*339*
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*3MB*

*English*
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*Year 1998*

Table of contents :

Abstract

Acknowledgements

1 Introduction

2 Tensor product representations of the Lie superalgebra p(n) and

their centralizers

2.1 The Lie superalgebra p(n) .......................

2.2 The commuting actions on the p(n)-module V®k ...........

2.3 Maximal vectors of p(n) in V®’c ....................

2.4 The algebra Ak as the centralizer algebra of p(n) ...........

2.5 The ring structure of A,‘ ........................

2.6 The decomposition of V” .......................

2.6.1 Decomposition of V82 .....................

2.6.2 Decomposition of V“ .....................

2.6.3 Non-complete reducibility of V“ ...............

3 The centralizer algebras of Lie color algebras

3.1 Preliminaries ..............................

3.2 Lie color algebras and Lie superalgebras ...............

3.3 Some simple Lie color algebras ....................

3.3.1 Special linear Lie color algebras ................

ii

iv

6

6

25

32

43

47

47

54

60

70

71

3.4

3.5

3.6

3.7

3.8

vii

3.3.2 Subalgebras of 91 (V, s) which leave invariant an e—symmetric

bilinear form on V ....................... 73

3.3.3 The family psq(V, e) ...................... 77

The Representation Theory of Lie Color Algebras .......... 81

End(V®") and the centralizer algebra ................. 85

Centralizer algebra of 91 (V, e) ..................... 89

The Centralizer Algebra of sq(V, e) .................. 97

Application: Decomposition of the free Lie color algebra ...... 101

Bibliography 105

Abstract

Acknowledgements

1 Introduction

2 Tensor product representations of the Lie superalgebra p(n) and

their centralizers

2.1 The Lie superalgebra p(n) .......................

2.2 The commuting actions on the p(n)-module V®k ...........

2.3 Maximal vectors of p(n) in V®’c ....................

2.4 The algebra Ak as the centralizer algebra of p(n) ...........

2.5 The ring structure of A,‘ ........................

2.6 The decomposition of V” .......................

2.6.1 Decomposition of V82 .....................

2.6.2 Decomposition of V“ .....................

2.6.3 Non-complete reducibility of V“ ...............

3 The centralizer algebras of Lie color algebras

3.1 Preliminaries ..............................

3.2 Lie color algebras and Lie superalgebras ...............

3.3 Some simple Lie color algebras ....................

3.3.1 Special linear Lie color algebras ................

ii

iv

6

6

25

32

43

47

47

54

60

70

71

3.4

3.5

3.6

3.7

3.8

vii

3.3.2 Subalgebras of 91 (V, s) which leave invariant an e—symmetric

bilinear form on V ....................... 73

3.3.3 The family psq(V, e) ...................... 77

The Representation Theory of Lie Color Algebras .......... 81

End(V®") and the centralizer algebra ................. 85

Centralizer algebra of 91 (V, e) ..................... 89

The Centralizer Algebra of sq(V, e) .................. 97

Application: Decomposition of the free Lie color algebra ...... 101

Bibliography 105

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Schur-Weyl dualities for Lie superalgebras and Lie color algebras Moon, Dongho ProQuest Dissertations and Theses; 1998; ProQuest

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entitled

dissertation

Schur-Weyl Dualities for Lie Superalgebr'as and Lie Color Algebras

submitted to the Graduate School of the University of Wisconsin-Madison in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

by

Readers‘ Page. This page is not to be hand-written except for the signatures

Dongho Moon

Date of Final Oral Examination:

April 29, 1998

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Approval Signatures of Dissertation Readers:

Signature, Dean of Graduate School

W:— [A4744 4%, L-Q LL J .s

3 04.4 N V

Wﬂ

SCHUR-WEYL DUALITIES FOR LIE SUPERALGEBRAS AND LIE COLOR ALGEBRAS

By Dongho Moon

A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

Doc'roa OF PHILOSOPHY

(MATHEMATICS)

at the UNIVERSITY OF WISCONSIN - MADISON

1998

To my Jade

ii

Abstract The study of centralizer algebras has been motivated by classical problems in

invariant theory. Let L be an algebra acting on a vector space V, and suppose there is a representation of L on the k-fold tensor product V“ of V. The problem of decomposing Ve" into indecomposable representations of L and of determining

the invariants can be studied using the centralizer algebra of transformations on V8" commuting with L. For the general linear Lie algebra gl(n) and the k-fold tensor V“ of its natural n—dimensional representation V, the symmetric group 5,; generates the centralizer algebra. This result is often called Schur- Weyl duality, and it is important for understanding the representation theory of gl (n).

