On the Combinatorics of Representations of Classical Linear Groups

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University M icrofilms International A Bell & Howell Inform ation C o m p an y 3 0 0 N orth Z e e b R oad. A nn Arbor, Ml 48106-1346 USA 3 1 3/761-4700 8 0 0/521-0600

Order Number 0104989

On the combinatorics o f representations o f classical linear groups Wei, Shiyuan, Ph.D. The Pennsylvania State University, 1990

UMI

300 N. Zeeb R A Ann Arbor, MI 48106

T h e P e n n s y lv a n ia S t a t e U n iv e r s ity T h e G r a d u a t e S ch o o l D e p a r t m e n t o f M a th e m a tic s

ON T H E C O M B IN A T O R IC S OF R E P R E S E N T A T IO N OF C LASSICAL L IN E A R G R O U P S A Thesis in M athem atics

by Shiyuan Wei

Copyright 1990 Shiyuan Wei

Subm itted in P artial Fulfillment of the Requirements for the Degree of D octor of Philosophy

August 1990

We approve the thesis o f Shiyuan Wei. D ate of Signature

( // v /9 d David M. Bressoud Professor o f M athem atics Thesis Advisor C hair of Com m ittee

'h u M

-

G ary L. Mullen Professor o f M athem atics

6kho R anee Brylinski Professor of M athem atics

R obert A. Hultquis Professor o f Statistics

£ / R ichard Herm an Professor o f M athem atics Head of th e D epartm ent o f M athem atics

i

/7

j o

A bstract In this thesis we investigate com binatorial proofs and extensions of identities for th e characters o f th e irreducible representations o f the classical groups S U (2 n ), Sp(2n), S O (2 n + 1) and SO (2n). It is known th a t the characters S p \ ( x f 1, . . . , a;*1) are associated with a finite set of sym plectic tableaux; the characters S O \ ( x f l , . . . , x ^ , 1) are associated with a finite set of 5 0 -ta b le a u x and the characters S O \ ( x f 1, . .., a:^1) are associated w ith a finite set of symplectic tableaux w ith some addi­ tional restrictions. O ur first task is seeing how to view the n-tuples of lattice p ath s as com binatorial objects representing the following deterihinants: 2 d et(^ A ,--i+ i + ^ A j - i - j + 2 );

- d et(h \.-i+ j + /iA j-i-j+2 + /lA j-i+ j-i + — d e t(/lA j-» + j + h A ( - i - j + 2

and

— h \i-i+ j-2 ~

We then apply the involution m ethod to show th a t these determ inants are the generating functions of special subsets of the universal sets of the n-tuples of these lattice paths. T he identity S p x ^ x f 1, . . . .a:* 1) = \ det(hAj_ l+J- + h xt_ ,_ j+2) follows im m ediately after we find the bijection between the special subset and the corresponding tableaux. In this thesis, we obtain tw o new com binatorial interpretations of the characters S O f a f 1, . . . , x ^ , 1) and S O ^ x f 1, .. . , x ^ ) and we also demon­ strate th e com binatorial equivalence of four classical definitions of th e Schur

function. This includes com binatorial proofs of the Jacobi-Trudi identity and det(/iA(- j + j ) = d e t(e v .-t+ j) by using n-tuples of lattice paths.

V

Table o f Contents A c k n o w le d g m e n ts ............................................................................................. viii

L is t o f F ig u r e s ..................................................................................................... vii C h a p t e r 1. P r e l i m i n a r i e s .................................................................................. 1 1.1 P artitions and Lattice P a t h .....................................................................1 1.2 Roots System and Weyl Group ..............................................................3 1.3 Symmetric Functions and the Schur Function ................................... 4 1.4 T he Involutionary M e th o d ...................................................................... 8

C h a p t e r 2. C o m b in a to r ia l E q u iv a le n c e o f D e fin itio n s o f t h e S c h u r F u n c tio n .............................................................................. 9 2.1 T he Definitions of the Schur Function .................................................9 2.2 T he Gessel-Viennot Correspondence .................................................. 11 2.3 Equivalence of Definitions 2.1.1 and 2.1.4 ......................................... 13 2.4 Equivalence of Definitions 2.1.2 and 2.1.3 ......................................... 15

C h a p te r 3.

N - tu p le o f L a ttic e P a th s a n d t h e C o rre s p o n d in g

D e te r m in a n ts ............................................................................................... 18 3.1 T he Determ inant \ det(/iA ;-i+j + ^ a ,-« -j+ 2)

• • 18

3.2 T he D eterm inant \ det(/i.\i _ i+j+AAl- i - j + 2+/iAj-i+ i-i+ /» A 1- i - i + i ) 20 3.3 T he D eterm inant \ det(hA i-i+j - /iA ,-.+i-2 + / i a .- .+ j- i “ ftA ,-i-j) 22

vi

C h a p te r 4 . C o m b in a to r ia l R e p r e s e n ta tio n s o f I r re d u c ib le C h a r a c te r s o f S p (2 n ), S O ( 2 n + l ) a n d S O ( 2 n ) ............................. 25 4.1 T h e Proof of Identity S p \ ( x f 1, . . . , x * 1) = | d e t ( / i A i - i + j + h \ i- i - j + 2) ......................................................... 2 6 4.2 A New Interpretation of the C haracter 50A (xjfcl, . • ., a?*1, *) • • • 43 4.3 A New Interpretation of the C haracter S O x fa f 1, . . . , 1 ^ ) ........ 54

C h a p t e r 5 . C o n c l u s i o n . ................................................................................... 6 1

B ib lio g ra p h y

63

Vll

List o f Figures F ig u r e 1. A 5 - tu p le o f L a ttic e P a t h s ........................................................ 12

F ig u r e 2. T w o N o n -O v e rla p p in g P a t h s

................................................. 14

F ig u r e 3. A 4 - tu p le C o r r e s p o n d in g to D[ ............................................ 29

F ig u r e 4 . A 4 - tu p le C o rre s p o n d in g to D'2 ............................................ 30

F ig u r e 5. A 4 - tu p le C o r r e s p o n d in g to D'3 ............................................32

F ig u r e 0. A 4 - tu p le C o r r e s p o n d in g to T w o P e r m u t a t i o n s

48

A cknow ledgem ents F irst and foremost, I would like to express my sincere g ratitu d e to m y advi­ sor, Professor David M. Bressoud, for suggesting my thesis problem and for directing me in com pleting my thesis. I would like to thank Professor Gary L. Mullen, Professor Ranee Brylinski and Professor R obert A. H ultquist for spending their valuable tim e to read my thesis and consenting to be on my thesis com m ittee. I would particularly like to express my heartfelt thanks to my parents, M r. and Mrs. T . C. Wei. W ithout their support and belief, certainly this thesis would not have been possible. Finally, I would also like to express my sincere appreciation to m y wife, W eizhen, for her understanding, confidence and love.

