Representation Theory Lecture 12

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 20 May 2013

Representation Theory Lecture 12

SO10= { gGL10 |detg=1, ggt=1 } 𝔰𝔬10= { x𝔤𝔩10| trx=0, x+xt=0 }


1=det(etx)= det ( eth1 ethn ) =et(h1++hn) =et·tr(x)


1=etx (etx)t= etx etxt= et(x+xt)


𝔰𝔬5= { ( 0 a12 a13 a14 a15 -a12 0 a23 a24 a25 -a13 -a23 0 a34 a35 -a14 -a24 -a34 0 a45 -a15 -a25 -a35 -a45 0 ) } ,dim(𝔰𝔬5) =5·42=10

Another choice is

SO10= { gGL10 |detg=1, gJgt=J } 𝔰𝔬10= { x𝔤𝔩10| trx=0, xJ+Jxt=0 }


J= ( 01 1 1 10 ) orJ= ( 0 10 01 10 01 0 ) .


( a11 a12 a13 a1-3 a1-2 a1-1 a21 a22 a23 a2-3 a2-2 a2-1 a31 a32 a33 a3-3 a3-2 a3-1 a-31 a-32 a-33 a-3-3 a-3-2 a-3-1 a-21 a-22 a-23 a-2-3 a-2-2 a-2-1 a-11 a-12 a-13 a-1-3 a-1-2 a-1-1 ) ( 000001 000010 000100 001000 010000 100000 ) = ( a1-1 a1-2 a1-3 a13 a12 a11 a2-1 a2-2 a2-3 a23 a22 a21 a3-1 a3-2 a3-3 a33 a32 a31 a-3-1 a-3-2 a-3-3 a-33 a-32 a-31 a-2-1 a-2-2 a-2-3 a-23 a-22 a-21 a-11 a-12 a-13 a-1-3 a-1-2 a-1-1 ) , 𝔰𝔬4= ( a11 a12 a13 a1-3 a1-2 0 a21 a22 a23 a2-3 0 -a1-2 a31 a32 a33 0 -a2-3 -a1-2 a31 a32 0 -a33 -a23 -a13 a21 0 -a-32 -a32 -a22 -a12 0 -a-21 -a-31 -a31 -a21 -a11 )


𝔥= { ( λ10 λ2 λ3 -λ3 -λ2 0-λ1 ) }

and if 𝔤=𝔰𝔬6 then

𝔤=𝔥+ a12Xε1-ε2 + a13Xε1-ε3 + a23Xε2-ε3 + a21Xε2-ε1 + a31Xε3-ε1 + a32Xε3-ε2 + a1-2Xε1+ε2 + a1-3Xε1+ε3 + a2-3Xε2+ε3 + a-21X-(ε1+ε2) + a-31X-(ε1+ε3) + a-32X-(ε2+ε3)

where, for i<j,

Xεi-εj= Eij-E-j-i, Xεi+εj= Ei-j- Ej-i Xεj-εi= Eji-E-i-j, X-εi-εj= E-ji-E-ij


[h,Xεi-εj] = [ h, Eij- E-j-i ] =(λi-λj) Eij- (-λj+λi) E-j-i = (λi-λj) ( Eij- E-j-i ) =(λi-λj) Xεi-εj, [h,Xεi+εj] = [ h,Ei-j- Ej-i ] =(λi+λj) Ei-(λj+λi) Ej-i = (λi+λj) ( Ei-j- Ej-i ) =(λi+λj) Xεi+εj,


h= ( λ1 λ2 λ3 -λ3 -λ2 -λ1 )

and εi:𝔥 is given by εi(h)=λi.

The root system

R = { ±εi±ej |1ij r } and R+ = { εi±εj |1i<jr } .

