Homology of the Lie algebra corresponding to a poset

Table of contents :
D E D IC A T IO N ............................................................................................................. ii
LIST OF FIG U R E S ................................................................................................... v
CHAPTER
I. Partially Ordered S e t s ........................................................................... 1
1.1 Definitions ........................................................................................ 1
1.2 The homology of a poset .............................................................. 4
II. Homology of Lie A lg e b r a ...................................................................... 10
2.1 The notion of Lie algebra ............................................................... 10
2.2 Homology of a Lie algebra ........................................................... 14
2.3 Main computational methods ........................................................ 22
III. Inclusion of Poset Homology into Lie Algebra Homology . . 32
3.1 Definitions ........................................................................................ 32
3.2 Insertion map ..................................................................................... 35
3.3 Example of the insertion ................................................................. 39
IV. The Formula for Laplacian of a Linear P o s e t ............................... 42
4.1 Simplification .................................................................................. 42
4.2 The Formula ..................................................................................... 48
4.3 E xam ple ............................................................................................ 51
V. The Representation Theory of the Symmetric G roup .............. 54
5.1 Young tableaux ................................................................................. 54
5.2 Specht modules .................................................................................. 55
5.3 Restricted and induced representations ....................................... 58
5.4 The Littlewood-Richardson rule .................................................... 59
iii
VI. The Eigenvalues of the L a p la cia n ..................................................... 63
6.1 Definitions .......................................................................................... 63
6.2 Embedding of the L-block in CSn ................................................ 67
6.3 The Laplacian Ly ................................................................... 69
6.4 Centerpiece Theorem for Ly ......................................................... 82
6.5 Example ................................................. 108
6.6 Adding the L x ....................................................................................... Ill
6.7 Homology ................................................................................................. 130
B IB L IO G R A PH Y ........................................................................................................... 137

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Order Num ber 9409711

H om ology o f th e Lie algebra correspon din g to a p oset Hozo, Iztok, Ph.D. The University of Michigan, 1993

UMI

300 N. Zeeb Rd. Ann Arbor, MI 48106

HOMOLOGY OF THE LIE ALGEBRA CORRESPONDING TO A POSET

by Iztok Hozo

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 1993

Doctoral Committee: Professor Philip J. Hanlon, Chairman Professor Andreas R. Blass Associate Professor Kevin J. Compton Professor M. S. Ramanujan Associate Professor John R. Stembridge

Iztok Hozo 1993 All Rights Reserved

To Sarajevo

TABLE OF CONTENTS

D E D I C A T I O N .............................................................................................................

ii

L IST O F F I G U R E S ...................................................................................................

v

CHAPTER I. P a r tia lly O rdered S e t s ...........................................................................

1

D e fin itio n s........................................................................................ The homology of a poset ..............................................................

1 4

II. H o m o lo g y o f Lie A l g e b r a ......................................................................

10

1.1 1.2

2.1 2.2 2.3

The notion of Lie a lg e b r a ............................................................... Homology of a Lie algebra ........................................................... Main computational m e th o d s........................................................

III. In clu sion o f P o se t H om ology in to Lie A lgeb ra H o m o lo g y

10 14 22

. .

32

D e fin itio n s........................................................................................ Insertion m ap ..................................................................................... Example of the insertion.................................................................

32 35 39

IV . T h e Form ula for Laplacian o f a Linear P o s e t ...............................

42

3.1 3.2 3.3

4.1 4.2 4.3

Simplification .................................................................................. The F o rm u la ..................................................................................... E x a m p le ............................................................................................

42 48 51

V . T h e R ep resen ta tio n T h eory o f th e S y m m etric G r o u p ..............

54

5.1 5.2 5.3 5.4

Young tableaux................................................................................. Specht m odules.................................................................................. Restricted and induced representations....................................... The Littlewood-Richardson ru le....................................................

iii

54 55 58 59

V I. T h e E ig en v a lu es o f th e L a p l a c i a n ..................................................... 6.1 6.2 6.3 6.4 6.5 6.6 6.7

63

D e fin itio n s .......................................................................................... 63 Embedding of the L-blockin C Sn ................................................ 67 The Laplacian Ly ................................................................... 69 Centerpiece Theorem for L y ......................................................... 82 ................................................. 108 Example Adding the L x .......................................................................................I l l H om ology................................................................................................. 130

B I B L I O G R A P H Y ........................................................................................................... 137

iv

LIST OF FIGURES

Figure 1.1

A Hasse diagram of P

...............................................................................

2

1.2

Another Hasse diagram of P .....................................................................

2

1.3

Unranked P o s e t .............................................................................................

3

1.4

Poset P , and the corresponding P ............................................................

4

1.5

The Boolean Algebra B 3 ............................................................................

