Tensor product multiplicities and convex polytopes in partition space

JGP - Vol. 5, n. 3, 1989

Tensor product multiplicities and convex polytopes in partition space A. D. BERENSTEIN, AN. ZELEVINSKY Scientific Council for Cybernetics Academy of Sciences

Moscow, USSR

Dedicated to I.M. Ge/land on his 75th birthday

Abstract. We study multiplicities in the decomposition of tensorproductof two lireducible finite dimensional modules over a semisimple complex Lie algebra. A conjecturalexpression for such multiplicity is given as the number of integral points of a certain convexpolytope. We discuss some special cases, corollaries andconfirmations of the conjecture. 0. INTRODUCTION

In this paper we shall deal with irreducible finite-dimensional modules over a semisimple complex Lie algebra g. The fundamental role in theory of these modules and their numerous physical applications is played by two kinds of multiplicities. The first is the weight multiplicity K~ of weight ~ in the irreducible g-module VA with highest weight )~.. The second one is the tensor product multiplicity c~ of V~ in VA 0 T/~. In fact, the weight multiplicity can be obtained as limit case of the tensor product multiplicity, so we shall mostly study multiplicities ct,. Our main <> is the conjectural expression for c~ as the number of integral points in certain convex polytope. For g s~ (or g~) such representation was given by I.M. Gelfand and one of the authors in [1], [2], [3]. Convex polytopes constructed there lie in the space of Gelfand-Tsetlin patterns. Here we present new approach replacing Gelfand-Tsetlin patterns by partitions into sum of positive roots. This language

Kcy-Words: SemisimplecomplexLie algebras,partition space, multiplicity, con vexpolyt opes. I98OMSC: 17B20,52A25.

454

A.D. BERENSTEIN, A.V.ZELEVINSKY

makes sense for any semisimple Lie algebra g. Our approach comes back to the idea of F.A. Berezin and I.M. Gelfand [4], who emphasized the fundamental role of the hierarchy P(i)

—4

.-

where P(’y) is the Kostant partition function [5]. The functions P(’y), KA/3 and c1~, depend respectively on 1,2 and 3 weights, and we have the following expressions due to B. Kostant: (0.1)

KA$= ~detw.P(w~+p)—~—p), WEW

(0.2)

c~, ~

det w

WEW

where W is the Weyl group of g, and p is the half-sum of positive roots. L let R be the root system of g, and R~ the set of positive roots. By definition, = card {(m 0)~ER, : m0 E Z~.,>mn a = ‘y}, i.e., P(-y) is the number ‘

of partitions of into the sum of positive roots. This number is naturally interpreted as the number of integral points of the convex polytope i~(‘y) in the <> space RR with coordinates (Zo)aER* . Namely, i~(‘y) is the intersection of the affine plane {~Z.~ a = ~ of codimension rkg with the <> {x~~ 0 for all a}. ‘~

-

Integral points of L~(~)will be called g-partitions of weight ‘y. lt is well-known that (0.3)

c~~ KA,~_~ P(~+ v

0

—

(see § 1 below). This suggests to look for expressions for multiplicities KAP and c’~, also in terms of g-partitions. Our conjecture is that there are convex polytopes z~ c C i~( )~+ i.’ ~) such that ~ is that number of integral points of i~, (and similarly for KA,~_U Note that P( )~+ v ~) is the <> corresponding to w = e in alternating sum (0.1) for KA,~_~ ; similarly, KA~_~ is the leading term in (0.2) for c~. It follows that once the polytopes i~ and are found, one can in principle prove the conjecture using formulas (0.1) and (0.2) by cancelling alternating terms. This method was used in [6] for g = ~ In this case the expression for c~,in terms of g-partitions (see §2 below) is equivalent to the one given in [1-3] and is also equivalent to the classical Littlewood-Richardson rule (see [7], [8]). Now we give a more precise version of our conjecture. Let fl C R~be the set of simple roots of g, and w0 (a E fl) be the fundamental weight corresponding to a. —

).

—

CONJECTURE 0.1.

There are linear forms ~s), ~t) ~ RR~ (where a E H and for each a s and t run over certain finite sets) such that for every highest weights

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TENSOR PRODUCT MuLTIPLICrFIE5 AND CONVEX POLYTOPES IN PARTITION SPACE

p, A

=

~

~

and

ii

=

~eH

~ n~w~ the multxplicity c~,is equal to the number nEIl

of g-partitions m of the weight A

+

u

—

~ satisfying the inequalities max 4~) ( m) <

L,~,maxn~(m)
Explicit (conjectural) expression of forms 4s), n(~ for all classical Lie algebras will be given in §2 (the case C2 will be considered in another publication). Conjecture 0.1 implies the following two expressions for weight multiplicities. COROLLARY. Jf 4s), n~ are linear forms from Conjecture 0.1 then for any highest weight A = ~ £0w,~and any weight /3 the weight multiplicity KAP is equal to the nED

numberofg-partitions m of weight A—~3suchthat max 48) ( m)
also equalto the numberofg-partitions m ofweight A —~3such that max n~( m) <£~ foraila. . Conjecture 0.1 is closely connected with the problem of constructing <> bases in the modules VA weight (see [1-3]). To explain thisweight relationship weany need someweight definitions. 3) be the subspace of VA of /3. For highest u = Let VA(/ >InEfl n~w~ let VA(/3,zI) = {v E VA(13) : e~1v = 0 for a E fl},where en is the root vector of weight a. It is well-known (see e.g. [9]) that (0.4)

c~, dim VA(~ z.’,v). —

Following [1-3] we say that a basis B in VA is good if each subspace VA(/3, v) is spanned byapartof B. CONJECTURE 0.2. For any g and A there exists a good basis in VA.

.

