Cohomology of Dowling Lattices and Lie Superalgebras

Table of contents :
Title
Abstract
Contents
1. Introduction
2. Posets and Poset Topology
3. Cohomology of the Dowling Lattice as a Wreath Product Module
4. Some Useful Algebraic Objects
5. The Isomorphisms
6. Whitney Cohomology
7. EL-labelings for the Dowling Lattice
8. The h,k-Equal Dowling Lattice
Bibliography

Citation preview

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

This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted.

Broken or indistinct print, colored or poor quality

illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted.

Also, if

unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back o f the book. Photographs included in the original manuscript have been reproduced xerographically in this copy.

Higher quality 6” x 9” black and white

photographic prints are available for any photographs or illustrations appearing in this-copy for an additional charge. Contact UMI directly to order.

UMI A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600

permission of the copyright owner. Further reproduction prohibited without permission.

R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.

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

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

UMI Number: 9905059

Copyright 1998 by Gottlieb, Eric Inness All rights reserved.

UMI Microform 9905059 Copyright 1999, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.

UMI 300 North Zeeb Road Ann Arbor, MI 48103

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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

Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.

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.

Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.

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.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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 }