Indivisual Research

My indivisual is about finding invariant representation (scalar) by using partition function(Hilbert series) and integral over group.

Starting paper is Hilbert Series for Adjoint SQCD.

Next page is Mathematica with FORM.

Prerequisite Knowledge

I used repersentation theory for Lie group.

To calculate, i also use residue theorem.

Method

They use Mathematica to calculate integral over the group.

Simple Mathematica Code example is here.


su3cfu[z1_, z2_] := z1 + z2/z1 + 1/z2 
su3caf[z1_, z2_] := z2 + z1/z2 + 1/z1
su3cad[z1_, z2_] := z1 z2 + z2^2/z1 + z1^2/z2 + 2 + z1/z2^2 + z2/z1^2 + 1/(z1 z2)
su3h[z1_, z2_] :=  1/( z1 z2) (1 - z1 z2) (1 - z1^2/z2) (1 - z2^2/z1)
Nf:=6
PE[z1_,z2_, t_, s_,o_] = E^Sum[(Nf*su3cfu[z1^n,z2^n]*t^n +Nf* su3caf[z1^n,z2^n]*o^n+su3cad[z1^n,z2^n]*s^n)/n, {n,1,6}]//Simplify
expan[z1_,z2_,s_,t_,o_]=Series[PE[z1,z2,t,s,o]*su3h[z1,z2],{o,0,3},{t,0,3},{s,0,6}]//Normal//Expand
g[s_,t_, o_] = SeriesCoefficient[SeriesCoefficient[expan[z1,z2,s,t,o]*z1*z2,{z1,0,0}],{z2,0,0}]//Expand
PL[t_,s_,o_] =Series[ Sum[(MoebiusMu[n]*Sum[-(1-g[s^n,t^n,o^n])^k/k,{k,1,10}])/n, {n, 1,10}],{o,0,3},{t,0,3},{s,0,6}]//Normal//Simplify//Expand

They use $1$ adjoint representation, $N_f$ fundamental representation, $N_f$ antifundamental representation.

Dimension of SU(N) tensor

Let's compute with Young Tableaux.

Highest order coefficient is always positive.

Order of $N$ is same number as sum of blocks.

Example

$[1,0,\cdots,0]=N$

$[0,1,0,\cdots,0]=N^2/2+N/2$

$[0,0,1,0,\cdots,0]=_NC_3=N^3/6-N^2/2+N/6$

$[0,0,0,1,0\cdots,0]=_NC_4=N^4/24-N^3/4+11N^2/24-N/4$

$[1,1,0,\cdots,0]=N^3/3-N/3$

$[1,0,1,0,\cdots,0]=$

$[1,1,1,0\cdots,0]=$

$[2,0,\cdots,0]=N^2/2-N/2$

$[3,0,\cdots,0]=N^3/6+N^2/2+N/6$

$[0,2,0,\cdots,0]=$

$[2,1,0,\cdots,0]=$

$[4,0,\cdots,0]=$

Application

2 adjoint

Let's think $2$ adjoint representation(in $SU(N)$ , $[1,0,\cdots,0,1]$ ), $N_f$ fundamental representation, $N_f$ antifundamental representation.

For $SU(2)$ , (under order 6)

$PL[g^{(N,SU(2))}(s_1,s_2,t)]=(s_1^2+s_1s_2+s_2^2)+[N_f(2N_f+1)]t^2+([N_f(2N_f-1)]+[N_f(2N_f+1)](s_1+s_2+s_1s_2))t^2-\cdots$

For $SU(3)$ , (under order 6)

$PL[g^{(N,SU(3))}(s_1,s_2,t,o)]=[N_f^2]to+(s_1s_2+s_1^2+s_2^2)+[N_f^2](s_1+s_2)to+[N_f(N_f-1)(N_f-2)/6](t^3+o^3)+(s_1^3+s_1^2s_2+s_1s_2^2)+[N_f^2](s_1^2+2s_1s_2+s_2^2)to+[N_f(N_f+1)(N_f-1)/3](s_1+s_2)(t^3+o^3)+s_1^2s_2^2+([N_f(N_f+1)(N_f-1)/3]s_1^2+[N_f(N_f+1)(5N_f-2)/6]s_1s_2+[N_f(N_f+1)(N_f-1)/3]s_2^2)(t^3+o^3) +[N_f^2](2s_1s_2^2+2s_1^2s_2)to+([N_f(N_f+1)(N_f+2)/6]s_1^3+[N_f^2(N_f+1)]s_1^2s_2+[N_f^2(N_f+1)]s_1s_2^2+[N_f(N_f+1)(N_f+2)/6]s_2^3)(t^3+o^3)+[N_f^2](s_1^3s_2+2s_1^2s_2^2+s_1s_2^3)ot+s_1^3s_2^3$