This conjecture was independently given by I.M. Gelfand and A.V. Zelevinsky (see [10]) and K. Baclawski [11]. In fact in [11] the conjecture was stated as theorem but, unfortuantely, the proof is incorrect. In [2] the conjecture is proven for g = a~ using the deep results of [12]. In [13] it is proven for g = sp 4 It is easy to see that conjectures 0.1 and 0.2 together imply the following CONJECTURE 0.3. Let 4~and n~ be linear formsfrom Conjecture 0.1. Then foreach highest weight A there is a good basis B = {Vm } in VA indexed by g-partitions m as follows: (1) For any weight ~3the subset B fl VA (/3) is indexed by g-pa.rtitions m of weight

A

—

/3

satisfying inequalities max

(2)ForanyaEll,vmEB

48)

( m)


min{n:e~vm=0}=maxn~)(m).

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A.D. BERENSTEIN, A.v. ZELEVINSKY

It was explained in [2] and [13]that the notion of good basis is closely connected with the problem of canonical definition of Clebsch-Gordan coefficients (or equivalently of canonical definition of tensor operators, see e.g. [14]and references there). We hope that the parametrization ofbasis vectors by g-partitions as in Conjecture 0.3 will be helpful for physical applications. There are two other general methods of computation of multiplicities. The first one comes back to D.E. Littlewood and is developed by B.G. Wyboume, R.C. King and others (see [15-18]). This is so called <> based on the theory of symmetric polynomials. This method enables one to express multiplicities KA$ and in terms of Young tableaux. Unfortunately, the answer in general is not completely combinatorial i.e., includes some negative terms (produced by so called modification rules). Thus, it is difficult to use this method for parametrization of basis vectors. Another approach to multiplicities and special bases is so called <> (see [19] and references there). This method gives an unified combinatorial expression for weight multiplicities and also the special basis in modules VA <> ([19]). Note that this basis is not <> in the sense of Conjecture 0.2. Recently P. Littelmann [20] obtained the unified combinatorial expression for c~,, for all modules of classical type. His result is based on the standard monomial theory. It would be very interesting to compare these two approaches to multiplicities with our approach. The paper is organized as follows. In §1 we collect some general results on multiplicities. In particular, we show here that multiplicities in reduction of g-modules to a Levi subalgebra g’ as well as tensor product multiplicities for semisimple part of B’ are special cases of tensor product multiplicities for g. In §2 we give the explicit form of Conjecture 0.1 for all classical Lie algebras (Conjecture 2.2). There are given also some special cases of Conjecture 2.2 which can be verified independently. In §3 Conjecture 2.2 is proven for classical Lie algebras of small rank. For g = sp4 we deduce it from the results of [131, and for 04, 05 and 06 use the exceptional isomorphisms D2 = A1 x A1, B2 = C2, D3 = A3. Finally, in §4 some reduction multiplicities including weight multiplicities are studied. For any classical Lie algebra B we introduce the notion of a B-pattern generalizing famous Gelfand-Tsetlin patterns, and establish the correspondence between g-patterns and B-partitions from Conjecture2.2. notion 02r+ The r and o of a ~P2 -pattern pattern is essentially due to D.P. Zhelobenko (see [9]); 2~-pattemsseem to be new although they are closely connected with generalized Young tableaux considered in [15].

1. GENERAL PROPERTIES OF MULTIPLICITIES Let B be a complex semisimple Lie algebra, /1 a Cartan subalgebra of B~R C 11” the

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TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES IN PARTITION SPACE

root system. We fix the set of simple roots U

=

{ai,...,ar}. Let q= ~

~ B(a) nCR

be the root decomposition of g. Choose standard generators e±n.E ~g(±a1),hn [en, e_n.] E Fl (i

=

1

,. . . ,

=

r). Let w1,... , w~E Fl~be the fundamental weights of g

defined by Wi(hn,) = Recall that A ~ h~is a highest weight for g (i.e. the highest weight of an irreducible finite-dimensional g-module VA ) if and only if A = ~ Lw1 with nonnegative integers I
L~.Integral linear combinations of fundamental weights are called integral weighLs~,it is well-known that each weight of a finite-dimensional B-module is integral. 3, ii) defined in Now consider a weight subspace VA(/3) C VA and its subspace VA(1 §0. PROPOSITION 1.1. a) Wehave 0
=

—

This proposition is well-known (see e.g. [9]). Having it in mind it is natural to introducethefollowingnotation: C~~(y)= dim VA(A—’y, 71) = ~ If A = ~ ii = ~

‘y

=

~g

1a~ then we shall sometimes write ~ (gi,... ,gr) for 3, ii) (and so that of CA~(’y))to the case cA~(~y) . We shall extend the definition of VA(/ when some of the coefficients n 1 in decomposition u = ~ n1w1 may be equal to co; this simply means that the corresponding condition en~ v = 0 is omitted. Since all en act on VA as nilpotent operators the infinite value of n~can be replaced by sufficiently large positive value. Since VA ® V~~ V~® VA it follows that cA~(’1) ç~(~y) For each fl’ C H we define the reductive Lie subalgebra g’ = ~(U’) C B to be B’~h~~(a), where the sum is over all roots a of the form ~

c1a~.The subalgebras ~q(11’) will

cz1fl’

be called Levi subalgebras of B- In particular, ~(0) = 11 is a Levi subalgebra. We denote by V~the irreducible B’-module with highest weight /3. PROPOSITION 1.2. The multiplicity of VA’~in therestriction VA g’ is equal to CA~(~)‘ where the coefficient n~in the decomposition ii = n1w1 is co for a1 ~ U’ , and 0 for a E H’. In particular, the weight multiplicity KA,A_,, is equal to cA~(’1),where all coefficients n~of v are oc. This follows at once from definitions since the subspace of highest vectors of weight A—1 in VAI,. isjust VA(A—~,v)={vEV~(A—’y):env=0 for aEH’}. •

458

AD. BERENSTEIN, A.V.ZELEVINSKY

With a subset U’ C H we also associate the semisimple Lie subalgebra Bo = Bo (11’) generated by e±n,hn for a E fl’. The Cartan subalgebra !i~of Bo is spanned by hn for a E [I’, so the natural projection p : Fr fi~sends w1(a~~ fl’) to 0 —~

Without loss of generality we can assume that U’

{a1,... a~}for some k < r.

