Cohomology of Dowling Lattices and Lie Superalgebras

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IN F O R M A T IO N T O U S E R S

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UNIVERSITY OF MIAMI COHOMOLOGY OF DOWLING LATTICES AND LIE SUPERALGEBRAS

By Eric Inness G ottlieb A D IS S E R T A T IO N

Subm itted to the F aculty of the U niversity o f M ia m i in p a rtial fulfillm ent of the requirem ents for the degree of D octor o f Philosophy

Coral Gables. F lo rid a July 1998

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UMI Number: 9905059

Copyright 1998 by Gottlieb, Eric Inness All rights reserved.

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UNIVERSITY OF MIAMI

A d issertation submit,ted in p a rtia l fu lfillm en t of th e requirem ents for the decree of D octor of Philosophy

C O H O M O L O G Y O F D O W L IN G L A T T IC E S A N D L IE S U P E R A L G E B R A S

Eric Inness G o ttlie b

A pproved:

M ic h elle Wachs G allo w a y

Steve U llm a n n

Professor of M a th e m a tic s

lortTRrim Dean of the G ra ilu a te School

C h a ir o f the D issertation C o m m itte e

__________ S h u lim K a lim a n Associate Professor of M a th e m a tic s

senberg Assistant Professor o!7C om puter Science

M a /v in M ie lk e Prcfessor of M a th e m a tic s

Thom as C 'urtright Professor of Phvsics

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G O T T L IE B . E R IC IN N E S S

(P h .D .. M ath em atics)

C O H O M O L O G Y O F D O W L IN G L A T T IC E S

(J u ly 1998)

A N D L IE S U P E R A L G E B R A S

A b s tra ct of a doctoral dissertation at the U niversity o f M ia m i. D issertation supervised by Professor M ichelle Wachs. N u m b e r o f pages in te x t: 139.

It follows from the w ork o f H anlon [H a l], Stanley [St 1], W i t t [W i]. and B ra n d t [Br] th a t the cohomology of n „ , th e lattice of partitions of sgn

3

{1

.......... a } , is isomorphic to

Lien, the sign representation tensored w ith the m u ltilin e a r com ponent of the

free Lie algebra on n letters, as representations of the s ym m e tric group S n. Wachs [Wa2] gave a new proof of this result by describing generators for the cohomology of n „ and for L ie n using ordered binary trees whose leaves are labeled w ith elem ents of

{1

.........n } . She found relations for these generators and produced a

m ap which respects the action of x and z > y. W e say th a t an upper bound r for x and y is a least upper bound fo r x and y if r < z' for every other upper bound z ‘ of x and y. T h e term s lower bound and greatest lower bound are defined sim ilarly. W h e n greatest lower bounds and least upper bounds exist, they are unique. W e denote the least upper bound of x and y by x V y and th e greatest lower bound of x and y by x A y.

W e say th a t P is a lattice if every p a ir of elem ents of P has a

greatest lower bound and a least upper bound.

T h e process o f ta k in g least upper

bounds and greatest lower bounds is associative, so we can ta lk about the least upper

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9

bound of any set of elements. A lattice P is geometric when every elem ent of P is the least up p er bound o f a set of atoms of P . and when the rank function for P satisfies

+

l { x)

l ( y) > /( x V y) + l ( x A y).

Now we give some examples of lattices. T h e Boolean lattice on a set 5 is denoted Bs and is defined to be the power set o f S where the order relation is given by inclusion. T h a t is. x < s _ y if and only i f x C y for x . y C 5 . W e define B n = Brn]w here [n] = { 1 ..........n } . T h e chain

0 < {3 } < { 1 .3 } < { 1 . 3 . 4 } < { 1 .2 . 3 .4 }

is m axim al in

B4.

N e x t we define the partition lattice.

A partition of a finite set N is a collec­

tio n of n o n e m p ty pairwise disjoint sets whose union is 5 . {B \

We denote the p a rtitio n

B k } of 5 by B i ! . . . / Bk- T h e sets B, axe known as the blocks of the p a rtitio n .