7

3.1

Poset P

..........................................................................................................

33

3.2

Poset Q

.........................................................................................................

34

3.3

Example: poset P

......................................................................................

39

4.1

Hasse diagram of P ......................................................................................

52

5.1

Example: Ferrers diagram for A = ( 4 , 4 , 2 , 1 ) ........................................

54

5.2

Example: A = (5 ,3,3,1), fi = (2,2,1)

.....................................................

55

5.3

Example: Littlewood-Richardson r u l e .....................................................

61

5.4

Example: Littlewood-Richardson r u l e ...........................................

61

6.1

Example: poset P .........................................................................................

65

6.2

Example: poset P .........................................................................................

66

6.3

Example: poset P .........................................................................................

73

6.4

Example: diagram of the L -b lo c k ...........................................................

74

6.5

Example: a poset tableau............................................................................

76

v

6.6

Example: poset tableaux for the poset P ...............................................

77

6.7

Example: poset tableau for the chain poset

.....................................

78

6.8

Example: poset P .........................................................................................

80

6.9

Example: the y-eigenvalues.........................................................................

81

6.10

Example: poset ta b lea u x ............................................................................

83

6.11

Example: poset ta b lea u x ............................................................................

83

6.12

poset tableau

...............................................................................................

84

6.13

poset tableau

...............................................................................................

84

6.14

poset tableau

...............................................................................................

85

6.15

poset tableau

...............................................................................................

85

6.16

Case 1 ............................................................................................................

88

6.17

Example: poset P

.....................................................................................

92

6.18

The structure of Yi and Y2 ........................................................................

92

6.19

Induction s t e p ...............................................................................................

93

6.20

All poset tableaux of Y\

............................................................................

93

6.21

All poset tableaux of Y2

............................................................................

94

6.22

Littlewood-Richardson rule I .....................................................................

94

6.23

Littlewood-Richardson rule I I ..................................................................

95

6.24

Case 2 ............................................................................................................

96

6.25

Example: poset P ............................................................................................101

6.26

Example: the subtree H

6.27

Eigenvalues of L y ............................................................................................102

...............................................................................101

vi

Example: multiple root .

103

Example: poset P . . . .

108

Example: a poset tableau

109

Example: poset tableaux

110

Example: poset P . . .

.

121

Example: poset tableaux

127

Example: poset tableaux

128

Example: poset tableaux

129

A diagram of the L-block

130

poset ta b lea u x .................

131

poset ta b lea u x .................

132

poset t a b l e a u .................

132

poset t a b l e a u .................

134

poset t a b l e a u .................

134

poset t a b l e a u .................

134

poset P

...........................

136

poset P

...........................

136

vii

CHAPTER I

Partially Ordered Sets

This chapter presents an introduction into the theory of the partially ordered sets. A more complete treatment of this subject area can be found in [31].

1.1

Definitions

A partially ordered set P (or p oset, for short) is a set(which by abuse of notation we also call P ), together with a binary relation denoted < (or< p when there is a possibility of confusion), satisfying the following three axioms: 1. For all x G P, x < x. (reflexivity) 2. If x < y and y < x, then x = y. (an tisym m etry) 3. If x < y and y < z, then x < z. (tran sitivity) We say two elements x and y of the poset P are com parable if x < y or y < a:; otherwise x and y are incom parable. We now list a number of basic definitions connected with partially ordered sets. Two posets P and Q are isom orphic if there is an order-preserving bijection \ P —►Q whose inverse is order-preserving; that is, x < p y •$=>■ (f){x) yi A zX3iV2 A • • ■A zXriVr. Note that £ cannot contain either zXlii or

Z i lV,

as a

constituent since those two elements were erased from the source. Suppose £ appears more than once. Our £ has to appear as a summand in dp, for some p - hence two of the intervals from the source were glued together, i.e., in the result £ we have had some t and m, such that d(z*i,vi A • • • A zXtim A • • • A zm , or in other words it is

span of all C-matrices with non-zero entries only in those positions, i.e. / '

0 0 0 * 0

\

0 0 * * * Lp —
c

0 0 0 0 0

\

0

0

0

0

o ;

where the * means that the entry might be any complex number. The only non-zero brackets in this algebra are [2 2 ,3 , 2 3 ,4 ] =

2 2 ,4 ,

and [2:2 ,3 , 2 3 ,5 ] =

*2,5The poset Q from figure 3.2 on the other hand, yields the following Lie algebra:

Figure 3.2: Poset Q L q = < 24>5, 31,2, «1,5) «1,3, «2,5> ^2,3 > , be-

0 L q =
c 0

0 0 0 *

0

0 0 0 0

-*■ L q defined by

: i

* 1,4

l-»

24,5

*2,3

h-»

21,2

-22,4

1—>

21,5

22,5

1—►

21,3 22,5

23,4 23,5

1—►

22,3

This is an isomorphism, since it preserves the bracket: ^(^2 ,4 ) = y$. The expressions A and C are two summands of the product L x L y , while B and T) are two summands of the product L yL x- As we can see, A + C = B + V. Thus L x ’ L y = L y • Ljf. ■ In view of Lemma VI.3, L x } Ly and L d are commuting linear transformations. So, to analyze the spectrum of their sum, we can compute the eigenvalues and eigenspaces of each separately. We will begin with Ly.

