical argument.
Consider the inclusion map
fibre lemma we need
only show that the fibres F a
ag are contractible. This is clearly
true if a 2 P k
a is a modular element with a unique
nontrivial block B of size 1: For notational convenience assume a is
the partition (with in which the elements are the
singletons. Viewing a as a partition of distinguished
element consisting of the block B; the fibre F a is poset isomorphic to the poset
Rn\Gammai+1 (S) obtained from b
\Pi n\Gammai+1 by removing the set S consisting of all modular
elements whose unique nontrivial block contains the distinguished element B; and
is of cardinality s;
The fact that these posets are contractible follows from the next lemma:
Lemma 2.2. Let k - 2; and let S be the subset of modular elements of \Pi n of type
such that n is in the unique nontrivial block of every element
of S:
Let Rn (S) be the subposet of \Pi n obtained by removing all elements of S: Then
(the order complex of) Rn (S) is contractible.
Proof Let ff n denote the partition in \Pi n consisting of exactly two blocks, one of
which is the singleton block fng: Note that ff n 2 Rn (S): Define a
Rn (S) 7!
\Pi n by
Here - denotes the meet operation in the lattice \Pi n : Note that the effect of taking
the meet of x with ff n is to fix x if n is a singleton of x; or else to produce a new
partition x 0 where x 0 is obtained from x by splitting the block B containing n into
two blocks so that n is a singleton. Now observe that
(a) f is order-preserving;
(b) the image of f is contained in
Rn (S); for this it suffices to note that - 0 is not
in the image of f; and this is ensured by the fact that S contains all the atoms
whose unique nontrivial block contains n;
(c) f(x) - x for all x:
Conditions (b) and (c) together imply that the fibres of f are contractible. Hence,
by Quillen's fibre lemma again, f is a homotopy equivalence between
Rn (S) and
the image of f: But the image of f clearly consists of all partitions in \Pi n in which
n is a singleton, except for the least element of \Pi That is, the image of f is
poset-isomorphic to -
where the - 1 is provided by the two-block partition
Hence the image of f is contractible.
This completes the proof of Theorem 2.1.
Remark 2.2.1. The conclusion of Lemma 2.2 is valid for more general subsets
S of modular elements, as long as S contains all the modular elements of type
(i.e., atoms) and that n is in the nontrivial block of all elements of S:
The special case of Lemma 2.2 when S consists only of atoms follows from [Wal1,
Theorem 6.1]; here S is the set of complements of the two-block partition ff n in
which n is a singleton (for elaborations of this principle see the references in [Bj3]).
Theorem 2.3. Let 2 - k - 1: The reduced integral homology of the posets P k
and
vanishes except in dimension (n \Gamma 4): This holds
more generally for the posets
In particular for n - 4 and k - 3; the (pure) posets Q k
are not Cohen-Macaulay.
Proof From Theorem 2.1 it follows that the two posets have the same homology.
n has rank (n \Gamma 2); its order homology vanishes for all degrees greater than
On the other hand, Proposition 1.4 says that Q k
can have nonvanishing
homology only in degrees 4: The result follows.
As one more application of these arguments, we also obtain the following
Theorem 2.4. The poset
is homotopy-equivalent to b
Hence the order complexes of P
n and
have the homotopy type of a wedge
of (n \Gamma 2)! spheres of dimension (n \Gamma 4):
Proof Consider the map f :
\Pi n as defined in Lemma 2.2. The image of
this map consists of all partitions in b
\Pi n such that n is a singleton, except for the
two-block partition ff n of Lemma 2.2; it is therefore isomorphic to b
(with respect to the image!) are contractible by the same argument as in the proof
of Theorem 2.1. More precisely, we consider only fibres F a
ag
for a in the image of f: (Note that the fibre of the two-block partition ff n of Lemma
2.2 is empty and hence not contractible.) The result now follows by Lemma 2.2
and Quillen's fibre lemma.