$[N_f^2]:\ [1,0,\cdots,0]\times [0,\cdots,0,1]$

$[N_f(N_f-1)(N_f-2)/6]:\ [0,0,1,0\cdots,0]$

$[N_f(N_f+1)(N_f-1)/3]:\ [1,1,0,\cdots,0]$

$[N_f(N_f+1)(5N_f-2)/6]:\ [3,0,\cdots,0]+4[1,1,0,\cdots,0]$

$[N_f(N_f+1)(N_f+2)/6]:\ [3,0,\cdots,0]$

$[N_f^2(N_f+1)]:\ 2[3,0,\cdots,0]+4[1,1,0,\cdots,0]$

Then. new terms arise!

Blended Casimir Operator: $s_1^2s_2$ , $s_1^3s_2^3$

Blended Adjoint Meason: $s_1s_2t\tilde{t}$ , $s_1^3s_2t\tilde{t}$

Blended Adjoint Baryon 1: $s_1s_2t^3$ :

Blended Adjoint Baryon 2: $s_1^2s_2t^3$ :

Interpretation: $s_1$ and $s_2$ have $SU(2)$ symmtery.

For $SU(4)$ , PL is complicated.

PL[t_,s1_,s2_,o_]=

Series[Sum[(MoebiusMu[n]*Sum[-(1-g[s1^n,s2^n,t^n,o^n])^k/k,

{k,1,Floor[6/n]}])/n,{n,1,6}],{o,0,4},{t,0,4},{s1,0,8},{s2,0,8}]//Normal//Expand

Goal is order $s^6t^4$ .

Order별로 나눠서 table로 대입?

For $Sp(2)$ , (under order 8)

$PL[s_1,0,t]'=s_1^2+s_1^4+(2N^2-N)t^2+(2N^2+N)s_1t^2+(2N^2-N)s_1^2t^2+(2N^2+N)s_1^3t^2+...$

$PL[s_1,s_2,0]'=(s_1^2+s_1s_2+s_2^2)+(s_1^4+s_1^3s_2+2s_1^2s_2+s_1s_2^3+s_2^4)+(s_1^4s_2^2+s_1^3s_2^3+s_1^2s_2^4)+(s_1^4s_2^3+s_1^3s_2^4)+(s_1^6s_2^3+s_1^5s_2^4+s_1^4s_2^5+s_1^3s_2^6)$

$PL[s_1,s_2,t]'=4N^2s_1s_2t^2+(6N^2+N)(s_1^2s_2+s_1s_2^2)t^2+((6N^2+N)s_1^3s_2+8N^2s_1^2s_2^2+(6N^2+N)s_1s_2^3)t^2+(4N^2s_1^4s_2+(10N^2+N)s_1^3s_2^2+(10N^2+N)s_1^2s_2^3+4N^2s_1s_2^4)t^2+((4N^2+N)s_1^5s_2+(8N^2+2N)s_1^4s_2^2+(12N^2+4N)s_1^3s_2^3+(8N^2+2N)s_1^2s_2^4+Ns_1^5+(4N^2+N)s_1s_2^5)t^2+(4N^2s_1^5s_2^2+8N^2s_1^4s_2^3+8N^2s_1^3s_2^3+4N^2s_1^2s_2^4)t^2+((2N^2+N)s_1^6s_2^2+(4N^2+2N)s_1^4s_2^4+(2N^2+N)s_1^2s_2^6)t^2$

Casimir invariants: $s^{2k}$ $u_{2k}=Tr(\phi^{2k})$ ( $k=1,2,\cdots,N_c$ ) $1$ dim

Measons: $t^2$ : $M^{ij}=J^{ab}Q^i_aQ^j_b$ ( $a,b=,1,2,\cdots,2N_c$ ) $N(2N-1)$ dim

