Weights of elementary representations in SU(5)-SO(10)-E6 chain

Pmm~n~_, Vol. 18, No. 4. April 1982, pp. 349-383. ~) Printed in India.

Weights of elementary representations in SU(5)-SO(10)-E6 chain

TAPAS DAS* and S K SONI *'rata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005,India Physics Department, University of Pennsylvania, Philadelphia, Pit 19104 USA

MS received 23 November 1981

Abstract. For Dynkin labels of weights in 5 and 10 of SU(5), 10 and 16 of SO(10) and 27 of Ee we find it instructive to (a) exploit the CTartan decom-'position-into a non- semisimple symmetric subalgebra and coset space, (b) introduce a set of annihilation anct creation operators, that may acron an mvariant 10), to represent suitably the shift-action and weight-vectors.

Keywords. Dynkin labels; elementary representations; annihilation operators; creation operators; symmetric subalgebra.

1. Introduction

The continuation of the simple-root-diagram for SU(5) in the manner

1 o o l o o suggests a chain terminating (see appendix 1) at E s. We are interested in the first three members of that chain:

SU(5)--SO(10)--E e chain. Furthermore we focus our attention on the elementary representations (defined in w 2) of those three algebras. Which, labelled by their dimensionality, are,

5 r of su(5)

10, 16 (16) of SO(10)

27 (27) of E0

excluding the adjoint of E s. We find it instructive to enumerate the complete set of their weights in a couple of ways as follows. (a) Through Cartan decomposition (see appendix 2) of a simple algebra into its non-semisimple symmetric subalgebera and the associated orthogonal complementary (coset) space. (b) By introducing a set of annihilation and creation operators, representing suitably the shift-action, that

*The Physics community has suffered a great loss in his passing away on 7 September 1981 at the age of forty-three.

349

350 Tapas Das and S K $oni

may act on an invariant 10). By definition, 10) is annihilated by all the generators, chosen as bilinears of annihilation and creation operators, of a maximal subgroup of the type SU(N)| SU(M)| . . . . i.e. with only unitary factor subgroups in it.

To begin with we explain some general results on semisimple Lie algebras poten- tially useful for SU(5)--SO(10)--E 8 chain. A dimensionality formula is stated in w 2. Expressed in terms of Dynkin labels of the highest weight of an irreducible representation, this formula is helpful for identification of the (first two) irreducible subeom- portents in the decomposition of the Kronecker product of a pair of irreducible representations. Another valuable tool is the Cartan decomposition of a simple algebra into a symmetric subalgebra and coset space. We are interested in that symmetric subalgebra which has a U(1) factor in it, thus making it non-semisimple. This is discussed in w 3 and illustrated for classical algebras in appendix 2.

Section 4 is a continuation of w 3, where we enumerate the complete set of weights in

5 of SU(5),

10 of SU(5),

10 of SO(10),

16 of SO(10),

27 of E e

respectively, by isolating the set of positive weights in the coset space of the following Caftan decompositions

su(5) @ u 0 ) c su(6),

su(5) | u(1) c SO(lO),

SO(lO) | U(l) c s0(12),

SO(lO) @ u(1) c Ee,

e e | 13(1) C E~

Here the decompositions are characterized by the parent algebra and the non-semisimple symmetric subalgebra.

Dynkin labels of the complete set of weights, as found for elementary representations from the above considerations in w 4, are useful in deducing subgroup content of those representations. However, independently of our knowledge of their complete set of Dynkin labels, w 5 is devoted to a discussion of the decompositions of products between elementary representations and subgroup content of the irreducible subcomponents in those decompositions. A simple identity is found very useful which may be applied equally well to find subgroup content of irreducible subcomponents in the decomposition of spinor • spinor for SO(2N).

Section 6 is the most amusing of all where we play with annihilation and creation operators. It is worth adding that we may play this game not only with SU(5)-

Weights of elementary representations 351

SO(10)-E s chain but also with SO(2N) with many a bonus. We designate against each simple root a~ in Dynkin diagram the shift-operator E+a~ represented suitably by anni-

hilation and creation operators. E~a t (E_ at = E, +) denotes the shift operator whose shift (or ladder) action is along (antiparallel) the simple root at. It is well-known that the successive action of E,,~ on a weight-vector yields the Dynkin labels of the associated weight. Therefore all we need to know are the weight-vector representatives (ww) expressed either in terms of annihilation and creation operators or by acting them on invariant 10 ~ without annihilating it. Though physical applications are not touched here, knowledge of w w may be helpful in understanding reflection under group-theoretic conjugation and in keeping track of Clebseh-Gordan coef - eients in the computation of group-invariant Yukawa couplings.

Our conclusions are stated in w As far as possible our notation conforms to Dynkin (1957). In appendix 2, however, we follow the notation of Gilmore (1974).

2. Review of the dimensionality formula /

We review definition of the n-tuples of non-negative integers (n being the rank), the Dynkin labels of the highest weight, which uniquely characterize an irreducible representation of a semisimple algebra. Stated in terms of them, the dimensionality formula may be used as explained at the end of this section to compute dimensionality of the first two irreducible components, 01 and 02, in the decomposition of Kronecker product of irreducible representations ~ and X,

4, x x = | -i- | -i- . . . , ( I )

The idempotent of a semisimple Lie algebra is a real linear vector space spanned by the roots as basis vectors. Caftan subalgebra is the minimal space that the idempotent generates over the field of complex numbers. The real dimension of the idempotent and the complex dimension of the Cartan subalgebra are both equal to the rank of the algebra. Ap, the complete system of weights of a representation p, is a finite system of vectors in the idempotent. Cartan scalar product (defined in appendix I, (A. I)) defines an Euclidean metric on the idempotent. Let at (i = I, 2, 3 . . . . n) denote simple roots. An arbitrary root may be expressed as a linear combination of the simple roots with integral coefcients; the non-zero coefficients are all of like sign; the root is called positive or negative depending on this sign is positive or negative. Let ,'z ~ 2al/(a, at) denote the normalized simple root. An arbitrary vector X in the idempotent may be expanded as

n

x = x, (2) i=I

This defines the dual basis and X~ are called covariant components. The dual basis {a't} (analogous to a direct lattice) is reciprocal to the Dynkin basis (analogous to a reciprocal lattice), {rq}, defined by the reciprocity relation,

i, j ---- 1, 2 . . . . , n (3)

352 Tapas Das and S K Soni

The contravariant components of X are

n

X= X Xi ~" ( 4 )

i = l

The contravariant co-ordinates A t of the weight A of a representation are called the Dynkin labels of the weight. An important result is that the Dynkin labels A s are all integral. Furthermore for the highest weight of an irreducible representation, the Dynkin labels, denoted by ~z in the following, are non-negative integers. The n-tuples, A 1, A2,. . . . . A,, are in one-to-one correspondence with the irreducible representations of a semisimple algebra.

From the defining relations (2), (3) and (4), we get (Dynkin 1957 a)

n

x ' = (x, ,,;) = ~ h '~ Xj, j=l

n

x , = (x , ,,,) = ~ h,, x J, (s) /=1

where h tJ is Cartan matrix,

h'~ = (a;, ~ )

and htj denotes the inverse of that matrix. They are both symmetric n • n matrices. The scalar product of two arbitrary vectors in the idempotent then becomes

n n

i = 1 i = 1

We also identify r as the highest weight of the i-th fundamental representation whose Dynkin labels ~j = (w~, a~) are defined by

A#=Sij. (7)

In other words, the Dynldn basis {,ri} is formed by the n highest weights of the n fundamental representations.

TJae dimensionality of an irreducible representation ~, uniquely characterized by the n-tuples ~1 ~ , ..., ~t,, is given by

N (P 01, '~, ..., ~ ) ) = 1I - - ~ - ( 8 )

2' it i--1

i=l

Weights of elementary representations '353

where Z + denotes the complete set of positive roots; the product has one term /1

corresponding to each given positive root �9 27+ expressible as 2' It ai with non- i----1

negative integral coefficients, l 1,/2, ..., 1,, whose sum is called the level of the root. The simple roots are each of level unity. Following Dynkin, the positive roots of level s > I may all be determined from a knowledge of (all) the positive roots of level less than s. The key result is that if A is a weight, so is A -- A l az (no summation); both belonging to the same representation. As also stated above, A ~ is an integer. In particular, we apply this result to the adjoint representation whose non-zero weights are positive and negative roots. If/3 is a positive root of level less than s, then / / - - (3, d,) a, is a positive root of level s if

~ , 4' 3 = - (s ' - level of p).

This inductive procedure may be continued to the root of highest level by beginning with (simple) roots of level unity. This enumeration of the complete set of positive roots in terms of simple roots is required in the dimensionality formula (8). This enumeration is carried out for SU(5), SO(10) and E s and the results are shown in tables 1-3, with some examples illustrating use of the dimensionality formula.

