properties of the exceptional g₂-lie group

Properties of the Exceptional G₂-Lie GroupAuthor(s): James McConnellSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 66 (1967/1968), pp. 79-92Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20488662 .

Accessed: 12/06/2014 22:34

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Royal Irish Academy is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of theRoyal Irish Academy. Section A: Mathematical and Physical Sciences.

http://www.jstor.org

This content downloaded from 185.2.32.21 on Thu, 12 Jun 2014 22:34:11 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=ria

http://www.jstor.org/stable/20488662?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


[ 79 1

6.

PROPERTIES OF THE EXCEPTIONAL G2-LIE GROUP

By JAMES MCCONNELL, M.R.I.A.

St. Patrick's College, Maynooth*

[Received, 22 SEPTEMBER, 1967. Read, 12 FEBRUARY. Published, 12 MARCH, 1968.1

ABSTRACT

The weight diagrams and basis vectors for lower-dimensional represent ations of the G2-group are studied. The irreducible 21-dimensional represent ation of GL(7) is reduced for G2 into the fundamental 7- and the regular 14 dimensional representation, and the irreducible 28-dimensional representation of GL(7) is reduced into a 27- and a one-dimensional representation of G2. Bilinear, trilinear and quadrilinear invariants for G2 are constructed.

1. INTRODUCTION

In the classification of semi-simple Lie groups Cartan distinguished be

tween three of rank 2, which he designated as A2, B2, G2. The group A2 is

the special unitary group in three dimensions and was well known, as was B2

which is essentially the symplectic group in four dimensions. The group G2

is less familiar. It is a subgroup of the orthogonal group in seven dimensions 07. A realization of G2 is the derivation algebra in a split Cayley algebra of traceless

elements (Jacobson 1962, p. 142). We shall study the G2-group by the more

elementary methods commonly employed for A2 (cf. McConnell 1965) using

weight diagrams and bilinear forms.

J_ z Lr(4

r rI) r7324 r(2

VI r(-6)4

rIG-2) rt- 3) ra-m for )

FIG. 1-The root diagram for G2.

* Now at Dublin Institute for Advanced Studies, Dublin, Ireland.

PROC. R.I.A., VOL. 66, SECT. A. [12]



80 Proceedings of the Royal Irish Academy.

G2 is the Lie group of rank 2 whose algebra has the root diagram of Fig. 1.

By applying Weyl reflections one deduces the formula for the highest weight

of the D(A, p) representation

m- ($1, o)?v(/2.3 1) (1)

The zero values of i and p give a single weight (0, 0) and the one-dimensional

representation D('1(0, 0). The weight diagram for the regular representation

is the root diagram with two zero weights at the origin (McConnell 1966).

The representation is 14-dimensional and its highest weight is (43/4, 1/4), so it is D14)(0, 1).

To discuss other representations we employ the general equations for Lie algebras

[Hi, HJ] = 0, [Hi, Ej = ri(oQ)Ea (2)

[Eax, Ej] = ri(ox) Hi, [Ea, E] =Nap Ea+p.

In writing down the values of Nap there is an arbitrariness in sign; we have in

fact five choices of sign for G2. We choose the positive sign for N13, N15,

N16, N26, N35 putting

13 N16 N26 35- 242

17 (3) N15=

V6 and deduce the other values of Nap from

Nap =Np,-ap =_N N-a--,a = _ -N_a sp-NpX. (4)

This convention of signs differs from that proposed in McConnell 1965. If one reflects in r(1) and then in r(2), the result is equivalent to a rotation

about the origin through an angle r/3. The weight diagrams are therefore

invariant under such a rotation. They are also symmetric about-the m1- and

ml-axes, since we can reflect each weight in these axes.

2. THE SEVEN-DIMENSIONAL REPRESENTATION

The weight diagram for the 7-dimensional representation D(7)(1, 0) is de

picted in Fig. 2. On affixing ket vectors to the weights as shown we deduce

that F2 - ro -

Hl=4l>/3 j 1 1_1 1| Hj1= _ -1H -I

44-v3 -2 ' 4~ 10

L -1~~~ -1 0



MCCONNELL-Properties of the Exceptional G2-Lie Group. 81

ID

13). ? 1l2>

14 m

14> 17> 1 1>

273

Ns> 16>

Fio. 2-The weight diagram for D(7) (1,0).

