Matrices Connected with Brauer’s Centralizer Algebras

Table of contents :
ACKNOW LEDGEM ENTS .......................................................................................................... ii
LIST OF FIGURES ....................................................................................................................... iv
CHAPTER
1. Introd uction .......................................................................................................................... 1
1. A Theorem of Schur ..................................................................................................... 1
2. Centralizer algebras for 0 n and Sp 2 n ..................................................................... 3
3. A tower of ideals in A^ ........................................................................................... 6
4. Matrices whose nullspaces encode the semisimple structure of AjX^ ................. 7
5. A combinatorial definition for 16
6. Some results about the matrices Y a / a « ..................................................................... 19
7. The algebra 20
2. Determinants of M x^ and .......................................................................... 25
1. Column permutations of standard matchings ........................................................ 25
2. Product formulas for and 31
3. Eigenvalues of Tk{x) and Tk{yi,..., yn) ................................................................. 32
4. The column span of P .............................................................................................. 34
5. Computation of det M x^ and 41
3. Combinatorial algorithms and the Littlewood-Richardson r u le ..................... 48
1. Robinson-Schensted-Knuth row insertion .............................................................. 50
2. Dual Knuth relations and equivalence ..................................................................... 51
3. Jeu de Taquin for standard tableaux ........................................................................ 53
4. The Littlewood-Richardson r u le .............................................................................. 54
5. A theorem of Dennis White ........................................................................................ 55
4. Jeu de Taquin for standard m atchings ....................................................................... 56
1. Definition of the algorithm ........................................................................................ 56
2. Jeu de Taquin preserves standardness .................................................................... 59
3. Dual Knuth equivalence with Jeu de Taquin for tableaux ................................. 65
4. The normal shape obtained via Jeu de Taquin .................................................... 70
5. Alternate statements of Theorems 2.23 and 2 .2 5 ................................................. 74
5. Remaining P roblem s .......................................................................................................... 75
BIBLIOGRAPHY ............................................................................................................................. 77

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M atrices C onnected w ith B rauer’s C entralizer A lgebras

by Mark D. McKerihan

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 1996

Doctoral Committee: Professor Philip J. Hanlon, Chair Professor Andreas R. Blass Associate Professor Kevin J. Compton Professor John R. Stembridge Associate Professor Berit Stens0nes

UMI N u m b e r:

9624683

UMI Microform 9624683 Copyright 1996, by UMI Company. AH 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

ACKNOW LEDGEM ENTS

I wish to th a n k several people for th e im p o rta n t roles th e y have played in m y g rad u ate education and in producing th is thesis. It has been m y pleasure to w ork u n d er th e direction of Professor P hilip H anlon, and I extend m y m ost heartfelt g ra titu d e to him . He has been an invaluable resource during m y tim e here, and I am p articu larly grateful for his unfailing patience. He has enriched m y experience as a g rad u ate stu d en t. I th a n k Professors A ndreas Blass and Jo h n Stem bridge for carefully reading m y thesis. Each pointed out errors and m ade suggestions which vastly im proved this m anuscript. Also, I th a n k Professor Stem bridge for several excellent courses he has ta u g h t. M y notes from his classes have been very useful to m e, and I expect th a t th ey will continue to be so. I w ant to th a n k W illiam Jockusch for several valuable conversations concerning th e subject of th is thesis. A conjecture of his was th e startin g point of th e research described here, and his insight was p articu larly useful to m e as I was beginning m y stu d y of B ra u er’s C entralizer A lgebras. For continually expressing th e ir confidence in m e, I th a n k m y p aren ts Jam es and B eth M cK erihan. T hey have always encouraged m e to pursue m y interests, and to th in k independently. Finally, I th a n k m y wife N avah Langm eyer for everything th a t we have together.

TABLE OF C O N TEN TS

A C K N O W L E D G E M E N T S ..........................................................................................................

ii

LIST OF F IG U R E S

iv

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

CHAPTER 1. I n t r o d u c t i o n ..........................................................................................................................

1

1. 2. 3.

A Theorem of Schur..................................................................................................... Centralizer algebras for 0 n and Sp 2n ..................................................................... A tower of ideals in A ^ ...........................................................................................