Chapter 1 Prelim inaries 1.1

P a r t i t i o n s a n d l a t t ic e p a t h

We sta rt w ith the following definition. A partition A o f a positive integer n is a non-increasing

D e fin itio n 1.1.1

sequence A = (Ai > A2 > • • •) such that (1) there is a £ > 0 such th a t A* = 0 for all k > I, (2)

E fe i

Ai =

n.

We w rite A I- n, or |A| = n, and refer to n as the size of A. We call the sm allest I satisfying (1), I = i{ A), the length of A. Also, each A,- is called a p a rt of A. E x a m p le 1 . 1.2

Let n = 10, then A = ( 3 ,3 ,2,2 ) is a partition of 10 where

/(A) = 4 D e fin itio n 1 .1 .3

The Ferrers diagram o f a partition A is a geom etric rep­

resentation o f A obtained b y drawing Ai boxes in the first row, A2 boxes in the second row below the first, . . . , with the rows left-justified. E x a m p le 1 .1 .4 T he p artitio n A = (3 ,3 ,2 ,2 ) of 10 has th e Ferrers diagram

0 0 0 0 D e fin itio n 1.1.5

0 0 0 0 O O

T he conjugate o f A b n is the partition, denoted A', o f

n, obtained b y reflecting the Terrors diagram o f A about its main diagonal. T hus A(- is the length o f ith column o f the Ferrers diagram o f A. E x a m p le 1 . 1.6

Let A = (3 ,3 ,2 ,2 ), then A' = (4 ,4 ,2 ,0 ).

Given a partition A o f n , A = (Ai > A2 > • • • > A„ > 0), a

D e fin itio n 1 .1 .7

Young Tableau o f shape A is a left-justified arrangement o f n rows o f positive integers such th a t i)

t h e j t h row contains Aj integers, all taken from the interval [1, n], with

repetitions allowed. ii) each row is w eakly increasing. Hi) each column is strongly increasing. Thus, i f A = (5 ,4 ,3 ,0 , . . . , 0 ) , n = 16, a possible Young Tableau is 3 4 8 9 D e fin itio n 1 .1 .8

L et

uq,

5 7 8 10

6 11 12 12 14 16 15

15

. . . , U( be a sequence o f lattice points in Z 2 such

th a t Uk+i =

for

0 < k< i,

where ctk is either (1 ,0 ) or (0, - 1 ) . Then Uq, .. .,U ( is called a lattice path and is denoted b y n = ( a o , . . . , a 0 the rth com plete sym m etric function h T

is the sum o f ail monomials o f total degree r in the variables x i , .. . , x n, so that

where the sum m ation is over all i j , io, . . . , in > 0 and i\ + 12 H D e fin itio n 1.3.3

1- i n =

t.

The Schur function, S \, indexed by a partition X with

£(X) < n is defined as

which can also be written as

_ E u X'n > 0),

is obtained from A by setting A'- = number o f parts o f A which are > j . Equivalently, Aj is the length o f t h e j t h column in a Young Tableau o f shape A. L et e k ( x i , . . . , x n) = e* be the kth elementary sym m etric function, that is to say, the sum o f all distinct monomials o f degree k in x i , . . . , x n which are themselves products o f distinct x; (eo = 1,

= 0, for k < 0). We then

have our final definition ••• i ®n) = det(eA'_,-+j).

2.2

T h e G e s s e l-V ie n n o t C o rre s p o n d e n c e

Gessel and V iennot established the equivalence of the first and th ird defini­ tions by first interpreting th e determ inant of complete sym m etric functions as sets of weighted lattice paths. Given a partition A of n , we define tw o sets of points in the first quadrant: Pi = (n — : + 1, n),

1 < i < n,

Qi = (n - * + 1 + A,-, 1),

1 < i < n.

We then consider all n-tuples of lattice paths consisting of horizontal steps to the right and vertical steps downward such th at i) the j t h p a th begins a t some P,- and ends at Q j. ii) no two p ath s start a t the same P,\ There is a natural perm utation on n letters associated to each n-tuple, namely th a t if th e path ending at Q j began a t P,-, then a { j ) = i. Let ( - l ) ff be the sign of th e perm utation a. We assign weight x t to each horizontal step a t height y = n + 1 - 1 and define the weight of our n-tuple to be ( - 1)" times the

product of the weights

of all the horizontal steps in

T hus if A

= (3 ,1 ,1 ,0 ,0 ), a possible n-tuple is shown in Figure

the n-tuple. 1, for which

a = 13245 and th e weight is - 2:3X4x$. Given

26

we get a new interpretation of the character of an irreducible representation of S O (2 n + 1 ) . In this section, we also introduce three known interpretations o f the characters of S O (2 n + 1) and m ention th a t there is no simple bijection be­ tween these three sets or the set we obtain in this section. In §4.3, we apply the sam e m ethod as the previous sections to obtain a new interp retatio n of the character of the special orthogonal group S O (2n). 4.1

T h e P r o o f o f I d e n ti t y

S p x i x f 1,.