The Dynkin diagram

The fundamental chamber is

(𝔥*)+ = { λ𝔥*| λ,α 0forα (R)+ } = { λ1ε1++ λrεr| λ1λ2 λr-1 |λr| }

This chamber is on the positive side of the hyperplanes

𝔥α= { λ𝔥*| λ,α =0 } forα (R)+


(R)+ { εi±εj |1i<je }


εi,εj =δij.

The walls of (𝔥*)+ are

𝔥ε1-ε2, 𝔥ε2-ε3,, 𝔥εr-1-εr, 𝔥εr-1+εr

and the Dynkin diagram is

ε1-ε2 ε2-ε3 ε3-ε4 ε4-ε5 ε4+ε5 for𝔰𝔬10

The Weyl group W0

W0 is generated by the reflection in the hyperplanes 𝔥εi±εj. Using the basis ε1,,εr for 𝔥*, the group W0 is generated by

sij= ( 1 01 10 1 1 1 01 10 1 1 )


sεi+εj= 1 1 i 1 1 j 1 1 1 1 -j 1 1 -i 1 1 ( 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 ) W0 { r×rmatrices with (a) exactly one nonzero entry in each row and column (b) nonzero entries are±1. (c)nonzero entriesaij=1 }

The character of the adjoint representation 𝔤

Let M be a 𝔤-module.

The character of M is

char(M)= μ𝔥* (dimMμ) eμ,where Mμ= { mM|hm=μ (h)mfor each h𝔥 }

is the μ-weight space of M.

The weights of the adjoint representation for 𝔤=𝔰𝔬10 are

ε1-ε2 ε1-ε3 ε1-ε4 ε1-ε5 ε1+ε5 ε1+ε4 ε1+ε3 ε1+ε2 ε2-ε3 ε2-ε4 ε2-ε5 ε2+ε5 ε2+ε4 ε2+ε3 ε3-ε4 ε3-ε5 ε3+ε5 ε3+ε4 ε4-ε5 ε4+ε5

their negatives and the weight 0:

𝔤0=𝔥and dim(𝔤0)= dim(𝔥)=5.

The character of 𝔤 is

sε1+ε2 = x1x2-1+ x1x3-1+ x1x4-1+ x1x5-1+ x1x5+ x1x4+ x1x3+ x1x2 x2x3-1+ x2x4-1+ x2x5-1+ x2x5+ x2x4+ x2x3 x3x4-1+ x3x5-1+ x3x5+ x3x4 x4x5-1+ x4x5 +5 +x1-1x2 +x1-1x3 +x1-1x4 +x1-1x5 +x1-1x5-1 +x1-1x4-1 +x1-1x3-1 +x1-1x2-1 +x2-1x3 +x2-1x4 +x2-1x5 +x2-1x5-1 +x2-1x4-1 +x2-1x3-1 +x3-1x4 +x3-1x5 +x3-1x5-1 +x3-1x4-1 +x4-1x5 +x4-1x5-1


xi=eεi, fori=1,2,,5.

In this example

p=12αR+ α=12 ( 8ε1+6ε2+ 4ε3+2ε4 ) =4ε1+3ε2+ 2ε3+ε4.

The Weyl denominator formula says

ap= wW0 det(w)w ( x14 x23 x32x4 ) = x14x23 x32x4+ x1-4 x23x32 x4++ x4-4 x3-3 x2-2 x1-1 = epαR+ (1-e-α)= x14x23x32 x4i<j (1-xi-1xj) (1-xi-1xj-1)

and the Weyl character formula says

sε1+ε2= aε1+ε2+pap = wW0 det(w)w ( x15x24 x32x4 ) x14x23x32 x41i<j5 (1-xi-1xj) (1-xi-1xj-1)

The crystal B(ε1+ε2)

𝔥* has basis ε1, ε2, ε3, ε4, ε5 and crystals are sets of paths in 𝔥*5 which are closed under the action of the root operators

e1, e2, e3, e4, e5, f1, f2, f3, f4, f5

corresponding the the wall of (𝔥*)+:

𝔥ε1-ε2, 𝔥ε2-ε3, 𝔥ε3-ε4, 𝔥ε4-ε5, 𝔥ε4+ε6 ε1-ε2 ε2-ε3 ε3-ε4 ε4-ε5 ε4+ε5

The weights of 𝔤 are ±(εi±εj), 1i<jr and these are some of the vertices of the 5 dimensional cube.