The final statement follows from the well-known fact that the order complex of
the partition lattice \Pi n is shellable ([Bj1, Example 2.9]), and hence (see [Bj2, Theorem
1.3], [BW, Theorem 4.1]) has the homotopy type of a wedge of (n \Gamma 1)! spheres
of dimension (n \Gamma (see [St2] for the M-obius (Betti) number computation).
From Corollary 1.3 we now have
Corollary 2.5. For
n denote the common dimension of the
unique nonvanishing homology of the posets P k
n and
1g: Then
In order to investigate whether or not P k
n is Cohen-Macaulay, we need to look
at proper intervals in the poset. Note that the obvious analogue of Theorem 2.1 is
false for arbitrary intervals of P k
For example, in Q 5
6 the interval J
is homotopy equivalent to a wedge of six spheres S 2 (it coincides with the same
interval in \Pi 6 ), whereas in P 5
6 the interval has rank 3. It is not
hard to see that J has the homotopy type of a wedge of 7 spheres of dimension 1.
To obtain information on intervals ( - 0; y) in P k
we need the following generalisation
of Lemma 2.2.
Lemma 2.6. Let S be the subset of the modular partitions in \Pi n as in Lemma 2.2
and let y 2 b
is in a nontrivial block of y: Then (the order
complex of) the subposet [ - 0; y]nS of the interval [ - 0; y] is contractible.
Proof Note that is simply the interval [ - 0; y] in the poset Rn (S) of
Lemma 2.2. Restrict the map f of Lemma 2.2 to the interval -
I
I : The image of f consists of all partitions in I such that n is a
singleton, except for the - 0: Also f(y) 2 -
I : this is because n is not a singleton in
and hence f(y) 6= y: Clearly f(y) is the (unique) greatest element of f(I); and hence
f(I) is contractible. Now by the arguments of Lemma 2.2, I is contractible.
Proposition 2.7. Let y 2 P k
denote the interval ( - 0; y) in P k
denote the subset of the interval ( - 0; y) in Q k
obtained by removing the set M y;k of
all modular elements whose unique nontrivial block coincides with a block of
has size - k: Then the inclusion J ,! J 0 induces a homotopy equivalence of order
complexes.
Proof This follows by checking that the fibres are contractible, as in Theorem 2.1,
except that now we make use of Lemma 2.6. Note that removal of the elements in
the set M y;k is ncessary in order to apply the lemma.
Proposition 2.8. Let be as in Proposition 2.8. Then the homology of J
(and vanishes in all degrees different from rank \Pi n (y) \Gamma 3;
the top dimension of the interval J of P k
Proof Proposition 2.7 says that J and J 0 have the same homology. There are two
observations. First, J 0 is obtained from the interval ( - 0; y) in \Pi n by deleting
an antichain. Hence by Theorem 1.1, the nonzero homology of J 0 can occur only
in degrees rank \Pi n 3: Second, the dimension of the order
complex of J is the smaller of these two degrees. The result follows.
be an interval in the poset P k
First assume there
are two nontrivial blocks of x which are contained in distinct blocks of y: In this
case it is clear that the interval [x; y] of P k
coincides with the interval between x
and y in \Pi n ; and is therefore Cohen-Macaulay.
Next suppose all the nontrivial blocks of x are contained in a single block of y:
Let a i be the size of the nontrivial block A i of s be the size
of the nontrivial block B of y which contains them. Note that r - 2: Let x 0 be
the partition of the set B induced by x (x 0 has r nontrivial blocks A i and
singletons). Then the interval [x; y] of P k
n is isomorphic to a product of the interval
together with a collection of partition lattices.
These observations and the preceding results show that P k
n is Cohen-Macaulay
if and only if all intervals of the form [x; - 1] have homology which vanishes in all
dimensions less than the highest. Although the analogue of Theorem 2.1 does hold
for such intervals, this fact is not as helpful in this case. The difficulty occurs
because there is no longer a shift in the dimensions of the order complexes of the
intervals J and J
Proposition 2.9. Let be an interval in P k
be the interval
in the poset Q k
the inclusion map J ,! J 0 is a homotopy equivalence,
and hence J and J 0 can have nonvanishing reduced homology only in dimension
denotes the rank function of
Proof The statements of the theorem are immediate if J 0 (and hence J) coincides
with the interval [x; - 1] of \Pi n ; i.e, if x is not smaller than a modular element of type
Hence we consider the other case.