In the ﬁrst part of this dissertation, we investigate the Lie superalgebra p(n). It has an ideal 5p(n) of codimension one, which is a ﬁnite-dimensional simple complex Lie superalgebra. We construct an associative algebra A,c and a representation of

Ak on V“, where V is the natural 2n-dimensional representation of p(n). We prove

that A, is the full centralizer of p(n) on VQ", thereby obtaining a “Schur-Weyl

duality theorem” for p(n). This result is used to understand the representation theory of p(n). In particular, using A, we decompose V‘g’k into indecomposable

p(n)-summands, for k = 2 or 3, and show that Va" is not completely reducible for any k _>_ 2.

The second part of this dissertation focuses on a generalization of Lie superalgebras, which are called Lie color algebras. We construct some new families of simple Lie color algebras, and we examine the centralizer algebras of these Lie color

iii algebras acting on tensor product space. We derive Schur-Weyl duality results for

these algebras from such results for their Lie superalgebra counterparts.

iv

Acknowledgements As I ﬁnish my dissertation, I remember the ﬁrst day I arrived in Madison. I had

arrived from Seoul, a city where it rarely snows, and I was anticipating the romantic quality of a winter with lots of snow. Since then, I have survived six bitter winters in Madison. I no longer look forward to winter, and I feel sympathy for my friends

in Minnesota who have experienced an even bitterer cold. Other than winters, Madison is a wonderful place. The diversity and the unique

mathematical atmosphere in Madison have been the best part of my life. During my ﬁrst year, I enjoyed the introductory graduate courses in algebra, analysis and topology. I would like to thank the professors for giving me a solid foun-

dation in mathematics. I especially remember Professor Louis Solomon’s algebra

course. I was very impressed by his lectures, which eventually led me to be an alge-

braist. I also learned much from algebra courses given by M. Isaacs, E. Zelmanov, R. Brualdi, P. Terwilliger and A. Ram. I thank them for their wonderful lectures. I am also grateful to Professor Arun Ram for his valuable insights into representation

theory, and to Professor Donald Passman for reading this dissertation. Now I would like to express my deepest gratitude to my advisor, Professor

Georgia Benkart. She has been a model mathematician and an excellent teacher. She has also been a good mentor and very understanding of my situation, and I

am indebted to her much. I could have not completed this dissertation without her gentle advice and generous support. Mere words cannot thank her enough,

but I do want to say that it has been my honor to be her student.

I also owe much to my parents for their constant support and encouragement. Before I had my daughter Yun, I did not realize what good parents they are. I would like to share this pleasure of a successful completion of the doctoral degree

with them. Yun has been a great source of pleasure during the past years, and I regret that I have not spent much time with her. I hope dedicating this dissertation

to her will pay off some of my debts to her. As a wife, a mother, and a graduate student, my wife Heejeon has been tremendously busy in Madison. Rather than just thank her for enduring all those respon-

sibilities, I want to take this opportunity to tell her I have always loved her during our past ten years together and will love her forever.

My six years in Madison will always remain very special to me. I thank all the good people I have met in Madison, and I extend my best wishes on everyone in Van Vleck.

Contents

ii

Abstract

iv

Acknowledgements 1

Introduction

2

Tensor product representations of the Lie superalgebra p(n) and

6

their centralizers

3

2.1

The Lie superalgebra p(n) .......................

2.2

The commuting actions on the p(n)-module V®k ...........