1 3 6

4. 5. 6. 7.

Matrices whose nullspaces encode the semisimple structure of AjX^ ................. A combinatorial definition for Some results about the matrices Y a/ a« ..................................................................... The algebra

7 16 19 20

2. D eterm in a n ts o f M x^ and

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

25

Column permutations of standard m atch in gs........................................................ Product formulas for and Eigenvalues of Tk{x) and T k { y i, . . ., yn) ................................................................. The column span of P .............................................................................................. Computation of det M x^ and

25 31 32 34 41

3. C om binatorial algorith m s and th e L ittlew ood -R ich ard son r u l e .....................

48

1. 2. 3. 4. 5.

1. 2. 3. 4. 5.

Robinson-Schensted-Knuth row in s e r tio n .............................................................. Dual Knuth relations and equivalence..................................................................... Jeu de Taquin for standard tableaux........................................................................ The Littlewood-Richardson r u l e .............................................................................. A theorem of Dennis W h ite........................................................................................

50 51 53 54 55

4. Jeu d e Taquin for stan d ard m a tc h in g s .......................................................................

56

1. 2. 3. 4. 5.

Definition of the a lgorith m ........................................................................................ Jeu de Taquin preserves standardness.................................................................... Dual Knuth equivalence with Jeu de Taquin for ta b le a u x ................................. The normal shape obtained via Jeu de T a q u in .................................................... Alternate statements of Theorems 2.23 and 2 . 2 5 .................................................

56 59 65 70 74

5. R em ain in g P r o b l e m s ..........................................................................................................

75

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

iii

77

L IST O F F IG U R E S

F igure 2.1

The sets A and B used in the Garnir relation....................................................................

36

3.1

A sequence of Jeu de Taquin moves for standard tableaux.............................................

53

4.1

A sequence of Jeu de Taquin moves for standard matchings............................................

57

4.2

Boxes near y' when y = d .........................................................................................................

61

4.3

The sets A, B, F and G in D .................................................................................................

62

4.4

The diagram D when 6(c) £ A ...............................................................................................

64

4.5

The sets A, B, F and G ..........................................................................................................

67

IV

CHAPTER 1

Introduction

1.

A T h e o re m o f Schur

B ra u er’s centralizer algebras were in troduced in 1937 by R ichard B rau er [Brr] in order to stu d y centralizer algebras of th e classical orthogonal and sym plectic groups on th e tensor powers of th e ir defining representations. In order to p u t B ra u er’s work in context, we will begin by describing related work of Schur concerning th e general linear group. D enote th e general linear group of n by n invertible m atrices w ith com plex entries by G L n (C) = G L n . Let V = C"' be th e n dim ensional com plex vector space w ith stan d a rd basis { e i ,. . . ,e n }. T h e general linear group of invertible linear tran sfo r­ m ations on V is denoted by G L ( V ) . A ny m a trix A £ G L n can be th o u g h t of as an invertible linear tran sfo rm atio n on V . Let p : G L n —> G L ( V ) be defined by

p (A )ei = Yj C j ( G L n) defined by

n-^A; where /(A) is the number o f parts in the partition A. In particular, (f> is an isom orphism i f and only i f n > f . Theorem 1.1is useful in th e proof of T heorem 1.2, th e double centralizer theorem for G L n .

3 T h e o r e m 1 .2 [Scu] A s an S f x G L n module, V® 1 ^ Q ) S X® W X AH/ where S x is the irreducible Specht module fo r S f indexed by X, and W x is the irre­ ducible Weyl module f o r G L n indexed by X. Theorem 1.2 shows th a t th e re is a pairing of th e irreducible representations for S f and certain irreducible representations of G L n , and is th e basis for th e b eau tifu l connection betw een th e represen tatio n theories of these tw o groups.

See [Wl] for

m ore inform ation on these theorem s and related topics. 2.