To establish the representation of ELa we calculate El and E6, and deduce the

others from the commutation relations

[Ea, E] = Nap Ea+p

with (3) and (4), and from

E-a -

The result may be expressed as follows:

2I6E > 12> 13> 14> 6 1> 6> 17>

2-l6 El 12 > /2 7 > 6 > V211 >

2721E?2 2 > - 1>

2V6E3 3> -42 7> 1> -/2J2> 272E4 _3> -2>

27/6E5 12> -4> 42 7> -1213> 2721E6 3 > 4>

(5) 276E- 1/217 > 3> 15,> /214> 21,V2E-2 - 5> 4> 246E-3 6> -42 7> 4> -1215> 24/2E-4 -6> - 5> 2V6E-5 1> -42 7> -l5> 1216> 212E-6 1 > 66>





3. PRODUCTS OF SEVEN-DIMENSIONAL VECTORS

Let la> be a basis vector of weight m(a) and lb> a basis vector of weight

m(b). Then in the product space D 7)(1, 0) ?D(7)(1, 0)

H1ja> lb> = (m1(a)+m1(b))la> lb>

121a > lb > -(M2(a) + M2(b))Ia > lb >, (6) so that the weight of I a> jb > is m(a) + m(b). The products of zero weight are

therefore 1t > 14>, 14> it >, 12> 15>, 15> 12>, 13> 16>, 16> 13>, 17> 17>,

and the most general bilinear form of zero weight is a constant times

17>17> +all>14> +bl4>11> ?cl2>15> +dl5>12> +eJ3>16> +fI16>13>. (7)

We examine whether this expression can represent the single-component vector of the one-dimensional representation D("(0, 0). If this is so, it must be

annihilated by the Ea's, since the infinitesimal transformation

1-igAF

where FA denotes H1, H2, Ea, must be the identity. On applying E and E2

successively to (7) and equating to zero we obtain

a=b=c=d= --1, e=f= 1.

The normalised ket 11}, 1 > for D(l'(0, 0) is therefore given by

1{1},t1> = 7-(17>17> -11>14> -14>11> -12>15> -15>12> + +13>16> +16>13>).

One immediately verifies that every Ea operating on {1}1, I > produces zero.

We write Ia> as x4 when it is in the first space and as y0 when it is in the

second space. The expression for I {I }, 1 > shows that we have an invariant

bilinear form

x7y7-x1Y4-X2yS+X3Y6-X4y1-X5Y2+X6Y3. (8)

In our representation we may define G2 as the subgroup of GL(7) that leaves

(8) invariant. We may express (8) as habXayb with

hab=j . .I

We define x0 as a fundamental covariant vector and associate with it the con

travariant vector Xa given by xa = hab Xb.

Then X

1 X4 X2 3_X5, X3-X6 X4 = -X15 X6 -X2 X6 = 73i X7 -X7




and (8) is expressible as x fYl+X2Y2 +X3 Y3 +X4 Y4+X5Y5 +X6?+X7 Y7

or more briefly xaya. The hab is a contravariant tensor of the second rank. It

is clearly symmetrical and the associated covariant tensor hab defined by

h hcb = b hachC a3

has elements identical with those of hab. We may refer to hab as the metrical

tensor for the 7-dimensional representation. We consider the symmetric product XaYb + XbYa, where b may be equal to a.

Let Eaxa = P Xc, Ex xb r Cxd,

where p and z are constant multipliers that may be zero. Then

E(XaYb + XbYa) =

P(XcYb + XbYc) + Z(XaYd + XdYa),

so that the effect of operating E4 on a symmetric function is to produce a linear

combination of symmetric functions. Thus, if any basis vector in any irre

ducible representation is expressible as a linear combination of symmetric

functions, every other basis vector is expressible in a like manner. A similar

result holds for antisymmetric functions, since

EX(aYb - XbYa) =

P(XcYb - XbYC) + T(XaYd - XdYa).

These results are true for the general linear group in n dimensions GL(n) and

may be obtained independently by employing the technique of Young dia

grams. Moreover

Hi(XaYb ? XbYa) = (mi(a) + mi(b)) (XaYb ? XbYa),

and the weight is therefore the sum of the weights of xa and Yb for both the

symmetric and antisymmetric combination. The weight of ELXaYb ? XbYa) is

m(a) + m(b) + r(cx).