The highest weight path in B(ε1+ε2) can be taken to be the straight line path from 0 to ε1+ε2.

Most of the time (in B(ε1+ε2)) the root operators are taking a straight line path to a straight line path. The only exceptions are

pε4+ε5 f5 (12p-ε4-ε5) (12pε4+ε5) f5 p-ε4-ε5 pε4-ε5 f4 (12p-ε4+ε5) (12pε4-ε5) f4 p-ε4+ε5 pε3-ε4 e3f3 (12p-ε3+ε4) (12pε3-ε4) e3f3 p-ε3+ε4 pε2-ε3 e2f2 (12p-ε2+ε3) (12pε2-ε3) e2f2 p-ε2+ε3 pε1-ε2 e1f1 (12p-ε1+ε2) (12pε1-ε2) e1f1 p-ε1+ε2

For the "standard model" in particle physics it is important to understand how this representation decomposes under the action of the subalgebras

SO10 SU5 SU3× U1× SU2

These restrictions are obtained by ignoring the operators

f6,for SO10 SU5,


f3,for SU5 SU3×U1 ×SU2.

The crystal graph B(ε1+ε2) is (all paths are straight line paths except the 5 exceptional ones listed above):

The crystal graph of B(ε1+ε2)

ε1+ε2 f2 ε1+ε3 f1 f3 ε2+ε3 ε1+ε4 f3 f1 f5 f4 ε2+ε4 ε1-ε5 ε1+ε5 f1 f1 f4 f2 f5 f1 f4 f5 ε2+ε5 ε3+ε4 ε2-ε5 ε1-ε4 f2 f4 f5 f2 f4 f1 f3 ε3+ε5 ε3-ε5 ε2-ε4 ε1-ε3 f3 f3 f4 f2 f3 f1 f2 ε4+ε5 ε4-ε5 ε3-ε4 ε2-ε3 ε1-ε2 f5 f4 f3 f2 f1 12(-ε4-ε5) 12(-ε4+ε5) 12(-ε3+ε4) 12(-ε2+ε3) 12(-ε1+ε2) +12(ε4+ε5) +12(ε4-ε5) +12(ε3-ε4) +12(ε2-ε3) +12(ε1-ε2) f5 f4 f3 f2 f1 -ε4-ε5 -ε4+ε5 -ε3+ε4 -ε2+ε3 -ε1+ε2 f3 f3 f4 f2 f3 f1 f2 -ε3-ε5 -ε3+ε5 -ε2+ε4 -ε1+ε3 f2 f4 f5 f2 f4 f1 f3 -ε2-ε5 -ε3-ε4 -ε2+ε5 -ε1+ε4 f1 f1 f4 f2 f5 f1 f4 f5 -ε2-ε4 -ε1+ε5 -ε1-ε5 f3 f1 f5 f4 -ε2-ε3 -ε1-ε4 f1 f3 -ε1-ε3 f2 -ε1-ε2

ε1-ε3 f1 f2 ε4-ε5 ε2-ε3 ε1-ε2 f4 f2 f1 12(-ε4-ε5) 12(-ε3+ε4) 12(-ε2+ε3) 12(-ε1+ε2) +12(ε4-ε5) +12(ε3-ε4) +12(ε2-ε3) +12(ε1-ε2) f4 f2 f1 -ε4+ε5 -ε2+ε3 -ε1+ε2 f1 f2 -ε1+ε3 3 bosons 1 photon 8 gluons of QCD w+,z,w-

Notes and References

This is a typed copy of handwritten notes by Arun Ram on 28/10/2008.

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