6.3.1

A p oset tableau o f ty p e (.X , Y ) p

D efinition VT.2 The diagram of the L-block, P [X ,Y ],

spanned by the sets

(X, Y )p , is the Hasse diagram of the subposet X U Y with order inherited from the poset P . Furthermore every vertex of P , which is in the intersection X fl Y is split into two nodes, with x-node above the y-node.

6.3.2

E xam ple of a diagram

Let P be the poset given in figure 6.3. And suppose that the sets X and Y are X = {1 ,1 ,2 ,4 } and Y = { 4 , 6 , 6 ,7}. Then the graph of the L-block V spanned by the (X, Y ) is presented in figure 6.4.

74 (y) - twice

7 (y )

4 (x)

4 (y )

1 (x) - twice

Figure 6.4: Example: diagram of the L-block D efin itio n VT.3 Given a node v in P[X , Y] define the r e p e titio n n u m b er o f v , k(v), to be the number of times that v appears in the multiset X if v is an x-node of P[X , Y \, or the multiset Y if v is a y-node of P [ X f Y\.

Let C(v) be the set of covers of node v in P[X, Y]. If v is a maximal node, than C (v) = 0.

D efin itio n V L 4 A p o se t ta b lea u o f ty p e (X, Y )p (or just of type ( X , Y ) ) is any labeling A of the diagram, P[X, Y], of the L-block V spanned by (X , Y ), where the labels are partitions A (y), such that A(u) is a partition of the number £ w>t, e(w)k(w), where +1

if w is a y-node

—1

if w is an x-node.

(

Given a poset tableau A we will define the m u ltip licity o f A, m(A), and the eigen valu es o f A, e(A). D efin itio n V I.5

• Let v be a y-node of the diagram P [X ,Y ], labeled with the

partition A(v) and with repetition number k(v). Let C(v) = {vi,V 2 , . . . ,u/} be the set o f covers of v. Let A,- denote A(vi), and let k{ denote the repetition

75 numbers, k(vi). The m ultiplicity of A at v is defined to be

m„( A) = c ^ iA|)fc(u)

• Let v be an x-node of the diagram P [X ,Y ], labeled with the partition A(u) and with repetition number k(v). Let C(v)

= {t>i,t> 2 , . . . , v /}

be the set of covers of

v. Let Xi denote A(vi), and let k{ denote the repetition numbers, k(vi). The

m ultiplicity of A at v is defined to be

mt»(A) =

CAi,...,A{CA(t,),l*('')*

If the multiplicity mu(A) = 0 then we know that that particular labeling is not valid.

6.3.3

E xam ple

Suppose the poset P is again given in the Hasse diagram given in the figure 6.3, with the sets X and Y , X = { 1 , 1 ,2 ,4 } and Y = {4 , 6 , 6 ,7}. Then the diagram of the L-block V spanned by the (X, Y) is presented in the figure 6.4. A detailed explanation how the procedure works is given in the figure 6.5, and all possible poset tableaux of positive multiplicity corresponding to this diagram of the L-block, are given in the figure 6 .6 . Note that the L-block V spanned by the sets (X , Y) has basis:

*1,4 A

Z ifi

A 22,6 A 24,7

*1,4 A 21,6 A 22,7 A 24,6 *1,4 A 2i,7 A 22,6 A 24,6 *1,6 A 21,7 A 22,4 A 24,6

76

Add these two partitions using L-R rule /

\

and obtain

I

(xj)| I—1< — take a square off (x)

Double node (4)

add a square (y)

take a square off (x) take the vertical strip of size 2 off

Figure 6.5: Example: a poset tableau Observe that there are 4 pure wedges in the basis of the L-block V - the same number as the number of the poset tableaux of this L-block.

6.3.4

Exam ple: Chain p oset T6

Suppose now that the poset P is a chain poset T«, i.e., P = {1 ,2 ,3 ,4 ,5 , 6 }, and 1 < p 2 < p 3 < p 4 < p 5 < p

6.