6

2.3 Maximal vectors of p(n) in V®’c ....................

25

2.4

The algebra Ak as the centralizer algebra of p(n) ...........

32

2.5

The ring structure of A,‘ ........................

43

2.6 The decomposition of V” .......................

47

2.6.1

Decomposition of V82 .....................

47

2.6.2

Decomposition of V“

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

54

2.6.3

Non-complete reducibility of V“ ...............

60

The centralizer algebras of Lie color algebras

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

3.1

Preliminaries

3.2

Lie color algebras and Lie superalgebras

3.3

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

Some simple Lie color algebras ....................

70

Special linear Lie color algebras ................

71

3.3.1

vii

3.3.2

Subalgebras of 91(V, s) which leave invariant an e—symmetric

bilinear form on V .......................

73

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

77

3.4

The Representation Theory of Lie Color Algebras ..........

81

3.5

End(V®") and the centralizer algebra .................

85

3.6

Centralizer algebra of 91(V, e) .....................

89

3.7

The Centralizer Algebra of sq(V, e) ..................

97

3.8

Application: Decomposition of the free Lie color algebra

3.3.3

Bibliography

The family psq(V, e)

...... 101

105

Chapter 1 Introduction

The study of centralizer algebras has been motivated by classical problems of invariant theory: Suppose 1r : L —) End(V) is a representation of an algebra L on

a vector space V which extends to a representation 1r“ on the k-fold tensor product space V9". The problem is how 7r“ decomposes into irreducible representations

of L, and what are the invariants of the action (see [28] or [29]). For the case L = gl(n) and its natural n-dimensional representation V, this problem was successfully solved by I. Schur using the centralizer, i.e. the algebra of linear maps 1' on Va"

such that

T1r‘9"(g) = 1r®"(g)'r

for all g e L.

In [25] and [26] Schur proved that the action of Sh, the symmetric group on k letters, on V“ by place permutations generates the centralizer algebra of L = gl(n). This result, which is often quoted as Schur» Weyl duality, connects the combinatorial and representation theories of gl(n) and $1.. The irreducible decomposition of

the gl(n)-module VQ" can be obtained from the decomposition of the group algebra (35,; into minimal left ideals which are labeled by standard Young tableaux.

After Schur’s papers [25] and [26], there have been various attempts to obtain analogues of Schur-Weyl duality in other settings. The orthogonal and symplectic

cases were studied by R. Brauer [9] using what are now called Brauer algebras.

The Brauer algebra has a basis consisting of k—diagrams, which are graphs with

two rows of k-vertices each, one above the other, and k edges such that each vertex is incident to precisely one edge. The product of two k-diagrams in the Brauer algebra is obtained by placing one diagram above the other diagram and identifying

the vertices in the middle two rows. The Brauer algebra itself has connections with combinatorics (Macdonald conjectures, see [11]), quantum groups, and knots and

links. The case when L is the quantized universal enveloping algebra (quantum group)

q (gl (11.)) was studied by M. Jimbo [13], and the quantum orthogonal and quantum symplectic group cases were also studied by J. Murakami [20] and J. Birman and

H. Wenzl [8]. The Lie superalgebras cases have also been investigated by a number of authors. The centralizer algebras of the general linear Lie superalgebra 91 (m, n) and the

decomposition of its tensor representations were obtained by A. Berele and A.

Regev [6]. This case was also studied by A.N. Sergeev [27] independently. In the same paper [27], Sergeev obtained similar results for the Lie superalgebras sq (n).

The centralizer algebras of the orthosymplectic Lie superalgebras spo(m,n) and characters of spo(m, n)-tensor representations appear in the work of G. Benkart,

C. Lee Shader, and A. Ram [5]. From the various results mentioned above, we now know all the centralizer algebras of the non-exceptional classical Lie superalgebras except for the family

sp(n). In this paper we will discuss the Lie superalgebras p(n). The algebras sp(n) = p(n) n sl(n, n.) are the only ones in Kac’s list [14] of classical simple Lie superalgebras whose centralizer algebras are not known (excluding the exceptional

algebras F(4), G(3) and D(2,1;a)). We obtain the full centralizer algebra of

p(n) in End (V‘Sk). The centralizer algebras are new associative algebras A; for k = 1, 2, . . .. The associative algebra A; has the same basis elements as the Brauer

algebra Bk(0), but the multiplication differs from that of Bk(0) by various minus signs. Furthermore we show some interesting properties of Ak. For example, AI, is not semisimple for any value of k = 2, 3, . . ..

