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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

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

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

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 �.

5

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

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

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

4Π

1234

14-2313-2412-34234134124123

3424142312 13

9

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

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

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

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

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

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

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

� 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

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

� 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

231424133412

14-2313-2412-34

Π <4, 2

Note:

~

H

i

(�

4;�2

;Z)

�

=

�

0; i 6= 0

Z

2

; i = 0:

20

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

� 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

~

H

1

(�

3

;Z)

�

=

Z

23

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

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

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

� 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

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

Theorem. Let � = e

2�i=n(n�1)

. Then

as S

n

-modules we have

ker �

n

(�)

�

=

W

n

:

29

Transparencies available at:

http://www-math.mit.edu/

�rstan/trans.html

30

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

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

115 (1996), 309{321.

20. D. Zagier, Realizability of a model in in�nite statistics,

Comm. Math. Phys. 147 (1992), 199{210.

32