Observe the L-block V spanned by the sets

X = {1,2,3} and Y = {4,5,6}. In that case V has dimension 6 , and the basis consists of six pure wedges:

*1,4 A 22,5 A 23,6

*1,4 A 2 2>6 A 2 3,5

*1,5

A 22,4 A 23,6

*1,5

A 22,6 A 23,4

*1,6 A 22,4 A 23(5

77

F>m

(y) □

(y)D

Figure 6.6: Example: poset tableaux for the poset P

78

□ □ n+i ’ UB 11M '' P 11l-l ’ P □3 ' HE

'□ 1mu

'

'

□

□

S P '1P 1 CB

P P I

■

□

'

’P '' P § '

'

'

'

0 Figure 6.7: Example: poset tableau for the chain poset Tq *1,6 A z 2,s A z 3f4

All poset tableaux are given in the figure 6.7. Note that again the number of poset tableaux is the same as the dimension of the L-block V corresponding to the same sets ( X ,Y ) . In general it is not difficult to show, that for the Chain poset, Tn the poset tableaux, with all of the y's and x ’s distinct and y’s above all x ’s, are in one-to-one correspondence with the pairs of standard Young tableaux of the same shape. The argument is that you simply build up a Young tableau while encountering the y’s, and then remove it while passing x ’s in the diagram. For example, the third poset

( 132 ’ 12

J‘

This defines two tableaux of the same shape. Using the Robinson-Schensted corre­ spondence we know that there are n! such pairs of tableaux. Thus in general in this situation, the dimension of the L-block is n!.

79 Now, we will define the y-eigenvalues for each y-node v of the diagram P[X, Y], We want to have as many y-eigenvalues as the multiplicity. From the representation theory of the symmetric group, we know that =

S

cAi,-,A|-

( 6 -1)

A ( v) / m ■ fc(ti)-horiaontal a trip

The n od e—eigen valu e, e„(A), for each node u, is the set of the sums of the content over all squares in A(v)//x above for a given fi minus the binomial coefficient

•

Recall that the content of a square is given by c(u) = k —i if the square is at position (i , k) in a partition ( i ^ row and k ^ column). For example, if the L-block V is defined by the relations 1 < 2 < 3 < 4 < 5 , 4 < 6 , and multisets X = {1 ,1 ,2 ,3 } and Y = { 4 ,4 ,5 , 6 }, one of the possible labelings is:

v /D A=

Y bpIp

v

B J3" But if we look at the vertex v, indicated above, we see that there are two different ways we could reach that particular partition: /

n and

jj+i+i

V

_____ y w

where the [+] marks just added squares. We will use [ f ] and [^] in our figures to clearly indicate which of the possibilities is considered. The sums of the contents of the added squares are 3 and 1. When we take the binomial coefficient into account, we get the node-eigenvalue, e„(A) = {2,0}. In general, this gives m„(A) eigenvalues at each y-node v. We now define yeigenvalue o f A, ev(A), to be the set of numbers obtained by taking a sum of one element of et/(A) for each y-node v. So |ev(A)| = ITy-nodes v m v(A).

80 5

Figure 6 .8 : Example: poset P In our centerpiece theorem later we will prove that each element r in ey(A) is an eigenvalue of Ly with multiplicity fl^-nodes v m«(A).

6.3.5

Example: The ^-eigenvalue

Return to our first example, when the poset was P = {1 ,2 ,3 ,4 ,5 } with the relations 1

• • ■> xn, where all > are covering relations.

2. Either Case 1 : There is more than one element covering x a. Case 2:

xa has unique cover in P[X, Y] but it is a y-element.

87 Let B = {xa, . . . , Xn}, and let G =Sym (B).

Lem m a VI.5 Let a £ G, and let ( = zXiitVl / \ z XiiM A • • • A z XiniVn be non-zero. Then C = i s a /s o

A z-\:

Figure 6.18: The structure of Y\ and Yi subtrees are given in figure 6.18. The induction step (what we know about smaller L-blocks) is given in figure 6.19. This follows from our theorem (which we assume holds for these small sizes). All poset tableaux for (Aj, Yi) and {Xi^Yi) are given in figure 6.20 and figure 6.21.

The Littlewood-Richardson

rule is then given in figure 6.22 and figure 6.23. We will denote by T the value of the eigenvalue ey(A), where A is a poset tableau. All together we get the eigenvalues of the Laplacian to be (with multiplicities): +3 (10), + 2 (20), +1 (10), 0 (40), -1 (10), -2 (20), -3 (10).