Even adjoint measons: $s^{2l}t^2 :$ (A_{2l})^{ij}=J^{a_1b_1}J^{c_1b_2}J^{c_2b_3}\cdots J^{c_{2l-1}b_{2l}}J^{c_{2l}a_{2l}}Q^i_{a_1}\phi_{b_1c_2}\phi_{b_2v_2}\cdots \phi_{b_{2l}c_{2l}Q^j_{a_{2l}}N(2N-1)$dim

Odd adjoint measons: $s^{2k-1}t^2$ : $(A_{2k-1})^{ij}=J^{a_1b_1}J^{c_1b_2}\cdots J^{c_{2k-1}a_{2k-1}}Q^i_{a_1}\phi_{b_1c_2}\phi_{b_2c_2}\cdots \phi_{b_{2k-1}c_{2k-1}}Q^j_{a_{2k-1}}$ $N(2N+1)$ dim

Blended Casimir invariants: $s_1^{k_1}s_2^{k_2}$ $(k_1,k_2)=(2,0),(1,1),(0,2),\ (4,0),(3,1),(1,3),(0,4),\ (4,2),(3,3),(2,4)\ (4,3),(3,4),\ (6,3),(5,4),(4,5),(3,6)$

Blended Adjoint measons: $s_1^{k_1}s_2^{k_2}t^2$

For $SO(3)$ , (under order 4)$

$PL=(s_1^2+s_1s_2+s_2^2)+(Ns_1+Ns_2)t+(N^2/2+N/2)t^2+((N^2/2-N/2)s_1+(N^2/2-N/2)s_2)t^2+(N^3/6-N^2/2+N/3)t^3+(N^3/6-N^2/2+N/3)(s_1^3s_2+s_1^2s_2^2+s_1s_2^3)t^3+(N^3/2-N^2/2)(s_1^3s_2^2+s_1^2s_2^3)t^3+(N^3/3-N/3)s_1^3s_2^3t^3+??? (N^4/24-N^3/4+11N^2/24-N/4)(s_1^3+s_2^3)t^4+(5N^4/24-3N^3/4+19N^2/24-N/4)(s_1^2s_2+s_1s_2^2)t^4+...$

For $SO(4)$ , (under order 4)

$PL[0,0,t]=(N^2/2+N/2)t^2+(N^4/24-N^3/4+11N^2/24-N/4)t^4$

$PL[s_1,s_2,0]=2s_1^2+2s_1s_2+2s_2^2$

$PL[s,0,t]'=(N^2-N)st^2+(N^2/2+N/2)s_1^2t^2$

$PL[s_1,s_2,t]'=2N^2s_1s_2t^2+(N^2+N)(s_1^2s_2+s_1s_2^2)t^2$

For $SO(5)$ ,

More complicated...

$PL[0,0,t]=(N^2/2+N/2)t^2+(N^5/120-N^4/12+7N^3/24-5N^2/12+N/5)t^5$

$PL[s_1,s_2,0]=(s_1^2+s_1s_2^2+s_2^2)+(s_1^4+2s_1^2s_2^2+s_2^4)+(s_1^4s_2^2+s_1^3s_2^3+s_1^2s_2^4)+(s_1^4s_2^3+s_1^3s_2^4)+(s_1^6s_2^3+s_1^5s_2^4+s_1^4s_2^5+s_1^3s_2^6)$

$PL[s,0,t]'=Ns^2t+(N^2/2-N/2)st^2+(N^2/2+N/2)s^2t^2+(N^2/2-N/2)s^3t^2+(N^3/6-N^2/2+N/3)st^3+(N^5/120-N^4/12+7N^3/24-5N^2/12+N/5)s^5t^5+(N^5/120-N^4/12+7N^3/24-5N^2/12+N/5)s^6t^5+(N^6/48-5N^5/48+5N^4/48+5N^3/48-N^2/8)s^5t^6+(9N^6/80-9N^5/16+17N^4/16-15N^3/16+13N^2/40)s^6t^6$