We use the same argument as in Theorem 2.1. We need to show that the fibres
ag for a 2 J 0 nJ of type (j; 1 n\Gammaj are contractible.
Let B be the unique nontrivial block of a:
Just as in the proof of Theorem 2.1, the fibre F a is isomorphic to a poset Rm (S)
as in Lemma 2.2, for a suitable choice of m (namely, the number of blocks of a)
and S: Hence it is contractible by the lemma.
The conclusion now follows from Theorem 1.1.
Fix an integer a between 2 and k: Define T -k
n;a to be the
subposet obtained from \Pi n by deleting all modular elements x of type (j; 1 n\Gammaj ); a -
such that the unique nontrivial block of x contains the a largest integers
n;a to be the subposet obtained from \Pi n by
deleting all modular elements x of type (k; 1 n\Gammak ); such that the unique nontrivial
block of x contains the elements
n be of rank
and assume x has at least one singleton block. Then it is easy to see that [x; -
is
poset isomorphic to T -k
is poset isomorphic to T =k
parameters m and a (viz., m is the number of blocks of x; and a is the number of
nontrivial blocks of x).
Hence Proposition 2.9 may be rephrased as follows:
1: The inclusion T -k
n;a is a homotopy equivalence.
Note that the order complexes of T -k
n;a and T =k
n;a both have the same dimension
hence by Theorem 1.1 we can only conclude that they both have
nonvanishing homology only in degrees 4: Moreover from Corollary
1.3 we have
dim ~
In particular, since the right-hand side is clearly positive, we are forced to conclude
that homology is nonzero in degree (n \Gamma 3):
Fortunately it is not hard to show that
Proposition 2.10. The posets T =k
n;a are (pure) shellable. Hence the posets T -k
n;a and
n;a are both homotopy equivalent to a wedge of (n \Gamma a)!
(n\Gammaa)!
spheres
of dimension (n \Gamma 3): Hence (the order complexes of) all intervals of the form [x; - 1]
and [x; y]; x 6= - 0; in P k
n have the homotopy type of a wedge of spheres.
Proof We shall use the following simple EL-labelling of the partition lattice due
to Wachs [Wac]. If u ! v is a covering relation in \Pi n ; so that v is obtained from
u by merging two blocks define the label of the edge v) to be
We shall show that this EL-labelling restricts to an EL-labelling of
With respect to this labelling, there is a unique strictly increasing chain c (x;y)
in every interval (x; y) of \Pi Proposition 2.8], it suffices to show that for
every x ! y in T =k
n;a ; the chain c (x;y) is a chain of T =k
We need only consider those elements
n;a such that the interval
contains elements forbidden in T =k
Such an element z must have a unique
nontrivial block B of size k containing the a largest integers
the unique strictly increasing chain c contains
the element z; since x 6= z; it must therefore have the label n on one of its edges.
This edge can only be the last edge of the chain, which implies that z = z
contradicting the fact that y 2 T =k
The remaining statements follow from the remarks preceding the proposition.
Putting together the work of this section, we have shown
Theorem 2.11. The poset P k
n is Cohen-Macaulay over the integers, and for all
the homology of the interval [x; y] is free.
Small examples suggest that all the intervals in fact have a nice topological
structure. We have
Conjecture 2.12. For the order complex of P k
n is shellable. More
generally, any interval [x; y] in the poset P k
n is either contractible or homotopy
equivalent to a wedge of spheres of the highest possible dimension rank(y)\Gammarank(x)\Gamma
(here rank is the rank function in P k
In particular P k
n is homotopy equivalent
to a wedge of (fi k
For
n is simply a rank-selected subposet of \Pi n ; hence its order complex
is shellable by [Bj1, Theorem 4.1]). It follows from the general theory of shellability
(see [Bj2, Theorem 1.3] and [BW, Theorem 4.1]) that the order complex has the
homotopy type of a wedge of spheres. The subposet P 3
(in fact the intersection
lattice of a codimension 2 orbit arrangement), was shown to be CL-shellable by
this author and V. Welker (1993, unpublished), and independently in recent far-reaching
work of Kozlov ([Kz]). However this argument seems to break down at a
can be seen that upper intervals (x; - 1) in P k
n are not
totally semimodular, making it difficult to show CL-shellability.