=

,

PROPOSITION 1.3. Let A,v be highest weights for B~and ‘y the weight of the form ~ g1a1. Let A’ = p(A), ii’ p(v), 1’ = p( 1) be the cormsponding Bo -weights. 1
Then

CA~(1) =

dent on

Lk+1,

..

cA,U,(’y’) -

.

In other words, ~

~k+1~-~

,

~

n~and equals

,0) isindepen(gi,...

~

‘9k)

-

Proof Let VA~ denote the irreducible Bo -module with highest weight A’, let Bo = ~ 1L~~ where n~j(resp. n~) is generated by en, ,e,~,,(resp. C~n,~••~ —n,,)’ and n! = ~ a), the sum over positive roots a involving some of the roots ,..

,

.

a,.. Our proposition follows readily from the next statement: The Bo -module (VA) ~eof j?-invariants in VA is isomorphic to VA and this isomorphism identifies the weight subspace VA (A’ ‘y’) with [VA(A The isomorphism of (VA)” with VA, is clear from the fact that (VA)” as a Bo -module has the unique highest vector. The inclusion [VA(A 1)]” C VA (A’ 1’) is evident, and converse inclusion follows from theequality VA, (A’ —1’) = U~VA (A’) where U is the weight subspace U(n~)(—’y’) ofthe universal enveloping algebra U(n~) ak+ 1’~

,

—

—

—

—

,

.

Proposition 1.3 shows that tensor product multiplicities for <> semisimple Lie subalgebras of B are special cases of these multiplicities for g. Next proposition will be used as a test to our conjectures on multiplicities. PROPOSITION 1.4. Let g be simple, and 8 be themaximalrootofg ([21]). Then c~,,= ce~(8) is equal to the number ofnonzero coefficients n~, in the expansion u = ~ Proof By definition, V0 g is the adjoint B-module, and V0(0) Fl. Therefore, c0~(O) = dim V0(0,v) = dim{h E 2(h) Fl : (ad en)”~(h) = 0, i 1,...,r}. But = 0 , which implies our statement. ad e~.( h) = at( h) and (ad Ca)

2. MAIN CONJECTURE In this section we give a precise form of Conjecture 0.1 for all classical simple Lie algebras. We use the standard numeration of simple roots a 1, a,. and corresponding fundamental weights w1,... ,w,. (sec [21]). Thus, the integral weights for each type B,., G,., D,. will be written as linear combinations of standard basis vectors e~, . ..

,

. . . ,

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TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES IN PARTITION SPACE

in )R (with integer or half-integer coefficients); for Ar_l the lattice of integral weights is identified with Z~/Z(s1+ + ...

Let m13 = rnE, rn~,= mE+E, m~= mE ,soa o2~~1-partition of weight ‘yE Z’~ isavector rn (m11(l <1<3< r), m~,(I <1<1< r), m1(1 <1< r)) such that all its components are nonnegative integers, and

>

m1~(e1— e,)

+

~m~,(c~+

e,)

+

>rn1c~ = ‘l-

8P2r~ and o It will be convenient for our purposes to realize ,sL,.-, 2,.-partitions in terms of o2,.~1-partitions Namely, we identify an sL,.-partition with an o2,~1-partition (m~,,m~,m1) suchthat rn1 = m~,= 0 for all Similarly, an o2,.-partition isan o2,.~1-partition (rn22, m~,m1) such that rn = 0 for all i. Finally, a sp2~-partitionis identified with a o2,.÷1-partition~ m~,m~) such that all m1 are even. It is clear that such identification preserves the weight of a partition. Now we give an expression for .

i,j.

= CAP(’y)

C~,,,

=

forg= .sL,.. Forany ,sLr-partition rn= (m1~) and anypair (2.1)

=

A13(m)

=

(see §1)

(9l’~• . , g,.~)

~

(1,5)

put

m~1 m~+1~~1 —

(with the convention that m21 = 0 unless I < I < j < r, so for example = rn1,.). We define the linear forms ~ = ~(8)() ~~t) = (1
L~

=

—

~

L\,~ (0
~

~t)

OP8

~

i.\~,

i~

=

(i < t ~ r).

tpi

The following theorem is the first motivation of Conjecture 0.1. THEOREM 2.1. The multi~plicity~

.,,,

...,~,

of sL,.-partitionsrn of weight ~ max n~t)(rn)
(g~,. g,...~)is equal to 1(m) the number
suchthatmax4~

The proof will be given in §4. REMARKS. 1. Let A,p,v E Z~/Z.(c~+ e,.) bethreehighestweights for aL,.. Choose representatives A, i7 E Z ~ of these weights so that [~I = I + I Dj, where ...+

~,

I/1iEi

+

..

+

p,.e,.~=

+

...

+

p,.. Denote by LR~ the coefficient given by the

460

A.D. BERENSTEIN, A.V. ZELEVINSKY

Littlewood-Richardson rule (see [8]), i.e. the tensor product multiplicity for gL,.. Then it is clear that c’~,, LR~V 2. Theorem 2.1 makes the statement of Proposition 1.3 for g = .sL,.~1‘Bo = evident: since g,. = rn1 ,~+ + m2 + + rn,.,..~i~ and all m13 are nonnegative, it follows that g,. = 0 implies m1,.~1 = = m,.,.1 = 0, and we are left with an sL,.-partition. 3. Consider the linear involution /3 on integral sL,.-weights such that ~ = —c,.+1_~,and extend it to sL,.-partitions by I = rn,.~1_1,.÷i_j. By definitions, n~ (m) = L(r:t)(,~) for all I, t. Therefore, Theorem 2.1 implies cA~(’y)= c~(~)(this equality follows at once from the tensor product interpretation of multiplicities and the fact that V~,= (VA)*, the dual module). Now consider other classical Lie algebras of types B,., C,. and D,.. Let I = I,. = {I,2,...,r,O,1,...,i} belinearlyorderedby O
...