93

Eigenvaluei of

L j d|envaluM on

s“

Eigenvalue! of

2

eigenvalue* ph sv

V

m

1 ,- 1

---

+1

-1

B

-2 -----

Figure 6.19: Induction step

m+i r~m □3

T-+2

□3

T--1

T- - 1

T-+1

T«-2

□3

T-+1

Lj

Figure 6.20: All poset tableaux of Y\

94

m I~F1

T- - 1

Figure 6.21: All poset tableaux of Yi repreionhtion X

eigenvalue «

■» t v_________with muHiplicily

n (+2) + (+1) = 3

(+1) + (+1) = 2

(-1) + (+1) - 0

(-2) + (+1) = -1

Figure 6.22: Littlewood-Richardson rule I

95

wpmenloion X

r r

eigmvriue " Ta + Tv_________with multiplicily

a (+2) + (-1) = +1

B

(+1) + (-1) - 0 (- 1) + (- 1) = -2

4

(-2) + (-1) - -3

1

Figure 6.23: Littlewood-Richardson rule II

• Now we have to decide what is the dimension of each of these eigenvalues. But that is something we will have to do in the second case too - so we will do it for both cases at the end.

6 .4 .3

Case 2

NB

Figure 6.24: Case 2

Let A = (j/n-fc+i) • • • >J/n} be the largest possible set so that

Xa

< P J/n < P J / n - l < P • • • < P

and there are no Xf’s with yn |j4|. Split the L-block

97 for (cii,. . . a\t) a sequence of length k of distinct elements from B and where y (a i, d j ,. . . , a*) is the span of all ( which are of the form

C

=

z x i,V i

A

• • • A

Z Xn_ klVn_ k

A

Za j,y n _ fc+i

A

• • • A

Zak ,yn

For the moment assume that all of the elements xa, . . . , x n are distinct and all of Vn—k+i »• *• >yn are distinct. Then we will look back at how we must modify the argument when some of the a;t’s and y f s are equal. Fix the sequence ( a i,. . . , a*), let B' — B \ { a i , . . . , a*} and let G' be the sub­ group G' = Sym (5') < G. Note that G' acts on V (a i,. . . ,a*).

L em m a V I. 1 0 V (a i,. . . , a*) is isomorphic to the L~block Vo given by the sets X 1 - X \{ a i,a 2 ,...,a fc } ,F ' = K\{j/n-* + i,. . . ,y n}.

Proof: The isomorphism if) : V (a i,. . . ,ajt) —►Vo is an obvious one

V *(

ZX1,VI

A

• • • A

Z Xn_ kiVn_ k

A

Z a u yn _ k+1

A

• • • A

Z a k ,y„ )

z x l ,yl

=

A

• • • A

Z Xn_ ktyn _ k .

It is clearly a bijective linear map.a • Now let’s examine the L-block given by X ' and Y ‘. Let A be the terminal F-set for X 1, Y '. Let B = B' U {new x ,’s} be the Zj’s below A. Let N B denote the set of those new x,’s. Let G = Sym(B). Note that G' < G. If A = 0 then there are no y ’s in the interval (yn-fc+i>^o]- In that case the A

A

group G is the group described in the case 1., i.e., G = C?i x G2 x • • • x G>, where Gi is acting on the subtree Tt- above xa. Now apply the induction hypothesis to X Y 1. This gives the decomposition of the L-block as a G-module. The theorem says that the irreducible summands

98 are indexed by the ( X \ Y') poset tableaux A of shape X (where A' h |B |), and ’’shape” means that the minimal element s of A is labelled with A'. Also the theorem tells us that the V-Laplacian Ly for (X \ Y') acts like the scalar ey(A) on this copy of the irreducible S x'. From this we can deduce by restriction from G to G' the following

L em m a V I. 1 1 As a G'-module, the space V { a i , . . . ,a*) decomposes as a sum over all ( X ^ Y 1) tableaux A of shape A' of the module

ig .

Moreover the Laplacian L y for (X Y ' ) acts like the scalar ey (A) on this entire restriction.

Let Ch(A', p') be the set of chains A' > Aj > ■*• > p' where the steps in the chain of the partitions are all of size 1 . Now using the fact [16] that

( S A') i g , =

we can rewrite this lemma to say that the sum is over all extensions of A to a labeling of the new x ,’s of S**' where the extension gives the label p' to t, the minimal element of the set N B . • Now we need one last lemma.

L em m a V I. 12 As a G-module V = indg,(V(a:n_fc+i , . . . , s n)).

Proof: For each sequence a = ( a i,...,a * ) let 7r„ be the permutation in G which maps a,- to x„-k+i and which leaves the elements of B \ { a 1}. . . , a^} in

99 increasing order. Then 7r« is a set of coset representatives for G \G '. Since there is one for every a this shows that as vector spaces

V “ V (x n-jtg+i, .. . yx n) ® g >C(G ). Let g € G, let ( a i ,. . . , a*) = or be a sequence, and let &,• = y(a;),

=

( 6 1 , . . . , bie). Then

9 ( z x i,y i

=

=

where

A ••• A

Z g ( x i) ,V i

Z x n - k ,V n -k

A •••A

9 ^ s ) z x *i< y i

A

z a i,y n - k + i

z g (X n - k ) ,y n . . k

A ••• A

A *• * A

z a k ty n )

A ^6i,y„_fc+1 A • • • A

Z x 'n _ k , y n . k

Z b k ,y n

A Zx„_fc+1 ,yn_ fc+i A • • • A 2In>Vn,

is obtained by replacing the elements of

by {ara, . . . , xn„fc} in order.