Similarly, we conjecture that
Conjecture 2.13. All intervals [x; y]; in the posets Q k
are either contractible or homotopy equivalent to a wedge of
spheres all of the same dimension d; where d is either the highest possible dimension
or one less. In particular
equivalent to a wedge of (fi k
3. The representation of the symmetric group Sn on the homology
In this section all homology is taken over the field of complex numbers. We shall
first compute the Sn-module structure of the unique nonvanishing homology of the
poset
: For this we need to recall some of the results of [Su]. For a finite poset Q
and a finite group G of autormorphisms of Q; we denote by Alt(Q) the Lefschetz
(G-)module of Q; i.e.,
Theorem 3.1. (See [Su, Theorem 1.10 and Remark 1.11]) Let P be a Cohen-Macaulay
poset of rank finite group of automorphisms of G-invariant
subposet of P:
Then as G-modules:
P\Omega ~
G c
where the sum runs over all representatives of G-orbits of chains c of elements not
in Q; and G c is the stabiliser of the chain c in P:
In the special case when PnQ is an antichain, this result simplifies to give
Theorem 3.2. Let P be a Cohen-Macaulay poset of rank r and G a finite group
of automorphisms of P: Let Q a G-invariant subposet of P such that PnQ is an
antichain. Then as a G-module, the Lefschetz module Alt(Q) of Q is determined
by
P\Omega ~
Another way to obtain Theorem 3.2 is to observe that all the maps in the exact
homology sequence of the pair (P; Q) are G-equivariant; consequently the proof of
Theorem 1.1 can be made G-equivariant to yield Theorem 3.2.
The hypotheses of the next theorem arise frequently in the study of subposets of
the partition lattice. The theorem is a general result on the homology representation
of upper intervals in posets of partitions, and was used extensively in [Su]. The
details of the proof are identical to the proof of [Su, Theorem 1.4].
Theorem 3.3. [Su] Let An ' \Pi n be a family of posets of set partitions and let
An be of type - where - is an integer partition of n with m i blocks of size
i: Assume that (x; - 1) An is poset isomorphic to a poset B r ; where r is the number
of blocks of x: There is an action of the symmetric group S r on the poset B r ; by
permuting the blocks of x: Let ff r denote the (possibly virtual) representation of S r
on the Lefschetz module Alt(B r Note that there is a copy of \Theta i Sm i in S r : Let G -
denote the stabiliser of x; thus G - is the direct product of wreath product groups
\Theta i Sm i [S i ]; where S a [S b ] is the wreath product group obtained by letting S a act on a
copies of
Finally assume that the restriction of the representation ff r to \Theta i Sm i can be
written (uniquely) as the following sum of irreducible modules:
ff r
- denotes the ordered tuple of partitions - (i) of denotes the the
irreducible Sm i -module indexed by the integer partition - (i) :
Then the (possibly virtual) representation of G - on the Lefschetz module of
An ) is given by
denotes the wreath product Sm i [S i ]-module of the irreducible V - (i)
with the trivial S i -module 1
The formula in the preceding theorem is more compactly expressed in terms of
the plethysm operation and symmetric functions; see [Su] for details.
For the purposes of this paper we shall only need to apply Theorem 3.3 to
the upper interval (x; - 1) of the partition lattice \Pi n ; when x is an element of type
In this case all the posets involved are Cohen-Macaulay. We write -n
for the representation of Sn on the top homology of \Pi n : The interval (x; - 1) is
isomorphic to the partition lattice \Pi n\Gammak+1 ; and hence in applying Theorem 3.3 we
need to compute the restriction of -n\Gammak+1 to Sn\Gammak \Theta S 1 : But by [St2] this is just
the regular representation of Sn\Gammak : Hence we have the following result, which was
also worked out in [Su].