C e n t r a l i z e r a lg e b r a s f o r On a n d S p 2 n

B rauer was in terested in proving analogues of Theorem 1.1 for other Lie groups. In a 1937 p ap er [Brr] he did th is for th e orthogonal group On and th e sym plectic group S p 2 n- In p a rtic u la r B rauer found algebras A ^ and B j 2n^ respectively th a t play th e sam e role for these groups th a t CS f plays for G L n . We will discuss th e algebra A ^

in detail here.

We have replaced n by x to

generalize B ra u e r’s definition allowing any real num ber a; as a param eter. Hanlon and W ales showed in [HW1] th a t B ^ is isom orphic to A^~x^, hence we only need to consider A ^ for all real x. T he algebra A ^ has a vector space basis consisting of 1-factors on 2 / points. A 1-factor is a graph in which each vertex is incident w ith exactly one edge. We draw a 1-factor

6

in two rows of / vertices. T he set of vertices in th e to p row is denoted

t(S) and th e set of vertices in th e b o tto m row is denoted b(S). Let F f be th e set of 1-factors on 2 / points. To m ultiply tw o 1-factors

6

\ and

6 2

, we begin by identifying th e vertices in b(S\)

w ith th e ir corresponding vertices in t(S 2 ) to o btain a graph B { 6 \, S2) on 3 / points. By

ignoring th e m iddle row of B ( 8 i, 8 2), and regarding p a th s w ith vertices in th e m iddle row as single edges, we obtain a new 1-factor /3(8i,82) on th e vertices £( i + 1 and I > i + 1, th en (2.35) holds as well since 7r has no effect on edges betw een rows k and I in th a t case. We conclude th a t if th e re is a difference betw een ttfr(e) and w

th e n it m u st occur in row i or

above, and we have already shown th a t th e lem m a holds in th a t case. B

T h eo rem 2.20 The set {e(£) : S G A x/^} fo r m s a basis f o r the vector space e(Vx/^). Proof. We already know th a t th e set of e(e) such th a t e is a row increasing m atching of shape X/p, spans

(see (2.20)). We can im prove th is slightly as follows:

e(VA/^) = (e(e) : e G F x ^ , e row increasing, e(e) ^ 0)

Define Wx/n = (e(6) :

8

(2.36)

G A x /f) . We will in d u ct on th e order -h £o(y)- Define th e sets A and B as in L em m a 2.19. Let such th a t Tfeh = e0. N ote th a t

C be th e num ber of p erm u tatio n s it £ Sym(v4 U B ) C ^ 0, since th e id e n tity clearly works.F urtherm ore,

note th a t for any m atching e we have

e(e) = e(e).

(2.37)

This follows from th e definition of e, and th e fact th a t e differs from e by a row perm u tatio n . Using th e G arnir relation (T heorem 2.18) we now ob tain Ce(e0) = -

e(7re0) •7rgSym(/luB),7^07^0

= -

X

e(Weo).

7reSym(yHJB),7reo/eo

(2.38)

By Lem m a 2.19, each Weo th a t appears in th e right hand side of (2.38) satisfies T(Wefi) -< T'(eo)) so by our choice of e0, we have e(?feo) = 0. Hence,

e(e0) = 0,

(2.39)

which contradicts our choice of eo. We deduce th a t th e re can be no such boxes x and y , and therefore e0 m ust be a sta n d a rd m atching, i.e. e0 £ 4 \//o a n d e(e0) € Now, if e is a row increasing m atch in g w ith T (e) y T ( e 0), such th a t e ^ A \ / ^ th en th e re is som e box x im m ediately above a box y w ith e(x) >h e(y). Define A

and B as before, and let C be th e num ber of p erm u tatio n s 7r £ Sym (A U B ) such th a t 7fe = e. N ote th a t C

/ 0. U sing th e G arn ir relation we o btain

Ce{e) = —

^2

e(7re).

(2.40)

7reSym(i4uB),7r£^e

By L em m a 2.19, every We th a t appears in th e right h an d side of (2.40) satisfies T(We) -< T (e), so by induction every te rm on th e right hand side is in W x / ^ which im plies th a t e(e) £ W x/^ as well. We have shown th e e(V y M) = W x / ^ It rem ains only to show th a t 8

th e set {e(5) :

€ A \ /n } is linearly independent. Suppose th a t th e re is a linear relation

£

We w ant to show th a t ag

— 0 for all

a se( 8 ) = 0.