IIs is the set o f partitions o f 5 ordered by x

0

.

n n is the set o f signed partitions on n letters w ith p a rtia l order determ ined by ( - . 7 ) ) of

so th a t B C B ' and ~;'\g = *-|g

• th ere is a nonzero block (B'.~f'\gi) of (jt 7. 7 7) so th a t B C B ' and ~;'\g = —~ \g

• B is contained in the zero block of (jt'.

W e denote the zero block {0. x \

7

x : ) of (x .

').

7

) by Oxi . . . x , . W e denote the nonzero

block ( B. ~f \ g) of ( ~ . - ) by ( y i . ~ f ( y \ ) ) .. . ( y p. ~’( yp)). where B = { y i we denote the p a ir ( y. ~f ( y) ) by y if

7

(y) =

1

yp}. For y € B

and by y if ~ ( y) = — 1 - L nder these

conventions.

0 / 1 / 2 / 3 / 4 / 5 < 0 /1 3 / 2 / 4 / 5 < 0 /1 3 /2 5 /4 < 0 4 /1 3 /2 5 < 0 4 /1 2 3 5 < 012345

is an exam p le of a m a x im a l chain of I I 5 . B n. n „ . and I I n are all finite, bounded, pure, and geom etric. T h e rank of x € Bn is |x|. T h e rank o f x £ n „ is n — jarj. where jx| is the n u m b er o f blocks of x . The rank o f x € EIn is n — |x| +

1.

T h e partition lattic e and th e signed p a rtitio n lattice

axe both exam ples of the D ow ling lattice, which w ill be defined in Section 3.1.

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B n. n n. and n n all arise as lattices o f intersections of C oxeter arrangem ents of hyperplanes.

1

B n. for exam ple, is th e la ttic e of intersections o f th e C o x e te r ar­

rangem ent {x , = 0 | i =

1.........n } .

S im ila rly . [ I n is the intersection la ttic e of the

C oxeter arran g em en t of ty p e .4. given by A n- \ = {x , — x: = 0 | 1 < i < j

< /?}.

and n n is th e intersection la ttic e o f th e Coxeter arrangem ent of ty p e B . given by B n = { x , ± Xj = 0 | 1 < i < j < n } U {x , = 0 | 1 < i < n }.

2.2

P o se t H om ology and C ohom ology

W e begin o u r discussion of poset hom ology and cohomology by defining a sim p licial com plex associated to a poset. A n n-simplex is the collection of subsets o f a set w ith n + 1 elem ents.

A simplicial complex is th e union of a finite n u m b er o f simplices.

E q u iva len tly, a sim plicial com plex is a fin ite collection of sets th a t is dosed under co n ta in m en t. A n elem ent of a sim p licial com plex is railed a fact. Faces o f c a rd in a lity one are called vertices. M a x im a l faces are called facets. T h e chains of any poset form a s im p lic ia l complex. T h e order complex A ( P ) of a hyperplane is an n — 1-dimensional subspace of real or complex n-space. i.e. the set o f real

or complex solutions o f a linear equation in n variables. A hyperplane arrangem ent is a finite set of hyperplanes o f the same dimension. A hyperplane arrangement is called a C ox ete r a rra ng e m e n t if the reflections about the hyperplanes in the arrangement generate a finite group. T h e intersection lattice o f a hyperplane arrangement is the lattice o f intersections of the hyperplanes in the arrangement ordered by reverse inclusion.

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fin ite bounded poset P is defined 2 to be the sim p licial com plex of chains of P \ { 0 . 1 }. T h e facets of A ( P ) are th e m a xim a l chains of P \ { 0 . 1 } . Som etim es A ( P ) has a nice structure th a t can be described directly. For exam ple, th e order com plex of B n is th e barycentric subdivision o f the boundary of an (n — l) sim p lex. Figure 2.1 shows the top view of th e b arycen tric subdivision of the boundary o f a 3-sim plex. T h e six facets on the b ottom are obscured from view. T h e vertices o f th e sim plicial com plex are labeled to reveal how each face corresponds to a chain in

\ {0 . i } .

T h e shaded facet corresponds to th e m a x im a l chain {4 } < { 1 .4 }