The set of transformations on a Dynkin diagram, that leaves invariant the length of simple roots and the angles between them, constitutes a group. This group is of order 2 for each of SU(5), SO(10) and E e. Its non-trivial element corresponds to the interchange (with labelling of the simple roots as shown in tables 1-3)

SU(5) : 4 x ~ % , 4 , ~ 4 3 ,

SO(lO) " 41, :, 3 -> % 3, 3, a , ~ a 5,

E s : ~ as , 4 2 ~ > % , aS, 4 -)" aS, 4, (9)

under which/7+ is left invariant.

Table 1. Positive roots and d,mensionali ty formula for SU(5). (2--1 0 hU = --1 2 --1 a t

0 --1 2 -- c

0 0 --1

QI~ O~, 0.8, "~, E + = a , + a : , a:-l-aa, a:+a.

az+a ,+ am. am+a:+a, a,+a,+a:+a,

N(~O,. ~,. ~,. ~)) = N(~(;~. ~,. ~, ~0) = (A,+ 1) (A,+ I) (A3+ I) (;%+ I)

1 X (A,+ ;q+2) (A:+ A,+2) (A,+ A4+2) 4--.' (A,+ A,+ A,+ 3) (A,+ A,+ A,+ 3)

(A,+ A,+ A3+ 3%+4) Useful examples N (~0, o, o, o)) = 5 = N (~(o, o, o, 1)) N (9(0, 1, 0, 0)) : 10 == N (p(0, 0, 1, 0))

(~0 , o, o, 1)) = 24.

(22 ~3 a4

354 Tapas Das and 3 K $oni

Table 2. Positive roots and dimensionality formula for SO(10)

hlJ --_

o o - - 2 - -1 0

--1 2 --1 -- a , 0 --1 0 o, 0 --1 0

~I, ~ , (g3~ a~, (15

a, q-an q-a,, a, q-aa + al, aa q-an + a~, aa q-.~-.u a~ al-}- a,c-[- a~.q- al, al--~-an-t-aaq- a~, al + as .-S al + a ~ -l + a~ q-aa + Cq + a~, ~ q- 2as q-a~ + a ~ al-~ a|-~ 2as q- a4 -~- a~, axq- 2alq- 2as ~- a4-~ a~

(I 4

0

~S

N (r (A,, A,, A~, &, &)) = N (~ (&, A,, A,, &, &)) (&+ I) (~,+ 1) (~+ I) (~,+ 1) (&+ I) �89 �89 �89 &+2) �89

= I(~,+~,+~,+3) �89 �89 �89 �88 As+ A~+4) �88 �88 �88 ~(A,+2Aa+A,+A~+5) �88 As+ 2A,+ ~+ A~+6) {(A,+2A,+2An+ A4+A,+7)

N (F(1, 0, 0, 0, 0)) = 10, N ($(0,1, 0, 0, 0)) = 45, N ($(0, 0 ,1 , 0, 0)) = 1 2 0 N ($(0, 0, 0, 0, 1)) = 16 = N ($(0, 0, 0, 1, 0)) N (~(0, O, O, O, 2)) =126 = N (~(0, O, O, 2, 0)).

Table ~. Positive roots and dimensionality formula for E,

hll = ( i - 1~176176 o - I 2 - I o o o ~ o. o,

--1 2 --1 --1 r r ,, 0 --1 2 0 0 --1 0 2 -- 0 0 0 -- 1 a ,

al~ al, r ~4, r ~6

a 5 a ~

~l+~s. ai+as, aa+a,, as+as, a6+a, a:+an+an, a . + a s + a , , an+as+~5, a3+a~+a6, an+a6+~8 a x + a i + a n + a , , ~:+ag+a3+as. a n + a 3 + a , + a ~ , az+a3+as+ae, as+a~+a6+a~ al"~-a2"~a$-~ai-~-as, ax-~ 2a3-~a4+as, al'~a2"~-aa+G5~-a6, as'~'an-~-a4-~'a5 -~-a6

Z + : , z x + a i + 2 a n + a i + a s , a : + a 2 + a s + a 4 + a s + a e , a , + 2 a s + n , + a s + a e a1-~-2a~-~2as--~-a4-~-a5 , ax-J-af~- 2as-~-a4-~-as-~-as, a~+ 2aa +a4 + 2as-~a6 a1-~ 2a2-~-2as ~-ai-~-as-~ as, a1~-a2 ~ 2aa ~-a4 ~- 2a~-~-a~ al-[-2a~+ 2an+ai+ 2a~+a n ax + 2an + 3az + a~ + 2a~ +a,

N (~(AI. A,. An. A~, As. A,)) = N(~(A~. A,. A,. A~, An. A,)) (A,q- I) (An+ I) (Azq- I) (Ar I) (Am+ I) (A,+ l) �89 + An + 2) �89 + As + 2) �89 + A~ + 2) �89 + A~ + 2) �89 + A, + 2) �89 �89 ) �89 �89 �89 �88 �88 �88 �88

�88 = �89 �89 A~+A,+6) ~(A,+A,+As+A~+A,+5)

~(~,+&+&+~+a,+s) �89 �89 Aa+A4+A6+ A,+6) ~(An+2A~+A~+ A~+Aa+6 ) ~(Ax+2A,q-2A,+A~+A~+7) ~(Ax+A,+2A,+A4+A,+A~q-7 ) ~(An-~ 2A,+A4+2A,+A,+7) &(A,+2A,+2A,+ h4+ Aaq-A~+8) &(A~+An+2An+A~+2As+A,+8) ~(A,+2A,+ 2A,+ AI+2A6+ A,+9) ,'e(A,+2At+ 3An+ At +2A~+A~+ I0) r~(A,+2A,+ 3A,+2A~+2A~+ A,+ I I)

N(I~ ( 1 , 0 , 0 , 0 , 0 , 0 ) ) = 27 = N(I~ ( 0 , 0 , 0 , 0 , 0 , 1)); N ( P (0 ,0 ,0 , 1 ,0 ,0 ) ) = 78 N (~ (0, I, O, O, 0, 0)) = ~1 = N (~ (0, O, O, O, I, 0)).


This renders manifest the identifies,

SU(5) : N(P(A 1A zA sA4) ) = N ( P ( A tA 3A sAx)),

SO(10) : N (~ (Ax A" A3 A4 As)) = N (~' 01 As A3 A5 A4)),

E 6 : N (~ (A 1 Ag. A 3 A 4 A 5 As) ) ----- N (~ (A s A 5 A 3 A~ A z Az)), (10)

wherein the pair of representations are related by complex conjugation. A certain subset of fundamental representations (defined by 7) is the focus of our

attention in the following sections. These are the elementary representations (Dynkin 1957). Let a t be a simple root corresponding to a terminal point in the Dynkin diagram of a simple algebra i.e. connected to no more than one simple root in the Dynkin diagram. To this corresponds the elementary representation characterized by A~ = 8, (i = 1, 2 . . . . . n). An arbitrary fundamental representation of the simple algebra may be obtained from an arbitrary elementary representation ~, by the opera-

tion ~{k} defined below (Dynkin 1957). Let R§ denote the carrier space in which operates and N be its dimensionality. Then the k-th antisymmetrization, ${k} of $,

operates in R~k}, of dimension (N), which is the linear closure of vectors

~i ~ , " " ~t = e i t i . . . . i , ' l i t ~?& "'" '~i,

Here '/1 ~/2 ... ~k denotes the vector to each system of vectors '71, */2 . . . . . '7k of Rr

(/1, is, ..., ik) denotes a permutation of the indices 1, 2 . . . . . k; eil i~ ... ik = 4- 1 for even

(odd) permutation (i x, i,., ..., it). This operation in general leads to a reducible 4, {k}.

The greatest component of ${k} is denoted by ~{k) : this is an irreducible representation contained in ${k} whose highest weight is A1 + Ae + ... + A, where Ax > A~ > ... > At are the first k highest weights of $.