. - , ® * 1 ) = £ d e t ( / i Ai_ i + i + h A (- » - j + 2 )

In §3.1, we proved th a t 4>(X>c) = | det(/i,\,_,+ j + h A ,-i-i+ 2) w ith respect to th e weight defined by wt{D) = sgn((Ac) = 5 d T h e o r e m 4 .1 .1

e

t

+ /ia.-—i—i-+-2)-

For all partition A o f length a t m ost n,

Ta

where T \ runs through all sym plectic tableaux o f shape A. D e fin itio n 4 .1 .2

W e define a weight for a sym plectic tableau T \ as follows

w t(Tx) = (®i)f t (a:i'1)/J- 1 • •

where (3i (or f l - i ) indecates that the letter i (or i) appears /?,■ times. Now, we prove th a t there exists a one to one weight preserving cor­ respondence between the elements of A c and the ways of filling a Ferrers diagram of shape A or an appropriate skew Ferrers diagram o f shape A with th e letters of alphabet l < I < 2 < 2 < * ” < n < n , where th e only restric­ tion is th a t th e entries of the diagram are weakly increasing along rows and strictly increasing down the columns. If th e n p ath s of a non-intersecting n-tuple (an element o f set A c ) sta rt from points P i,P 2 , • • • , Pn• Clearly, the ith p ath m ust sta rt from Pi and end a t Q i, which contains A,- horizontal unit segments. We fill th e letters of the alphabet 1, 1, 2, 2, • • •, n, n increasingly from left to right in th e i th row such

28

th a t these letters indicate the horizontal levels of the ith p a th ’s horizontal segment. E x a m p le 4 .1 .3

If A = (3 3 2 2 ), the n-tuple D \ corresponds uniquely to

the Ferrers diagram D [:

1 2 2 1 2 3

2 3 3

3

D \ is shown in Figure 3. Since the first path (from Pi to Q \) contains Ai = 3 horizontal segm ents, so the first row of the diagram D contains 3 letters of the alphabet.

T he second path of n-tuple D \ (from P2 to Q%) contains

A2 = 3 horizontal steps, so the second row of the corresponding diagram D contains 3 letters of the alphabet and so on. T he letter i denotes th a t there is a horizontal step which lies on the x,- level and i means th a t the step lies on xj"1 level. Hence, if n paths of an n-tuple s ta rt from P i, P i , . . . , P n , the corresponding Ferrers diagram has th e shape A. If there is only one path of an n-tuple startin g from some P(, th a t is, the n p ath s sta rt from P i,P 2 , . . . , P i- i,P i+ i,. . . , Pn,P f, then the first p ath is from P{ to Q 1, which contains [Ai - (i - 1)] horizontal steps, the second path is from P\ to Q 2 which contains A2 - 1 horizontal steps and so on, the ith p ath contains A,- —1 horizontal steps and the ( i+ j') th p a th contains A,+j horizontal steps for j = 1 , 2 , . . . , n — i. E x a m p le 4 .1 .4

If A = (3 3 2 2) and the four path s of n-tuple D 2 sta rt from

P \i -f*2» P it P it then the n-tuple Do corresponds to the skew Ferrers diagram

29

iV

Pz

Pi

Pi

Pi

pi

rZ

N/VW1

1

PI

1 --

2

l v w \ - -

k v s/v s

K w ,

kV N /V \

I 'w / v s

K w \

1W A A A A A

4 --

Qa

-I

1-

Figure 3.

Qz

Qi

Qi

-4—

A 4-tuple Corresponding to D[.

X

30

1 1

2

2

3

3 4

4

F igure 4.

A 4-tuple Corresponding to D'2.

31

^2

X x x 1

X 2 2 3

1 2

Z?2 is shown in Figure 4. W e denote the letter in the ith row and j t h colum n by a y . Generally, if only one p a th o f the n-tuple sta rts from P j, then th e corresponding skew Ferrers diagram will miss the first row of letters from a n to a i,i_ i and the first colum n of letters from a n to a y . If there are tw o paths of an n-tuple which s ta rt from P( and P j, where i < j , then th e first p ath is from Pj to Q \ which contains Ai - ( j — 1) horizontal steps. T he second p ath goes from P ■ to Q 2 which contains A2 —i horizontal steps. T he third path goes from Pi to Q 3 which contains A3 — 2 horizontal steps and so on to the (i + l)s t p ath which goes from P ,_ i to Qi+i and contains Al+i - 2 horizontal steps. From th e (i + 2)nd p a th to the j t h p a th , the path s contain Xk - 1 horizontal steps, where i + 2 < k < j and from ( j + l)s t p a th to the n th path, all those p a th s contain \ t horizontal steps, where j + 1 < I < n. E x a m p le 4 .1 .5

If A = ( 3 3 2 2 ) and th e four p ath s sta rt from P4 , P i, P j,

P j, then the n-tuple D 3 corresponds to th e skew Ferrers diagram D '3 x x x

x x x

2

2

1 2

Z?3 is shown in Figure 5 In general, th e startin g points carry the labels Pjn , P jn_ , , . . . , Pjk+1, Pjk , ■■ P j,, where j i > j 2 > ■■■> jk > 2, 1 = j k+i < jk+2 < • • • < j „ , and

32

Pz

F ig u re 5.

pf r3

Pi

(?3

Qi

Qi

-4-

-4-

-4-

Pi

Pi

Qa

Pi

A 4-tuple Corresponding to D'3.

33

{ jt I k + 1 < t < n} and { jt | 1 < t < &} are disjoint sets whose union is the set of integers from 1 through n. T he num ber of horizontal steps in the first p a th is Ai + n — (n + j i — 1) = Ai —j i + 1. For 1 < t < k , the num ber of horizontal steps in the tth p a th is

At + n — t + 1 —(n + j t —■1) = Aj —j t + 2 — t.

For k + 1 < t < n , the num ber of horizontal steps in th e fth p a th is

At + n — t + 1 — (n —j t + 1) = At + j t — t.