We also construct maximal vectors of p(n). Then we use the centralizer algebra

of p(n) to decompose the tensor product spaces V“2 and V‘83 into indecomposable summands. This decomposition enables us to ﬁnd dimension formulas for some highest weight p(n)-modules. As far as the author knows these are the only known

dimension formulas for the irreducible p(n)-modules. It will also follow from the

decomposition of V“2 and V8’3 that V9" is not completely reducible for any It 2 2.

We hope that the technique developed to decompose V‘82 and V83 might be used for higher values of 1:.

While Lie superalgebras are Lie algebras graded by Z2, the Lie color algebras are Lie algebras graded by a ﬁnite abelian group G. In his paper [23], M. Scheunert initiated the study of Lie color algebras, and he proved that there is a bijection

between G-graded representations of a Lie color algebra and G-graded representa-

tions of the Lie superalgebra associated with the Lie color algebras. The centralizer algebras of the general linear Lie color algebras were also studied by S. Montgomery and D. Fischman in [10], using cotriangular Hopf algebra

constructions. In Chapter 3, we show that we may prove their result without con-

structing cotriangular Hopf algebras, and that there exists a general way to relate

the centralizer algebras of Lie color algebras with those of the corresponding Lie

superalgebras for the other cases.

In Section 3.3 we construct some families of simple Lie color algebras from the families of classical simple Lie superalgebras. In Section 3.4 we see that the weight space decompositions of the Lie color algebra modules and the corresponding Lie

superalgebra modules are essentially the same. In Section 3.5 we develop a. tech-

nique to relate the centralizer algebras of the Lie superalgebras and the Lie color algebras we constructed in Section 3.3. We obtain a vector space isomorphism be-

tween the two centralizers. Therefore, if we know the centralizer algebra of the Lie superalgebra, then we may obtain the centralizer algebra of the corresponding Lie

color algebra by this isomorphism. Finally, we derive Schur-Weyl duality results

for the general Lie color algebra gl(V, e) in Section 3.6 and for 5q(V, a), a family of simple Lie color algebras constructed in this paper, in Section 3.7. We also see that

the irreducible decompositions of VQ" are the same for the Lie superalgebra and the Lie color algebra. While it is not shown explicitly in this thesis, the centralizer

algebras of another family of simple Lie color algebras p(V, a) can also be obtained using our techniques. As an application of our method in this paper, we obtain the irreducible decomposition of the kth homogeneous component of the free Lie

color algebra (XV);c in Section 3.8. Let’s add a word on the notation. Let L be a Lie algebra or Lie superalgebra and suppose M is an L-module. From examining what commutes with the action

of L on M we obtain a representation of a certain algebra A, say A —) EndL(M).

From a group theoretical viewpoint, it is more natural to have the algebra A acting on M from right. On the other hand, it is traditional that the Lie algebra or Lie superalgebra L acts on M from left. We will keep these two concepts together in

this paper. For example, notation such as the following

1: - (v1 (8 ’Uz)(12),

for a: 6 gl(n),v.- 6 V, and (12) 6 52,

does not create any diﬂiculty in its meaning because the actions of gl(n) and its centralizer algebra commute with each other.

On the other hand, in Chapter 3 we will keep the tradition that every action on a space M is from left. Chapter 2 and Chapter 3 were prepared as separate

papers ([18], [17], [19]). These papers apply results out of [2], [4], and [5] in the proofs, and maintaining consistency with the conventions in the references they quote required using the right action of the centralizer for Chapter 2 and the left

action for Chapter 3. It is hoped the switch from right to left is less burdensome to the reader than having to translate all the results in the references.