$PL[s_1,s_2,t]'=Ns_1s_2t+N(s_1^2s_2+s_1s_2^2)t+N(s_1^3s_2+2s_1^2s_2+s_1s_2^3)t+N(s_1^4s_2+2s_1^3s_2^2+2s_1^2s_2^3+s_1s_2^4)t+2Ns_1^2s_2^2t+N(s_1^4s_2^2+s_1^3s_2^3+s_1^2s_^4)t+N(s_1^5s_2^2+s_1^4s_2^3+s_1^3s_2^4+s_1^2s_2^5)t+N^2s_1s_1t^2+$

13N^3s_1^6s_2^3t^4

65N^4s_1^5s_2^4t^4

767N^4s_1^6s_2^4t^4

27N^5s_1^5s_2t^5

661N^5s_1^6s_2^2t^5/120

241N^5s_1^4s_2^4t^5/120\

...

For $G_2$ , (under order 6)

...

symmteric tensor and that's conjugate

Let's think $1$ or $2$ symmetric tensor(in $SU(N)$ , $[2,0,\cdots,0]$ ), $1$ or $2$ conjugate of symmetric tensor(in $SU(N)$ , $[0,\cdots,0,2]$ ), $N_f$ fundamental representation, $N_f$ antifundamental representation.

Character is $\frac{1}{2}(\chi(fun^2)-\chi(fun)^2)$

For example, $SU(2)$ symmetric tensor is $[2]$ , which is same as adjoint representation above.

For $SU(3)$ , (under order 6)

Weight diagram of symmetric 2 tensor is

$\chi_{symm}=z_1^2+z_2+z_2^2/z_1^2+z_1/z_2+1/z_1+1/z_2^2$

$\chi_{ants}=z_2^2+z_1+z_1^2/z_2^2+z_2/z_1+1/z_2+1/z_1^2$

For 1 symm, conj. symm,

$PL[s_1,s_2,0,0]=s_1^3+s_1s_2+s_1^2s_2^2+s_2^3$

$PL[s,0,t,0]=s^3+s^2t^2(N^2/2+N/2)+t^3(N^3/6-N^2/2+N/3)$

$PL[s,0,t,o]-PL[s,0,t,0]-PL[s,0,0,o]-PL[0,0,t,o]+PL[s,0,0,0]+PL[0,0,t,0]+PL[0,0,0,o]=to^2s^2(N^3/2-N^2/2)+t^2os(N^3/2-N^2/2)-\cdots$

$PL[s_1,s_2,t,0]-PL[s_1,s_2,0,0]-PL[s_1,0,t,0]-PL[0,s_2,t,0]+PL[s_1,0,0,0]+PL[0,s_2,0,0]+PL[0,0,t,0]\\ =t^2s_1s_2^2(N^2/2+N/2)+t^3s_1s_2(1+s_1s_2)(N^3/3-N/3)+t^3s_1^3s_2^3(N^3/6+N^2/2+N/3)$

$PL[s_1,s_2,t,o]-PL[s_1,s_2,t,0]-PL[s_1,s_2,0,o]-PL[s_1,0,t,o]-PL[0,s_2,t,o]+PL[s_1,s_2,0,0]+PL[s_1,0,t,0]+PL[s_1,0,0,o]+PL[0,s_2,t,0]+PL[0,s_2,0,o]+PL[0,0,t,o]-PL[s_1,0,0,0]-PL[0,s_2,0,0]-PL[0,0,t,0]-PL[0,0,0,o]\\ =to(s_1s_2+s_1^2s_2^2)N^2+to^2s_1^2s_2(s_1+s_2^2)(N^3/2+N^2/2)+to^2s_1s_2^2N^3+t^2os_1s_2^2(s_1^2+s_2)(N^3/2+N^2/2)+t^3o^3s_1^5s_2^5(2N^6/9-N^5-4N^4/9+2N^2/9)+t^3o^3s_1^6s_2^6(11N^6/36-N^5/6-13N^4/36-N^3/3-4N^2/9)$

2 symmetric&conj. symm.