We fill in a skew Ferrers diagram of o uter shape A. For 1 < t < k , th e right­ m ost At - i t + 2 - 1 places of the tth row are filled w ith the horizontal levels o f the t p a th in increasing order. For k + 1 < t < n , th e right-m ost A t+ it —t places of th e tth row are also filled with th e horizontal levels of th e tth p ath in increasing order. This leaves us w ith a diagram of em pty space, fi, where the p arts o f/i a r e i j - l , i 2, . . . , j k + k - 2 , k + l - j k + i = k, k + 2 - j k + 2 , • • •, n - j n . W e observe th a t jn = n - \ { s \ j a > n + l - s } | ,

i„ _ ! = n — 1 — |{s | i* > n - s } | ,

j t = t - |{s |i a > t + 1 - s } | ,

ifc+1 = k + l - \ { s \ j a > k +

2

- s } \ = k + l - k = l.

34

It follows th a t Hk+i = k and for k + 2 < t < n, Pt = \ { s \ j * > t + 1 - s}| = |{« I js + s ~ 2 > t - 1}| = num ber of rows of p of length a t least t — 1 = length of the (t - l)s t column of p. This implies th a t for 1 < t < k, the num ber of rows below th e Durfee square of length a t least t is one more than the num ber of columns to the right of the Durfee square of length a t least t. Therefore, the length of the ith column is one more than the length of the ith row. If we consider the skew diagram corresponding to n-tuples described above which have at least one starting point of the form P j and call these C n skew diagram, then we have proved the following theorem . T h e o r e m 4 .1 .6

A C„ skew diagram has the shape \ / p , where the partition

p satisfies /i; + 1 = p\ (pi is the ith largest part, p\ is the ith largest part in the conjugate partition; equivalently pi is the length o f the ith row, p\ is the length o f the ith column.) Clearly, since all the elements of the set A c are non-intersecting, the corresponding Ferrers diagram of shape X and C n skew diagram s of shape A/ p m ust satisfy the first restriction of symplectic tableaux, th a t is, all entries are weakly increasing along rows and strictly increasing down the columns. D e fin itio n 4 .1 .7

We define the sign o f a Ferrers diagram to be + 1 , the

sign o f a C n skew diagram X /p to he ( —1)M /2 where |/i| is th e num ber o f entries in p .

35

T h e o r e m 4 .1 .8

The sign o f a C„ skew diagram X /p is the sam e as the

sign o f the perm utation o f the corresponding n-tuple o f lattice paths. Proof. The sign of the perm utation

j l h ' ' ' j k j k +1 • •' jn i

3l > 32 > • • • > jk > jk+ l = 1 < jk +2 < ••• < jn is (—i ) ( i i -1 )+0'a-1 )+",+ (j* -1). The num ber of entries of p on or to the right of the main diagonal is (j i - 1) + f a - 1) +

h (jk - 1). Since the

tth column is one longer than the t row, the to tal num ber of entries in p. is 20*1 “ 1) + 2(ia - 1) + - - - + 2 (jfc - 1). Now we consider the set of all Ferrers diagram of shape A and C n skew diagrams of shape \ / p , where p satisfies /t,- + 1 = p \, as the universal set U. The set T c of symplectic tableaux of shape A is a subset of U. By the involution m ethod, since the generating function of the set T c with respect to the weight is S p \ ( x f 1, . . . , x * 1) and the generating function of the set U is the determ inant | det(/iA(—*+j +

i+ 2)» the rest of the proof of

identity Spx(xf\...,x ^ ) =

d et(hAi-i+ j + A ^ - i- i+ a )

is just to prove th a t the generating function of the complement set U — T c is zero. Given a C n skew tableau and a pair of em pty places (places inside p ), say a ,j and aj+ i,i, j > i, for which either (a) i = j and neither a,,j+i nor a ,+2,< are inside p; or (b) i < j and none of the places

a,-+i,i+i, Oj+2,i are inside p.

36

the following algorithm , ADD, inserts letters into a y and

a j+ i,i

and produces

an element of U whose weight is the negative of the weight of the original C n skew tableau. D e fin itio n 4 .1 .0

L et E be a place in our tableau o f shape A. R ( E ) is the

place im m ediately to the right o f E . R ( E ) = 0 means th a t R ( E ) is outside the shape A. D ( E ) is the place im m ediately below E , unless there is no letter there, in which case it is the second place below E . t ( E ) is the letter in the place E . I f there is no letter in the place o f E , then we write 1(E) = oo. How to play Schiitzenberger’s [9] jeu de taquin: Let E be a place in the tableau for which 1(E) = oo (1) If R ( E ) = 0 and D ( E ) = 0, then we are done. (2) If R ( E ) = 0 or l ( D ( E ) ) < £( R( E) ) , then move the letter £( D( E) ) into th e place E and replace E by D( E ) . (3) If D ( E ) = 0 or l ( D ( E ) ) > £( R( E) ) , then move the letter l ( R ( E ) ) into th e place E and replace E by R( E) . (4) Iterate. To play jeu de taquin on the place E up to the letter m , we follow the algorithm as above bu t stop when either R ( E ) = D ( E ) = 0 or the next letter we would have to move is greater than m . A lg o r ith m A D D . (1) L et E i be the place aij and F\ be the place aj+i,,\ Initialize r = m = 1. ( 2 ) P lay jeu de taquin on the place Fr up to the letter m . (3) I f (i)

l( R ( E rj) = m

or

i ( D ( E r)) = m,

37

and (ii)

£(R (F r)) = m

or

£(D (F r)) = m ,

(iii)

£ ( D( E r)) = m

or

£( D( Fr)) = rh,

and

then let p

_

r+1 p

/

D {Er),

\ R ( E r), _ J D (F r)t

r+ 1

\

R ( F r ),

i f £ ( D ( E r)) =

m ,

otherwise; i f £ ( D ( F r ))

= rh,

otherwise.

W e e m p ty the places E r+1 an d F r+ i and then replace r by r + 1 , m b y m + 1 . Itera te this step

as long as i t applies. Since we have eliminated the facter

from the weight o f our tableau, we have not changed the absolute value o f its weight. (4) I f (i)

D ( E r) = 0

or

£ ( D( E r) ) > m ,

= 0

or

£ ( D ( F r ))

and (ii)

D (F r)

> fh,

then we insert m in to the place E r and m into the place Fr . Again, this does n o t change the absolute value o f the weight o f our tableau. I f r = 1, we are done. Otherwise, replace r by r — 1. (5)

P lay jeu de taquin on the place o f Fr up to the letter m .

