multiplicities in weight diagrams

Multiplicities in Weight DiagramsAuthor(s): James McConnellSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 65 (1966/1967), pp. 1-12Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20488644 .

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PROCEEDINGS OF

THE ROYAL IRISH ACADEMY

PAPERS READ BEFORE THE ACADEMY.

MULTIPLICITIES IN WEIGHT DIAGRAMS.

By

JAMES McCONNELL, M.R.I.A.

St. Patrick's College, Maynooth

[Received, 28 SEPTEMBER, 1965. Read, 28 FEBRUARY. Published, 6 MAY, 1966.]

ABSTRACT

The properties of weight diagrams of the B2 Lie group are studied in detail, and rules for finding multiplicities are derived. It is shown how they are related to Wigner's rule for multiplicities in SU(3) weight diagrams.

1. Introduction

The semi-simple Lie groups of rank two denoted by A2, B2, G2 are being

used for the classification and interaction of elementary particles. Particles

constituting a multiplet are placed on a weight diagram corresponding to an

irreducible representation of the group, and the isotopic content of the

multiplet may be read off when one knows the multiplicities of the weights.

A2 is the special linear group in three dimensions, which for physical reasons

is taken to be -unitary and is written SIJ(3). Nearly thirty years ago its weight diagrams were studied by Wigner (1937) who obtained the multi

plicities by rather complicated counting processes. More recently Gasiorowicz

investigated (1963) the same problem using the familiar equations of Lie

algebras but he did not complete the proof of the main theorem on

multiplicities. We shall study the general properties of the weight diagrams for the B2

group, which has applications both to leptons (McConnell 1965a) and to

hadrons (Harari 1965) Then we shall indicate briefly how the same method

may be applied to the SU(3) diagrams. Finally a few, remarks about the G2

weight diagrams will be made.

We begin by writing down the commutation relations for semi-simple Lie

algebras adoptinig the notation and conventions with regard to phase factors.

found in McConnell 1965b.

[Hi, H1] = 0, [Hi, Ea] = r, (a) Ea

[En, E_] = ri (a) Hi, [E., E1] Nae Ea+s.

PROC. R.I.A., VOL. 65, Ssar. 3. [1]



2 Proceedings of the Royal Irish Academy.

it() 42 Z?.

/ \S2(32 (2)

rT6) 7-t) ,/6 ZIS/37

X&2) ?&3) T r-4)

FiG. 1.-The root diagram for A2. FIG. 2.-The root diagram for B2.

{(-z) 27-3) )Qs) X(- 6)

FmG. 3.-The root diagram for G2.

For the groups of rank two the values of r, (a) may be read off the root diagrams in Figures 1, 2, 3. We may say that, apart from those depicted, there are in each case two zero roots. We may justify this by writing

[H,, H1] = 0 H1, [H, H2]= 0 H2

and comparing with

[H,, EJ = r, (a) Ea. This will give us strings of roots like (- r(l), 0, r(l)). More generally there will be a zero root whose multiplicity is equal to the rank of the algebra.

An interesting and useful theorem, which we shall derive, is that the weight diagram for the regular representation is identical with the corresponding root diagram when the zero roots are included. Let the order of the algebra be n and the rank be /; let Latin letters run from I to 1, Greek letters from 1+1 to n and small capitals from 1 to n. Equations (1) have been established in the regular representation where



MCCONNELL-Multiplicities in Height Diagrams. 3

Hi=-Gi, Ea=-Ca and

PA = , O? CFQ} 0S tari(a) 8a0 i - 0, QJ= 0, 0.

The weights in the regular representation are n sets of numbers (nl, n2, .. ., nMI), where mi is an eigenvalue of Hi, i.e. of - Ci. Now CQ is clearly a diagonal matrix with I zero elements and the rest equal to the ith. components of the

non-vanishing roots. The minus sign is of no consequence, since for every root there exists an equal and opposite root. Hence the weights of the

regular representation are the roots, including the vanishing ones. The only multiple weight is the zero one, and its multiplicity is equal to the rank. The

weight diagram for the regular representation of A2 is Fig. 1 with a double

weight at the origin, so the dimension is 8. Similarly the dimensions of the

regular representations of B2 and G2 are 10 and 14 respectively.