$PL[s1,s2,0,0]=2 \text{s1}^4 \text{s2}+4 \text{s1}^3 \text{s2}^3+4 \text{s1}^3+9 \text{s1}^2 \text{s2}^2+2 \text{s1} \text{s2}^4+4 \text{s1} \text{s2}+4 \text{s2}^3$

$PL[s1,0,t,0]=\frac{1}{6} N^3 \text{s1}^6 t^3+\frac{2}{3} N^3 \text{s1}^3 t^3+\frac{N^3 t^3}{6}+\frac{1}{2} N^2 \text{s1}^6 t^3+\frac{3}{2} N^2 \text{s1}^2 t^2-\frac{N^2 t^3}{2}+\frac{1}{3} N \text{s1}^6 t^3-\frac{2}{3} N \text{s1}^3 t^3+\frac{3}{2} N \text{s1}^2 t^2+\frac{N t^3}{3}+4 \text{s1}^3$

$PL[s,0,t,o]'=+\frac{11}{6} N^3 o^3 s^5 t^2+N^3 o^2 s^5 t+2 N^3 o^2 s^2 t+2 N^3 o s^4 t^2+N^3 o s t^2+N^2 o^2 s^5 t-N^2 o^2 s^2 t+N^2 o s^4 t^2+2 N^2 o s^3 t-N^2 o s t^2$

$PL[s1,s2,t,0]'=\frac{5}{3} N^3 \text{s1}^4 \text{s2} t^3+5 N^3 \text{s1}^2 \text{s2}^2 t^3+N^3 \text{s1} \text{s2}^4 t^3+\frac{4}{3} N^3 \text{s1} \text{s2} t^3+N^2 \text{s1}^6 \text{s2} t^2+\frac{15}{2} N^2 \text{s1}^4 \text{s2}^2 t^2+3 N^2 \text{s1}^4 \text{s2} t^3+2\ 4 N^2 N^2 \text{s1}^3 \text{s1} \text{s2} \text{s2}^5 t^2 t^2+10 N^2 \text{s1}^2 \text{s2}^3 t^2+3 N^2 \text{s1}^2 \text{s2}^2 t^3+3 N^2 \text{s1} \text{s2}^4 t^3+4 N^2 \text{s1} \text{s2}^2 t^2+2 N \text{s1}^6 \text{s2}^2 t-N \text{s1}^6 \text{s2} t^2+4 N \text{s1}^5 \text{s2} t+20 N \text{s1}^4 \text{s2}^3 t-\frac{9}{2} N \text{s1}^4 \text{s2}^2 t^2+\frac{4}{3} N \text{s1}^4 \text{s2} t^3+8 N \text{s1}^3 \text{s2}^5 t+12 N \text{s1}^3 \text{s2}^2 t+13 N \text{s1}^2 \text{s2}^4 t-2 N \text{s1}^2 \text{s2}^3 t^2-2 N \text{s1}^2 \text{s2}^2 t^3+2 N \text{s1}^2 \text{s2} t-2 N \text{s1} \text{s2}^5 t^2++2 N \text{s1} \text{s2}^4 t^3+4 N \text{s1} \text{s2}^3 t+2 N \text{s1} \text{s2}^2 t^2-\frac{4}{3} N \text{s1} \text{s2} t^3$

$PL[s1,s2,t,o]'=...$

General Number of Symm.