(6 )

P lay jeu de taquin on th e place o f E r up to the letter rh.

(7) Replace m b y m + 1 an d return to step 3. T he level of this algorithm is the final value of m .

38

In th e following exam ples, 0 ’s m ark th e spaces E r and

E x a m p le 4 .1 .1 0

Fr, V > l . T he steps th a t have been used are in parentheses above th e arrow . T he current value o f m is below the arrow . N ote th a t we can always apply the algorithm ADD to any C „ skew tableau. X X X X X 1 X X X 1 0 1

(«)

2

X X X x 0 1 x 0 x 1 X 1

(b)

X X x x x

(4) —>

m =

2

X x X X X

(4) —► m = 3

x X

0

1 2

0

to

X

0

—► m = 3

X 1 X 0 1 3

0 3

X X X X 1

(4) ----► 771 = 1

X 2

X 1

1

X 2

X X X 1 I

1

—y m = 2

X 1 2 2 3

1 2

X 2

X 3

(2 -6 )

—► m = 1 1 2

(5 -6)

^ m =3 —

X

1

1

2

X 1

0

X 2

1

0 0 0

X X X x x

(5 -6 )

2 2

X 1

X x X x X

(3) ----► m = 1

0

1 2 (4)

1

0

X 1 2 2

X 2

X 0

X x 1 0

1

0

1

X x r 0 2

1

1 2

0

(3) ► m = 2

2

—

0

0

0

x x 1

3 0

1 2

3

1 0

(3) ► m = 4 —

x x 1

3 4

1 2

3

1 4

39

o 0 1 1

00

(2-3) m = 1

2 2

(5-6) m = 3

0 0 1 2

(«)

P> “ ^‘

m = 3

(4) 771 = 5

O O 3 3 O O

D e fin itio n 4 .1.11

0 0 3 3 4 4

m =4

m = 4

1 1 0 33 o0 q 0 1 J 5

2 3

0

~~*A m =4

(5-6) 771 = 5

1 0 0 0

1 0 4 4

1 1 3 5 4 0 4 0

%

~ \ 777 = 3

0 3 3 3

(4)

1 4 _ 5 5

771 = 5

(3)

0

0 3

O

3

m=2

O

3

1 3

1 5

(4)

~ *

3

5 5

0

3 3 4 4 0 0

(5-6)

0 m=

1 1 ! 0 3 0 4 ■ ;= 4 5

4

m= 2

2 2

—

0 0 0 0 0 0

(3)

(4)

1 3

1 2 3 3 1 00 o0 Q 0

O O O O

1 0 0 4

(5-6) ► m = 4

(4) * 771 = 6

1 3 4 4

1 3 4 0 1 5 6 6

1 0

1 5

A C n sA'ew diagram is said to be deletable a t a p a ir o f

places dij and aj+ i.i i f the entries at these tw o places are 1 and 1, respectively, or i f we can find a m inim al pair o f letters o f our alphabet, m and fh, at places to the right o f a n d /o r below a ,j and aj+i,i, respectively, such th a t after we delete these two letters and invert the algorithm A D D , we g et a new skew diagram which is e m p ty at a ,j and aj+ i.i.

40

E x a m p le 4 .1 .1 2

is deletable a t

Let A = (3 3 2 2), the C„ skew diagram x x

1 2

1 2

3 3

1 3

and 031. We can find 2 and 2 a t 041 and 022 (where,

I = k = 2) and after deleting these two letters, we get the new diagram x x x

1

x

1

1 3 3

3

If a C n skew diagram is deletable at a pair of places a ,j and aj+ i,i, then we define th e level of this deletion to be the level of the inverse algorithm, ADD, which restores these places. Equivalently, if our deletion begins by deleting th e letters m and m , then it has level m . A Cn skew diagram m ay have more than one pair of places to which we can apply algorithm ADD, and it may be deletable at one or m ore pairs of places. To establish a bijection between pairs of elements of U — T c which are skew diagram an d /o r deletable, we choose those pairs of places for which th e corresponding algorithm has minimal weight. Among these, we choose the pair of places a,j and aj+i,< which minimizes i. E x a m p le 4 .1 .1 3 (a)

For th e C n skew diagram D of shape A = (3 3 2 2): x

x

X

X

x x

x 4

x 2

~

41

We can apply ADD to two pairs of places (013, a 4i) or (022, 032). We choose the smallest i = 1, add 1 and 1 a t 013 and a4i , th e added diagram is D': x x x

(b)

x x x

1 2

I 4 The Cn skew diagram D \ of shape A = (3 3 2 2): x X x 1

x

1

1

2

1 4

is deletable a t two pairs of places (013, a4j) and (022, 032 ), we delete the pair of leters with smallest t = 1. T he deleted diagram is D[: x x x 1 2

X X X

(c)

1 4

Example 4.1.12 shows th a t we can apply algorithm ADD to the C n

skew diagram X x

O

0

2

1

1 2

1

2 to get an added C n skew diagram x x 1 3 4

1 2 3

1 4

The level of this addition is 3. Actually, our skew tableau is also deletable at places 013 and a44, the deleted skew diagram is x x x x x x

1

1 2

1

42

T he level of the deletion is 2 and if we apply algorithm ADD to this deleted skew diagram , we get the original C„ skew diagram:

x x X

x 0 1 2

0

1

x ( 2- 6) x —► x

x 1 2

1 0

m = 1 1

x (2) x —► X

m= 2 O

1

x

1

1 2

2

1 2

Therefore, we pair the original C —n skew diagram with th e deleted diagram since this has a lower level: x x x x 1 1 X 2 x 1

a nd

x x x 1 x 2 1

1

2

2

are a p air of diagram s which have same absolute value of weight bu t different signs. T h e o r e m 4 .1 .1 4

F or each Ferrers diagram E o f shape A which satisfies

only the first restriction o f the definition o f symplectic tableau, there exists a uniquely defined Cn skew diagram E ' such that w t(E + E ') = 0. D istinct Ferrers diagrams o f shape A which satisfy only the first restriction correspond to distinct C n skew diagrams. Proof. Let E be a diagram of shape A which satisfies only th e first restric­ tion of symplectic tableau. Assume a;i is the first letter such th a t a ,i < i (we need only consider the first column, since if the entries of th e diagram satisfy both restrictions of symplectic tableau, then definitely, the diagram is a symplectic tableau.) Since all the entries of th e diagram are strictly increasing down the colum n, a n m ust be

43

t — 1 (otherwise

will be the first letter which is not greater or

equal to * - 1) and th e a j_ i,i m ust be i — 1, we delete these two letters and move all entries higher th a n the place

down tw o

units an d change the letters of alphabet if necessary as described in definition 4.1.12 to get a C „ skew diagram E '. T h e o r e m 4 .1 .1 5

$(17 —T c ) = 0.