Chapter 2 Tensor product representations of

the Lie superalgebra p(n) and their centralizers

2.1

The Lie superalgebra p(n)

Let V = 0”" be a Zz-graded (m + n)-dimensiona1 vector space over C, with V = V5 63 Vi, where Va = C’" and Vi = C". The general linear Lie superalgebra

gl (m, n) = gl(m, 105 69 gl(m, n)i is the set of all (m + n) x (m + n) matrices over (C, which is Zz-graded by A

0

91(m,n)a = 0

D

0

B

C

0

gums "h =

A 6 Mm(C),

D e MnXﬂ(C)} ’

B 6 mn(C),

C 6 Mnxm(C)} :

together with the super bracket

[1" y] = my - (-1)“"y$ for :1: e gl(n,n)a,

y e gl(n,n);

a, b = 0,1.

We deﬁne the supertrace str on gl(m, n) by,

str(:z:) = TrA - TrD,

for a: =

B

e gl(m, n), where Tr is the usual matrix trace. The special

0 D linear Lie superalgebra sl(m, n) is the subalgebra, sl(m, n) = {:z: 6 gl(m, n)|str(:z:) = 0}, of gl(m, n) of matrices of supertrace zero.

There is a. natural action of gl(m, n) on V by matrix multiplication, which extends to an action on the k-fold tensor product VQ" of V. More precisely,

1" (”1®112®"-®vk) I:

= Z(-1)“°‘+'”+“"‘-1 m o - - - e um o m- o vi+1 ® - - - e '01,, i=1

where a: e gl(n,n)a, and vi 6 V5,, a,b,- = 0 or 1. The symmetric group 5,, on k-letters acts on Va" by graded place permutations.

So for (ii+1) 65k,

(v1®---®vk)'(ii+l)=(—1)b‘b‘+’vi®'-'®vi-1®vi+1®Ui®m®vks where 'Uj e Vbj. The actions of S,c and gl (m, n) on vol: commute with each other

(see for example, [6] or [27]). For the rest of this paper we restrict our considerations to the case dimVa =

dimVi = n. Let ( ,) be a nondegenerate bilinear form on V x V such that

(i) (v,w) = (—1)“b(w,'v) for v 6 Va, and w 6 V5.

(ii) (v,w) =0 ifv,w€ Va orv,weVi. Then we deﬁne the homogeneous spaces of the Lie superalgebra p(n) as follows. Fora=00r1,

p(n)a = {1: e gl(n,n)a ($11,111) + (-1)“b(v,zw) = 0

(2.1.1)

VveVg, b=Oor 1,Vw6V}. Then p(n) = p(n)5 EB p(n)i is a subsuperalgebra of gl (n, 17.).

Since the bilinear form is nondegenerate on V, there exists a basis B = Bo U 31 for V such that Bo = {e1,. .. ,en} is basis for V5 and B1 = {en+1, . .. ,ezn} is basis

for Vi, and

(€n+i,6j) = (ejaen+i) = 5m,

(31': 81') = (emu en+j) = 0,

for i, j = 1, 2,. . . ,n. In other words e; and en“ are dual to each other with reSpect to the bilinear form. So we will use the notation

61" 2= en+i

and

67.4.; I: ei,

(2.12)

fori=1,...,n.

The matrix of the bilinear form relative to the basis B is given by

f3 = ((eiaej))15i,j$2n =

0

In

In

0

The choice of the basis B of V affords a realization of the elements of gl(n, n) and p(n) as matrices. The standard matrix units Eij deﬁned by EiJ-e; = (SJ-lei, for 1 g

2', j,l 3 2n determine homogeneous basis elements of gl(n, n) with Eij e gl(n, 11);; if e;, ej 6 Bo or ei,e,- 6 BI, and Eij E gl(n,n)i ifei 6 Bo and e,- e B; or vice versa.