$PL[s1,s2,0,0]=\frac{17 \text{s1}^3 \text{s2}^6 \text{Ns}^9}{4320}+\frac{17 \text{s1}^6 \text{s2}^3 \text{Ns}^9}{4320}+\frac{11}{360} \text{s1}^3 \text{s2}^6 \text{Ns}^8+\frac{1}{18} \text{s1}^4 \text{s2}^4 \text{Ns}^8+\frac{11}{360} \text{s1}^6 \text{s2}^3 \text{Ns}^8-\frac{31}{144} \text{s1}^3 \text{s2}^6 \text{Ns}^7+\frac{1}{15} \text{s1}^2 \text{s2}^5 \text{Ns}^7-\frac{1}{12} \text{s1}^4 \text{s2}^4 \text{Ns}^7-\frac{31}{144} \text{s1}^6 \text{s2}^3 \text{Ns}^7+\frac{1}{15} \text{s1}^5 \text{s2}^2 \text{Ns}^7+\frac{\text{s1}^6 \text{Ns}^6}{120}+\frac{17}{72} \text{s1}^3 \text{s2}^6 \text{Ns}^6+\frac{\text{s2}^6 \text{Ns}^6}{120}-\frac{11}{60} \text{s1}^2 \text{s2}^5 \text{Ns}^6-\frac{5}{18} \text{s1}^4 \text{s2}^4 \text{Ns}^6+\frac{17}{72} \text{s1}^6 \text{s2}^3 \text{Ns}^6+\frac{5}{36} \text{s1}^3 \text{s2}^3 \text{Ns}^6-\frac{11}{60} \text{s1}^5 \text{s2}^2 \text{Ns}^6+\frac{39}{160} \text{s1}^3 \text{s2}^6 \text{Ns}^5+\frac{1}{12} \text{s1}^2 \text{s2}^5 \text{Ns}^5+\frac{7}{12} \text{s1}^4 \text{s2}^4 \text{Ns}^5+\frac{1}{12} \text{s1} \text{s2}^4 \text{Ns}^5+\frac{39}{160} \text{s1}^6 \text{s2}^3 \text{Ns}^5-\frac{1}{6} \text{s1}^3 \text{s2}^3 \text{Ns}^5+\frac{1}{12} \text{s1}^5 \text{s2}^2 \text{Ns}^5+\frac{1}{12} \text{s1}^4 \text{s2} \text{Ns}^5-\frac{\text{s1}^6 \text{Ns}^5}{24}-\frac{\text{s2}^6 \text{Ns}^5}{24}+\frac{\text{s1}^6 \text{Ns}^4}{8}-\frac{29}{90} \text{s1}^3 \text{s2}^6 \text{Ns}^4+\frac{\text{s2}^6 \text{Ns}^4}{8}+\frac{1}{12} \text{s1}^2 \text{s2}^5 \text{Ns}^4-\frac{5}{18} \text{s1}^4 \text{s2}^4 \text{Ns}^4-\frac{29}{90} \text{s1}^6 \text{s2}^3 \text{Ns}^4+\frac{5}{36} \text{s1}^3 \text{s2}^3 \text{Ns}^4+\frac{1}{12} \text{s1}^5 \text{s2}^2 \text{Ns}^4+\frac{1}{4} \text{s1}^2 \text{s2}^2 \text{Ns}^4-\frac{7}{216} \text{s1}^3 \text{s2}^6 \text{Ns}^3-\frac{3}{20} \text{s1}^2 \text{s2}^5 \text{Ns}^3-\frac{1}{12} \text{s1} \text{s2}^4 \text{Ns}^3+\frac{\text{s1}^3 \text{Ns}^3}{6}-\frac{7}{216} \text{s1}^6 \text{s2}^3 \text{Ns}^3-\frac{1}{3} \text{s1}^3 \text{s2}^3 \text{Ns}^3+\frac{\text{s2}^3 \text{Ns}^3}{6}-\frac{3}{20} \text{s1}^5 \text{s2}^2 \text{Ns}^3+\frac{1}{2} \text{s1}^2 \text{s2}^2 \text{Ns}^3-\frac{1}{12} \text{s1}^4 \text{s2} \text{Ns}^3-\frac{7 \text{s1}^6 \text{Ns}^3}{24}-\frac{7 \text{s2}^6 \text{Ns}^3}{24}+\frac{11 \text{s1}^6 \text{Ns}^2}{30}+\frac{1}{18} \text{s1}^3 \text{s2}^6 \text{Ns}^2+\frac{11 \text{s2}^6 \text{Ns}^2}{30}+\frac{1}{10} \text{s1}^2 \text{s2}^5 \text{Ns}^2+\frac{\text{s1}^3 \text{Ns}^2}{2}+\frac{1}{18} \text{s1}^6 \text{s2}^3 \text{Ns}^2+\frac{2}{9} \text{s1}^3 \text{s2}^3 \text{Ns}^2+\frac{\text{s2}^3 \text{Ns}^2}{2}+\frac{1}{10} \text{s1}^5 \text{s2}^2 \text{Ns}^2+\frac{1}{4} \text{s1}^2 \text{s2}^2 \text{Ns}^2+\text{s1} \text{s2} \text{Ns}^2+\frac{\text{s1}^3 \text{Ns}}{3}+\frac{\text{s2}^3 \text{Ns}}{3}-\frac{\text{s1}^6 \text{Ns}}{6}-\frac{\text{s2}^6 \text{Ns}}{6}$