Proof. It follows immediately from theorem 4.1.15 and definitions 4.1.10 and 4.1.12. T h e o r e m 4 .1 .1 6

$ (T C) = £ det(/iAj- i+ i + h A i-i-j+ 2)-

Proof. $ (T C) = $ ( ! /) - $ ( tf - T c ) = ^ d e t(h xi-i+ j + T h e o r e m 4 .1 .1 7

S p \(x f* , . . . ,

1) = \ det(hA,—i+i +

Proof. It follows from Theorem 4.1.1 and Theorem 4.1.18. 4 .2

A N e w I n t e r p r e t a t i o n o f t h e C h a r a c t e r S O f a f 1, . . . ,

X»1)

Before giving a new com binatorial interpretation of the characters of the special orthogonal group SO (2n + 1), we would like to introduce some well known results. R. C. King proved the following theorem. T h e o r e m 4 .2 .1

[10] Fill the Ferrers diagram o f A with entries 1 < 1
t(r) where th e sum m ation is taken for all T satisfying ( K T 1 ) - ( K T 3 ) , shape(T)= X, and u>t(T) as defined above. Before introducing the theorem proved by S. Sundaram , we need the following definition:

45

D e fin itio n 4 .2 .3

R everting to the alphabet o f 2 n + l sym bols 1 < 1
The num ber of entries of fi to the right of the m ain diagonal is (ji - 1) + (72 - 1) H

h (jk — 1) which is exactly

Since (it — p 'f

equal to

The following theorem is proved by Littlewood in Theory o f Group Char­ acters. T h e o r e m 4 .2 .0 S O x ( x f 1, . . . . x * 1, 1) = ^ d et(h Ai-i+ j+ /iA 1-i--j+ 2 +/iAj-«+ j-i+ /»A (-»-i+ i)C o ro lla ry 4 .2 .1 0

\ det(hA i-i+j + h xi-i-j+ 2 + ^Aj- i+ j- l +

54

Proof. Since A'B = A s - A'b , we have th a t

9(A'b) =(9(Ab)-*(A"b) -

2 d e t( h A ,- « + i +

h x i - i - j + 2 + A * ,— i + j - i + A a ,— , - j + i ) .

C o ro lla ry 4 .2.11

T

Where T consists o f all Ferrers diagrams o f shape A and all B„ skew diagrams o f shape X /p where p is a self conjugate diagram. Corollary 4.2.11 comes directly from Theorem 4.2.8, Theorem 4.2.9 and Corollary 4.2.10, and gives a new com binatorial interpretation of the char­ acter S O \ ( x f ~*, . . 4 .3

1).

A N e w I n t e r p r e t a t i o n o f t h e C h a r a c te r S O f a f 1, . . ^ x ^ 1)

For the special orthogonal group G = S O (2 n ), King and El-Shankauby show, th a t the tableaux and generating function of Theorem 4.2.1 also give a com­ binatorial interpretation for the character of the irreducible representation of S O (2n) indexed by A: S O x i x f 1,. . . , x ^ ) , i ( X ) < n . If t(X) = n, we have T h e o r e m 4 .3 .1

[13] S O x{ x f \ . . . , x t 1) = J 2 2 ^ w t ( T ) , T

where v(T) = { 10

“ 1

i f m (T ) > °» otherwise.

Where m ( T ) is as in Theorem 4.2.1, and the sum ranges over all tableaux T o f shape A satisfying ( K 1 ) - ( K 3 ) and in addition

55

(K 4) i f m ( T ) = 0, ie, i f the entry in the first column o f row j is j o r ] for all j = 1 , . . . , n , then th e num ber o f j such th a t (j , l)-e n try o f T is j is ( even i f i = 1, \ odd i f i = 2; and S O x i x f 1, .

= SO* + S O \.

In §3.3, we proved ^

d e t ( h \ . - i+j

h xi-i+ j-2

-

+ /tA i-i-j+ 2 -

A a j-i-j)

where $ ( P d ) is the generating function for th e set V u which contains ail possible n-tuples of p ath s with startin g points labeled as P j = ( n —j + l,2 n ) ;

Pj = (n + j - l,2 n ) ; Pj'

= (n - j + 3 ,2 n ) or

Q i = (A,- + n — i + 1 ,1). We denote

Pj" = (n + j + l ,2 n ) to

a subset A d of T>d in which each

n-tuple contains no intersecting p ath s and, following the argum ent given in C hapter 2, we have th a t $ (*

4d )

= ^ d e t(h A ,-i+ j +

-

h xi-i+ j-2 - h x , - i - j ) .

T he set {P j, P j, P j', P j" \ 1 < j < 4} contains 2n + 1 distinct points: ( l ,2 n ) , ( 2 , 2 n ) , . . . , ( 2 n + l,2 n ) . M ost of these points can be described in m ore th an one way. For 2 < j < n — 1, P j-1 = (» - j + 2 , 2n) = P j'+ 1,

P j+1 = (n + j , 2n) = P j'U .