...

—

...

=

(2.3)

A

—

U

A U

—

m~3— rn~,

<

L~çy=

—

J m~1÷~ m~~1~÷1 forj°< r

—

1~m~

—

—

m1~1

forj

=

r

(with the convention that rn11 = m,~•= 0 unless 1 < 1
,

—

. -

A

(2.4)

I I

—

O~<8

—

(tO) = (t I) =

A~, A~,

+ ~÷i<~<~ 1~
(to)

+ ~

where in each sum p runs over the indicated interval in I, and empty sums are understood as 0. We shall also need the involution m on the set of all o 2,.-partitions defined by —~

(2.5)

=

{

We define the forms

fory
rn ~

(a E I) and

e~8)

~

~(rI)

{

m’ forj
separately in each of the cases B,.

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TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES INPARTITION SPACE

C,. , D,.

D,. :

4~~(m) = 4_l(m),

B,. :

~(a)

=

—

C,.:

~

=

—~-

~r~I)(m)

~IAp,.,

~r,1)

( rn)

=

=

r4~~(fn)= ~

m,.

(2.6) >A~,.,

n~.~’~(m) = m,./2.

O
(here ~

means that the summands A1,., A2,.,

...

are taken with coefficient 2).

MAIN CONJECTURE 2.2. Let g be one of the classical simple Lie algebras of type B,., C,., or D,. (i.e. g = 02,.+i, ap2,. or °2,.)~ Then ~ (g1,. .g,.) is equal to the number of g-paititions m of weight ~ ga1 such that max ( m) < .

4~

max ~

( m)

< n,, for all i, 5

=

1,.

.

. ,

r, where maximums are over all possible

e with the only exception that in case D,. the forms n~’~(I to n~ ,.,0) are not taken into account.

i ~ r — 1) equal

REMARKS. 1. Actually, the linear forms contributing to Conjecture 2.2 are the following: B,.,c,. :L~ (1

<5<

r, 0
<5), and
,~t~E)(li~r—l,itrfore=0,

1)

4rI)

D,.:L~8)(lj~r_l,0~s<5), L~(O
(ri)

n~_1, n,.’

-

2. Conjecture 2.2 is compatible with Proposition 1.3 in the same way as Theorem 2.1 (cf. Remark 2 to the Theorem 2.1). Namely, putting g,. = 0 we obtain the multiplicity for aL, predicted by Proposition 1.3. Similarly, putting 9i 0 we obtain the multiplicity for the subalgebra g~({a2, . .. a,.)) i.e. the previous member of the same series as g. 3. It is easy to see that the expression for c~~(’y)given by Conjecture 2.2 in case C,. can be reformulated as follows: ,

(8P~r) ~

i

~

462

AD. BERENSTEIN, A.V. ZELEVINSKY

is equal to the number of ~

~2,.+

1-partitions m = (rn13 m~,m1) contributing to 2g,.)

(°2 +,)

(gi,~-~g,._i,

and such that all m~are even. 4. In case D,. let A —f A be the linear involution on weights such that f~,= for I < r I = —s,. (clearly, it is compatible with the involution m—* th on —

~,.

,

o 2,.-partitions defined by (2.5)). Then Conjecture 2.2 easily implies ct,, = c~ (which is of course evident since the involution A A is an automorphism of the root system D,.). —~

Applications of Theorem 2.1 and Conjecture 2.2 to Lie algebras of small rank will be given in §3. Now we give some confirmations of Conjecture 2.2. First we show that the Theorem 2.1 and Conjecture 2.2 imply Proposition 1.4. It suffices to prove PROPOSITION 2.3. Let B be one of classical simple Lie algebras of rank r, and 0 = ~ n~w~ be its maximal root. Then there are exactly rh-partitions mm,. ,m~ ‘~) of l
Theorem 2.1 and Conjecture 2.2).

The proof is straighforward. We shall only give the list of ~ not mentioned are understood to be 0). A,. : 0

=

1 =

m~ (all parts

w

=

—

. . . ,

(mjk+1

+

= mk+lk+2

=

...

=

m,.,.~1= 1).

m23

=

m~3= 1),

mk+2k÷3=

1= (m B,. :0

=

+

m~

=

=

(ml,k+I

rn~= (m1

C,.: 0

2e1

=

m~

=

w2

=

=

m~’ mzk+i

=

rn2

=

=

=

l)(2 ~ k ~ r— 1),

1). m~ = (mlk+1

2w1

(m1

=

12

=

mik+1

l)(1

=

~k


—

1),

2). 1~ are the same as for B,.,

Dr(r

~

4) : 0 =

= ~i

(mi,.

+ 62 = =

w2

rn 2,.

=

1).

mm,... , m~’~

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TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES IN PARTITION SPACE

PROPOSITION 2.4. Conjecture 2.2 is true for the case when

£2 = £3 =

...

=

=

£,.

0.

The proof is based on the <> of [6]. We postpone it till another publication. Finally, consider the case = n 2 = = 0. ...

COROLLARY 2.5 (OF CONJECTURE 2.2). a) For (02,.,)

~

,.

m1

—

—

...

>

mm (n,., 2( g,.

—

g,._1)) we have

— —

(at) Ct,..’~,_,;m2_m,

=

£,.

—

m,—m,

,(91

—rn1,g2 —m1 —m2,...,g,._1—

m,.1)

t such that 0 < rn the sum over all (m1,... , m,.) E Z m1+...+rn,.=g,.. b) ForE,. min(n,.,2g,.