2. The B12 Diagrams

For the B2 group equations (1) give

[El, Et1] = H1 [F2, E-] 6 H1 + - H2

[E33 E-3] = H2, [E4J 1EL]= 641 II+ H2 V/6 V6 V/6

[E1, Fs] = VT E2, [E1, 14] - - V Ea

[F1, E-2] =- V iE_3, [E,, E-3] V -4 (2)

[F2, E.] _- V E3, [14, l3] = - v 1,

[E,31 E V-7FF4, [14212 = V E 1E1, [F3, JE3 ] = V Et

[E4, E?] =- - j -

[E-,, E-3] =-2, [E-Ll, E4 = Vj E_3,

all the other commutators of theE's vanishing. The fundamental highest weights 1 1

have (ml, M2) coordinates (1, 1) and (1, 0), and the highest weight

for the irreducible representation D (A, ,t) is

\6 1. ) + - (11, ).(3

The regular representation is D(10) (2, 0). The equivalent weights are obtained by Weyl reflections in the root vectors of Fig. 2. If we reflect in r (3) and then in r (4), the result is a rotation about the origin through an angle i/2. Since the multiplicities are unaltered by Weyl reflections (Racah 1965, p. 47), the weight diagrams are invariant under a rotation through ir/2.

Let M (M1, M2) be the highest weight of an irreducible representation. Then (Racah 1965, p. 48) it is simple and let its normalised ket vector be

1M>, so that H

IhM > =

M1 IM H2J IIM > = M2J MW>.

PROC. R.I.A., VOL. 65, SECT. 3. [2]




Since the weights IV + r (1), M + r (2), M + r (3), M + r (-4)

do not exist in the representation, E M > = 2 M > _= E3 I M> =_4 1 M > O.

* 9 0 9 0 X *

* 9 9 . 9 0 9 9 0 9

(a} () C ( d) Ce ) FIG. 4.-Possible configurations of weight diagrams near the highest weight.

In the neighbourhood of the highest weight the configuration may be as shown in Figure 4. Case (d) may be excluded for the following reason: it presumes that E. I M > = 0, E-2 I M> 0 - and therefore that

0 = [E3, E-2] I M > = i/ -E1 I >, (4) which contradicts our supposition that M - r (1) exists. Similarly case (e) is excluded. Actually it is easy to show from (2) that (4) is obeyed by a highest weight only in the one-dimensional representation D1) (0, 0) and that M = 0 then.

Since the only allowed cases in Figure 4 are (a), (b), (c) and since the diagrams are unaltered by a rotation through an angle a/2, the boundaries are either square or octagonal. They are therefore convex. In case (a), E4 j M>

E_4 f 1 > 0= and the relation 1 1

[F4, JE 4 ]-E u s HI + usH2

gives M1- = M.. According to (3) we are in a D (A, 0) representation. The boundary is a square with sides parallel to the axes and situated symmetrically with respect to them. In case (b), E3 J M > =E-3 J M > = yield M2A == 0

We have a D (0, vz) representation and the sides of the square make an angle 21/4 with the axes. In case (c) the representation is D (A, v) with A, Z positive integers, the octagonal boundary is placed symmetrically with respect to the axes and alternate sides are equal in length.

Turning to the question of multiplicities we show that the weights on the boundary are simple. To illustrate the procedure suppose that the boundary

FIG06-St o weghs o bda

FIG. 5,-String of weights on boundary.



MCCONNELL-Multiplicities in Weight Diagrams. 5

goes from M in the direction of r (4). The ket for the neighbouring A0 in

Figure 5 may be obtained by successively applying the displacement operators

EQ to J K > in a variety of ways corresponding to the various paths from K to AO that do not cross the boundary. We may have, for example,

E41 IM >, E3s L1 I M >, E3 E31 B2 I

M >, E 1 El1 E1L E1 1M >,

E3 Es IL1 E-83 m >.