The elementary representations of SU(5)--SO(10)--Er chain are, labelled by both dimensionality and {Al},

SU(5) : 5_ (1000), 5 (0001),

SO(10) : 10 (10000), 16 (00010), 16 (00001),

: 27 (100000), 78 (000100), 27 (000001), O0

su(5) : 1o ( o l o o ) = s_f2} = i 3},

where the simple roots are labelled as shown in tables 1-3 and complex conjugation is denoted by bar on dimensionality. (78 which is adjoint of Es is not considered any more for enumeration of the complete set of weights.) Those fundamental representations, which are not elementary, may be obtained as,


SO(IO) : 4_5 (010(30) ---- 1_0 {2},

0(00100) = 10(3) = 16(2) -- ]_6(2r;

Eo :351 (010000) = 2_7{2},

292..._5 (001000) = 27 (3} = 2_7 (3}',

351 (000010)---- 2-57 {2}, (12)

where each antisymmetrization considered leads to an irreducible representation. Before going to the next section, we state two rules (Dynkin 1957b) for securing the

n-tuples characterizing qh, called the greatest component, and ~2 in the decomposition • X [equation (1)]. Let M and M' be the highest weights of (the irreducible

representations) ~ and X (of a simple algebra) with their Dynkin labels {A,~ and {AI}. Then, (I), Or, has the highest weight M -4- M' with Dynkin labels {~i + A~}, (II) O2 has the highest weight M + M' -- (% + %+1 + ... + %+k) with Dynkin labels {A l + A' l -- 2(a~, % + %+1 +. . . + %+k)/(% ai)} where the chain %, %+1,.. ' , %+k of the simple roots is required to satisfy the following conditions.

(a) (a , ,a ,+x)#O b u t ( a , , a j ) = O f o r [ j - - i I > 1,

(b) ~ p # O but Ap+I= ~p+s . . . . . Ap+k=O,

' ' AJ+I "" ' (c) ~ + k # O b u t ; ~ = ' = = '~ ,+k-x=0 .

Dynkin proves that O1 (O~) corresponds to one and only one irreducible component in the decomposition ff • X- With the knowledge of their n-tuples, (8) may be used to compute their dimensionalities, N(Ox) and N(q~).

3. The caftan decomposition into non-semisimple symmetric subalgebra and its coset space

An important tool that sharpens our understanding of the elementary representations (equation (11)) is the Cartan decomposition of a simple algebra into a symmetric non-semisimple subalgebra and the orthogonal complementary coset space, discussed and illustrated in appendix 2. We state the key result from there.

Let G be a simple algebra and G its proper maximal semi-simple subalgebra which satisfies the following condition (i), or equivalently, (ii).

(i) G has a symmetric (non-semisimple) subalgebra given by G0)u (1).

(ii) Dynkin diagram for G results by deleting a simple root from Dynkin diagram of G and there exists no regular maximal subalgebra (also defined in w 4) G' satisfying

cG'ca. Then the adjoint representation ~G of 17 admits of the Caftan decomposition

§ = i -i- 4- (13)


into the adjoint representation 4~ + 1 of (~$ U(1) and the orthogonal complemen-

tary space, 4o + 4o, where 4o is an irreducible representation of ~ Furthermore, as shown in appendix 2, the subcomponent | in the decomposition of 4| • 40 as well as in that of 4o • 4o is readily identifiable: in the former case, barring SU(N) for G, it is 4~,, the adjoint of G; in the latter case, when G is simple, it is 4~ 2} which is

irreducible (when G is simple). Thus

4o X 4'o = greatest component (@x) + $~ + 1, (14)

and,

$o • 40 = greatest component (01) + ~2} _]_ 03. (15)

In (14), 1 is the scalar product 4o ~e. Following rule (I), w n-tuples of greatest component in (14) ((15)) are given by n-tuples of 4o and 40 added together (of 6o multiplied by 2). See also (27).

We illustrate (13) for the following maximal embeddings

(a) SU(5) | U(1) C SU(6) (b) SO(lO) | U(1) C S0(12)

(c) SU(5) | U(1) C SO(IO) (d) SO(IO) | U(1) C E e

(e) E e | U(1) r E~

which all meet the condition (i), or equivalently (ii). We have

(a) 4 o = 5; SU(6) : 3 5 = 1 + 2 4 + 5 + ~underSU(5) |

(b) 4 o = 10; SO (12): 6 6 = 1 + 4 5 + 10-[- lOunderSO(lO) | U(1),

!

(c) 4 o = 10; SO(10): 4 5 = 1 + 2 4 + 10+ 10 under SU(5) | U(1),

(d) 4o= 16; E 6 : 7 8 = 1 + 4 5 + 16 + 1 6 under SO(10) | U(I),

(e) 4o = 27; E~ : 133 = 1 + 78 + 27 -t- 27 under Ee | U (1). (16)

Use of the Cartan decomposition in enumeration of weights

In this section, we use (16) to systematically arrive at the Dynkin labels for the complete set of weights of the following representations and explain their reduction under regular maximal semisimple subalgebras.

5, 1_0 (1-~ and .~ of SU (5),


I0, 16 (and i"6) of SO (I0),

27 (and 27) of E 6.

These representations contain only triplets, anti-triplets and singlets under the regular embedding of colour SU (3) (see w 5.3). The discussion begins with projection matrices (of which many interesting examples are given by Slansky 1981) whose use is natural for the following considerations.

Let G be a regular subalgebra of G i.e. (Dynkin 1957 a) there exists a basis for the

subalgebra G consisting of elements of Cartan subalgebra of G and root-vectors of G.

The simple roots of (~, a t, (p, = I, 2 . . . . , rank G), are expressible as linear combina- tions of the simple roots of G, a~ (i = 1, 2 , . . . , rank 63 (the coefficients being all integral). Let (for each t~)

rank G

i=1

(17)

In all the cases of interest to us

(zt`, ~t`) = (~,, ~,), (18)

for each value of subscripts /~ and i. Dynkin labels of a given weight A are

(equation 5) l"kt` ~ 2 (A, a't`)/(a'~,, ~t`) and A t ~ 2 (A, al)/(r a~) for the projection

onto Gand G respectively. From the definition (17) and the given condition (18) we get,

rankG

/kt̀ ---- ~ Pt`, (G C:: 63 A', p, = I, 2...rankG. I=I

(19)

This identity expresses Dynkin labels of the projected weight A as linear combina-

tions of Dynkin labels of A by means of the projection matrix P (G c:: 63. As shown by (I 6), the embeddings (a) through (e) generate all the representations in

whose weights we are interested. The following projection matrices project the (non- zero) weights (in the adjoint) of the parent algebra onto the weights of subalgebra

(a) iv (SU (5) C SU (6)) = (14 i 0),

(c) P (SU (5) c SO (10)) = (14 i 0),

(e) P (E6 C ET) = (le ! 0),

(b) P (SO (I0) C SO (12)) = (l 5 i 0),

(d) e (SO (10) C E s) ---- 05 ! 0),

(2O)

where In is n • n identity matrix, 0 an n-component null column vector and each matrix projects out the simple root labelled (by subscript) rank G i.e. an. Let us follow


Dynkin's algorithm for enumerating complete set of positive roots as linear combina- tions of simple roots (w 2, discussion following equation 8). Of interest are the additional positive roots which result by adding an to the Dynkin diagram of the sub-

algebra G. For their projections onto G are the weights of~0 (13) and (16). Those projected Dynkin labels are got by acting the appropriate projection matrix (equation (20)) on Dynkin-labels of the an- dependent positive roots of G. Tie weights

of~ 0 are minus the weights Of~o. By acting P (SO (10) c: E 6) on (Dynkin labels of) weights of 27, we get

E e : 27 = 1 + 10 + 16 under SO (10). (21)

By acting P (SU (5) C SO (10)) on weights of 16, we also get

SO (10) : 16 = 1 + 10 + 5 under SO(5). (22)

We have already seen in appendix 2 that 10 of SO (10) reduces under SU(5) as 5 + 5" Table 4 gives Dynkin labels of 27 and, using (21) and (22), those of 10 and 16 of SO (10) and 10 and 5 of SU (5) may be ready off easily (read also caption to table 4).

Having discussed the relevance of one type of maximal embeddings, we turn to regular maximal semisimple embeddings which are all generated by the following two-step procedure due to Dynkin: (i) Let the parent algebra G be simple and its simple roots be labelled by ol, ~ .... a,. Let 8 be the shortest root of G. 8 = -- A, where A is the root of highest level. (The level of a root is defined in w 2). The first step is to add 8 to the system of simple roots of G and describe the resulting n + 1 roots by the so-called extended Dynkin diagram. (ii) Pluck the root ak from the extended Dynkin diagram such that this results in a Dynkin diagram (of n simple roots) distinct from that of G. The resulting subalgebra, denoted by G(ak), is a regular maximal semisimple embedding. The completeness also holds: by ranging ak through all the simple root of G, all such embeddings may be generated.