While the application of Ea to a bilinear expression with a certain sym metry produces another with the same type of symmetry, repeated applications will not necessarily produce all such expressions. Indeed we have seen that the linear combination of symmetric functions

1(x7y7-x1y4-X4y -X2y5-X5y2 + X3Y6 + X6Y3)

is the only member of the one-dimensional representation. It cannot be pro duced by operating any number of times on xly2 + x2y1, for example, because this combination does not belong to the one-dimensional representation and

repeated applications of E4 keep the basis vectors within the same irreducible

representation. From the basis vectors of the 7-dimensional representation

X1 X2 X3 X4 X5 X6 X7

Yi Y2 Y3 Y4 Y5 Y6 Y7




we construct the symmetric functions

2x1yj, X1Y2+x2Y1, X1y3+X3y1, ... Xly7+x7y1

2X2y2, X2y3+X3Y2, ..* X2y7+X7Y2 (9)

2x6y6, x6y7+X7Y6

2x7y7.

There are 28 such functions, and there are 21 antisymmetric functions XaYb - XbYa (b $ a). There are 28 independent combinations of symmetric functions and

of these one belongs to D(1)(O, 0). We shall now investigate whether all the

other combinations constitute a 27-dimensional representation.

4. THE TWENTYSEVEN-DIMENSIONAL REPRESENTATION

The expression x1y1, or II > I I>, has weight

and the weight is higher than that of any other of the expressions XaYb+ XbYya. It is therefore the highest weight for the functions (9), and equation (1) shows

that 11 > 1I > belongs to the representation D(2, 0). On making Weyl reflec

tions and noting that neighbouring weights differ only by the root vectors of Fig. 1, we see that the weights for D(2, 0) are situated at the points numbered

on Fig. 3.

The weight at 1 being the highest weight is simple, and we shall show that

the weight at 2 is also simple. Let ui be the ket vector for 1. We may obtain

the ket vector for 2 by displacing from 1 to 2 directly, or from 1 to 2 via 12.

These give, respectively

E5u, E4E-3u.

m2

S i 4 3

Ito

6. IS',IS w iht da 14" 2

7 Ib',Ib'' IS' 13" 13', 13"

/3 S* 17117" 18 '8& 12.

3 '10 .11 FIG. 3-The weight diagram for D(27) (2,0).




Since, however, E4u vanishes,

E4E-3u=[E4,&E3]U= N4,. E5 U,

and this shows that there is a unique ket vector for 2. As the weight diagrams are invariant under a rotation of angle n/3 about the origin, we conclude that

the weights on the boundary are simple. We may proceed from 1 to the point marked 13', 13" either directly or by

passing through 2 or 12, and these give

E1ju, E5E-3u, E.3E5u.

Since,

E5E- 3U-E_ 3E5U = [E5, E_ 3]U =N5,_ 3E- lu,

there are two and only two independent ket vectors. The weight is therefore

double and so are all on the second layer. It is a little more complicated,

though elementary, to show that the zero weight is triple. The representation is therefore 27-dimensional. This together with the one-dimensional repre sentation accounts for all the symmetric functions in the product space

V(7)(1 Ox(D(_7)(l 0).

We denote by J{27}, i> the normalized basis vector corresponding to the

point labelled i. In terms of products of 7-dimensional vectors

1{27}, 1> = 11>11>.

The other basis vectors are deduced from this by displacing, employing (5)

and, when multiplicity occurs, choosing independent linear combinations that

are mutually orthogonal. This procedure gives the following results:

{27}, 1 > -= 11>11>, 1{27},2> = 27(11>12> + 12> I1>)

1{27},3> = 12>12>, 1{27},4> = --(12>13> +1312>)

42 127},5>

= 13>13>, 1{27}16> =- 1, (13> 14> + 14> 13>)

1{27},7> = 14>14>~ 1{27},8> = - /(14 >15 > + 15 >14>)

4-2

I{27},9> = 15>15>, I{27},10> = (15> 16> + 16> 15>)

I{27},11> =16>16>, i{27}, 12> = __ (11>16> + 6>1 >)

42

J{271, 13'> = (I>I17 >+I17> I1l>), I{27},13Y'> = (12>16> +j6>12>)




f{27},14'> =I(j2 >17> +17>12>) 1{27},14" > =?(12>13> +13>11>) 12 42

1 1

j{27}, 16'> =, 1(f4>f7> + 1>l4>), 1{27}, 16"> -2(J3>j5> +15>13>)

1{27}, 17'> = i(15> 17> + 17> 15>), 1{27}, 17"> = 12(14>16> +16>14>)

12

12 It271,16' = = l(j4>j7>+j7>j4>), 1{27},16 > =_z(j3>j5>+j5>j3>)

I{27},19'> = g1 (217>17> +11>14>?+14>I1> +4212>15> + ?1215>12> +4213>16> +42j6> 1>)

1

{27}, 19"> = --(-(2 > 17 >-1+211 >14>) -1I214> I >- 12> 15> + 414 l>/213>16>-4216>13>)

I 27}, 19 > =1= (217>17> +?21>14> +1 24>1 12>1212> 15> 414 - 4215 >12>-13>16>-16>13>).

