The White House - MIT Mathematicsrstan/transparencies/whouse.pdf · The White House This photo...
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The White House
This photo shows the South side of the Mansion; visitors tour the the ’South Lawn’. To the west (left ofphoto) is the Old Executive Office Building (not shown); to the south is the National Mall (also not
shown).
Photo Courtesy of the Washington, D.C. Convention & Visitor Center
go to the White House page
-or-
go to main tour page
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SarahWhitehouse, �-(Co)homology of
commutative algebras and some related
representations of the symmetric group,
Ph.D. thesis, Warwick University, 1994.
2
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The module Lie
n
Let V be a complex vector space with
basis x
1
; : : : ; x
n
. Let Lie
n
be the part
of the free Lie algebra L(V ) that is of
degree one in each x
i
.
dimLie
n
= (n� 1)!
Basis: [� � � [[x
1
; x
w(2)
]; x
w(3)
]; : : : ; x
w(n)
];
where w permutes 2; 3; : : : ; n.
3
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The symmetric groupS
n
acts on Lie
n
by permuting variables.
(1; 2) � [[x
1
; x
3
]; x
2
] = [[x
2
; x
3
]; x
1
]
= �[[x
1
; x
2
]; x
3
] + [[x
1
; x
3
]; x
2
]
4
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For any function f : S
n
! C , recall
that
ch f =
1
n!
X
w2S
n
f (w)p
�(w)
;
where if w has �
i
i-cycles then
p
�(w)
= p
�
1
1
p
�
2
2
� � � ;
with p
k
=
P
i
x
k
i
.
In particular, if �
�
is the irreducible
character ofS
n
indexed by � ` n, then
ch�
�
= s
�
;
the Schur function indexed by �.
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C
n
= subgroup of S
n
generated by (1; 2; : : : ; n)
Theorem. As an S
n
-module,
Lie
n
�
=
ind
S
n
C
n
e
2�i=n
:
Hence
ch(Lie
n
) =
1
n
X
djn
�(d)p
n=d
d
:
6
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Theorem. Let �
�
be the irreducible
character of S
n
indexed by � ` n.
Then
hLie
n
; �
�
i = #SYT T of shape �;
maj(T ) � 1 (mod n);
where
maj(T ) =
X
i+1 below i
i:
42
1
4
2313
ch Lie
4
= s
31
+ s
211
7
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Other occurrences of Lie
n
�
n
= lattice of partitions of [n],
ordered by re�nement
~
H
i
(�
n
) = ith (reduced) homology group
(over Q , say) of (order complex of) �
n
As S
n
-modules,
~
H
i
(�
n
)
�
=
�
0; i 6= n� 3
sgn Lie
n
; i = n� 3:
8
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4Π
1234
14-2313-2412-34234134124123
3424142312 13
9
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T
0
n
= set of rooted trees
with endpoints labelled 1; 2; : : : ; n;
and no vertex with exactly one child
Note: Schr�oder (1870) showed (the
fourth of his vier combinatorische Prob-
leme) that
X
n�1
#T
0
n
x
n
n!
= (1 + 2x� e
x
)
h�1i
;
where F (F
h�1i
) = F
h�1i
(F (x)) = x.
For T; T
0
2 T
0
n
, de�ne T � T
0
if T
can be obtained from T
0
by contracting
internal edges.
10
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213
312
32
321
1
Theorem. As S
n
-modules,
~
H
i
(T
0
n
)
�
=
�
0; i 6= n� 3
sgn Lie
n
; i = n� 3:
11
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A \hidden" action of S
n
on Lie
n�1
(Kontsevich)
Let L
n
be the free Lie algebra on n
generators x
1
; : : : ; x
n
, and let
h ; i = nondegenerate inner product
on L
n
satisfying
h[a; b]; ci = ha; [b; c]i:
For ` 2 Lie
n�1
and w 2 S
n
, let
h`; x
n
i
w
= h`
w
; x
w(n)
i
straighten
�! h`
0
; x
n
i:
So the map w : Lie
n�1
! Lie
n�1
de-
�ned by w(`) = `
0
de�nes an S
n
action
on Lie
n�1
, the Whitehouse mod-
ule W
n
for S
n
or the cyclic Lie op-
erad. (Explicit description of action of
(n� 1; n) by H. Barcelo.)
12
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dimW
n
= dimLie
n�1
= (n� 2)!
W
n
�
=
ind
S
n
S
n�1
Lie
n�1
� Lie
n
:
chW
n
=
p
1
n� 1
X
dj(n�1)
�(d)p
(n�1)=d
d
�
1
n
X
djn
�(d)p
n=d
d
hW
n
; �
�
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
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s
2
; s
111
; s
22
; s
311
s
42
+ s
3111
+ s
222
s
511
+ s
421
+ s
331
+ s
3211
+ s
22111
� � � + 2s
422
+ � � �
Getzler-Kapranov:
W
n
M
n�1;1
= Lie
n
;
whereM
�
is the irreducibleS
n
-module
indexed by �.