—

1 <

g,._1) we have

(sp2,)

— —

.t~oo ~

C11

~ m,. ~ n,. and

...

) ..4~;m2 —m1

(at ~

= —

m1

—

...

—

m,—m,_1

(gi

—

rn1 ,92

—

m1

—

m2,.

.

,

9,.—i —

rn,.1)

the sum over all (rn1,... ,m,.) E Z~ such that 0

m1 <

...

<

m,. ~ 2r,,,.,

m1

—

rn1+...+ m,.2g,.,and rn1,...,rn,. are even. c) ForE,.~min(r~,.,g,.+g,._1 (02,)

— —

~

=

~

~

(at

)

m,_m,_,(9i

+ —

m,._2,g,._1

the sum overall (rn1,... m,.

= 2g,.

and rn,.

=

wehave

9,.—2)

—

..

.

—

rn1, .

..

,9,.—2

—

. .

.

—

+

2

m,.) E Z” such that 0
~

...

<

.

We shall outline the proof of a). Consider an o2,.~1-partition m = (m13, m~,rn1) contributing to ~ (g~,...,g,.) by Conjecture 2.2. It is easy to see that the inequalities max r4t~(w) < n~imply that m~ = 0 for all and 0 < rn1 < i,j,


464

AD. BERENSTEIN, A.V. ZELEVINSKY

m as an sE,.-partition m’. One can verify that the conditions of Conjecture 2.2 except those related to are equivalent to the fact that m’ contributes to (at ) £,.

Cti..~,i;m

2_mi

m~

—

— ...

—

mm(9i

—

m1,g2

—

m1

—

m2,..

-

m,.1)

by Theorem 2.1. It remains to observe that the inequality max £~8)( rn) < follows from each one of the inequalities n~and E,. 2(g,. g,._i) The proof of b) is entirely similar, and the proofof c) follows the same lines although is more complicated: in this case the parameters rn are chosen so that m,._2k = m,._2k_I = mr_2k1,,.. Consider now the special case of Corollary 2.5 when n,. = + oo. According to Proposition 1.2 the multiplicity in question is the reduction multiplicity from B = °2,.+I sp2,. or ~2,. to the Levi subalgebra gE,.. Let A = A1 e~+ + A,.~,.,[3= [3161 + + /3,.e,. be partitions in sense of [8], i.e. A1, /3~ ~ Z, A1 A,. 0, [3~ 0. Let VA be the irreducible g-module with highest weight A (note that for g = 02,.+ and 02,. this excludes spinor modules), and V~the gE,.-module with highest weight [3. £,.

£,.

..

—

-

...

..

PROPOSITION 2.6. For B =

.

..

~2,.+ 1

>

/3,.

the multiplicity of V~in VA Igt~ is equal to ~ LR~

(see Remark 1 to the Theorem 2.1), where p runs over all partitions Pi ... t’,. 0 such that I~I= IAI — 1/31. For B = ~P2,. this multi~plicityisequal to ~ LR~V, the sum overall partitions v with evenpartssuchthat IziI = IAI— 1/31 .Finally, forg= °2,. the multiplicity is equal to and

ii’

>j~LR~V,,the sum over the same partitions

u as above,

stands for the conjugatepartition [8].

This follows atonce from Prorposition 1.2, Corollary 2.5 and the well-known symmetry LR~ = LR~ where is the involution defined in Remark 3 to Theorem 2.1 (the coefficient LR~ should be understood as the multiplicity of V~in Vi). On the other hand, Proposition can be deduced from the results of [17], which gives an independent affirmation of Conjecture 2.2. Using Corollary 2.5 for £1 = £2 = = £,._~ = 0 , = +oo and Proposition 1.2 we obtain ,

/3—p

/3

VA’

...

®

£,.

COROLLARY 2.7 (OF CONJECTURE 2.2). Let B be one ofthe algebras °2,.~For any

°2,.+1, sp2,., or

n ~ 0 the restriction ofthe B-module V,..~ to the Levi subalgebra 91,. is isomorphic to ~ V1~,where /3= /3~ + + f3,.c,. runs over all weights satisfying: 2 /3i /~2 i3,. _n/2, n/2 /3~E 7. a) ForB= °2,.+l~! b) For B ~P2,. n ~ 13~ /3,~ —a, n— /3~E 2Z. c) FOrg = 02,. f/2 [3~ /3~ [3,. —n/2, n/2 i3~ E 7, /3,. = = /3,._2k_1 (with the convention /3~= n/2). - ..

—

-.-

..~

..

—

465

TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES IN PARTITION SPACE

REMARK. Part a) is equivalent to [8, ch. I, §5, Ex. 16, formula (2)].

3. CLASSICAL LIE ALGEBRAS OF SMALL RANK The next theorem summarizes the statements of Theorem 1.2 and Conjecture 2.2 in some small rank cases. THEOREM 3.1. In each of the following cases the multiplicity ~ is equal to the number of g-partitions m satisfying the conditions:

(9i,.-.

,

g,.)

a) A

1,g=sE2:

(3.1) ~E1 ~ m12

r~~ m~2 g1rn12

b) A2,g= sE3

(3.2)

£~ m12 £2 m13 £2 m13 + m23

n1 ~ m13 ~i ~ —

m12

~2

9i

rn12

+

m13

92 =

m23

+

m13

rn~3+ m12

—

rn23

> m23

c) A3,g= &E4

£~ m12 £2 > m13

(3 3)

£2

1n13

+

rn23

—

£3

rn14 rn14 rn14

+

rn24

—

£3 £3

m12

n1 > m14 n1 ~

+

m13

n1 ~ rn14

+

m13 — rn24

~2

+

—

9i 92

rn13 rn13 + rn34

m12 + rn13 = m23 + m13 = m3~+ m24 =

m23

+

m14

+ +

m14

m14

713

+

m~

> m24

~ ~ rn~+ m23 —

—

—

m34

> m34

m24

d) B2,B=o5:

£2

(3.4)

£2 £2

rn1 rn1 rn2

—

1~1

m~ + m1 m12 + m1

—

~2

rn2

2(rnt + +

V2 ~

2 rn12) 2(rn~2 m12) 92

m~+ rn~2+ m1 = 2mT2 + m1 + m2

— —

m2 m2

+

m12

—

rn23

466

A.D. BERENSTEIN, A.V.ZELEVINSKY

e) C

2,B= .sp4: (m3, rn2 are even)

£1rn12 £2 rni!2 2+m~ £2 m1! 2—rni2 12+ (rn~ £2 rn21 2—m12)

(3.5)

91 = 92 =

m~2+ m1 rn2 rni~+mi —rn2 V2 m2!2 fl1 ~

—

V1

m12 + m~2+ m1 rn~2+ m1/2+ m2,/2

I) D2,B=o4:

(3.6)

(£i_

rn~2

n~~ rn~

~ £2

rn12

V2

rn12

g) D3,B=o6:

£1rn12 £2 rn13

—

rn~2— rn12

£2

rn13

£2

m13 + m~2 m12 + m23 rn~3 m~3+ m~2 m12

£3

(3.7)

£3 £3

+

—

—

—

m13

> rn~3+ rn~2 rn12 + rn~3 rn13 —

=

=

Parts a),

—

—

—

92 =

Proof

Thrn~ ~i ~ m~+ m~3 n~> m~+ m13 n1 ~ rn~+ m13 n1 ~ m12 + rn13 V2 m23

—

m23

rn~3 rn~3+ rn~3 rn23 m~3+ mt3 rn23 —

—

rn~3

713

m12 + m~2+ rn13 + rn13 rn~2+ m13 + rn23 rn12 + m~3+ m~3

b), c) are special cases of Theorem 2.1. To prove e) we use the result

of [13]: PRoPosmoN 3.2 ([13], THEOREM 3). ~ (91,92) is equal to the number of 8tuples (P2, p~-,Pi-~P12, Poo ,PT2 p~, Po) ofnonnegative integers satisfying the condi,

tions: (1) If a and t are two nonadjacent vertices of the graph

~ 12

00

12

TENSOR PRODUCT MIJLTIPL1CITIES AND CONVEX POLYTOPES IN PARTITION SPACE

then min(pa,pt) (2)

467

0.

=

J1Eip2+p~pi+2po P1POOf2~Pi~ 1 9i P2 Py~2Po

2Pr2~Poo V1P2~Pr~ V2 PPo~Pj~Pj~

£2

=

+

+ 2p

1-+ 2Pi7: 2131-2 + p~ + 21317: g2P~PoPlr2~Poo~ By this proposition to prove e) it suffices to construct a bijection between 8-tuples (p 8)

satisfying (1) and (2), and

3P4 -partitions

m satisfying (3.5). It is straightforward

to verify that the desired bijection can be defined by formulas m12 rn12

P2 + 73~+p~+2P~ Py + Poo

= =

rn1 = 2(pj7:+ 2(P~-~ PP~2) o +p rn2= 1~+pj7:)

mln(m12,rn12) p~= min(rn1!2,m2!2) p-j-—

Poo

=

[_A12]~

P12 = P2 = [min(A12,A12 + p1~-=[min(_A1~72,—A1~,/2 —A12)]~ P0

=

p~-

where A12

=

[min(—A1~/2,A1~-!2+ A12)]~ [min(A12,—A12—A1~)]~,

rn12

—

rn~2,A1~= rn1

Proof of d). Since the root systems

exceptional) isomorphism between CA~(’y) = C~A(’7) (see §1) imply (05)

~-

—

—

£2 05

rn2 (see (2.3)), and [x]~

=

max(0, x)

and C2 are isomorphic we have a (so called and sp4. This isomorphism and the property

(.~p.,)

C11fl,.~~‘.91~~2) — C,,2,,,.tt

,.

‘.92,91

-

into account the already proven part e) it remains to construct a bijection between 05-partitions (rn12 m~2,m1, m2) satisfying (3.4) and 8134-partitions (~12, m12, rn1 rn2) satisfying Taken

,

Li

rn2/2

£2

rn12

£2

rn12

n~~ V1 mi!2+m12 2÷mi

+

rn1 91 92

—

V1m2! V2 m12

rn2

-m~~ —

m12

+

m12

+

m1

m12

2m12

468

A.D. BERENSTEIN, A.V. ZELEVINSKY

(cf. (3.5)). Evidently, the desired bijection can be defined by (rn12, rnT2, rn1 m2) ,

=

(~2!2,~l!2,~i2,~l2)

Part f) is evident from the isomorphism D2 A1 x A1 exceptional isomorphism D3 = A3 which implies (06)

-

To prove g) we use the

(844)

Ct,t2t5(9l,92,93)=C44t.~ny.~(92,9I,93)

Using c) it remains to construct a bijection between o6-partitions (m12, rn23, rn13, rnT2 rn~3,rnT3) satisfying (3.7) and .sL4 -partitions (~12,rn13, rn14, rn23, rn24, rn34) satisfying ,

Thrn24 £1

m13

£3

rn14 rn14 + in24 m14 + m24

£3 £3

+

in23

—

rn12

m13 — m13 + rn34

92 = =

rn13

—

rn23

m14

+

m23

m12 + rn13 m1~+ rn24

+

rn14

+

rn34

+

71124 + 7fl23

V2

rn14 + rn13 — m24 m14 + m13 —m24 + m12 m34

V2

—

=

V1

V1 +

—

rn34

m23

rn24

(cf. (3.3)). One can verify that the following formulas define the desired bijection: + [—A12]~ min(rn24 rn13 + [A.12]~) rn14 + [~I3 + [A12]~]÷

~l2

=

rn23

=

rn13 rn13

m23 = m~3= rn34

rn34 m13

=

rn23

m13 = min(~12,~23) rn~3= rn14 + [—A13— [A12]~]~

rn24 in14

rn12 m12

(39)

where A12 A13 =

= =

= —

m13

,

+ +

[—A12+ [A12 —

min(rn12,m12 m~2+ [A1~]~ = min(m~3

=

[—A11I~)

+

=

,

m12 — m~2, A1~- = rn~3— m23 (cf. rn24 (cf. (2.1)).