The last three are readily seen to be orthogonal to 1{1}, 1>.

We express these results as

1 {27}, c > = La ,bxj

where ac assumes progressively the 27 values 1, 2, ..., 19", 19"'. Each LJjnb is

symmetric in a and b, and with hab they constitute the 28 independent symmetric

tensors of the seventh rank. We define the mixed tensor Labc by

tAC-heaEab

and from the mixed tensor set up the metrical tensor in twenty-seven dimensions

Yafi' where

Yapf= cLfc L?b

For the general linear group in seven dimensions GL(7) the 28 symmetric

expressions XaYb + XbYa constitute the basis of an irreducible representation.

We may take 28 linearly independent combinations of XaYb+XbyXba as the basis.

Of these only one, viz. habXa,yb, transforms into itself under G2 because no other

is annihilated by H1, H2 and all the Ea's. Hence the 28-dimensional irreducible

representation of GL(7) reduces under its subgroup G2 into one-dimensional and 27-dimensional irreducible representations.



MCCONNELL-Properties of the Exceptional C2-Lie Group. 87

To consider the reduction more fully take the linear transformation

XC- A c. x.4 d (c, d=1 , 2,. ..., 7)

or more concisely,

x' = Ax. (10)

Then

hab Xayb = XahabYb = hy,

where * is the transpose of x. We may describe G2 as the group of transforma

tions that leaves Thy invariant for arbitrary x, y. This requirement imposes a

condition on the matrix A. As a result of (10)

Thyx -4hy' XAhAy

and, since x, y are arbitrary, we see that

AhA = h. (11)

On taking determinants of both sides we deduce

det A det h det A = det h,

so that

det A=+l.

Let us now take a tensor Til i2i3 o of order r. Then

his2T ii2i3v**lr-h,li2Ail j6 i2j2 i3j3-* AirjrTjlj2j3 .*r

=A i3 ... A ijrhJjjxTjIj2j3 * jr by (1 1),

= (hjlj2TJl%2j34'5r)

= (hi1 2Til!23 ...

and this shows that the linear transformation commutes with the contraction

operation of summing over i1 and i2 in h1i 2Ti1i2i3 .. i.. Thus as a result of the

transformation the components of the (r-2)-dimensional tensor hj,j"Ti,hi3 ...ir transform among themselves. Since the procedure holds for summation over

any two of the suffixes of T, we have altogether

r(r-1) 2

subspaces of (r - 2)-dimensional tensors. When

Tilhh s2 i3r @ rXilyh)

we have one invariant subspace of dimension zero, that is, with the sixngle

invariant member hi,i2xilyi2.




m2

5 4j4 *2 4

6 13 14 12 m2

2J3

Cb

9'

FIG. 4.-The weight diagram for D(14) (0,1).

5. THE FOURTEEN-DIMENSIONAL REPRESENTATION

We mark in Fig. 4 the position of the weights in the diagram for D(14)(0, 1).

The weight at position 1 is equal to the sum of the weights of I I> and 12 > in

D(7)(1, 0). The corresponding basis vector I{14}, 1 > expressed as a bilinear form must be orthogonal to 1{27}, 2>, and so

1 1{14), 1>= - I > 12 l> 2-12

> I I>). 42

Employing the displacement operators as before we obtain altogether

1{14},I> = 2(j1>j2> j2>j1>) 42

1 1{14},2> =-(11>13>-13>11>-4212>17>+4217>12>) 46

1{14}93> = 2(j2>j3> -j3>j2>) 42

1{141,4> = 1(12>14> -14>12> + /213>17> -f/217>13>) J6

1{141,5> 1=(j3>j4>-j4>3> 42




j{14},6> = 6(-13>15>+15>13>+/214>17>-V217>14>

{1417=> - (-14>15> +15>14>)

1{14}, 8> = -(-14> 16> + 16> 14> +1 215>17> -1217>15>) 16

1{14},9> ,21 1>-?5?

{14},11> = >16> -16> 1>)

12

1{14}, 12> = >>6(- 12> 16> + 16> 12> -/211 >17> +-217> 11>)

211>14> ?214>11>)

V2 IfI41,12> = -(-12>16> +16>12> -,21? 16>17> )/171>

We write

1{14}, A > DjDA3XaYb> where a, b==12, ... 7 and A=1X,2, ..., 14. DAab is antisymmetricitna and band from it we construct the mixed tensor DA^, viz.