=

{rca ,

. . . , xn} \ { a i , . . . , a*.}

This computation shows that the vector space

isomorphism above is a G-module isomoprhism.a Now putting all the claims together with the fact([16]) that

h C $ > ( 5 “') =

(6-3)

shows that as a module for G, V decomposes as a sum over all ( X , Y ) poset tableaux A of shape A of a copy of S \

In the expression 6.3 we know that

Ly> — L y — Lo, where L q is the sum of the switches involving the terminal y -se t, acts like the scalar ey(A'), where A' is the labeling as far as the point t (the minimal x above A). The switches involving the terminal K-set A must be studied. But those y,- in A can either be switched with each other or with yj that are above an a?,- € B. By the choice of the terminal set A, y; is comparable to yj for all y,- € A. It follows that L q acts on V = © V (o i,. . . , a*) by the sum of all transpositions (a;,1, Xj) for x,- £ B , xj € { 0 1 , . . . , a*,}. In terms of our

100 induced module,

Lq

acts on

in d g ,^ ') = S»' ®c a' C G like left multiplication on CG by But J2 x i , x j G B ' ( x

ii

x j)

- Erj,xjeB>(xi>xj)-

passes through

® c g > to

act on S*' like the scalar

[24]. And Ex,,xj6 B(x«) x j ) acts on each G-irreducible S x in S*' °v'

The result is that

Lq

Y lx tn ' cx

® c g > CG

like

acts on each copy of S x in ind^/C.?^) like

EveV^' ° v This explains the scalars ey(A) and their multiplicity. In order to be able to add the eigenvalues of L y and L q, we need the following lemma.

Lem m a VI. 13 Lq’

Ly

=

Ly

•Lq

Proof: Let (xi,Xj) be a transposition in Sym(-B), where £,• g B', and let (xi,Xk) be a transposition of Lyi. By the choice of B we know that (xj,Xk) is also transposition in L y , and since

the lemma is clear.■

6.4.4

Example: M ultiple root

Let P be the poset given in figure 6.25, and observe the L-block V spanned by the sets X = { 1 ,2 ,3 ,4 ,5 }, and Y = { 6 , 6 ,7 ,8 ,9 }. Thus node Y with repetition number fc(6 ) = 2.

6

is a root of set

101 9

8

y i> 6 P

=

5 n 4

ii

3

n

2

H1 Figure 6.25: Example: poset P 8

» =

9

\ j /

Figure 6.26: Example: the subtree H By induction we know all of the eigenvalues and irreducible eigenspaces of the L-block spanned by the subsets Xo = {1,2,3} and Vo — {7,8,9}. The subtree H is given in the figure 6.26. So

Lq

as described above is

JLo = (6,7) H- ( 6 , 8 ) + (6,9) and Ly/, the part of the Laplacian dealing with everything else is Lyi = (7 , 8 ) 4 * (7,9). L y acts on the L-block spanned by the sets (X q, Vo) and the eigenvalues of this Laplacian L y (of H) are given in figure 6.27. Our main result tells us that we must add a horizontal strip of size 2 to those three partitions. The calculations are given in figure 6.28, where the multiplicity on the right of each eigenvalue is in fact the dimension of the corresponding irreducible S x. The Laplacian Ly acts as a scalar on that irreducible, so the multiplicity of

102 eigenvalue*on s'* l " l ... 1 1

2

1

—

1,-1

-2

—

Figure 6.27: Eigenvalues of Ly> each eigenvalue is at least the degree of that irreducible, f \ = deg(Sx), i.e., the number of Standard Young tableaux of shape A. The multiplicity of an eigenvalue could of course be larger, if the same eigenvalue occurs in several irreducibles. Note that L q, the part of the action of L y dealing with the root, acts only on the trivial irreducible

= S^.

That explains why we are

applying the Littlewood-Richardson rule only with the trivial representation on the right side (the horizontal strip). The four summands on the right of each tableau A above are respectively c(A), ^2

j j c(p), and the eigenvalue(s) of Ly-, h^. Thus in this example we find that

Laplacian has eigenvalues (with multiplicities): 8(1), 5(4), 3(8), 2(5), 0(22), -2(11), -4(5) and -5(4). A computer calculation has verified that these are the correct eigenvalues and multiplicities.