The last expression is present only for an octagonal boundary. We shall prove that all the kets are the same apart from a constant multiplier, so that the weight AO is simple. Since Es I M > = E1 I

K > = 0,

E31E1IM> = [E3, EK] > = i E4KM >,

E343 E-2 M >=EE3,; 2] M >== Es E1 M > iE4 K >,

Es El E1 E-1 M > Es (H; g+ E-lEl E-1 M >

is Al -_~E L1 1KM > + Es I-1Q6j H + -1 E1) K >

1 1 _ _ ,g_) E4 |lM >,

E3 E3sE_1 E13 M > = A2 E4 1 M >. 3

We have not employed the relation E-4 K M > = 0, so the same method shows that Bo in Fig. 5 is a simple weight. Proceeding along the side we get a simple weight at each point. Similarly the weights on the side coming into

M are simple and by reflecting we deduce that the weights on the boundaries

of the diagrams for D (0, M) and D (A, g) are simple. By taking a boundary that goes from M in a horizonal direction we may establish the theorem for

ID (A, 0).

* co Bo Ao . M

C B A * .1 *i .

* 2 K h

* (4 3 B3 A3

FIu. 6.-Weights on the D(A, 0) diagrams.




To find the multiplicity of weights within the boundary we examine separately the different types of diagram.

(a) D (A, 0) diagrams. We shall prove that the weights on the first layer inside the boundary

are double, those on the next are triple and that the multiplicity increases by one for each layer until we end in a square or a point. The distinguishing relation for the highest weight is E4 I M > = 0. The ket vector for A2 in Figure 6 may be expressed as

K-2(M>, EKE&1jIM>, 11E.BJIM> (5)

or as one arising from a roundabout set of displacements, e.g.

E;g E_3 E_ B1 1M >. Since

E13 EL1 M > - E-1 E13 I M > - V L2 I M >,

only two of (5) are independent. Besides, the more complicated ones reduce to linear combinations of these ; thus for the above example

W1~ ~ ~ ~~; -I

E_ h/_ X --6h-l1- -F tA3 -3 -a E-1 I M >.

To obtain the ket vectors for B1 we note that any transition from M to B. through A1 will just give B-1 before (5), and of these two are independent.

Roundabout transitions as before will give nothing new, so B1 and similarly Co etc. are double weights. We deduce by reflection that every member of the layer is a double weight.

The ket vectors for B2 may be derived by placing E1, E-3 and E 2 before two of (5). To put EIL E-1 before them is just to make linear combination of these operations. We therefore have

E-1E31E-2 1M >, IE-1I31L E-3 IM >, J-L2 1W >,

L2 E-l1 E3( M > (6) Since

E1E-3I IME2 | > = E1tE-1IL E3 Ilvi> =IE2E-1IE3 M >

and since roundabout transitions yield nothing new, B2 is a triple weight. As before C2 and all the weights on the layer have multiplicity tbhee.

We observe what is happening in (6). We have three independent kets corresponding to the three methods of going from M to B2, viz. by two E-2,'s one EI2 and EI1 and EI3 in any order, two BE1's and two E-3's in any order. In going to C3 we can employ three, two, one or zero E-2's, so the multiplicity increases by one as we go to the next layer. In a roundabout transition each Ba will cancel out the corresponding EIa, and the product of EI1 and B-3 will be equivalent to E2. This will not affect the multiplicity. Finally we shall come to a square, if A is odd, and a point, if A is even.



MCCONNELiL-Multiplicities in Weight Diagrams. 7

(6) D (0, i) diagrams

* a * a a * P

*A * A' A *A4* A3 A2 Al M

0 6

FIG. 7.-The D(O, ,) configuration.

In the case of D (0, t) diagrams we shall show that the layer next to the boundary consists of simple weights, the next two layers consist of double

weights, the next two of triple weights, etc. For all values of /i we shall clearly end in a point.

Since we have elaborated the general method, we treat this case more concisely and omit roundabout transitions. The special feature of D (0, a) is that El3 I M > = 0 and this has the consequence that the ket vectors for A1 of Figure 7

E1 IM>, E3 E4j 1>, E3E-1%M >

are all equivalent, so that A1 is simple. Similarly

E]1E4 I M>, E4E-1[ M >, E3EJ1jE1 I M >

are equivalent, so B1 is a simple weight as are all on the first layer within the boundary.

For A2 we can have

E-1 E-1 IM >, E ?3E4E-j1 IM >, E3 E2 E1 I M>,

E3-E-1E4 I M > E3Ej JE2 I M >, E4JEL2 M >, E-2E4 I M>

but they are all linear combinations of

EL1 E1 M >, E E j2 jM >, (7)

so A2 is double. The independent ket vectors for A3 are E.1 operating on

(7), so A3 and similarly B3 etc. are double. When, however, we go to A4, we have not only EL1 ELb acting on (7) but also E-2 E_2 E4 E4 I M > for a

transition through the point P on the boundary. Thus the layers including A4 and A5 consist of triple weights, there being no increase in multiplicity till we come in line with Q, namely at A6. Hence the multiplicity is as stated.