Table 4. This gives Dynkin labels for the additional 27 positive roots (i.e. positive weights in the adjoint of E0 obtained as a result of appending E6 by the simple root an

~t Q~ o3 05 QO Q; o~ Q$ 05 Qe Q~ o o ~ ~ ~ c ~, ~ ~ ~ o

! ! Q4 aa

For Dynkin labels of the complete set of weights in (a) 5 of SU(5): read rows 24-28, columns 1-4 (b) 1_0 of SU(5): read rows 4-13, columns 1-4 (c) 10 of 80(10): read rows 19-28, columns 1-5 (d) L6 of SO(10): read rows 3-18, columns 1-5 (�9 2_7 of Et: reads rows 2-28, columns 1-6

The fermionic quantum numbers are shown alongside. Nt (SO(10) singlet), Ng ((10, 5) under (SO(10), SU(5))) and N:, (1_0 (~) under (SO(10), SU($))) are all self-conjugate.

P.--$


Table 4. The set of integers 2 ( A , ai)/(at, ai) = A i

A ~ A' A 8 A' A 5 A ~ A'

1 NI 0 0 0 0 0 --1 1 2 j,c 0 0 0 0 --1 +1 0 3 e + 0 0 --1 0 + 1 0 0 4d3 0 --1 + 1 --1 0 0 0 5 da --1 + 1 0 --1 0 0 0 6 dt + 1 0 0 --1 0 0 0 7u8 0 --1 0 + 1 0 0 0 8 u~ --1 + 1 --1 + 1 0 0 0 9 u~ + 1 0 --1 + 1 0 0 0

10u~ --1 0 + 1 0 --1 0 0 11 u~ + 1 --1 + 1 0 --1 0 0 12u e 0 + 1 0 0 --1 0 0 13 d~ - 1 0 0 0 + 1 - 1 0 1 4 d l + 1 - 1 0 0 + 1 - 1 0 15 d~ 0 + 1 --1 0 + 1 --1 0 1 6 e - 0 0 + 1 --1 0 - 1 0 17 v 0 0 0 + 1 0 --1 0 18 Dg --1 0 0 0 0 + 1 --1 19 D~ + 1 --1 0 0 0 + 1 --1 2 0 D e 0 + 1 --1 0 0 + 1 --1 21 E- 0 0 + 1 --1 --1 + 1 --1 22N3 0 0 0 + 1 --1 + 1 --1 23 N~ 0 0 0 --1 + 1 0 --1 24 E + 0 0 --1 + 1 + 1 0 --1 25 Ds 0 --1 + 1 0 0 0 --1 26D~. --1 + 1 0 0 0 0 --1 27 D1 ~-1 0 0 0 0 0 --1

Tables 1, 2, 3 show the root of the highest level for SU(5), SO (10) and E e respectively. The shortest root is

SU(5) : 8 = - - ( ~ i + ~ + % + ~ , ) ,

SO(10) : 8 = - - ( ~ z + 2 % + 2 % + a 4 + a ~ ,

E6 : ~ = - - ( ~ z + 2 % + 3 % + 2 a 4 + 2 a 6 + % ) .

Note that in each case (8, 8) = (~,, at) where ai is any simple root. The extended Dynkin diagrams (step 1, Dynkin's procedure) are

SU(5):

sO0 O):

E 6 :

a 2 (Z 3

~ 02 Q3 I1~ Om,,~,mO,

1:. ~

c~ 6

o


The (proper) regular maximal semisimple embcddings (step 2, Dynkin's procedure) are

SU(5) : none,

SO(10) :G(%) = G(%) = suPS(4) | SU L (2) | SU R (2),

: G (%) = G (~4) = G (as) = SU (6) | SU (2)

G (%) = SU c (3) | SU L (3) | SU R (3),

which may be characterized by the following projection matrices.

f 1 0 0 0 O1 t

0 1 0 0 0

--1 --2 --2 --1 --1

0 0 0 1 0

0 0 0 0 1

= P (G(%) ~ SO(10)),

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 0 1 0

0 0 0 0 0 1

--1 --2 --3 --2 --2 --1

= P c Ee),

1 0 0 0 0 0"

0 1 0 0 0 0

0 0 0 I 0 0

--1 --2 --3 --2 --2 --1

0 0 0 0 1 0

0 0 0 0 0 1

t I = e c

t (23)

Here the sub-(super-) scripts L, R, IS, C on the subgroups stand for left, right, Pati- Salam, colour, as has become customary to denote those subgroups in physical applications. These matrices may be applied onto the c~lumn-vector of Dynkin labels of the weight in the parent algebra. The arrows indicate how in the resulting column vector, the Dynkin labels are classified u n ~ r the simple factor subgroups in the semisimple embedding. By acting the above projection matrices on weights in the elementary representations, we may verify


~o = (6, 1, D + (l, 2, 2)- SO(lO):

16 = (4, 2, 1) + (4~, 1, 2) under SUPS(4) | SUL(2 ) | SUR(2),

27 = (15, 1) + (6, 2) under SU(6) | SU(2)

27 = (_I, _3, 3_)+(_3, 3, _I)+(3~, 1, 3) under SUe(3) | SUL(3 ) | SUR(3 ). (24)

The identification of representations, under the simple subgroups in the semisimple embedding, is not hard because they belong to the set of fundamental representations of the simple subgroups with readily identifiable n-tuples (equation (7)). Finally, it is easy to guess the subgroup of the adjoint 78 of E e.

78 = (3_5, 1) + (1, _3) + (20, 2) under SU (6)| SU (2),

78 = (8, 1, 1) + (1, 8, 1) + (1, 1, 8) + ~, 3, 3) + (3, 3, 3),

(3, 3, 3)C(I, 3, 3) • @~, I)C2._7 • 2__7,

(3, 3, 3)c(1,3, 3) x (3,b 3)c2 3 • 2-~

under SU C (3) | SU L (3) | SU R (3). (25)

The first reduction scheme in (25) is an illustration of Cartan decomposition under the symmetric subgroup SU (6)| SU(2) of E e.

5. Subgroup content of irreducible components in products

The subgroup content of the elementary representation of SO(10) and E s was given in w 4, exhausting all their maximal regular embeddings. In this section we give the subgroup content of the irreducible components in the decompositions: 1_6 • 16, 16 • 16, 27 • 27, 27 / 27. We shall require (a) the decompositions of products taken between fundamental representations of SU(5). (b) the following identity for k-th anti-symmetrization of a reducible representation @1 5~ 4's :

(~1 r +@>--#~ r ~k-,> • ~: 4 # - 2 ) • ~2~ ~ ...

�9 .. 4 ~2~ • ~-2~ ~ ~1 • 4~-'> 4 ~ . (26)

(c) the decompositions

16 • L6 = (10 + 12_0. + L20. so0o):

I_6 x 16----- I + 4_5 + 21_0,


p

27 x 27 = (27 + 351')~ + 351,

27 X 27 = 1 + 78 + 650. (27)

Note that the subgroup content of 27 under SUe(3) | SU(3) | SUR(3 ) makes it clear

that 27 • 27 C 27. In the decompositions in (27), rules I and II in w 2 were followed to identify t-he first two components. See also the discussion of (14) and (15).

5.1 SU(5) products

Let ct,~M 8 be a 5-dim. totally antisymmetric tensor. Tensor representations for the fundamental representations of SU(5) are (,~ for "transforms as")

1 4 ~', ,~,pa8 4 ~' ~ 5; 4 [~''] , ~.. ,~,,p;~8 ,/,b,,] ,..., 1_0,

4[~,,p~, ] ~', 1

eta 'P ] , . c~.p~a 4 [~ 'p ] ~ 1-6; 4 [~ 'p;q, 4-~ ~t.,,p~.~ " " _

where the repeated indices are all summed up over their full range i.e. 5

,,t,,p~8 4 ~' = ~ . ,,~,,,0~8 4 t' - 4[,,paB] # = 1

5 5 1 4[~, ] 1 2~ "t,,pa8 = ~. ~'~ ~ "t,,pa8 r D''] ------- r etc.

# = 1 v = l

Let us agree to denote a traceless tensor which is totally anti-symmetric in p contra- ,[~! This is the variant (or upper) indices and q covariant (or lower) indices by ~,[qj.

tensor representation of an SU(5) irreducible representation with dimensionality

( ; ) . ( ~ ) _ ( p 5 1 ) ' ( q 5 1 ) = ( ; ) ' ( : ) ' 6 - - p - - q

The products, taken among the fundamental representations, admit of the following decompositions

't'tx] ~-r i.e. 5 • 5 = 15 + 10, ~[1 ] X ~ [4 ] = "r'[4] . . . .

,/,tz] 3 r 4is] i.e. 5 • 10 = 40 + 1-0. ~[11 X r : "r[3] . . . .

t~ [1] X 4[2] : dill] ~ 411] i.e. 5 x 1--0 : 45 + 5, r[s] . . . .