W e elim inate the labels P{ and P " and partition these 2n + 1 points into n pairs and one single point: { (l,2 n ),

( 2 n + l,2 n )}

{ (2 ,2n),

(2n, 2n)}

{ (n ,2 n ),

(n + 2,2n)}

56

and a single point (n + l ,2 n ) = PJ = P J \ We now divide the set A d into tw o non-intersecting subset A'D and A'b such th a t (1) A'd contains all n-tuples in which the n startin g points of th e path s come from n distinct pairs as above (2) A 'b contains all n-tuples in which a t least two startin g points of each ntuple come from the sam e pair of points listed above or ( n + l , 2n) = PJ = PJ' is a startin g point for som e lattice p ath in the n-tuple. T h e o r e m 4 .3 .2

$ ( A p ) — 0.

Proof. By the same argum ent as given for Theorem 4.2.5, we can prove th a t if there are two (or more) startin g points for the lattice path s of a n-tuple which come from the sam e pair of points listed above, then corresponding to this n-tuple, then by changing labels on two o f these startin g points we can pair such n-tuples into pairs th a t have the sam e absolute value of the weight b u t different signs. If an n-tuple contains a lattice p a th startin g a t the point (n + l,2 n ) = PJ = P J', we recall th a t a lattice p a th startin g a t PJ con­ tributes a positive weight while a lattice p ath startin g at PJ' con­ tributes a negative weight. By changing the label on this p ath s ta rt­ ing, we do not change the perm utation b u t we do change the num ber of lattice p ath s of negative weight by 1, thus changing the sign of th e weight of th e n-tuple. Now, we consider all possible n-tuples of th e set A 'D. As we proved in th e previous sections, there exists a one to one corre­ spondence between the elem ents of the set A 'D and th e Ferrers diagram s of

57

shape A o r th e skew Ferrers diagram s if we place th e letters of our alphabet l < l < 2 < 2 < - - - < n < n increasingly from left to right and the ith path becomes th e ith row o f a Ferrers diagram or a skew Ferrers diagram . Since th e n sta rtin g points of all n-tuples of set A 'D m ust come from n distinct pairs of points and the n subscripts m ust form a perm utation of ( 1 , 2 , . . . , n ), we have to choose one startin g point from P i or P{" (we elimi­ nated P{ and PI1), choose one point from P 2 or P 2" (because P 2 = P 2 can not be chosen). Since we have already chosen P j o r P j" , we can no t choose P3' or P 3 and have to choose one point from P3 or P j" and by th e same argum ent, we have to choose P4 o r P j" , P 5 or P " ' and so on. We have the result th a t the n startin g points of all n-tuples of A 'p have labels of the form P j or P j" . If all n starting points have labels o f the form P j, then the corresponding diagram is a Ferrers diagram of shape A. If th ere is only one label of the form P j", asy P /" , then the corresponding diagram will miss th e entries in the first row from a n to

and th e entries

in the first column from a n to a ,i. If these are two startin g points w ith labels of th e form P j", say P /" and P j" where i < j , then th e corresponding diagram will miss the entries in the first row from a n to a j j + i , in the second row from a 2i to a 2i{+2 and miss the entries in first colum n from a n to aj i , and in second column from a j 2 to a j+ ii2. In general, the sta rtin g points carry the labels Pj a , Pj n_ , , . . . , Pjk+1, P]k

P j , , where j i > j 2 > • • • > j k > 1, 1 < jk+ i < jk +2 < ‘ " < j n , and

58

{j t | k + 1 < t < n} and {j t | 1 < t < k} are disjoint sets whose union is the set o f integers from 1 through n. T he num ber o f horizontal steps in th e first p a th is Ax + n - (n + j i + 1) = Ai —j i - 1. For 1 < t < k, the num ber of horizontal steps in the tth p a th is

At + n - 1 + 1 - (n + j t + 1) = A< - j t - 1.

For k + 1 < t < n, th e num ber of horizontal steps in the ith path is

At + n - 1 + 1 - (n - j t + 1) = At + j t - t.

We fill in a skew Ferrers diagram of outer shape A. For 1 < t < k, the right­ m ost At —j t ~ t places of the ith row are filled w ith the horizontal levels of the t p a th in increasing order. For k + 1 < t < n , the right-m ost At + j t - t places of th e ith row are also filled with the horizontal levels of the ith p a th in increasing order. This leaves us with a diagram of em pty space, fi, where the parts of (i are j i + 1, j 2+ 2, . . . , j k + k, k + 1 - j k+i = k, k + ^2 - j k+2 , . . . , n - j n . We observe th a t in = n - |{s \ j a > n + 1 — s}|, i n —1 = n - 1 - |{s I j a > n - s}|,

j t = t ~ |{* I ja > t + 1 ~ -S}|,

ifc+1 = k + 1 - |{« I js > k + 2 - s}\.

59

For k + 2 < t < n, fit = |{s | j , > t + 1 - s}\ = \ { s \ j s + s > t + 1}| = num ber of rows of p of length a t least t + 1 = length of the (t + l)s t column of p. T his implies th at for i < p(p), we have m =

+ 1.

It proves the following theorem: T h e o r e m 4.3.3

A D n skew diagram has the shape X /p where p satisfies

the condition pi = p[ + 1. Equivalently, the ith row o f p is one longer than the ith column. T h e o r e m 4.3.4 = Z i - i ^ v K T ) T

where T is a Ferrers diagram o f shape X or a D n skew diagram, m is the num ber o f entries o f p and k is the number o f entries o f main diagonal o f p . Proof. T he sign of the perm utation

j l h ' ' 'jkjk+l ‘ ' ' j n ,

j l > j i > ' • • > jk > jk +1 > 1 < jk+2
The num ber of entries of p to the right of the main diagonal is (ji - 1) + ( j 2 - 1) + -----1- (jk - 1)* Since the ith row is one longer than the i column, the total num ber of entries in p is 2(j i - 1) + 2 (j2 - 1) H

h 2(jk - 1) + 2k. This complete the proof.