(c) D (A, /z) diagrams When the boundary is octagonal, the multiplicity is not the same for

all points on a layer. We shall investigate the multiplicities of weights on

the same horizontal level as the highest weight. The method will then be clear for weights on other levels, many of whose multiplicites can also be

found by Weyl reflections. In contrast with the last case EL3 DM > : 0

and we must be careful about taking over results.




* S

* * , * *V

9~ AIA3 ?A2 *Al

* S 0 0 0

FIG. 8.-The D(A, it) configuration. In going from W to Al in Figure 8 we obtain the kets

EL I M >, AL3 E4 WM>, E4 EJ3 I M >, E3 E_2 M >,

E3 E_] E_3 I M >.

The last three are linear combinations of the first two, so Al. is a double weight. Then for A2 we can have

E1E I M >I E>,L EA3E4 I M >, E3?E4EA fI M >, E-3E4E3E4 f M >,

E4 E-E4E3 LIM >, E2E4 IM >, E4E2 {IM >

We can express these as inear combinations of

ELE1 {IM >, E1E3?E4 {M >, EA3A1E4 [M >

and either EA3 AL3 ?4 ?4 l M > or E4 ?4 EA3 EL3 WI >. If either vanishes, we have a triple weight. This condition signifies that there are only two weights on the slant side or on the vertical side of the boundary. When there are at least three weights on both sides, A2 has multiplicity 4. An understanding of this may be obtained by referring to Figure 8. In addition to transitions equivalent to a displacement from A1 to A2, which do not increase the

multiplicity, there is a transition via M Bo A2 and another via M P A2, when P and its corresponding weight on the vertical side exist. Similarly on going from A2 to A3 we obtain an increase of multiplicity one corresponding to the path WI Q A3, if there are four or more weights on both sides ; otherwise there is no increase. When we go from A3 to A4, we obtain an increase of

multiplicity one for a transition M P A4, if P and its corresponding weight exist, and an additional increase of one for a transition M R A4, if both R and its corresponding weight exist. When we reach the m2-axis, we can stop.

To illustrate the theorems we draw the diagrams with multiplicities for D (2, 1), which has highest weight

(A~/6 ' V6)

and dimension, that is the number of independent ket vectors in the representation, equal to 35 and also for D (2, 2) with highest weight

3 4*



MCCONNELL-Multiplicities in Weight Diagrams. 9

m z

! / 2I0 /S 2 V16

/6

__________ '___ ' 3 *2

9/ ' / 0,6 61 n

FIG. 9.-The D(35) (2, 1) weight diagram. The isotopic content is 2,2,2,1,1,

1,1, 1, 1, O, O.

46

76,

., .Z Z ,3 .2 */

.Z .4. 1 4. .4 2 *

Jt __o3 .; t- 0 3 -1 m

*2 *4L " 4 2

*/ *2 3 *

FIG. 10.-The D(81) (2, 2) weight diagram. The isotopic content is 3, 3, 3,

2,2, 2,2, 2, 2, 1, 1, 1, 1, I1, 1, 1, 1, 1,0, 0,0.




and dimension 81. We note in both cases the isotopic content, viz. the isotopic multiplets, interpreting n? as

1

76gI3. 3. The A2 Diagrams.

Equations (1) and the root diagram of Figure 1 give the commutation relations

1 1 1 [El, EJ]- HI [E2, E-2]-2V3 HI + - H2

1 1 [E3, E-31 -2g [ + - H2

1 1 [El, El] -- 6 EB2, [El1, ?E2] - B---3 (8)

[E?2

E1]

= - E3, [?2, E>3]

= ?1 V ~~6 <62

I

[X3' E-2l -76 E]1>, [EI1, I_] E -- X 2.