(2.3)), A12

=

rn12

—

rn23

4. REDUCTION MULTIPLICITIES AND GENERALIZEDGELFAND-TSETLIN PATTERNS THEOREM 4.1. Let B be a simple classical Lie algebra of rank r, and A

=

~

1
a highest weight for g. Then the weight multiplicity KAs is equal to the number of B-pa.1z’itions m of weight

A

—

/3 satisfying max

the L~/~ were defined in (2.2), (2.4), (2.6).

( rn)

£~8)

<£, for

5

=

I

,...

,

r, where

TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES IN PARTITION SPACE

469

According to Proposition 1.2 this Theorem is a special case ofTheorem 2.1 and Conjecture 2.2 (cf. Corollary to Conjecture 0.1). We shall reformulate Theorem 4.1 in terms of generalized Gelfand-Tsetlin patterns which will be defined separately for each type. First we recall familiar Gelfand-Tsetlin patterns. A GT-pattem (or gE,.-pauern) is an array A = (As,) (1 i A1, for all 1 ~ 1<5 ~ r. It is usually drawn as follows: A12

A11 A22

A1,. A23

A2,.

A,.,. The vector A = (A11, A12,... A1,.) E Z~will be called the highest weight of A andtheweight /~=(f3I,--•,/~,.) ofA isdefinedby /32=1A11—1A2+11,where 1A11= A11+A~~1+...+A2,.,IA,.+11=0. We define an sE,.-pattem as an equivalence class of gEr-patterns modulo the following equivalence relation: A A’if and only if A1, = A~,+ C for some C c Z The highest weight and weight of an sL,.-pattem will be thought of as elements of Z~/Z (1, 1,..., 1) i.e. as integral sL,.-weights. We define an °2,.+1 as an array of numbers (A17, i~.,,) (1 I j < r) satisfying the following conditions: ,

.

,

(1) For all i,j we have ~

E Z

-

~-;

moreover, these numbers except

~,.,

,‘r~,.,. either all lie in Z orall lie in + Z (2) min(A~~_1,A1+1~1) ~ max(A11,A2÷1,) for 1 ~ 1<5< r,and min(A1,.,A1~1,.) m~ 0 for I <1< r (with the convention A4~12= +oo). It is convenient to draw an o2,.~1-patternas follows: ~-

A12

A11 ‘lii A=

A1,. ‘112

‘li,.

A22...... A2,. A,.,. ‘l,.,.

An sp2,.-pattem is an o2,.÷1-pattemsuch that all A1, and

~

are integers.

470

AD. BERENSTEIN, Ay. ZELEVINSKY

Finally, an o2,.-pattem is an array (A1, (1 < I < .f < r) satisfying.

~ (1 < 1

<5 < r 1)) —

(1) Either all A11 and are integers or all lie in +Z (2) min(A~,1_1,A1+1~_1) ~ max(A11,A1~1,,) for 1 < I < 5 < r, and A1,. + A1~1,.+ min(A1,._1,A~~1,.~1)~ for 1 < i < r I (withtheconvention A1~1~ = +oo). The highest weightof each of these patterns is the weight A11e1 + A 12c2 A1,.r,., andtheweight /3 = /31e1 + .+/3,.e,. ofapatternisdefinedby /3~= IA1l—2lmH-IA~+iI. It is easy to see that the highest weight of a g-pattem A is a highest g-weight, and the weight of A is an integral g-weight. It is also convenient to identify a gL,.-pattern A = (A13) with an °2,.+1 or o2,.-pattern(A11,’113) ,where ‘11j = A1~,1~1(1 < 1
—

+..

.+

. -

-,

—

THEOREM 4.2. For any simple classical Lie algebra B the weight multiplicity KAfi is equal to the number of all B-patterns A ofhighest weight A and weight /3. To show that Theorems 4.1 and 4.2 are equivalent it suffices to construct a bijection between B-partitions rn from Theorem 4.1 and B-patterns A from Theorem 4.2. It is easy to verify that such bijection in each case can be defined by formulas (4.1)

rn11

= ‘1i,j_l

rn~=

— A~5,

‘1i.j_I

—

in1 =

A~~11,

2~,..

For B = sE,. the statement of Theorem 4.2 is well-known (the simple proof using only (0.1) is given in [6]). Before considering other algebras we shall outline the proof of Theorem 2.1. Proof of Theorem 2.1. For B = sE,. the correspondence (4.1) takes form m13 A11. It is easy to see that it transforms the fl~t)( m) from (2.2) into

(4.2)

n~(A) = A1+1~ A1~+

~

—

=

A 1+1 ~

—

(2A1÷1~— A1~ ~ —

1p~,. t+

Therefore, Theorem 2.1 is equivalent to THEOREM 4.3. Let A,p and

ii =

~

n,w 1 be highest .sL,.-weights. Then c~ is

1
equal to the number of sL,. -patterns A of highest weight A, weight that ~t)(A)


1<1<

r—

I, i< t< r.

(~ v) —

and such

TENSOR PRODUCT MULTIPLICITIES AND CONVEX POLYTOPES IN PARTITION SPACE

471

But Theorem 4.3 is essentially proven in [2], [6] (note that definition of the weight of a GT-pattern used there differs from the present one by the transformation (/3,... ,Ø,.) (/3,.,... ,/3~)).