Corollary 3.4. (See [Su, Example 2.11]) Let x be an element of type (k; 1 n\Gammak ) in
The representation of Sn\Gammak \Theta S k on the top homology of the interval (x; - 1) is
ae
where ae n\Gammak denotes the regular representation of Sn\Gammak :
It is now easy to compute the homology representation of Q k
Theorem 3.5. Let 2 - k - 1: The representation of the symmetric group Sn
on the unique nonvanishing homology ~
n ) is given by
(ae
Proof Let x 0 denote any partition of type (k; 1 n\Gammak Theorem 3.2 gives the following
equality of Sn-modules:
~
)\Omega ~
Now use Corollary 3.4 and the fact that ( - 0; x 0 ) is isomorphic to \Pi k :
Our next goal is to compute the homology representation of P k
We indicate two
approaches. The first is a straightforward application of Theorem 3.2, and uses the
same arguments as in the proof of Theorem 3.5.
Theorem 3.6. Let 2 - k - As an Sn-module the unique nonvanishing
homology ~
n is given by
here ae n\Gammak denotes the regular representation of Sn\Gammak :
Proof We proceed by induction on k: The result holds for by [Su, Theorem
2.10 and Example 2.11]. Assume it holds for all parameters
is the subposet of P
obtained by deleting the elements of type (k; 1 n\Gammak
if x 0 is a partition of type (k; 1 n\Gammak ); then using Theorem 3.2 (with
n and
we have the equality of Sn-modules
~
j\Omega ~
The interval
in P
n is isomorphic to a partition lattice, and the (Sn\Gammak \Theta
)-module structure of its homology follows from Corollary 3.4. The interval
in P
n is clearly isomorphic to P
induction hypothesis the
structure of the homology of P
k as an S k -module is given by the representation
It follows that as an (Sn\Gammak \Theta S k )-module, the homology of ( - 0; x
is
given by 1
Now by routine manipulations the result follows.
Corollary 3.7. Let 2 - k - 1: The character values of the representation of
the symmetric group Sn on the unique nonvanishing homology of P k
n and of Q k
for an element in Sn of cycle-type oe; are
d -(d)
d
d
d -(d)
d
d
d
0; otherwise:
Proof By a well-known result of Hanlon (see [Ha1, Theorem 4.1], [St2, Lemma
7.1]), the character values of the representation -n on an element of cycle-type oe
in Sn are given by
d
-(d)
d
d
d
0; otherwise:
Now the result follows from formula (3.1).
By Theorems 3.5 and 3.6, the posets P k
have Sn -isomorphic homology.
In fact we can show that the homotopy equivalence of Theorem 2.1 is an Sn -
homotopy, thereby establishing the result in another way. First we state the group-
equivariant version of Quillen's fibre lemma.
Theorem 3.8. (See, e.g., [Be, Chapter 6]) Let P and Q be bounded posets, let G
be a finite group of automorphisms of P and Q; and let
Q be an order-preserving
G-map of posets. For a 2 -
a denote the stabiliser of a: Assume
that for all a 2 -
Q; the fibre F a
ag is G a -contractible (i.e.,
the fixed-point subposet F Ga
a of points in F a fixed by G a ; is contractible). Then f
induces a G-homotopy equivalence of the order complexes \Delta(P ) and \Delta(Q): (The
same conclusion holds if the fibre F 0
ag is G a -contractible for
all a 2 -
In order to show that the homotopy equivalence of Theorem 2.1 is group equi-
variant, we need to show that the fibres F a in the proof of the theorem are G a -
contractible, where G a is the stabiliser of the element a of type (j; 1 n\Gammaj (Thus G a
is isomorphic to the Young subgroup Sn\Gammaj \Theta S j :) This in turn will follow from the
group-equivariant version of Lemma 2.2.