The highest weight for the irreducible representation D (A, jk) is

(6 6) 6 '6) and the regular representation is D (8) (1, 1). It may easily be proved that the weight diagrams are unaltered by a rotation about the origin through and angle 27r/3, that the boundaries of D (A, 0) representations are equilateral triangles with vertices downwards, that the boundaries of D (0, M) are equilateral triangles with vertices upwards and that the boundaries of D (A, p) are

hexagonal, the angle between adjacent sides being 2Xr/3. The hexagon is regular for A = v.

The multiplicity rule given by Wigner (1937) is that the bounding hexagon consists of simple weights, the next hexagon of double weights, the next of triple weights and so on until the hexagon reduces to a triangle. After this there is no further increase in multiplicity. Since an explicit proof of this important theorem based on (8) does not appear to have been published,

we shall establish it briefly. The highest weight M is simple and for a hexagonal boundary I M >

satisfies El. f M > E= ? I 1W> _ E3 I M > = 0-.

The ket vectors for A( in Figure I1 are E3 1W > and E? E, IN >, but these are equivalent. It is evident that a ket arising from a roundabout transition, e.g. E2 ?3 12 1 M >, is proportional to E3 j M >. We shall in future omit roundabout transitions, since it may be verified in each case that they do not upset the conclusions. Thus AO, and similarly Bo and all the weights on the boundary, are simple. The kets for Al are



MOCONNELL-Multiplicities in Weight Diagrams. 11

* . B, , A0

* 0 6S A

* *

FIG. 11.-The A2 weight diagram with hexagonal boundary.

The difference of the second and third is proportional to the first, so we have

two independent vectors which we take to be

E_1 M>, E.2E3s IM >.

The ket vectors for B2 are obtained by prefixing E-1 or E_2 E., since E3 E-2

gives nothing new, and we have

E& E-, DIM>, E-12E12E3 M >, EJ2E3E-1 I M >,EL2E3LE2E3 i M >.

Since IL-1 E2E3 j M >, _ E2E-,E3 I M > - E2E3E-1 IM >,

we have a triple weight at B2. On referring to (5) and (6) we notice that the multiplicity problem is mathematically the same as that for the D (A, 0)

T P

R a

FG. 12.-The triangular layers of weights of an A2 diagram.




diagram of the B2 group. Hence the multiplicity increases by one when we go to the next layer, as long as we are proceeding from a layer of hexagonal shape.

When we reach a triangle, which without loss of generality,we may take with vertex downwards, we let the highest weight be at P in Figure 12. If the multiplicity of the layer including P is s, the independent ket vectors of P are

E-1 E-1 ... E l1l >,

there being s-I operators E_, and those obtained by replacing one or more of these by E-2 13. We denote them generically by j P >. The ket vectors of Q are E.1 B-2 1 P > and B,2 E_1 j P >, and these are equivalent since Et1 and EB2 commute. Thus Q has multiplicity s If we move towards the left, we obtain for R a ket E-l E-1 EL2 [ P >. If we proceed more circuitously, we obtain for example

E3 11 1-k E-2 | P > =: Ek1 E4B Ar E2 E3J)Es I P >

= vt6 E 1 E-1 L-2 I P > +-p 1E2(VB E-+ E-2 E3) P>.

Now E_2 E3 iP > being a ket vector for T is just EL1 acting on one of the

I P >'s, so altogether we obtain a multiple of EL1 EL1 E_2 I P > and Q has multiplicity s. The same is true for every point on the layer and on all inside layers

4. The G2 Diagranms The highest weight of the irreducible representation D (A, ,u) of the ex

ceptional group 02 is

A(6 n ? + '4 (4, 4)

The weight diagram of the regular representation D(14) (0, 1) is that of Figure 3 with two weights at the centre. The boundary is not convex. It may be proved that the weight diagrams are invariant under a rotation through an angle ir3 about the origin and that the weights on the boundary are simple.

However there seems to be no easy way of formulating rules for multiplicities of weights inside the boundary.

REFERENCES

Gasiorowicz, S. 1963 A Simple Graphical Method in the Analysis of SU^ Argonne National Laboratory?6729.

Harari, H 1965 Phys. Rev. Lett. 14, 1100.

McConnell, J. 1965a Canadian Journal of Physics, 43, 705. -19656 Introduction to the Group Theory of Elementary Particles.

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

Racah, G 1965 Springer Tracts in Modern Physics 37, Berlin, Springer, pp. 28-84

Wigner, E 1937 Phys. Rev. 51, 106.



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