$PL[s1,0,t,0]=\frac{5}{36} \text{Nf}^3 \text{Ns}^3 \text{s1}^3 t^3-\frac{1}{12} \text{Nf}^3 \text{Ns}^2 \text{s1}^3 t^3-\frac{1}{18} \text{Nf}^3 \text{Ns} \text{s1}^3 t^3+\frac{\text{Nf}^3 t^3}{6}+\frac{1}{15} \text{Nf}^2 \text{Ns}^5 \text{s1}^5 t^2-\frac{1}{6} \text{Nf}^2 \text{Ns}^4 \text{s1}^5 t^2+\frac{1}{12} \text{Nf}^2 \text{Ns}^3 \text{s1}^5 t^2+\frac{1}{12} \text{Nf}^2 \text{Ns}^3 \text{s1}^3 t^3-\frac{1}{12} \text{Nf}^2 \text{Ns}^2 \text{s1}^5 t^2-\frac{1}{4} \text{Nf}^2 \text{Ns}^2 \text{s1}^3 t^3+\frac{1}{4} \text{Nf}^2 \text{Ns}^2 \text{s1}^2 t^2+\frac{1}{10} \text{Nf}^2 \text{Ns} \text{s1}^5 t^2+\frac{1}{6} \text{Nf}^2 \text{Ns} \text{s1}^3 t^3+\frac{1}{4} \text{Nf}^2 \text{Ns} \text{s1}^2 t^2-\frac{\text{Nf}^2 t^3}{2}-\frac{1}{60} \text{Nf} \text{Ns}^5 \text{s1}^5 t^2+\frac{1}{8} \text{Nf} \text{Ns}^4 \text{s1}^4 t+\frac{1}{6} \text{Nf} \text{Ns}^3 \text{s1}^5 t^2-\frac{1}{4} \text{Nf} \text{Ns}^3 \text{s1}^4 t-\frac{1}{18} \text{Nf} \text{Ns}^3 \text{s1}^3 t^3-\frac{1}{4} \text{Nf} \text{Ns}^2 \text{s1}^5 t^2-\frac{1}{8} \text{Nf} \text{Ns}^2 \text{s1}^4 t-\frac{1}{6} \text{Nf} \text{Ns}^2 \text{s1}^3 t^3+\frac{1}{4} \text{Nf} \text{Ns}^2 \text{s1}^2 t^2+\frac{1}{10} \text{Nf} \text{Ns} \text{s1}^5 t^2+\frac{1}{4} \text{Nf} \text{Ns} \text{s1}^4 t+\frac{2}{9} \text{Nf} \text{Ns} \text{s1}^3 t^3+\frac{1}{4} \text{Nf} \text{Ns} \text{s1}^2 t^2+\frac{\text{Nf} t^3}{3}+\frac{\text{Ns}^6 \text{s1}^6}{120}-\frac{\text{Ns}^5 \text{s1}^6}{24}+\frac{\text{Ns}^4 \text{s1}^6}{8}-\frac{7 \text{Ns}^3 \text{s1}^6}{24}+\frac{\text{Ns}^3 \text{s1}^3}{6}+\frac{11 \text{Ns}^2 \text{s1}^6}{30}+\frac{\text{Ns}^2 \text{s1}^3}{2}-\frac{\text{Ns} \text{s1}^6}{6}+\frac{\text{Ns} \text{s1}^3}{3}$