4 [zl • 4[1] = ,t, it] ~ ,/,[01 i.e. 5 • 5 = 24 + 1, "r[1] w[0] . . . .

d, ts] _~ ,/,[zl ~ 4[z] i.e. 10 • 10 = 50 + 45 q-- 5, 4 [*] X 4is] = "rts] "rts] . . . . .

#,[s] ~_ 4 [z] ~ ,/,[01 i.e. 10 • 1--6 = 75 + 24 + 1. (28) ~ 1 X ~[a] = -rpq (z] - - "r[o] . . . . .


We have used

ar , i ~ ~]t[P] , L [ P ' ] ,LIP P ] _~ ,J,[Pq'P - - 1 ] _J.. [P+P'--S] "r'[,~] • v'[r = ~[a+,J'] W[q+,~'-Xl - - ~ [ ~ + q , - S ] " " "

I" ~b[ p+p'-~-a']

I a.[o] 'e[o]

~[~+a'-p-p']

when p + p' > q + q'

w h e n p + p ' = q + q'

when p -~- p' < q + q'. (29)

The script (0) has been omitted wherever its omission does not lead to eonfBsion. The generalization of the above considerations to SU(N) is straightforward.

5.2 SO (lO) products

The k-th antisymmetrization of N-dimensional fundamental representation ~b N of SU(N), with n-tuples A, = 3~1 (i = 1, 2 . . . , N -- 1), is the (N)-dimensional funda-

mental representation, with n-tuples A, = 3,k. Here k = 1, 2 . . . N - 1. The N-th antisymmetrization of ~b N is SU(N) invariant, with n-tuples ;~l = 0 (i = 1, 2, ..., N-- 1). For SU (5), this means

5f = 5, 5_(2}= lo, 5[3>= Y__o, 5 4>= 3, 5(5}= 1,

~{1}=~ , ~{2}= 19, ~{3}___ 10, _5{4}= _5, ~{5}_ _1" (30)

The k-th (k ~< N -- 1) antisymmetrization of 2N-dimensional fundamental representation ~N of SO(2N), with n-tuples Al = 8,1 (i = I, 2, ..., N), is the (~V)-dimensional fundamental representation, with n-tuples

(k = I, 2..., N- -2 )

)~i ----- 8,, N-I nt- 8~, N (k = N - I)

For k = N, the k-th antisymmetrization of ~b N is reducible into the pair of �89 (2~r)-dimensional irreducible representations, which are complex conjugate of each other for odd N but are real and inequivalent for even N, For k > N, the k-th antisymmetrization of ~N is the (2ff)-dimensional fundamental representation,

with n-tuples:

~' = St, N-1 --}- St, N' k = N + 1 ;

A l ~ 81~ tiN-k, k = N + 2, N + 3..., 2 N - - 1;

At --= 0 k = 2 N


The 2Noth antisymmetrization of •N is SO (2N)-invariant. For SO (10) this means,

10 O} = 10 10 (2} = 45 10 (3]- = 120,

10 (4} = 210 10 (5) = 126 + 126 10 (6} =- 210,

10 (7]- = 120 10 (8]- = 45 10 (9]- = 10,

10 (10} = 1. (31)

For SO(6) ~- SU(4) this means,

6 {1]- -- 6 6 (2]- = 15

6 (4) = 15 6 (5) = 6

6 (3) = lO + i-6,

6 (6) = 1. (32)

For S0(4) - SU(2) | SU(2) this means,

(2, 2) (1) = (2, 2) (2, 2) (2]- ---- (3, l) + (l, 3),

(2, 2) (3]- = (2, 2) (2, 2) (4]- ---- (1, 1). (33)

We are all set to study the subgroup content of SO(10) representations. Under su(5):

10 = 5_ + ~,

4# = (_5 + _g)(2) = 5_v~ + (5 • ~) + ~v), !

= LO + (_24 + 9 + 1o,

120 = (5 + ~)(3} = 5(3} _[_ (532} • --_5) -[- (5 X ~{2}) + _~{3},

= 1-~0 + (4-~ -[- 5) + (4__5 + -_5) + lO,

210 =: (5+5) (4} = 5 ~ + (5_{3}• 5_) + (5{2}x 5 -(2)) + ( 5 • (3}) + ~(4},

-5 + (~-O-FIO) + (7_5+24+ !) + (4()_ +10--) + 5. (34)

Under SO(b0 | SO(4) ~ suPS(4) | SUL(2) | SUR(2):

10 -- (6, 1, 1) + (1, 2, 2),

4# -- (6, ! , !) (2} + (6, 1, 1) • (1, 2, 9 + [(b 2, _2)(2)],

= O_5, ], !) ~ (6, 2, 2) + [(1, 3, 1) + (1,1, 3)1,


120 = [(6, 1, 1)(3)] +(6 ,1 , 1) (2) x (1, 2, 2 )+ [(6, 1, 1) x (1, 2, 2){2)] + (1, 2, 2){3),

= [(10, 1, 1)+(10, 1, 1)1+(15 , 2, 2)+[(6, 3, 1)+~, 1, -3)1+(1, 2, 2_),

210 ---- (6, 1, 1) (4) + [(6, 1, 1) {3} • (1, 2, 2)] + [(6, 1, 1) (2) x (1, 2, 2) {2)

+ (6, 1, 1) x (1, 2, 2) {3]" + (1, 2, 2)(4},

= (15, 1, 1) + [(10, 2, 2) + (10, 2, 2)1 q- [(15, 3, 1) + (15, 1, 3)]

+ (6, 2, 2) + (1, 1, I). (35)

The subgroup content of.126 follows by substitution of the subgroup content of 16 and 10 into

(1_6 x 1_6)~ = 1.0 + 126,

(see (30)). The subgroup content of (1_6 x 1_6), follows from that of 1_6 x 16 on deleting that of 120 ----- (16 x 16)a. Finally,

126 = 1 + 5 + 19 + 1~ + 47 + 50 under SU(5),

126 = (6, 1, 1) + (10, 3, 1) + (10, 1, 3) + (15, 2, 2)

under SU Ps (4) @ SU L (2) (~ SU R (2) (36)

The above considerations based on use of the identity given by (26) are readily generalizable to reduction under the SU(N) and SO(2m)| m + n = N, subgroups of SO(2N). The input is the subgroup content of the vectorial i.e.

2N = _N + N under SU(N)

2N = (2m, 1) + 4 , 2n) under SO(2m)| SO (2n)

5.3 E e products

We briefly discuss E e. The subgroup content of 27 yields that of 351, which is the second antisymmetrization of 27, and of 351' from the decomposition of (27 x 27),. The additional input of the subgroup content of 78 yields that of 6_50, using (27), the decomposition of 27 x 27. The results are given in appendix 3.

Before closing this section, we remark how a well-known result stems directly from (26). Suppose SU c (3) is embedded in SU(N) by means of the projection matrix

P(SUc(3)c:SU(N))----( 1 01 " 0 )


Then the k-th antisymmetfization of the N-dimensional defining fundamental representation of SU(N) contains only colour singlets, triplets and antitriplets. The same is also true of the 27 of E e, 16 and 10 of SO(10) and the fundamental representations of SU(5), which contain only different antisymmetrizations of 5 of SU(5) C SO(10) C Ee.

6. Annihilation and creation operators for shift-operators and weight-vectors

The shift-action of SU(N) may be represented (or realized) by introducing N annihilation operators b~ and their adjoints b + satisfying the algebra,

(b , , = (b , , - 1 8, , = 0, i , j = 1, 2, . . . , N . (37)

By acting the N creation operators on SU(N)-invariant vacuum 10), the fundamental representations of SU(N) may all be generated one by one, by the following 'vectors' which represent weight-vectors.

b, 10> = 0 i = l , 2...N,

N or ~[11 = b~ i0) , ~ or ~b[1 ] = (0[ b,,

N {2} or ~b [2] + + , _ = b, b; I O) ~{2} or ~b[2 ] ---- <~0 1 ba b, (i < j),

N (3} or ~[3] + + _ = b, b, b + ] 0) , ~{3} or ~[3] = (0 J b~ bj b, (i < j < k),

etc. This is discussed in w 6.1. In the following, we consider a maximal embedding with only unitary factor sub-

groups. For each such factor subgroup, a set of annihilation and creation operators is introduced as above which also commute with those corresponding to other unitary factor subgroups. The vacuum 10) is invariant under the action of all those simple factor subgroups. We shall illustrate our strategy in detail for E e in w 6.2 and very briefly for SO(10) in w 6.3.