8

(2.41)

£ Ax/p. If n o t, th en

m atching th a t m axim izes T ( 8 0) am ong those

8

let

8 0

be th e stan d ard

£ A x / Mw ith ag ^ 0 .

In th e proof of L em m a 2.21 below, we show th a t for any sta n d a rd m atch in g 8 , e( 8 ) has a non-zero

8

coefficient in V x /^ and th a t if Si and

w ith T ( 8 i) -< T ( 8 2), th en th e In p a rticu lar, for any zero. Since th e

8 0

8

8 2

8 2

are tw o stan d a rd m atchings

coefficient of e{ 8 \) is zero.

£ Ax/n such th a t T ( 8 ) -< T ( 8 0), th e So coefficient of e( 8 ) is

coefficient of e(50) is no t zero, we m u st have ag0 = 0, co ntradicting

our choice of £0. Hence, we m ust have ag = 0 for all

8

£ A x /fl. H

A nother way to s ta te T heorem 2.20 is th a t th e colum ns of P form a basis for th e space eiVx/n). 5.

C o m p u t a t i o n o f det M x^ a n d d e t M x^

T he com putations of det M and d et M. are v irtu ally id entical, so we will only com pute det M explicitly, and m erely s ta te th e corresponding resu lt for det M..

42 Suppose th a t v € V2v is an eigenvector of Tk( x ) w ith eigenvalue h 2 u{x). Since Tk(x) com m utes w ith th e action of Sx/^, we have Tjb(®)(e(u)) = e(T k(x)v) = e{h 2 v(x)v) = h 2 u{x)e{v), (2.42)

i.e. e(v) is also an eigenvector of Tk( x ) w ith eigenvalue h 2 l/(x). F urtherm ore, it is not h ard to see th a t th e h 2 v(x) are all d istin c t, so we m ust have e(i>) G V2v. We have shown th a t

(2.43)

e {V2u) Q V2u.

It now follows th a t e(Vx/^) = e

®

V*

■2vh2k

= ® e(V2l/). 2i/(-2fc (2.44)

Let dv = d*u be th e dim ension of e{Vv) (thus, if v is not even th en dv = 0). For every even p a rtitio n

2

v b

2

k, le t £?2l/ = {e ( uj " ) , . . . , e ( u ^ )} be a basis for e(V2u).

Let B \ / M= ( J B 2v. We now have two bases for

nam ely

and B \ / M.

Let Q be th e Fx/^ x Bx/n m a trix whose colum n indexed by e{v2v) is th e expansion of e ( v fu) in V * i n term s of th e basis F

x Let S be th e Ax/fj x Bx/n m a trix such

th a t

P S = Q.

(2.45)

43 T he m a trix S is an invertible tra n sitio n m a trix from th e basis e(Ax/n), f° th e basis Bx/ti

e(V^/M).

T he im portance of th e m a trix Q is th e fact th a t its colum ns are eigenvectors for Tfc(x). Thus,

(Tk( x ) Q ) e(v?v) = h 2 v{x )Q e(v2iuy

(2.46)

T his is equivalent to saying th a t Tk (x) Q = Q D , w here D is a Bx/fj, x Bx/n diagonal m a trix whose diagonal e n try in th e colum n indexed by e(v?"), is /i2„(a:). We can use these facts to stu d y th e p ro d u ct form ulation of th e m a trix M as follows: S lM = S tP tT k ( x ) J = Q lT k{ x ) J = D Q 'J = D S tP t J. (2.47)

From th is we obtain det M = d et D d e t( P o — n n — 1 . . . 2 1.