14
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Other occurrences of W
n
� Nonmodular partitions (Sundaram)
�
n
= f� 2 �
n
: � has at least two
nonsingleton blocksg
Theorem. As S
n
-modules,
~
H
i
(�
n
)
�
=
�
0; i 6= n� 4
sgnW
n
; i = n� 4:
15
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12345
234
12453
14253
15243
13452
14352
23451
24351
25341
15342
15
134 13524
Σ5
42513
43512
52314
52413
53412
14253451213
24523
14534
12515
23435
235124
16
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� Homeomorphically irreducible trees
(Hanlon, after Robinson-Whitehouse)
T
n
= set of free (unrooted) trees
with endpoints labelled 1; 2; : : : ; n,
and no vertex of degree two
For T; T
0
2 T
n
, de�ne T � T
0
if T
can be obtained from T
0
by contract-
ing internal edges.
17
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34
21
43
21
4
3
2
1
43
21
Theorem. As S
n
-modules,
~
H
i
(T
n
)
�
=
�
0; i 6= n� 4
sgnW
n
; i = n� 4:
18
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� 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
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231424133412
14-2313-2412-34
Π <4, 2
Note:
~
H
i
(�
4;�2
;Z)
�
=
�
0; i 6= 0
Z
2
; i = 0:
20
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Theorem (Sundaram):
~
H
i
(�
n;�k
;Z)
�
=
�
0; i 6= n� 4
Z
(n�2)!
; i = n� 4:
Moreover, as S
n
-modules,
~
H
n�4
(�
n;�k
; Q )
�
=
sgnW
n
:
21
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� Not 2-connected graphs (Babson et
al., Turchin)
G = loopless graph without multi-
ple 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 con-
nected.
�
n
= simplicial complex of not 2-
connected graphs on 1; : : : ; n.
22
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~
H
1
(�
3
;Z)
�
=
Z
23
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Theorem (Babson-Bj�orner-Linusson-Shareshian-
Welker, Turchin):
~
H
i
(�
n
;Z)
�
=
�
0; i 6= 2n� 5
Z
(n�2)!
; i = 2n� 5
Moreover, as S
n
-modules,
~
H
2n�5
(�
n
; Q )
�
=
W
n
:
24
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General technique:
G acts on �; w 2 G
�
w
= fF 2 � : w � F = Fg
Hopf trace formula =)
~�(�
w
) =
X
(�1)
i
tr(w;
~
H
i
(�))
| {z }
character value at w
of G acting on
~
H
i
(�)
Use topological or combinatorial tech-
niques such as lexicographic shellability
to show that
~
H
i
(�) vanishes except for
one value of i.
25
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Note: Explicit S
n
-equivariant iso-
morphisms (up to sign) between the cyclic
Lie operad, the cohomology of the tree
complex T
n
, and the cohomology of the
complex �
n
of not 2-connected graphs
were constructed by M. Wachs.
26
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� A q-analogue of a trivial S
n
-action
(Hanlon-Stanley)
For w 2 S
n
and q 2 C , let
`(w) = #inversions of w
�
n
(q) =
X
w2S
n
q
`(w)
w 2 CS
n
�
n
(q) acts on CS
n
by left multiplica-
tion.
Theorem (Zagier, Varchenko):
det �
n
(q) =
n
Y
k=2
�
1� q
k(k�1)
�
n!(n�k+1)=k(k�1)
27
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Proof (sketch). Let
T
n
(q) =
n
X
j=1
q
j�1
(n; n�1; : : : ; n�j+1):
Easy:
�
n
(q) = T
2
(q)T
3
(q) � � � T
n
(q):
Let a � b and
[a; b] = (a; a� 1; : : : ; b) 2 S
n
:
Let
G
n
=
(1�q
n
[n�1; 1])(1�q
n�1
[n�1; 2]) � � � (1�q
2
)
H
�1
n
=
(1�q
n�1
[n; 1])(1�q
n�2
[n; 2]) � � � (1�q[n; n�1]):
Duchamps et al.: T
n
= G
n
H
n
:
28
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Theorem. Let � = e
2�i=n(n�1)
. Then
as S
n
-modules we have
ker �
n
(�)
�
=
W
n
:
29
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Transparencies available at:
http://www-math.mit.edu/
�rstan/trans.html
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REFERENCES
1. E. Babson, A. Bj�orner, 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 Mil-
nor �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, Interna-
tional Press, Cambridge, Massachusetts, 1995, pp. 167{201.
5. P. Hanlon, Otter's method and the homology of homeo-
morphically irreducible k-trees, J. Combinatorial Theory
(A) 74 (1996), 301{320.
6. P. Hanlon and R. Stanley, A q-deformation of a trivial sym-
metric group action, Trans. Amer. Math. Soc., to appear.
7. M. Kontsevich, Formal (non)-commutative symplectic ge-
ometry, in The Gelfand Mathematical Seminar, 1990{92
(L. Corwin et al., eds.), Birkh�auser, 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, Sonder-
forschungsbereich 343, Universit�at Bielefeld, preprint 92-
083, 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 hyper-
planes and Lie algebra homology, Inventiones math. 106
(1991), 139{194.
31
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12. E. Schr�oder, Vier combinatorische Probleme, Z. f�ur Math.
Physik 15 (1870), 361{376.
13. S. Sundaram, Homotopy of non-modular partitions and the
Whitehouse module, J. Algebraic Combinatorics, to ap-
pear.
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 2-
connected graphs and the graph-complex of trees, Amer.
Math. Soc. Transl. (2) 185 (1998), 147{153.
17. A. Varchenko, Bilinear form of real con�guration of hyper-
planes, 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 re-
strictions of representations of �
n+1
, J. Pure Appl. Algebra
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