—~

REMARK. Correspondence (4.1) enables one to translate Conjecture 2.2 to the language of B-patterns. We leave this reformulation to the reader. The proof of Theorem 4.2 becomes easier if we generalize it to some reduction multiplicities. Fix one of the series B, C, D and denote the corresponding Lie algebra of rank r by g,.. For k = 0,... ,r consider the subset of simple roots H’ = { a~,... ca~,.}.Clearly, the corresponding semisimple Lie subalgebra go(’1’) is iso morphic to B,.k , and the Levi subalgebra B( H’) naturally decomposes as ~ k ~ B,.k (see §1). A~(fl’)-weightwillbethoughtofasapair (/3’, A’) ,where /3’ = (/3~,...,f3~), and A’ = (A’k÷l,... A~) is a g,._~-weight. Denote by ~ the multiplicity of an irreducible B(I1’)-module ~ in the reduction VAlIn~)(see §1); for k = r this is just the weight multiplicity K~. Forany k= 0,1,...,r byatnincatedg,.-patternA wemeananarray (A 11 (1 < i < k + 1, 1 < 5 < r) ~ (1 < I < k, I < 5 < r)) which can be extended to a ,

,

,

B,.-pattern. For example, for k = 1 a truncated B,.-pattern looks as A11 A12 A1,. ~lii

‘112

A22

A2,.

and for k = r it is a usual g,.-pattem. The highest weight of a truncated g,.-pattern A is A = 2I~~1 (A11,A12,... ,A1,.) and the weight of A is the pair (f3’,A’) ,where /3~= + IA~+ A1~— 1I 1 <1< k) and A’ = (Ak+Ik+1,Ak+1k+z,...,Ak+1,.) ,

THEOREM 4.4. The reduction multiplicity K~j~~ is equal to the number of truncated

B,.-patterns of highest weight

A and weight (/3’, A’).

An obvious induction shows that it is sufficient to prove Theorem 4.4 for k = 1 i.e. to compute reduction of irreducible modules from B,. to ~ B,.~ The reduction from ~P2,. to (ff~~ ~P2,._2) was computed by D.P. Zhelobenko [9], and the case of orthogonal Lie algebras can be deduced from the results of [17]. Another way to prove Theorem 4.4 for k = I is to use the involution method of [6]. Details will be given in another publication. ~

.

REMARKS. 1. The similar result holds for g = aE,.. In this case it is well-known (see e.g. [6]). 2. It is easy to verify that Theorem 4.4 is compatible with Conjecture 2.2.

472

AD. BERENSTEIN, Ay. ZELEVINSKY

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72-75. [2] I.M. GELFAND, A.V. ZELEVINSKY. Multiplicities and regular bases for g4. in <>, Proc. of the third seminar, Yurmala, May 22-24, 1985, v.2. 22-31. Moscow, Nauka 1986. [3] I.M. GELFAND, A.V. ZELEVINSKY. Canonical basis in irreducible representations of 923 and its applications, ibid, 31-45. [4] F.A. BEREZIN, I.M. GELFAND. Some remarks on the theory of spherical functions on Riemannian symmetric manifolds. Proc. Moscow Math. Soc. (in Russian), 1956,5, 311-352. [5] B. KOSTANT. A formula for the multiplicity ofa weight. Trans. Amer. Math. Soc. 1959,

93, 53-73. [6] A.D. BERENSTEIN, A.V. ZELEVINSKY. Involutions on Gelfand-Tsetlin patterns and multiplicities in skew GL,,-modules. Doklady Akad. Nauk USSR, 1988, 300, No. 6, 12911294. [7] D.E. LrrrLFwooD. The theory ofgroup characters. 2nd edn. Oxford Univ. Press., 1950. [81 I.G. MACDONALD. Symmetric functions and Hall polynomials. Clarendon Press, Oxford, 1979. [9] D.P. ZHELOBENKO. Compact Lie groups and their represent.ations. (in Russian). Moscow, Nauka, 1970. [10] A.V. ZELEVINSKY. Multiplicities in finite dimensional representations of semisimple Lie algebras. Uspckhi Math. Nauk, 1989. [11] K. BACLAWSK1. A new rule for computing Clebsch-Gordan series. Adv. in AppI. Math., 1984, 5, 416-432. [12] C. DE C0NCINI, D. KAZHDAN. Special bases for S,, and CL,,. IsraeLJ. Math., 1981, 40, 275-290. [13] V.S. REFAKI1, A.V. ZELEVINSKY. Base affine space and canonical basis in irreducible rep-

resentationsof Sp4. Doklady Akad. Nauk USSR, 1988, 300, No. 1, 31-35. [14] J.D. LOUCK, L.C. BIEDENIIARN. Some properties of the mt erwining number of thegeneral linear group. in <>, Adv. in Math. Suppi. Studies, 1986, 10, 265311. [15] R.C. KING, N.G.I. EL - SHARKAWAY. Standard Young tableaux and weight multiplicities of the classical Lie groups. J. Phys. A., 1983, 16, 3153-3178. [16] G.R. BLACK, R.C. KING, B.G. WYBOURNE. Kronecker productfor compact semisimple Lie groups. J. Phys. A., 1983, 16, 1555-1589. [17] R.C. KING. Branching rules for classical Lie groups using tensor and spinor methods. J. Phys. A., 1975, 8,429-449. [18] K. KOIKE, I. TERADA. Young-diagrammatic methods for the representation theory of the classical groups of type B,,, C,, D,,. J. Algebra, 1987, 107, No. 2,466-511. [19] V. LAKSI-IMIBAI, CS. SESHADRI. Geometry of C/P V. J. Algebra, 1986, 100,462-557. [20] P. LITFELMANN. A generalization of the Littlewood-Richardson rule, Dec. 1987, Preprint. [21] N. BOURBAKI. Groupes et Aigèbres de Lie. Ch. IV-Vl. Herman, Paris, 1968. Manuscript received: October3, 1988