103

eigenvalue

(2)

lepreecnution X

n

+ +

c(X)- ( 2) -c• • • ) =

( # of poset tableaux labelings from p! up to A then down to 0, which add a horizontal strip of length k for every y-vertex of repetition number k and subtract a vertical strip of length k for every x-vertex of repetition number k.)

This completes the proof of the theorem.*

C orollary V I.17 The eigenvalues of the Laplacian Ly are integer numbers.

108

P a

V V4

Figure 6.29: Example: poset P C orollary V I. 18 The dimension of the L-block V spanned by sets (X, Y )p is equal to the number of poset tableaux of type (X , Y ) p .

6.5

Example

For example, let the poset P be given by figure 6.29 and let the L-block V be spanned by the sets X = { 1 ,2 ,3 ,5 , 6 } and Y = {4 ,7 ,7 ,8 ,9 }. The analysis of typical poset tableau with the evaluation of the eigenvalue is given in figure 6.30. The complete list of the rest of the eigenvalues and the poset tableaux is given in figure 6.31. So the eigenvalues are (with the multiplicities):

+ 4(3),+ 2(9), 0 (9),-2(15)

109

Adding double y = 7 (this y appears twice)

By restricting we get one of these two partitions

□3

Right side is occuring twice, and thus twice the multiplicity

Figure 6.30: Example: a poset tableau

110

□

s

□

□

m rra

5

-G)-
- © ■

F31F3 fF mg

033 W

i i-i

□3

HE

B

B B

-2

-2 -2

B

D

™ FmEP F |

F F | g 11— i g i i-i

p p

f f i /

g

B

B

B

B

-2 -2

-2

+2 +2

+2 +2 +2 +2 +2 +2 +2

B

B

Figure 6.31: Example: poset tableaux

B

B

Ill

6.6

Adding the Lx

Consider the Laplacian L x - Since we have identified the L-block V with a sub­ space of the symmetric group algebra C S n, by fixing the order on the x ’s, every time the Laplacian L x switches a pair of x ’s, it is actually putting a minus sign in front of the corresponding basis element, with x ’s ordered. Since L x acts as a sum of transpositions, every eigenvalue we obtain from the L x , will have a minus sign. Recall the multiplicity of the x-node. Let v be an x-node of the diagram P [X , F], labeled with the partition A(t>) and with repetition number k(v).

Let C (v) =

{t>i,U2 , . . . , v/} be the set of covers of v. Let A,- denote A(ut), and let k, denote the repetition numbers, k(vi). The m u ltip lic ity o f A at v is defined to be m «(A) = Z rfCAi,....A|CA(«),i*(*)*

The n o d e—eigen valu e, e„(A), for each node v, is the set of sums of the content over all squares in p /A (v ) above for a given p minus the binomial coefficient

•

This gives mv(A) eigenvalues at each x-node v. We now define x -eig en v a lu e o f A, ea,(A), to be the set of numbers obtained by taking a sum of one element of e„(A) for each x-node v. So |ex(A)| = FU -nodes v ^v(A ). T h eo rem VT.19 (L ^ -C en terp iece) Let P be a linear poset with a minimum el­ ement, 0. Let X and Y be two (multi-)sets, subsets o f P . For every labeling A of positive multiplicity, each element in e^A ) is an eigenvalue o f LxProof: The proof of this theorem will be by induction on the sizes of the (multi)-sets X and Y . So let n — \X\ — \Y\ (counting multiplicities). If n =

1

— there is nothing to prove - the Laplacian L x has no pairs to switch,

and the only x-node is the minimal element for the diagram of the L-block. The

112 Laplacian L x is the one-by-one zero matrix and the eigenvalue of this unique pair is zero. Now, we will treat the general case n > 2.

• Let P \ X \ 1Yi\i P [ X ^ Y ^ . . . i P \X 0^Y^ be connected components of P [ X ,Y \. In that case the L-block V is the tensor product of the the L-blocks of the P[Xi, Yi\, The Laplacian L x switches only comparable x ’s, and two x ’s from different connected components are not comparable. Thus C

Lx{vi ®

® vc) = 5 3 vi ® ' •' ® { L x i ® “ 'vc. i=i

So if v i , . . . , v e are eigenvectors of Lx with the eigenvalues e i , . . . , e „, then vi ®

® vc is an eigenvector with eigenvalue e\ H

\-ec. Thus, by induction,

we can label each of the components P [X , Y] to get the eigenvalues of L x on the total L-block V. • Suppose that P [ X , Y] is connected. In that case, there must be a maximal element amongst the x ’s of p y x , n Call that maximal element Xj. Then define X2 , X3 , . . . , x* by:

113 1.

xi > X2 > • • • > £ * > t>o, and all > are covering relations.

2, Either — There is more than one element covering Vq. — Xib is unique cover of v0 in P[X, Y] but vQis a y-element.