T h eo rem 3 .7 For any a 6 S n , the follow ing properties o f R S K hold: ( 1 ) (P(lo 0 ctujo),Q(ll!0 ctuj0)) = (P ((r)evac, Q(cr)evac)), where P evac is obtained fro m P by a process known as evacuation (see [SciilJ). ( 2 ) (P e v a c )* =

( P ‘)e v ac

( 3 ) [ S C U 2 ] ( P e v a c ) ev ac — P •

(4) shape Pevac = shape P . (5) (P(oru 0 ),Q (cru0)) = (P(o-)‘,Q (h s'{. H T he claim ju s t proved is an obvious contradiction because

it im plies th a t th e sizes

of A and G are infinite. So, S'(c),S'(d) ) = shape P ( tcs)- In p a rtic u la r, suppose th a t S' is a sta n d a rd m atching of norm al shape V2 - In th a t case v 2 = shape S' = sh ap e P ( tts>) = shape P(7Tfi) =

(4.30)

Here, th e second equality comes from th e fact th a t since S' has norm al shape, T h e­ orem 3.11 im plies th a t we m ust have P ^ s 1) = S'. ■

72

C orollary 4 .1 0 I f the standard m atching S is taken to a standard m atching 8 ' fo r a norm al even shape v via J d T , then

8

' is independent o f the sequence o f J d T moves

chosen. In fact, v = shape P(tts)> and

8

' is the unique standard m atching o f shape

v. Proof. T his follows from T heorem 4.9 and C orollary 2.9. ■ B ecause of Corollary 4.10, we can m ake th e following definition.

D efin itio n 4.11 I f

8

is a standard m atching, then let v ( 8 ) denote the shape o f the

standard m atching o f norm al shape obtained fro m

8

via J dT.

For th e norm al shape v h 2n, by g*v we denote th e num ber of tim es v appears as v ( 8 ) for som e

8

6 A \ /^. T h e following theorem shows th a t g*u is th e Littlew ood-

R ichardson coefficient c*„. Its proof m akes heavy use of theorem s from ch ap ter 3.

T h e o re m 4 .1 2 For any skew shape X /p and any norm al even shape v with \X/p\ = \v\, we have g*v = cj„.

Proof. We find a bijection , betw een th e set of stan d a rd m atchings

8

w ith u ( 8 ) = u ,

and th e set of LR fillings T of shape X /p such th a t h x has weight v. Suppose

8

€ A\ f n, and v{ 8 ) = v. T h en h$ is a fixed point free involution of

w here X / p has size 2n. F urtherm o re, hs = ttsu>q. Taking inverses, we o btain hs = ojottJ 1.

(4-31)

Since 8 is sta n d a rd tab leau , by T heorem 3.16, we have lp(P(7T5)) = h x for som e LR filling T of shape A/ p . By C orollary 4.10, shape P{tts) = v , and it follows th a t h r has w eight u. We define (f)(8 ) = T. Now, if T is a LR filling of shape X / p such th a t h x has weight v, th en let P be th e stan d a rd tab leau of shape v such th a t lp (P ) = h x . Let 7r £ Sin be th e p e rm u tatio n

73 such th a t P { * ) = P, Q(ir) =

(4.32)

P evac.

Using Theorem 3.7 (5), (2) an d (3), we obtain P(7ro;o) = P \ and

Q( TTWo) = (Pevac)Lc =

(4-33)

Since P l consists entirely of even colum ns, Theorem s 3.5 and 3.6 im ply th a t t^ujq is a fixed point free involution in

S211

•

T hus,

TTWo = (tTCUo)-1

Uq7T_ 1 .

=

(4.34)

We have lp(P(7r)) = h?, so by T heorem 3.16, too^r-1 = hg for som e stan d ard tab leau S of shape X/fX. S i m

M oreover, since a;07r_1 isa fixed

fi isa m atching of shape X/ fi, and hence

8

€

point free involution of

To show th a t u( 8 ) = v , we

com pute v(fi) = shape P(irg) = shape P(hgto0)

= shape P(cuo7r_1a;o) = shape P ( 7 r _ 1 ) e va c = shape P(7r-1 ) = shape Q ( tt) = v. ( 4 . 35 )

74 Clearly, (5) = T . In fact, it is no t h a rd to see th a t is a bijection, which finishes th e proof of th e theorem . B

5.

A lte r n a te sta te m e n ts o f T h eo re m s 2.23 and 2.25

Theorem 4.12 allows us to re s ta te T heorem 2.23 in term s of Jeu de Taquin for m atchings as follows.

