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Sarah Whitehouse, -(Co)homology of
commutative algebras and some related
representations of the symmetric group,
Ph.D. thesis, Warwick University, 1994.
2
The module Lien
Let V be a complex vector space with
basis x ; : : : ; xn. Let Lien be the part
of the free Lie algebra L(V ) that is of
degree one in each xi.
1
dim Lien = (n 1)!
Basis: [ [[x ; xw ]; xw ]; : : : ; xw n ];
where w permutes 2; 3; : : : ; n.
1
(2)
3
(3)
(
)
The symmetric group Sn acts on Lien
by permuting variables.
(1; 2) [[x
= [[x
1
; x3]; x2]
2
3
1
1
; x2]; x3
1
3
2
4
= [[x ; x ]; x ]
] + [[x ; x ]; x ]
For any function f : Sn ! C , recall
that
1 X
ch f =
f (w )p w ;
n!
w2Sn
where if w has i i-cycles then
1 2
p w = p p ;
(
(
P
)
1
)
2
with pk = i xki .
In particular, if is the irreducible
character of Sn indexed by ` n, then
ch = s;
the Schur function indexed by .
5
= subgroup of Sn
generated by (1; 2; : : : ; n)
Cn
Theorem. As an Sn-module,
n e i=n:
Lien = indS
Cn
Hence
1X
n=d
(d)pd :
ch(Lien) =
n
djn
2
6
Theorem. Let be the irreducible
character of Sn indexed by ` n.
Then
hLien; i = #SYT T of shape ;
maj(T ) 1 (mod n);
where
X
maj(T ) =
i:
i+1
1 3 4
2
below
i
1 2
3
4
ch Lie = s + s
4
31
7
211
Other occurrences of Lien
n = lattice of partitions of [n],
ordered by renement
~ i(n)
H
= ith (reduced) homology group
(over Q , say) of (order complex of) n
As Sn-modules,
0
; i 6= n 3
~
Hi(n) =
sgn Lien; i = n 3:
8
1234
123
12
124
13
134
234
23
14
12-34
13-24
34
24
Π4
9
14-23
Tn = set of rooted trees
0
with endpoints labelled 1; 2; : : : ; n;
and no vertex with exactly one child
Schroder (1870) showed (the
fourth of his vier combinatorische Probleme ) that
X
xn
#Tn = (1 + 2x ex)h i ;
n!
n
where F (F h i) = F h i(F (x)) = x.
For T; T 0 2 Tn , dene T T 0 if T
can be obtained from T 0 by contracting
internal edges.
Note:
1
0
1
1
1
0
10
2
3
1
3
1
1
3
2
1 2 3
Theorem. As Sn-modules,
0
; i 6= n 3
~
Hi(Tn ) =
sgn Lien; i = n 3:
0
11
2
A \hidden" action of Sn on Lien
(Kontsevich)
1
Let Ln be the free Lie algebra on n
generators x ; : : : ; xn, and let
h ; i = nondegenerate inner product
on Ln satisfying
h[a; b]; ci = ha; [b; c]i:
For ` 2 Lien and w 2 Sn, let
h`; xniw = h`w ; xw n i ! h`0; xni:
1
1
straighten
(
)
So the map w : Lien ! Lien dened by w(`) = `0 denes an Sn action
on Lien , the Whitehouse module Wn for Sn or the cyclic Lie operad. (Explicit description of action of
(n 1; n) by H. Barcelo.)
1
1
12
1
dim Wn = dim Lien
Wn
1
= indSSn Lien
n 1
ch Wn =
p1
n
X
1
dj n
1X
(
n
djn
= (n 2)!
1
Lien:
n
(d)pd
(
=d
1)
1)
n=d
(d)pd
hWn; i = #SYT T of shape ,
maj(T ) 1 (mod n 1)
#SYT T of shape ,
maj(T ) 1 (mod n)
(Not a priori clear that this is 0.)
13
s2;
s111;
s22;
s311
s42 + s3111 + s222
s511 + s421 + s331 + s3211 + s22111
+ 2s + 422
Getzler-Kapranov:
Mn
;
= Lien;
where M is the irreducible Sn-module
indexed by .
Wn
1 1
14
Other occurrences of Wn
Nonmodular partitions (Sundaram)
n = f 2 n : has at least two
nonsingleton blocksg
Theorem. As Sn-modules,
0
; i 6= n 4
~
Hi(n) =
sgn Wn; i = n 4:
15
123
45
124 134
35 25
234 125 135
15 34 24
235 145 245
14
23 13
345
12
25 24 23 15 14 13 15 14 12 15 13 12 14 13 12
34 35 45 34 35 45 24 25 45 23 25 35 23 24 34
1 1 1 2 2 2 3 3 3 4 4 4 5 5 5
Σ5
16
Homeomorphically irreducible trees
(Hanlon, after Robinson-Whitehouse)
Tn = set of free (unrooted) trees
with endpoints labelled 1; 2; : : : ; n,
and no vertex of degree two
For T; T 0 2 Tn, dene T T 0 if T
can be obtained from T 0 by contracting internal edges.
17
1
3
1
2
1
2
2
4
3
4
4
3
1
2
3
4
Theorem. As Sn-modules,
0
; i 6= n 4
~ i(Tn) H
= sgn W ; i = n 4:
n
18
Partitions with block size at most k,
(n 1)=2 k n 2 (Sundaram)
n;k = poset of partitions of f1; : : : ; ng
with block size at most k
Assume (n 1)=2 k n 2.
19
12
14-23
13-24
12-34
34
13
24
14
23
Π 4,<2
Note:
~ i( ; ; Z ) H
=
4
2
20
0; i 6= 0
Z ; i = 0:
2
Theorem (Sundaram):
0; i 6= n 4
Z n
; i = n 4:
Moreover, as Sn-modules,
~ n (n;k ; Q ) H
= sgn Wn:
~ i(n;k ; Z ) H
=
(
4
21
2)!