Now the relation in (2.1.1) may be translated to the matrix equation sTFBw + (—1)“v.7-'3$w = 0,

where “T” denotes the transpose and a: E gl(n,n)a, v 6 V7,. Using this matrix

equation we can easily verify that p(n) can be represented as

p(n) =

A

B

6 M2nx2n(c)

A, B, C e 91(n),

BT=B,

0 -AT

C’F=—C

Here AT denotes the usual matrix transpose of A.

In [14] Kac showed that

59(11) = Mn) 0 sl(1W) A

A 6 sl(n),

B

B, C e gl(n),

e M2nx2n(C)

0 —AT

BT = B,

CT = -—C

is a simple Lie superalgebra provided n 2 3. So p(n) = CI 63 sp(n), where

Note I 6 5p(n)6 does not commute with p(n)i, while it commutes with p(n)0.

10 We will denote homogeneous basis elements aij, bij, qj of p(n) by

“ij ‘= Eij — Ej+ni+n 6

A

0

0

—AT

0 B

bij== ij+n+Eji+n€

Cij3=

Haj—EHM-E

0

0

0

0

A e gl(n)

for 1 S i,j S n, .

Begl(n),BT=B

CEgl(n),CT=—C

forlﬁiﬁygn,

forlSi V‘,

v I—-—> fu by fu(w) = (11,112), for 11,111 E V.

Note that

1

ifj =1? ,

0

otherwise.

f¢i(ej) = (eiuej) =

Thus fei = fi+n and fei- = f;, for each i = 1,2... ,n, and hence we see f: V5 —>

(V‘)i, and f : Vi —> (V*)5, i.e., f has parity 1.

15

Then for any homogeneous v, w 6 V and 1: 6 13(1).)a

(3 - fu)(W) = -(-1)“"’(f"fu($w) = _(_1)a((p(v)+p(f)) (v, mm) = (_1)a-p(f) (212, w)

= (-1)“'1fx-u(W)Thus f is p(n)-module isomorphism of parity 1, i.e.,

x - ft! = (-1)“’1fz-m 2: 6 Mn);There is also a p(n)-module structure on End(V) deﬁned by

(z - ¢)(w) = a: - P$2K, 01112 = x4 02211 = X, P102 = X, 15201 = X We want to ﬁnd the complete reduction system 3.43 so that all the ambiguities

are resolvable. We start with the initial reduction system S from Deﬁnition 2.2.2 consisting of the following relations:

xi = 11 ”i = 01 ”£5 = —Ui, $10!: = ”i,

for i = 1121

F2F131 = $132131, 010201 = -01, 020102 = —02, 1110201 = -F201, 0201F2 = ’02F1-

There are some ambiguities which we cannot resolve from the initial reduction

system S. For example, the ambiguity 3:131!)a is not resolvable.

24

(2311310201

31(F1l0201)

101201

P1(-|F201)

0201

-P1Z~‘201

Thus we add a new reduction relation 3:11:21” = —t)21)1 to our S. Repeating this

procedure, we obtain a reduction system 3A3 on the generators 1:1, 32, 01,92 so that

all ambiguities of 3,43 are resolvable. In this case the reduction system 543 consists of the following relations:

Pi = 1:

01' = 0:

301': 01': for i = 1,2,

011% = —0i,

020102 = —02,

332331131 = mm,

010201 = —01,

0201332 = —02F1,

$121201 = -0201.

5233102 = 0102,

$20102 = 33102,

0102M = 01332,

33201F2 = 13102231-

3310201 = -F201,

02231212 = ‘0201,

02P102 = 02,

01F201 = "01,

(22-10)

01F2P1 = 211232,

Using this reduction system, we easily get the products of the 3-diagrams in A3.

For example, X * X =(3102P1X0132) = £102(F101)Fz

#103201») = -z:102zc1

=_>K,

25

2.3

Maximal vectors of p(n) in V“

In this section we construct maximal vectors of p(n) in the tensor product space VQ" using the commuting action of Ak. Note that we regard CS,c as a subalgebra Of .4".

Deﬁne the mapping r.1M by

cM := o’lmo.

where a 6 CS,: is such that (1)0 = p, and (2)0 = q. Then \II(cM) is the contraction map c applied to the p and q tensor slots. It is not difﬁcult to show that cM is well-deﬁned:

Lemma 2.3.1. The contraction map cM is independent of the choice of a.