60

T h e o r e m 4 .3 .5

$(*4r>) = g d e t^ A i- i+ j - f a i - i + j - 2 + * a 4—*—i+2 - J»A.—i-j)>

Proof. This follows immediately from Theorems 3.3.1 and Theorem 4.3.2. T h e o r e m 4 .3 .6

= ^ d e t ( h x . - i+j - h x i - i + j - 2 + h x , - i - j + 2 -

C o ro lla ry 4 .3 .7

*

«

) =

i )“ f 2 i “"(r ),

T

where T is a Ferrers diagram o f shape X or a D n skew diagram, m is the num ber o f entries o f diagram p and k is the num ber o f entries o f main diagonal o f p. Proof. It follows from Theorem 4.3.6, Theorem 4.3.5 and Theorem 4.3.4 directly. Theorem 4.3.7 gives an interpretation of the character of the special orthogonal group SO (2n).

Chapter 5 Conclusion We need th e following definitions to summarize the results in this thesis. D e fin itio n 5.1 Let A and p be two partitions, we define p < A to mean th a t p /X is the shape of a skew diagram. D e fin itio n 5.2 We define p = p(p) to be the num ber of entries on the main diagonal of p and define P j , j = 0 ,± 1 to be three sets for which /*,• = p\ + j when i < p. Then the theorem 4.1.10, 4.2.8 and 4.3.4 which we proved in chapter 4 are equivalent to the following theorem s respectively. T h e o r e m 5.3 S p x( x f \ . . . , a « ) =

£

( - l ^ l / ^ / ^ f 1, . . . , ^ 1)

T h e o r e m 5.4

SO>(xf\. ..,* " ,1 )=

Y

(" 1

T h e o r e m 5.5

SOxixf1

a") =

Y ( - 1)(M“3p)/2W * ? 1. - . * " ) M£Pj,/iCA

where S \/n is defined to be the skew Schur function, th e sum over all skew tableaux T of shape X /p of w( T) . Theorem 4.1.1 states

S p x ( x f 1, . . . t x * 1) =

Y wiT)' TX

62

where T \ runs through all symplectic tableaux of shape A. In this thesis, theorem 4.1.16 establishes th e correspondence between sym plectic tableaux and the tableaux and skew tableaux counted in the theorem 5.3. This gives a com binatorial proof of the identity

S p x ( x f l , . . . , x ^ 1) = I d e t ^ A . - . + j + h A , - , - , ^ ) . which is an open problem listed in S undaram ’s thesis. It is still an open problem to find the correspondence betw een the var­ ious sets o f com binatorial objects defined by King, Koike and T erada, and Sundaram in interpreting S O x i x f 1, . . . , x ^ , 1). T he signed skew tableaux counted in the Theorem 5.4 m ay provide a n atu ra l bridge. It is also still open to find the correspondence between th e signed skew tableaux counted in theorem 5.5 and the type of tableaux defined by King in interpreting S O A ^ f 1,...,® ;!11).

63

B ibliography [1] H. W eyl, The Classical Group; Their Invariants and Representations, 2nd. ed., Princeton University Press, Princeton, N J, 1946. [2] D. E . Littlewood, The Theory o f Group Characters, Oxford, 1940. [3] I. Gessel, Tournam ents and Vanderm onde’s D eterm inant, J. Gragh The­ ory, 3(1979), 305-307. [4] D. M. Bressoud, Colored Tournam ents and W eyl’s D enom inator Formula, Europ. J. Combinatorics (1987) 8, 245-255. [5] D. Zeilberger and D. M. Bressoud, A P roof o f A ndrew ’s q-Dyson Conjec­ ture, Discrete Math. 54(1985), 201-224. [6] I. Gessel and G. V iennot, Determ inants, Paths and Plane Partitions, P reprint. [7] P. G oulden, Directed Graghs and the Jacobi-Trudi Identity, Can. J. Math 37(1985), 1201-1210. [8] It. C. King, Weight M ultiplicities fo r the Classical Group, in Lecture Notes in Physics 50, 490-499, New York: Springer, 1975. [9] M. P. Schiitzenberger, La Correspondance de Robinson in “Combinatoire et Representation du Groupe Sym etrique”, Strasbourg, 1976(D. Foata, ed.) Lecture Notes in M athem atics, No. 579, Springer-Verlag, Berlin, 1977. [10] R. C. King and N. G. I. El-Sharkaway, Standard Young Tableaux and Weight M ultiplicities o f the Classical Lie Group,J. Phys. A : M ath Gen., 16(1983), P P. 3153-3177.

[11] K. Koike and I. T erada, Young-Diagrammatic Methods fo r the Repre­ sentation Theory o f the Classical Groups o f Type B n,Cn,D n, J. o f Algebra, 107,No.2(1987), P P . 466-511. [12] S. Sundaram , Orthogonal Tableaux and an Insertion Scheme fo r S O (2 n + 1), Subm itted to J. o f Combinatorial Theory.

V ita N a m e : Shiyuan Wei D a te o f B i r t h : M arch 14,1945 E d u c a tio n : M.S. in M athem atics, W estern Illinois University, Macomb, IL 61455, Aug. 1984; B.S. in M athem atics, Fudan University, Shanghai, China, Dec. 1968. E m p lo y m e n t: A ssistant Professor, D epartm ent of M athem atics, Penn State at Mont Alto, Aug. 1990-; G raduate A ssistant/ G raduate Lecturer, Penn S tate, University Park, PA 16802, Aug. 1984-May 1990; G raduate A ssistant, Western Illinois University, M acomb, IL 61455, Aug. 1982- May 1984; Instructor, Shanghai Worker’s University, Shanghai, China, Jan. 1973June 1982. H o n o rs : 1. H. GLENN AYRE SCHOLARSHIPS, 1983 at W estern Illinos Uni­ versity. 2. HASKELL B. CURRY FELLOW SHIP, 1984-1985 a t Penn State. M e m b e r s h ip in P ro fe ss io n a l O rg a n iz a tio n s : M ember of American M athem atical Society. P u b lic a tio n s : 1. (w ith D. Bressoud) Com binatorial Equivalence of Classical Defini­ tions of Schur Function. (Subm itted) 2. A Com binatorial Proof of Identity S P \ ( x f 1, . . . , x ^ 1) = \d e t ( h \ .- i +j + h x i- i- j+ 2 ) ( To be subm itted )