T h eo rem 4.13 det M =

\ Rs \ \ Cs \ hv(s){x).

feAx/p Sim ilarly, we can re sta te T heorem 2.25.

T h eo rem 4 .1 4

det A4 =

JJ s €A>.ln

\R s \ \ C s \ Z ( i u{s)){ y u . . . , y n).

CHAPTER 5

Remaining Problems

In th is ch ap ter we give a list of problem s th a t rem ain open concerning B rau er’s centralizer algebras. We list th ese problem s m ore or less in order of increasing diffi­ culty. P r o b l e m 1: T heorem s 4.13 and 4.14 give a nice com binatorial form to th e d eterm in an t for­ m ulas for

and M . I t

would be nice to have a m ore direct proof of these

theorem s th a t b e tte r illum inates th e role of th e Jeu de Taquin algorithm . P r o b l e m 2: F ind a d e term in a n t form ula for Y P r o b l e m 3: F ind a d eterm in a n t form ula for y x^ . P r o b l e m 4: F ind th e nullspaces of th e m atrices M x^ a t all integer values of x. P r o b l e m 5: a t n-tuples y i , . .. , y n th a t are roots of

F ind th e nullspaces of th e m atrices

th e ir d eterm in an ts (so th ey will be roots of zonal polynom ials).

75

P r o b l e m 6: F ind th e nullspaces of th e m atrices Y

a t all integer values of x. A solution

to th is problem would give com plete inform ation ab o u t th e sem isim ple stru c tu re of

P r o b lem 7: Find th e nullspaces of th e m atrices

a t all n -tu p les y i , . . . , y n - A solution

to this problem would give com plete inform ation ab o u t th e sem isim ple stru c tu re of Aj [k).

B IB L IO G R A P H Y

78

B IB L IO G R A P H Y

[Brr]

R. Brauer, On algebras which are connected to the semisimple continuous groups, Ann. of Math. 38 (1937), 857-872.

[Brn]

W. P. Brown, An algebra related to the orthogonal group, Michigan Math. J. 3 (1955-1956), 1- 2 2 .

[G]

H. G. Garnir, Theorie de la representation lineaire des groupes symetriques, Mem. Soc. Roy. Sci. Liege 10, No. 2 (1950), 5-100.

[HW1] P. Hanlon and D. Wales, On the decomposition of Brauer’s centralizer algebras, J. Algebra 121 (1989), 409-445. [HW2] P. Hanlon and D. Wales, Eigenvalues connected with Brauer’s centralizer Algebras, J. Al­ gebra 121 (1989), 446-476. [HW3] P. Hanlon and D. Wales, Computing the discriminants of Brauer’s centralizer algebras, Math. Comput. 54, No. 190 (1990), 771-796. [HW4] P. Hanlon and D. Wales, A Tower Construction for the Radical in Brauer’s centralizer algebras, J. Algebra 164 (1994), 773-830. [Jal] A. T. James, The distribution of the latent roots of the covariance matrix, Ann. Math. Statist. 31 (1960), 151-158. [Ja2] A. T. James, Zonal polynomials of the real positive definite symmetric matrices, Ann. of Math. 74 (1961), 456-469. [Ja3] A.T. James, Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Statist. 35 (1964), 475-501. [Jo]

V. F. R. Jones, Index for Subfactors, Inv. Math. 72 (1983), 1-25.

[Joe] W. Jockusch, Private communication. [K]

D. E. Knuth, Permutations, matrices and generalized Young tableaux, Pacific J. Math. 34 (1970), 709-727.

[LR]

D. E. Littlewood and A. R. Richardson, Group Characters and algebra, Philos. Trans. Roy. Soc. London Ser. A 233 (1934), 99-142.

[M]

I. G. Macdonald, “Symmetric Functions and Hall Polynomials,” Clarendon, London, 1979.

[Me]

M. McKerihan, Matrices connected with Brauer’s Centralizer Algebras, Electronic J. Comb. 2, #R 23 (1995), 40pp.

[Sa]

B. Sagan, “The Symmetric Group,” Wadsworth k Brooks/Cole Math. Series, Pacific Grove, CA, 1991.

79 [S]

C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13 (1961), 179-191.

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