Proof. Let 0,1' 6 S], such that (1)0 = (1)-r = p and (2)0 = (2)7 = q. We want to show that

0—1010 = T-I‘JIT, which is equivalent to showing

91 = 07-1017'0‘1.

Note that (1)1'0‘l = (p)a‘1 = 1 and (2)7’0‘l = (q)a'l = 2. Since TO'_1 ﬁxes 1 and 2 we can write ro‘l as a product ofg- for z' = 3, 4, . . . ,k- 1. But 01 commutes

with r,- for i 2 3. Therefore

O'T—lilU-l = 0101" 1 70— 1 = 91.

26 Lemma 2.3.2. FOTl = 1,2, . . . , k - 1, Cu.” = 01.

Proof. By Lemma 2.3.1 it is enough to show (1 l)(2 l + 1)l)1(2 l + 1)(ll) = 9,.

As we have seen in (2.2.10), we have (i i+1)|),-+1(2' z'+1) = (i+1i+2)l),-(i+1i+2), for i = 1,2, . .. ,k — 2. Also note that

(21+ 1) = (23)(34)---(l— 1 l)(l l + 1)(l - 1 l) - -- (34)(23).

Thus (21+1)m(2l+1)

=(23)---(ll+1)---(23)t)1(23)---(ll+1)---(23)

=(23)---(ll+1)---(12)172(12)---(ll+1)~-(23)

=(23)- - -(1 1+ 1) ---(12)(34)92(34)(12) - .-(z 1+ 1) - - . (23) =(23).--(z—1z)(12)---(z—1z)m(z—1z)..-(12)(z—1z).-.(23). Let 0' = (l — 1l)---(12)(l— 1 l) - - - (23)(11). Then and (i)0' = i for allz' Z l. Thus we may write a as a product of x,- for z' < l —- 1. Since 9; commutes with g,- for i< l — 1, we have that

(ll)(2 1+ 1)t)1(21+ 1)(1z) = 0—1010 = Ina—10’ = mC]

If 2 = {p1, . .. ,pj} and g = {q1,... ,q,-} are two disjoint ordered subsets of

27 {1,... ,k} such that p,- < qi, for alli= 1,...,j, then we set

. 3:11-“1

(Eag:=cph¢11..'cpjﬂj!

k E

1

CM) 2 = id.

Let (2,9 = {(p1,q1),... ,(p,-,q,-)}, and denote by P(j) the set of all such (2,2). [5]

Also we set P = O P(j). i=0

We will view VQ" as a right Ale-module via the representation ‘11 : Ak -—-)

End(V®"). Since the image (V82): is a 1-dimensional trivial p(n)-module and 9,acts as [Sh-’1) ® c ® [SW-1’, we have

Proposition 2.3.3. V®kcm 2 V®("‘2l for all p, q = 1,2,. .. ,k and p aé q. Let A be a partition of l S k. Then we denote by ((A) the length of A, which is the number of nonzero parts of A. For each partition A = (A1,/\2,/\3, . .. ,Azn)

with length [(A) 3 211., we associate a weight of p(n) in the following way; /\ = A1€1+'--+ /\2n€2nA standard tableau r of shape /\ is obtained by ﬁlling in the frame of /\ with elements of 1, . . . , I: so that the entries increase across the rows from left to right

and down the columns. We set 3(1) 2: ((A). We associate two subgroups in the symmetric group 3,; to r. The row group R, consists of all permutations which permute the entries within each row. Similarly, the column group C, is the group consisting of all permutations permuting the entries within the columns. Deﬁne

3,, an element of the group algebra CS,“ by

s, := (2 u) (2: sgn(¢)¢). lﬁERr

¢€Cr

(2.3.4)

28

Then 3, has the property that there is some h(A) e Z+ that only depends on the shape of 1' such that 3,2 = h(/\)s,. (see [29]). Now the Young symmetrizer determined by 1' is the idempotent deﬁned by c—

1

(2.3.5)

s

”T " he) "

Let 1'1, 7'2 be two standard tableaux of same shape A. Compare entries of 7'1

and 1'2 starting at the left end of the ﬁrst row and moving from left to right. If the ﬁrst nonzero difference j; - jg is positive for corresponding entries jl in 1'1 and jg in 1'2, we say 1'1 > 7'2. If all corresponding entries in the ﬁrst row are equal, we

proceed to the second row, and so on. In this way we may give an ordering on the standard tableaux of the same shape. With respect to this ordering, we have the following result.