Not 2-connected graphs (Babson et
al., Turchin)
= loopless graph without multiple edges on the vertex set f1; : : : ; ng.
Identify G with its set of edges.
G is 2-connected if it is connected,
and removing any vertex keeps it connected.
G
n = simplicial complex of not 2connected graphs on 1; : : : ; n.
22
~ 1(3; Z )
H
23
= Z
Theorem (Babson-Bjorner-Linusson-ShareshianWelker, Turchin):
0; i 6= 2n 5
~
Hi(n; Z ) =
Z n
; i = 2n 5
(
2)!
Moreover, as Sn-modules,
~ n (n; Q ) H
= Wn:
2
5
24
General technique:
acts on ;
w = fF 2 :
G
w
2G
= Fg
w F
Hopf trace formula =)
X
w
~ i())
~( ) = ( 1)i tr(
w; H
| {z }
character value at w
of G acting on H~ i()
Use topological or combinatorial techniques such as lexicographic shellability
to show that H~ i() vanishes except for
one value of i.
25
Explicit Sn-equivariant isomorphisms (up to sign) between the cyclic
Lie operad, the cohomology of the tree
complex Tn, and the cohomology of the
complex n of not 2-connected graphs
were constructed by M. Wachs.
Note:
26
A q-analogue of a trivial Sn-action
(Hanlon-Stanley)
For w 2 Sn and q 2 C , let
`(w ) = #inversions of w
X `w
q
w 2 C Sn
n (q ) =
w2Sn
n(q ) acts on C Sn by left multiplication.
Theorem (Zagier, Varchenko):
(
det n(q) =
n Y
k =2
1
27
)
q k (k
1)
n
!(
n k +1)=k (k
1)
Proof (sketch). Let
Tn ( q )
Easy:
=
n
X
qj
1
j =1
(n; n 1; : : : ; n j +1):
n(q ) = T2(q )T3(q )
Tn ( q ) :
Let a b and
[a; b] = (a; a 1; : : : ; b) 2 Sn:
Let
Gn =
(1 qn[n 1; 1])(1 qn [n 1; 2]) (1
1
(1
qn
=
[n; 1])(1
Hn 1
1
Duchamps et al.:
qn
Tn
28
2
q 2)
[n; 2]) (1 q[n; n 1]):
= GnHn:
Theorem. Let = e i=n n
as Sn-modules we have
ker n( ) = Wn:
2
29
(
1)
. Then
Transparencies available at:
http://www-math.mit.edu/
rstan/trans.html
30
REFERENCES
1. E. Babson, A. Bjorner, S. Linusson, J. Shareshian, and
V. Welker, Complexes of not i-connected graphs, MSRI
preprint No. 1997-054, 31 pp.
2. G. Denham, Hanlon and Stanley's conjecture and the Milnor gre of a braid arrangement, preprint.
3. G. Duchamps, A. A. Klyachko, D. Krob, and J.-Y. Thibon,
Noncommutative symmetric functions III: Deformations of
Cauchy and convolution algebras, preprint.
4. E. Getzler and M. M. Kapranov, Cyclic operads and cyclic
homology, in Geometry, Topology, and Physics, International Press, Cambridge, Massachusetts, 1995, pp. 167{201.
5. P. Hanlon, Otter's method and the homology of homeomorphically irreducible k-trees, J. Combinatorial Theory
(A) 74 (1996), 301{320.
6. P. Hanlon and R. Stanley, A q-deformation of a trivial symmetric group action, Trans. Amer. Math. Soc., to appear.
7. M. Kontsevich, Formal (non)-commutative symplectic geometry, in The Gelfand Mathematical Seminar, 1990{92
(L. Corwin et al., eds.), Birkhauser, Boston, 1993, pp. 173{
187.
8. O. Mathieu, Hidden n+1-actions, Comm. Math. Phys. 176
(1996), 467{474.
9. C. A. Robinson, The space of fully-grown trees, Sonderforschungsbereich 343, Universitat Bielefeld, preprint 92083, 1992.
10. C. A. Robinson and S. Whitehouse, The tree representation
of n+1, J. Pure Appl. Algebra 111 (1996), 245{253.
11. V. Schechtman and A. Varchendo, Arrangements of hyperplanes and Lie algebra homology, Inventiones math. 106
(1991), 139{194.
31
12. E. Schroder, Vier combinatorische Probleme, Z. fur Math.
Physik 15 (1870), 361{376.
13. S. Sundaram, Homotopy of non-modular partitions and the
Whitehouse module, J. Algebraic Combinatorics, to appear.
14. S. Sundaram, On the topology of two partition posets with
forbidden block sizes, preprint, 1 May 1998.
15. V. Turchin, Homology of the complex of 2-connected graphs,
Uspekhi Mat. Nauk 52 (1997), no. 2, 189{190; English
transl. in Russian Math. Surveys 52 (1997), no. 2.
16. V. Turchin, Homology isomorphism of the complex of 2connected graphs and the graph-complex of trees, Amer.
Math. Soc. Transl. (2) 185 (1998), 147{153.
17. A. Varchenko, Bilinear form of real conguration of hyperplanes, Advances in Math. 97 (1993), 110{144.
18. S. Whitehouse, -(Co)homology of commutative algebras
and some related representations of the symmetric group,
Ph.D. thesis, Warwick University, 1994.
19. S. Whitehouse, The Eulerian representations of n as restrictions of representations of n+1, J. Pure Appl. Algebra
115 (1996), 309{321.
20. D. Zagier, Realizability of a model in innite statistics,
Comm. Math. Phys. 147 (1992), 199{210.
32