Let Ax = { x i , . . .

Call Ax "terminal X-set of V ” Let B = Y\ be the set

of the y-elements above Ax, and let G = Sym (5). Note that \B\ > |j4*|.

Lemma VI.20 Let to act on 5'*' like the scalar

cx

[24]. And £ WlV>6 B(yi,yj) acts on each G-irreducible S x in S^' ® cg >CG like £ v€a °v‘ The result is that L0 acts on each copy of S x in indg,(5M- By the choice of B we know that (yj,yk) is also a transposition in Lx>, and since

the lemma is clear .■ • Now we want to consider the case where some of the elements of the sets

A x,

B , and N B = {new y's} are equal. So let’s write A

=

£*1 U

B

=

ftUftU.-.UA*,

NB

=

7i

&2

U . •. U Otl,

U 7 2 U . . . U 7 „,

where |a,-| = a,, |/?;| = &,• and |7 ,| = Cj, and the y’s in each of the a ,• are equal, the x ’s in each of the /?,• are equal, and the x ’s in each 7 ,- are equal. Let nA

=

5^

nB

=

£

=

(II Sgn( = 2 + 3 + 3 + l = 9 . Thus :

0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0

As we can see, the matrix L y* is block-diagonal with respect to this special ordering of the basis of L-block V.

124

0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 .0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0

Ly

=

Lyi +

Lq

125

0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 Lx

=

0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 1 0

The Laplacian L is L = Ly — L* + Ld , where Ld is the diagonal matrix with 9

126 on the diagonal.

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 L y —L x

=

—

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

The eigenvalues of L y — L x are then 1 and -1 , both with multiplicity 9. If we add the diagonal entries, e©, we get that the Laplacian L has eigenvalues 8 and 10 on the L-block V , both with multiplicity 9. On the other hand, if we try poset tableaux , the result is as shown in figures 6.33

Figure 6.33: Example: poset tableaux through 6.35. Thus, we get the same eigenvalues +1, and -1 with the multiplicities (9), without having to go through high dimensional linear algebra with matrices.

128

□ 3

-l

l

l

□ 3

+i

+i

Figure 6.34: Example: poset tableaux

+t

129

□

B3 P

EP □a

□ +i

+1

+i

+i

+1

m+i

□ 3

Figure 6.35: Example: poset tableaux

130

6.7

Homology

The object of the thesis is naturally to get a step closer to evaluating the homology of any Lie algebra corresponding to a linear poset, using only combinatorial properties of the poset. In these two small cases (n = l and n=2) we had no dificulty. For larger n, we need some extra results.

6 .7 .1

Hi

For example, if we want to evaluate the homology H i(L p) of a Lie algebra Lp corresponding to a linear poset P , with 0, our construction gives an immediate answer. An L-block V of size 1 , is determined by the sets (X , Y ), X = {x }, and Y = {y }. Obviously, if we want V to be non-zero, x < p y. So the corresponding diagram of this L-block is given in figure 6.36. «' □ n

Q

Figure 6.36: A diagram of the L-block

Both indicators ey and ex are zero, so the eigenvalues are given by ep>- But A ( X ,Y ) = 0 too, since X fl Y — 0. Thus L(zXiV) = w (X , Y )z XiV = \(x,y)\zXiy. In other words, the eigenvectors of the Laplacian L\ are the basis vectors zx>v, and the corresponding eigenvalues are |(x, y)|, i.e., the number of the vertices in the poset P , between x and y. The dimension of the homology is the number of zero eigenvalues, i.e., the number of the intervals z*(V, such that y covers x.

131 □

■□

rm □3 □ T-0

'B □ T.O

Figure 6.37: poset tableaux Thus d im (H i(L p)) = ( # of covering relations in P ).

6 .7 .2 H 2 In this case the L-block V in question is spanned by the (m ulti-)sets (X , Y ), each of size 2, i.e., X = { x i ,x 2} and Y — { 3/ 1 , 3/ 2 }- There are several possibilities for the L-block.

1.

All four elements are comparable, and x ’s are below the t/’s.

x i < x2 < yi < y2.

All possible poset tableaux are shown in figure 6.37. As we can see, both ey and ex eigenvalues are zero. So we don’t have to worry how to add them up - we will always get zero. A is also zero, since the sets X and Y are disjoint. Thus again, the Laplacian is L {zXuyi A z X2tV2) = w ( X yY ) ( z Xuyi A zX2tV2). But in this case, both intervals contain at least one element, so w (X , Y ) > 0. Thus in this case we never get a zero eigenvalue, which might contribute to the homology H2.

Figure 6.38: poset tableaux □ < B El

■□ T-0

Figure 6.39: poset tableau 2. y's are not comparable. Vi X i< X2