Denote as v 1 the linear vector space (LVS) spanned by vectors. Denote as v~ the LVS consisting of all the linear transformations in v 1. The shift operator E acts as follows.

fi ~ v2--> [E, n] E vn. (38)

Knowing a weight-vector I ~:) E v x or fi E v~., the end points defined by

(E+a,) ag R~ E R~+ag% but (g+a,) ''+z R~ = 0,

(E_at) r' R~ E R~ -ptai but (E_a,)~' +1 R~ ---- 0, (39)


may be determined as

( e + j , I o

(e_a,)', I # 0

but (E+a,)', +1 [ ~:> ---- O,

but (E_a,)r, +x ] s r = O, (4o)

if the weight-vector is represented by I f> E v 1, or, the end points may be determined a s

[E§ [E+a,, [E+a, ... [E+at, n] # 0, (q, times),

but [E+a,, [E+a,, [E+a, ... [E+a,, [E+a,, ~2] = 0, (q, + 1 times),

[E_a,, [E_a,, [E-a, [... [E_a,, ~] # 0, (p, times),

but [E_a,, [E_a,, [E_a, [... [E_a,, [E-a,, o ] = 0, (p, + 1 times), (41)

when the weight vector is represented by n E vs. In (39), R~ k is a subspace of the carrier space R~ (in which a given representation ~ acts) which is spanned by weight- vectors of weight A. E+a t (E_a,) is a shift-operator with its shift-action or ladder- action along (antiparallel to) the simple root a,. An important result is the relation- ship between the end-points, Pt and q , and the i-th Dynkin label A' of the weight A.

A t = p~ - - q~. (42 )

6.1 SU(N)

The operators defined by (37) may be used to represent the N - 1 shift-operators of SU(N) corresponding to its simple roots a x, a s, ..., aN_ 1 as

E+at = g + a t = b + b l+ l , i = 1, 2, . . . , N - 1.

o---.-~-,'_-.--o-- m o o (43) (~I (22 (~3 aN--2 (~N--I

They represent N -- 1 weight-vectors with their Dynkin labels given by,

E a l ~ ( 2 - - 1 0 0 . . . 0),

E a l ~ ( - - 1 2 - - 1 0 . . . 0),

g a . § 2 - - 1 0 . . . 0),

.EaN_ 1 ~ (0 0 0 . . . "0-12). (44)

To get this result, we have substituted the end-points in each case via (41), into (42). The i-th now in (44) coincides exactly with the i-th now in the N - 1 • N - 1


Cartan matrix for SU(N). Hence the representation (43). The construction of fundamental representations using creation operators acting on SU(N) invariant vacuum is illustrated for SU c (3) and SU(5).

suC(3)

~,b~ b;b~ 0

~ .:;,o>

u3b~lO>

t " ' # u~ b~ b 2 tO>

r "t. +

Ul bEb310> 4 - E-a t u~ b3b t + .c- I0 >

E . t E + = + b +b2; E = = = E + = b +b 3 - -=1 - -=2

4- +105 E_=,b~-105=-t- 1. b +10~> E_=,b +b + 1 0 > = - 1 . b sb 1

E_c~b +105=- t -1 . b8 +10> E-axb + b ~ l O ) = - l . b~b +10~>.

This also explains our notation, (E_=,_=, ~ [E_=: E_a=]). The associated Dynkin labels are

b, + [o5 +-->- ( + 1, o) b+ b+ t 05 < ~ (-- 1, 0),

4- 4- b~lO 5 , > (--1, +1) bsb z lOS .<--~ (--l-l, --1),

+ + (0, + 1 ) . (45) b + l o ) , > ( o , - 1 ) bl b, Io) <--->

suo)

Eal --- b~ b 2 E== = b + b 3 Ea3 ----- b + b~ E~, = b"s b 5

3"/0 Tapas Das and S K Soni

o o o - - ~ - - ~ - -

~ + ~ +

I +

+~ ~ +~

"f +I gl+ + A Q

+ ~ ~ ~ + ~

+ ~

A 0

@ --

+~

~ + ~

A

T

A O

]

I, A

A 0

+~

1

A o -- A +~

A O

+~

f~

A O

-l-q

t

A O

+~

+~

+,~

]

f

,i # A O

+~

+~


+ F._,, ,8,.+ r E~__~,C~b+lO) 5 ~lb I Io> z---~r -~w ~810) z-% - + + + " E ~ E a, E "s

E-a4 + + + +

These may be more compactly written as

3-! 2 ! r b~ b~ b~ [ O); 2 ! b~ ,

1 # ,vo,~s b + + ~. b~lo ) and 4-~. b ~ ~ b~lO).

6.2 E~ We find it convenient to "paramet r ize" in terms of the SUC(3) | SUL(3) | SUR(3)

embedding. For the extended Dynkin diagram we have

here, (bo bs]- = .(b +, bs)" -- ~u = 0 similarly for 1, and r,

) i , j= 1, 2, 3

and b's commute with rs and r's, l's with r's.

F . ~ ~ 4" 4' ~" § "[' "l' ,a-b, b2 F-~e~b a b 3 E+a3: (t, b2b, r,'~b3%rzt,+b3tzt3r ,) Easrtr2 Ene:r~r3 O, 0 ~) 0 0

Ee4= I,~ L,

%-" L;L,

The Dynkin labels of the weight-vectors, E~, . . . . , E~ e, ES, shown above are

E,1 <-.-> (2 , -- 1, 0, 0, 0, 0),

Ea~ <---> (-- 1, 2, -- 1, 0, 0, 0),

E,,a <--~ (0, -- 1, 2, -- 1, -- 1, 0),

Ea, <--,-> (0, 0, -- 1, 2, 0, 0),

Eas <-.-~ (0, 0 , - 1 , 0 , 2 , - 1),

E,~ +.-> (0, 0, 0, 0, -- 1, 2), ES < ~.~ (0, 0, 0, -- 1, O,O).

Caftan matrix (a',, aj) for E e, where a~ = 2ai/(ai, a~), is correctly reproduced. The weight-vector of 27 with highest weight (1, 0, 0, 0, 0, 0,) is represented by

l+a b+ [ 0). The complete set of weight-vectors in 27 is given as follows.

fl+' 13+ b[{O); 3~ '+a . bx + b~ b~ Ix + l~ r+10) , i =, 1, 2, 3

372 Tapa,r Da~ and $ K Soni

A 0

+,,4

A

I

I

A ~ ~ I A A

A I

A A A

t I I

A 0

A +,.4 O " "

+~ I

Weights of elementary representation~ 373

where the first twleve fields transform as (6, 2) under SU (6) | SU (2) c E e and the last fifteen fields as (15, 1) under that embedding. Knowing the above representatives for weight-vectors, the associated Dynkin labels are readily accessible: we only have to substitute the endpoints determined via (40) into (42). In fact, A t A z A 4 A 5 and A s are trivial in view of the extended Dynkin diagram and (45). The complete set of Dynkin labels tallies with those given in table 4. The association with fermionic quantum numbers goes as

a # = ds

'P = Dl

~bujl = ~sj~ u~

i , j - - 1,2,3

a ~b4,5,6=(e +,_vC, N t )

3 $ 4, 5, 6 = (E+, N'a, - - v)

C CDC) ~b [,4], [ss], (ss] ---- (u,, ds,

~(4~i, [4s], [~] = (e-, E- , N c )

N a is SO (10) singlet v c is SU (5) singlet

N a [N, c] belongs to (1_0, ~) [(10, 5)] under (SO (10), SU (5)).

The weight-vectors of 27 are obtained from those of 27 through the substitution:

b + ~ b + r + <_ ~> 1 +

Or, we may take the dual (~ I of each vector ] ~) in 27 to get the complete set of + +

wctors in 27, e.g. the dual of b x I t I0) is (0 [/1 bt.

6.3 so(lo)

Let (b , +, b s ) - B , s = { b , , b , ) = O i , j = 1,2 . . .5

SO (10) Cartan matrix is correctly reproduced if

Eat ---- E+al = b~ b~

Eo.=E+_aa=b+ab, <=.~

(2 - 1 ooo) ,

( - 1 2 - 1 oo),

(0 - 1 2 - 1 -1 ) ,

(0,o-12o),

(OO-lO2).

374

Thus

Tapas Das and S K $oni

~ 4-

E(I i b z b= . + Ca- b, b~, (D-- C~

§ +

Ecls= b s b,=

+ Ea,= b4b.

The weight-vectors in 1_0 are

5 | = ~ § b; + §

p=l

Their Dynkin labels are readily found by substitution of the endpoints secured via

(41) into (42). The weight-vectors in 16 and I_6 are

1 e~.,oA, ~ + ~o ~,I,/,A,~ + + + b~10 )

respectively. Their Dynkin labels follows from (40) and (42). For the fields in the former, fermonic quantum number identification goes as

~,,6 = (e-,-v), r

~.~J] = eljl u~, ~ ' ] , ['s] = (ul, dr), 4~u] --- e+

where all the fermion fields are left-handed. See also Mohapatra and Sakita (1980).