DA1 = hca DA

Then a metrical tensor tUB is defined by

=AC D Dab:

The DAabXaYb constitute 14 linearly independent combinations of XaYb - XbYa. Since there are altogether 21 such combinations, we examine the significance

of the remaining 7. Consider the point

(+ o

on a weight diagram. The 14-dimensional basis vector at this point is

J{14}, 12>, viz.

-1 (-1/211>17> +1/217>jl>-12>16> +16>12>). 16




Another linear combination of antisymmetric functions orthogonal to this and related to the same point is clearly

t (11>17>-17>1t>-4212>I6>+42i6>12>). (12)

If now we apply the displacement operators to this expression we find that El, E2, E3, E4, E-4, E_5, E6 annihilate it but that E., E6, E_, E2, E3 do not.

Fig. 2 shows that these are properties of I I >; in other words, (12) appears to be

a basis vector {7}, 1 > of the 7-dimensional representation corresponding to

the weight

(2t3' o). To go further with this idea, we define 1 {7}, 2 > as the result of operating with

2Q6 E5 on (12). This gives

1{71,2> = -1(-12>i7>+i7>12>->,/211>13>+.,/213>11>),

4v6

and by successive displacements we obtain the relations

{71,3> = (-4/212>14>+4/214>12>+13>17>-17>13>)

(74,4> = 21(-42 3>15> +4V2>15>13> -14>17> +17>14>)

1 {(7},5> =>--(42l24>16>-4216>14>?15>I7>-_l7>15>)

1(7},6> = k(-V211>I5> +?215>11> -16>17> +17>16>) 4,,6

1{7}7> = 1?-(11>14>-14>I1>-12>15>+15>12>--13>16>+ 46 16>13>).

It may be verified that these satisfy all the relations of (5).

We have obtained 7 combinations of xayb - Xbya that are the basis vectors of

the space of xa This result for the G2-group is unexpected; it is not found in

the more familiar semi-simple Lie groups of rank 2, A2 and B2. The result

may be derived, by using the property of G2 that it is a subgroup of 07, in a

more complicated way outlined by Behrends, Dreitlein, Fronsdal and Lee (1962).

If we express the last equations as

1{7},a> = Ba xcyd,




Bad is a mixed tensor antisymmetric in c and d. The seven Bcd

and the fourteen

DAcd constitute a basis for the antisymmetric seven-by-seven matrices. More over it is now easy to construct an invariant trilinear expression. If we write

then

xa XanBed habncd X X YcZd = XbBa YcZd a ba Bcd ba.cd

= hbGBCdXbYcZd- Xbyczd,

where

Bbcd = hba Bcd

Thus Bbcd XbYc Zd is a trilinear invariant, and similarly a quadrilinear invariant

may be expressed as BeabB'dXaYbZcUd

6. REDUCTION oF D(7)(1, 0)?D(7)(1, 0)

We use the foregoing results to reduce the direct product

D(7)(1, 0) (D (7)(1, 0)

into its irreducible representations. On displacing the weight diagram of Fig. 2

successively in the direction of each weight we obtain the 49 weights shown in

Fig. 5. On referring to Figures 2, 3 and 4 we identify the decomposition as

D(7)(1, 0)?D 7)(1, 0) D(27)(2,0)?D(v4)(0, 1)?D(7)(1, 0)D'D()(0, 0).

M2

)4 0 1 ; 1 If II

/3

I w d l

FIG. 5-The weight diagram for D(7) (1,0) & D(7) (1,0).





For the group GL(7) the reduction of the product is represented by

L lD = El ElDE,

or 7? 7-+28(21.

In the case of its subgroup G2 the final representations reduce, as we have

seen, like

0E0- hab XaYb

0+ E@ Xa Yb,

that is

28 -+ 1 Q27,

and ol

B cd

cd DA X, >1a XYOA cYd,

that is 21 -+ 7b14.

References

Behrends, R. E., J. Dreitlein, C Fronsdal and W. Lee 1962 Rev. Mod.

Phys. 34, 1.

Jacobson, N. 1962 Lie Algebras, New York, Interscience Publishers.

McConnell, J. 1965 Introduction to the Group Theory of Elementary Par

ticles, Communications of the Dublin Institute for Advanced Studies, Series A, No. 16.

McConnell, J, 1966 Proc. R. Ir. Acad. 65 A, 1.



properties of the exceptional g₂-lie group

Documents