It is not hard to see that the homotopy equivalence of Lemma 2.2 is an
homotopy, where we identify with the subgroup of Sn which fixes n: For any
subgroup H of is easy to check that the map f restricts to a homotopy
equivalence on the fixed point subposet Rn (S) H consisting of points fixed by H;
and that the image remains contractible. Hence the posets Rn (S) are in fact
contractible.
Proposition 3.9. The inclusion c
more generally, for any subset
I the inclusion
c
induces an Sn -homotopy equivalence of the corresponding order complexes.
These observations also imply that the homotopy equivalence between
and -
\Pi n\Gamma1 in Theorem 2.4 is an -homotopy. Because the case
particular interest, we state it separately:
Corollary 3.10. The posets P
n and Q
more generally, the posets
are -homotopy equivalent to -
\Pi have homology modules that are Sn -
isomorphic to the representation
This is the representation of Sn computed by Sarah Whitehouse [W] on the tree
complex of Alan Robinson (see also [R], [RW] and [Ha2]). It follows from Corollary
3.8 that the restriction to Sn\Gamma1 is the representation
Denote by -
-n the lifting of -n\Gamma1 given by the representation (3.2). Let V (n\Gamma1;1)
denote the irreducible Sn-module indexed by the integer partition (n \Gamma
basic manipulations one sees that
a formula which appears in [GK].
4. Conclusion
In this final section we discuss some questions raised by the phenomena exhibited
in this paper for the partition lattice \Pi
Let Mn denote the subposet of \Pi n consisting of the modular partitions in -
together with the elements - 0 and - 1: Clearly Mn is just the truncated Boolean lattice
of subsets of an n-set, with the subsets of size 1 (i.e., the rank one elements) deleted.
It follows from Stanley's theory of R-labellings ([St3]) that the M-obius number is
On the other hand by Theorem 2.4, we know that P
number (\Gamma1) n\Gamma4 (n \Gamma 2)!
Hence we have, at the level of M-obius numbers, the equation
We also have the topological result that
The formula (3.3) of the preceding section further suggests that the factorisation
carries over to the homology, at the level of Sn-modules, with the introduction
of a sign twist. For by a result of Solomon ([So], see also [St2]), the representation
of Sn on the homology of Mn is precisely the irreducible indexed by the integer
partition (2; 1 n\Gamma2 Hence (3.3) says that as modules over the integers,
Mn
)\Omega ~
and as Sn-modules,
Mn
)\Omega ~
It would be interesting to see if these phenomena, e.g., (4.1) and (4.3), occur
for other instances of removing modular elements from a supersolvable geometric
lattice. For example, the analogues of (4.1) and (4.3) hold trivially for the Boolean
lattice, where every element is modular. The analogue of (4.2) however is clearly
false.



--R



"Representations and Cohomology, "


"Handbook of Combinatorics,"

"Geometry, Topology and Physics,"
The fixed-point partition lattices
Otter's Method and the Homology of Homeomorphically Irreducible k-Trees



"Elements of Algebraic Topology,"
Topological Results in Combinatorics
Homotopy properties of the Poset of Nontrivial p-Subgroups of a Group
The space of fully grown trees
The tree representation of
A decomposition of the group algebra of a finite Coxeter group

Some aspects of groups acting on finite posets
"Enumerative Combinatorics,"
The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice
A basis for the homology of the d-divisible partition lattice
Topology and Combinatorics of Ordered Sets
Homotopy type and Euler characteristic of partially ordered sets

--TR
Enumerative combinatorics
Otter''s method and the homology of homeomorphically irreducible <italic>k</italic>-trees
Topological methods

--CTR
Sarah Whitehouse, The Integral Tree Representation of the Symmetric Group, Journal of Algebraic Combinatorics: An International Journal, v.13 n.3, p.317-326, May 2001
Federico Ardila , Caroline J. Klivans, The Bergman complex of a matroid and phylogenetic trees, Journal of Combinatorial Theory Series B, v.96 n.1, p.38-49, January 2006
group representation;poset;homology;homotopy;set partition