7. Coadm/eas

We would like to stress the following points:

w w w w w

Rules I and II, and the dimensionality formula. Decompositions given by (16). Table 4 based on those decompositions. The identity given by (26). Representatives for shift-operates E,., and weight.vectors, with their Dynkin labels determined by (40) through (42)~

Appendix 1 Appendix 2

Weights of elementary representations

The argument for termination of the chain. The illustrations based on classical algebras.

375

Appendix 1

In the following, the chain mentioned in the beginning of w 1 is shown to terminate at E s by exploiting the positive definiteness of the Cartan-kiiling metric restricted to the Cartan subalgebra (or the idempotent). This trick is originally due to Dynkin. We first define Cartan scalar product (x, y) between arbitrary elements x and y of a semisimple algebra, with representation matrices P (x) and P (y) in a linear representation p. By definition

(x, y) ---- Tr {P (x) P (y))/lp (A. 1)

Tr stands for trace and 1~ is called index of the representation. Given two linear representations P1 and Pz, then Tr {~1 (x) ~ (y)} / Tr {~2 (x) ~2 (Y)} = l ~ [ l~, is a proportionality constant, independent of the choice of elements x and y. [This is reminiscent of Wigner-Eckart theorem: the ratio of matrix elements, between a given pair of states, of irreducible tensors with identical transformation properties is independent of the choice of that pair of states.] Dynkin has also given a simple expression to compute l~ from a knowledge of the Dynkin labels for the highest weight.

Consider the following hypothesis. Examine a Dynkin diagram with n + 3 simple roots, n = 1, 2, 3 ... which are labelled as below

n n*3 n+1 C ~) O * * , �9 o 0 Q 0 0

1 2 3 n - 2 n-1

r~+2

All the roots are of equal length and the angles between them are as shown, either 90 ~ (no hne connecting) or 120 ~ (connected by a simple line). Let us evaluate norm of

n+3 the vector 27 a, a, in the Cartan subalgebra (the idempotent) with the choice

al=i i = 1 , 2 , . . . , n + 3.

The final expression is

n+3 n+3 2 ( X i a , , ,= s j % )

(aa, al ) = �89 n (n + 1) + 7 (3 -- n). (A.2)

n+3 (el, al) > 0. From (A. 2), we see that norm of the vector 27 i a, becomes negative

t = l

semi-definite for n ~> 6. Thus our hypothesis has led us to contradict the theorem that for a semisimple algebra the Cartan scalar product defines on Euclidean metric on Cartan subalgebra (or the idempotent). Hence the promised result, since n = 5 corresponds to E s.

P.m6


Before concluding this appendix, we give the canonical form for the metric, which is defined by the Cartan scalar product taken between the bases of the algebra. For a complex semisimple algebra, the bases are/-/1, H~..., H, belonging to Cartan subalgebra and E• (where a is positive root) belonging to the root-subspace V~. The canonical choice of the metric in the basis is

(Hi, Hj) = 8,j (E+a, E_/~) = 8~#, (A.3)

and the complex semisimple algebra is spanned by those bases multiplied by complex coefficients. A real semisimple algebra is spanned by bases X 1, X~ ..., Xd (d = dimension) which close with real structure constants, multiplied by real coefficients and the canonical choice for the metric is

(X,, Xj) : 4- 8 o. (A.4)

A generator is called compact (noncompact) if (Xl, X~) equals -- 1 (q- 1). The difference between the number of non-compact and compact generators is called the character of the real algebra. The so-called compact real form has all compact generators, or, character (compact form) = -- dimension of algebra.

Appendix 2

The complex simple Lie algebras are classified into the classical algebras, A, (n >~ 1), B, (n >~ 2), (7, (n >~ 3) and D, (n/> 4), and the exceptional algebras, G~, F 4, E e, ET, and Ee. In this note we first discuss, closely following Gilmore (1974) the role of Cartan decomposition in generation of all real algebras (defined in appendix 1) with a given complex simple algebra as their extension when the real coefficients (multiplying the bases of the real algebra) are continued analytically into the field of complex numbers. The starting point is the compact real form (defined in Appendix 1) whose generators X1, X 2 . . . . . zY d (where d is dimension of the algebra) satisfy (X~, Xj) = - - ~ i, j = I, 2 . . . . . . d. Let T be an involutive automorphism (i.e. T 2 -- identity) such that TX, T -1 ----- -4- X~. Let t denote the set of generators which do not flip sign under T and p denote the rest. The automorphism leaves invariant the (real) structure constants

d

tx , , x , j - - xk k = l

(B.1)

Furthermore (from the definition (A. 1) it follows that) (Xl, Xj) is proportional d d

to 2 :27 C,~ C~. Thus the automorphism leaves the metric invariant i. e. k = l 1=1

(TX,, TXj) = (X~, Xj). (B.2)

From (B. 2) it follows that the real compact form, denoted by g, admits of the decomposition


g ~ f t ~ p ,

(t,p) = O,

It, t] _ t [t,p] - ~ p [p,p] __ t. (B.3)

Next construct the real algebra g* (following the Weyl's unitary trick) spanned by bases t and p* = / p (i s = -- 1). From (]3. 3) we have for g*,

g* = t ~ p*,

(t, p*) -----0,

[ t , t ] _ t It, p*] = p * [p*,p*] c _ - - t . (B.4)

Since for g*, the metric restricted to p* is positive definite, the character of real algebra (Appendix 1) g* equals dimension of p minus dimension of t. g and g* enjoy a common complex extension. The automorphism T has thus associated the real form g* with the compact real form. The completeness also holds: by ranging T through all the involutive automorphisms, all the associated real forms with the same complex extension, as has the compact form may be generated from the latter by the decomposition (B. 3) followed by the Weyl's unitary trick. The decomposition g = t ~ p (g* = t (~ p*) is called Cartan decomposition ofg (g*) into the symmetric subalgebra t and the orthogonal complementary co-set space p (p*). The diagonal generators whose sign is flipped by the involutive automorphism T generating the symmetric subalgebra t, their number is the rank of the coset space.

Let us note (w 3) that the symmetric subalgebra need not be semisimple. In fact, this may have at most one abelian u(1) factor subalgebra (Gilmore 1974). In that case (i.e. when t is non-semisimple in the presence of a u (1)) the coset space is even- dimensional and the Cartan decomposition assumes the form given by (13).

This is the basis for the condition (i) in w 3. Case by case, its equivalence with condition (ii) will be shown after illustrating Caftan decomposition with following examples.

As a first candidate for the (involutive) automorphism, consider complex conjugation. Let us work in the faithful representation of a given simple algebra as provided by the structure constants ((B.1)). Since they are all re~l, complex conjugation of the generators is clearly an automorphism. The generators,

{iH~, iH~, ..., iH~, i (Ea + E_a)IV~ and (E~--E_a)/V~

~vhich close under commutation with real structure constants, are also all cdmpact as follows from (A.3), where a in the above set ranges over all the positive roots. This set therefore gives a basis for the compact real form. Since the ~representation matrices are real, the symmetric subalgebra is generated by (E~--E-a)/V~ and the

[ i(B" orthogonal complementary coset space is generated by iHt, ~ V~ }) for the

P.--V


compact form and by {Ht, (EaJrE_-)/V'2} for the associated noncompact form (the so-called normal form). Since all the diagonal generators flip sign, rank of the coset space equals rank of the simple algebra being decomposed. Dimension of the symmetric subalgcbra is one half the difference between dimension and rank of the algebra. Referring to Gilmore, this subalgerba is SO(N), SO(n)| SU(n) | U(1), SO(n) | SO(n), SO(3) | SO0), Ca | SU(2), C o SU(8) and SO(16) for SU(N), B,, C,, D,, G~, F4, Ee, E~ and E s respectively.

In the remainder of our illustrations, we focus only on such (involutive) automorphisms as generate a non-semisimple symmetric subalgebra of a simple classical algebra. O) SU(N) | U 0 ) c SO(2N). A faithful representation for the compact real form of SO(2N) is provided by 2N X 2N antisymmetric real matrices,

~ ( A C) AT-~ - A BT= - - -C T B

acting on 2N-component column vector

where ,4, B, C are all N x N and x, y are both N-component column vectors Z preserve the scalar product

xTX+yTy

( (.4 + n)/2 (c + cT)/2\ I preserves in addition xTy yT x,

- (C + cT)/2 (,4 + B)/2 ]

or, equivalently, (x T ~= i yT) (x • i y). We are following Wilczek and Zee (1982).