Lemma 2.3.6. [2.9, Theorem 4.3D] Let 1'1, 12 be two standard tableaux of the same

shape. If 1'1 > T2 then ynyn = 0. Example 2.3.7. Assume n = 8, k = 14, and A l- 10. Then

V9331? —'”—> V(51) —-—> 0. Therefore V9331? = V(61) EB ker galveayEF.

57

Now we will show C = ker nglvoayg}! = U(L)05. It is clear that u(L)05 Q C. To show the other inclusion note that as a gl(n)-module, C decomposes as

c =N1 EBA/2 63M 69M eMeNs, where JV,- is the irreducible gl(n)-modules generated by the maximal vector 10,- of weight /\,- such that

U11 = (61 ® 61 ® €3)yE!I.

102 = (e; ® e;-1® 3321?, ws = (61 ® 61 8’ 3;)9321:

m; = (6; ® 621 ® tidy-31$,

1.05 = (e; 8) 61 ® e2)yglyg}2h

we = (61 ® 6; ® 63313315521. and A1 = 261 + 52, A2 = -2€n - En—h A3 = 251 - an, /\4 = 61 - 5n - 511-11

As =51 +52 -6n,

A6 = 61 — 25". We may compare the dimensions to assure this. The dimension of V8331? is 2nS2n—1!!2n+12

3

. Thus

dim C

=2_n(_22-_1) 3 £2n_+1_)_ 2n.

58 We know the dimension of the irreducible gl(n)-modules so that dil = dimM =

n(n — 1)(n+ 1) , 3

dim/V3 = dimNE; = n(n2+1) x n — n,

dimM =dimJV5= ”(Tl- 1) xn—n, and we can check that

2n(2 —1)(2 +1)

( —1)( +1)

n( +1)

3 +2(n(n2—1)xn—n). Then we may easily show that 10,- e Ll(L)05 for i = l, . .. ,6, and C Q Ll(L)95.

Now if U(L)05 = ker nzlvmyag is not an irreducible p(n)—submodule, there

should be another p(n)-maximal vector in u(L)95. But from (2.6.7) we know there is no such a maximal vector since (01 + 202 + 303)t)2 aé 0. Thus U(L)05 is the

irreducible p(n)-module of highest weight El + 52. So we have the irreducible decomposition of Val/BF" V9331? = “(L)05 $u(L)(01 + 292 + 93),

u(L)05 E V(2£1 + 52)

and

U(L)(01 + 202 + 03) E V(51) E V.

We also obtain the dimension of the irreducible highest weight module V(251 +52),

dim V(2£1 + 52) = dil(L)05 = dim V9311? — 2n _ 2n(2n + 1)(2n - 1) _

‘

3

2n.

59

. VeayEm .

V931}? is isomorphic to Vaayag as p(n)-modules. So way? is completely reducible.

o V®3yg is an indecomposable p(n)-module which is not irreducible.

There are two maximal vectors in this submodule, 07 and —01+92—03. Applying

similar arguments to the ones used in Section 2.6.1 we may show the vector 07 generates the whole module V®3yl, and -—01 + 92 — 03 is a maximal vector of

weight 51. So Ll(L)(—01 + 02 — 03)Uis an irreducible module which is isomorphic to V = C“. There are no other submodules in V®3ya Therefore, we obtain the

following diagram:

V®3yg = U(L)07

Z“LN-91 + 92 — 93) I217.

(0), and

“(LN—91 + 92 ’ 03) g V031) 1; V,

V®3yg /