(O--l) This gives us the We must have [J, 27t] = 0 or J 27t J-~ = 27 t where J = 1 0 "

decomposition

z=,~, + z,

(.4 - - B)12 (C -- cT)/2 I

~'~ -- ~ ( C - cT)/2 -- (.4 - - B)I2 I

] Z o : - ~ = - ~o

The change of basis from (~ to U (~) makes manifest the validity of 03. 3), where

1

u ( ~ ) - -

( x - ty)/r


U , ~ t U -1 ~_.

U ~tp U -1 =

A B f C C T TI + , . _ + o

A~+2 B + i C+2C 0 A--B+ i ~ _ _ l

2

A--B . C--C T 0

This form for Cartan decomposition shows that (a) the SO(2N) vectorial reduces as N + N under SU(N), (b) ~r, are generstors of U(N) (c) the generators 2"j, span- ning the complementary space reduce as �89 N (N--1) and �89 N (N--1) under SU(N).

(2) SO(2N- 2) | U(1) c SO(2N)

Another faithful representation for the compact real form of SO(2N) is given by 2N y-matrices satisfying the Clifford algebra {V~, yp} = 28~p 1,/~, v---- 1, 2 . . . . 2N. The generators are �88 [7~,, 7p]- The automorphism 71 7~ generates the symmetric subalgebra spanned by �88 [Yl, Y~] ~ �88 [Y~, Y~] a, fl = 3, 4...2N and the 2(2N-- 2) dimensional orthogonal complimentary space spanned by Yl ya O y~ r,,.

(3) SU(N) | U(1) C Sp(2N) _ C N

This case resembles closely the case (1). The compact real form of Sp (2N) is re-

presented faithfully by the defining 2N-dimensional representation (x)that preserves

xTy _ yT x. The generators 2' are 2N x 2N real matrices

l = B = B T C = C T C --A T

The decomposition is 2" = 2"t + 27p, where,

~t = ; J Zt J-Z = + Zt ; J= \-(B-C)~2 (A--AT)/2]

~t in addition preserves x T x + yr y, or equivalently, preserves (x r:t: iyr) (x 4- i.1,)

((A+Ar)/2 (B+C)[2 ~

2"= \(B+c)12 - (A+f fV2/ ; . . - z ,

f 1 11 i) Cartan On making a similarity transformation by means o ~ [1 --i_' decompo-

tion in the transformed basis clearly shows that (a) Sp (2N) defining representation


reduces as N + N under SU(N) (b) ~t are generators of U(N) (c) the complemen- L

tary space spannect by Z'p reduces as �89 N (N+ 1) and �89 N (N+ 1) under SU(N).

(4) SO(2N--1) | U(1) c SO(ZN+I)

This case may be treated just as the Case 2 with 2N replaced by 2N + 1.

(5) SO(M) | SU(N--M) | U(1) r SU(N)

The involutive automorphism that generates the symmetric subalgebra in this ease

JMN= "-(0 M 0 )where 1M (IN)is an M x M (NxN) identity matrix. The /

is 1N

complementary space spanned by ~'p : JMN '~p -x _ _ J~N-- "~z, reduces under SU(M)|

SU(N--M) as (M, N--M) ~ (M, N--M) The cases (1) through (5) exhaust the nonsemisimple symmetric subalgebras of

the classical algebras. Those for the exceptional algebras are only SO(10) | U(1) r E e and E 8 | U(1) c E~. This case by case consideration leads us to the observation that this list of nonsemisimple symmetric embeddings is exactly the list of maximal regular embeddings, table 5 (taken from Dynkin 1957a). Hence the condition (ii) stated in w [The latter embeddings correspond to a proper semisimple subalgebra, whose Dynkin diagram is got from that of the parent algebra by deleting a suitable (simple) root, which is not contained in any regular maximal semisimple subalgebra (defined in w of the parent algebra.] In table 5, we show the n-tuples for the irreducible piece r of the orthogonal complementary coset space in Cartan decomposition (see 03)) corresponding to a non-semi-simple symmetric subalgebra. We give n-tuples for the greatest component ~1 in the decomposition ofr •162 as well as for the greatest component of r (2:~. Equations 05) and (14) are observed to hold with the exceptions stated in w 3.

Appendix 3

We give here the subgroup content of 65_0 (greatest component of 27 • 27), 351 (the second antisymmetrization of 27) and 351' (greatest component of 27 • ~ ) under the subgroups SO(10) | U(1), SU(6) | SU(2)and SU C (3) | SU L (3) | SUR (3) of E e, the only maximal regular known to us.

Table 5. MRS'S* of simple groups

Parent, G MRS,

1 .~ E~ 2. E0 Ds 3a. Dn (n ;. 4) (a) Dn-1 3b. (b) An-1 4. C. (n ~ 3) An-x 5. Bn (n > 2) Bn-t 6. An A~"]- A.-x-r162

*~mts (Maximal regular subalsebra ) is defined,

Weights of elementary representatio~ 381

r ~

0

o

~

.o

~

R ~'~ + ~ + ~ ~

~1 ~ l ~ ~ •

o.-o -o ?

r~ N ~ 0 0

o

0 0 o~O

J i

= A ~ A

~5 �9 ~ ~: v~ ~5

.2

E

o

•


(A) SO(I0) | U(1) c: Ee 27 ---- 1 + 10 + I_6 under SO(10),

SO(10) I~00010) ~ I0(10000) ~ 144(I0010) + I-'~ ] decompositions: 10(10000) x 1_0(10000) = 54(20000) + 4_5-+ _1 ,

23 x 23 = ! + (54 + 4_5 + _1 ) + (1 + 45 + 210) +

650 =

+ 2(1_0) + (16 + 16_--') + (1~ + 144 + 16 + 1_44) under SO(IO),

1 + 10 + 1_{) + 16 + 16 + 4_5 + 54 + 1_44 + 144 + 210 under S0(10),

351 = 27 {2} n

= 10 {2} + 16 {2} + 10 + 16 + 16"+ 144

= 45 + 12_0 + 1_0 + 1_6 + 1-6 + 1~, 351' = 1 + 54 + 126 + 10+ 16 + 144.

(B) s u ( o | su(2) c E~

27 = (6, 2) + (15, 1).

Using the SU(6) decompositions,

we get

351 =

6 • 6 = 1 + 1 5 + 2 0

6 x_~= 1+ 3.s

_6x 1_5= 8-4 + ~ 6 x 1 5 = 7 0 + 2 0

1 5 x 1 5 - - - - 1 + 3 5 + 1 8 9

15 x 15 = 15 + 105 + 105',

27 x 27 = (1, 1) + (35, _1) + (1, 3) + (35, 3) + (70, 2) + (20, 2)

+ (7_0, 2) + (20, 2) + (1, 1) + (35, 1) + (189, !),

650 = (!, !) + (35, 3) + (3_5, 3) + (2o, 2_) + (79, _2) +

(7o, 2) + 1_89, D, 27{2} = (!, !) + (2. 0,1) + (15, _3) + (I05_~", 1) + (.M, _2) + (~, _2),

351' ---- (1, _3) + (20, ..3) + (15, 1.) + (105;, 1) + (84, 2),

(c)

27 =

650 =

su c (3) | su L (3) | su R (3) c E~,

(I, ~, 3_)+ (L_3, I_)+ (-_3, .~, ~),

0,!,D + (i, _I,D + @,_LD + (_I, _8,9 + q, !,0 + -{- (8,8, 1.) + ~ , 1, 8) + (1, _8, 8) + (~, 3, ..3) + ~ , 3, .~)

Weights of elementary representations

+ o_. _~, _3) + ~, 6, ~) + ~, 3, 3) + ~. 3. 3-) + + ~, 3, 3_) + ~_3, _3. 3-) § ~, _3, _~ + ~3, ~ 6)

351 = (_1, 6,-_3) + (1,_3,_6) + (6,_3,1) + (_3,__6,1) + + ~, !, 3) + ~, _~, ?,) + c_3, s,_3) + ~, b_3) + ~, 3, ~) + 0_, 3, _~) + ~_3. 3_, 9 + ~- 3_-, 9,

3 5 r = ~, ~, 6) + ~!, 3_, 3_-) + ~,_6,1) + 6,_3. 9 + + ~,_1, _~ + ~3_, 1,3_) + ~, _8,3) + ~s_,_3, 33 + ~-, 3- 9.

383

Rderene~

Dynkin E B 1957 a, b Am. Math. Soc. Trans. 2 6 111, 327 Gilmore R 1974 Lie groups, Lie algebras and some of their upph'cations CIL 9 (New York: Wiley) Mohapatra R N and Sakita B 1980 Phys. Rev. D21 1062 $1amky R 1981 Phys. Pep. 79 1 Wflcz~ and Zcc 1982 Phys. Rap. D25 553

Weights of elementary representations in SU(5)-SO(10)-E6 chain

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