$PL[s,0,t,o]=\frac{1}{18} \text{Nf}^2 \text{Ns}^6 o t s^6-\frac{1}{6} \text{Nf}^2 \text{Ns}^5 o t s^6+\frac{1}{18} \text{Nf}^2 \text{Ns}^4 o t s^6+\frac{1}{6} \text{Nf}^2 \text{Ns}^3 o t s^6-\frac{1}{9} \text{Nf}^2 \text{Ns}^2 o t s^6+\frac{11}{240} \text{Nf}^3 \text{Ns}^5 o^2 t s^5-\frac{1}{240} \text{Nf}^2 \text{Ns}^5 o^2 t s^5-\frac{1}{24} \text{Nf}^3 \text{Ns}^4 o^2 t s^5+\frac{1}{8} \text{Nf}^2 \text{Ns}^4 o^2 t s^5+\frac{5}{48} \text{Nf}^3 \text{Ns}^3 o^2 t s^5-\frac{7}{48} \text{Nf}^2 \text{Ns}^3 o^2 t s^5-\frac{5}{24} \text{Nf}^3 \text{Ns}^2 o^2 t s^5+\frac{1}{8} \text{Nf}^2 \text{Ns}^2 o^2 t s^5+\frac{1}{10} \text{Nf}^3 \text{Ns} o^2 t s^5-\frac{1}{10} \text{Nf}^2 \text{Ns} o^2 t s^5+\frac{3}{16} \text{Nf}^3 \text{Ns}^4 o t^2 s^4+\frac{5}{48} \text{Nf}^2 \text{Ns}^4 o t^2 s^4-\frac{1}{8} \text{Nf}^3 \text{Ns}^3 o t^2 s^4-\frac{1}{8} \text{Nf}^2 \text{Ns}^3 o t^2 s^4+\frac{1}{16} \text{Nf}^3 \text{Ns}^2 o t^2 s^4+\frac{7}{48} \text{Nf}^2 \text{Ns}^2 o t^2 s^4-\frac{1}{8} \text{Nf}^3 \text{Ns} o t^2 s^4-\frac{1}{8} \text{Nf}^2 \text{Ns} o t^2 s^4+\frac{29}{144} \text{Nf}^4 \text{Ns}^2 o^3 t s^4-\frac{1}{16} \text{Nf}^3 \text{Ns}^2 o^3 t s^4-\frac{5}{36} \text{Nf}^2 \text{Ns}^2 o^3 t s^4-\frac{1}{8} \text{Nf}^4 \text{Ns} o^3 t s^4+\frac{1}{8} \text{Nf}^3 \text{Ns} o^3 t s^4+\frac{1}{12} \text{Nf}^4 \text{Ns}^3 o^2 t^2 s^3-\frac{1}{12} \text{Nf}^2 \text{Ns}^3 o^2 t^2 s^3-\frac{1}{4} \text{Nf}^4 \text{Ns}^2 o^2 t^2 s^3+\frac{1}{4} \text{Nf}^2 \text{Ns}^2 o^2 t^2 s^3+\frac{1}{6} \text{Nf}^4 \text{Ns} o^2 t^2 s^3-\frac{1}{6} \text{Nf}^2 \text{Ns} o^2 t^2 s^3+\frac{1}{3} \text{Nf}^2 \text{Ns}^3 o t s^3-\frac{1}{3} \text{Nf}^2 \text{Ns} o t s^3+\frac{1}{6} \text{Nf}^4 \text{Ns}^2 o t^3 s^2-\frac{1}{6} \text{Nf}^2 \text{Ns}^2 o t^3 s^2-\frac{1}{6} \text{Nf}^4 \text{Ns} o t^3 s^2+\frac{1}{6} \text{Nf}^2 \text{Ns} o t^3 s^2+\frac{1}{2} \text{Nf}^3 \text{Ns}^2 o^2 t s^2-\frac{1}{2} \text{Nf}^2 \text{Ns} o^2 t s^2+\frac{1}{2} \text{Nf}^3 \text{Ns} o t^2 s-\frac{1}{2} \text{Nf}^2 \text{Ns} o t^2 s$

$PL[s1,s2,t,o]=...$

For $Sp(2)$

Same, because $Sp(n)$ adjoint. rep. is symmetric tensor.

For $SO(4)$

$PL[s_1,s_2,0]=2 \text{s1}^4 \text{s2}^4+\text{s1}^4 \text{s2}^3+2 \text{s1}^4 \text{s2}^2+\text{s1}^4+\text{s1}^3 \text{s2}^4+2 \text{s1}^3 \text{s2}^3+\text{s1}^3 \text{s2}^2+\text{s1}^3 \text{s2}+\text{s1}^3+2 \text{s1}^2 \text{s2}^4+\text{s1}^2 \text{s2}^3+3 \text{s1}^2 \text{s2}^2+\text{s1}^2 \text{s2}+\text{s1}^2+\text{s1} \text{s2}^3+\text{s1} \text{s2}^2+\text{s1} \text{s2}+\text{s1}+\text{s2}^4+\text{s2}^3+\text{s2}^2+\text{s2}$

$PL[0,0,t]=\frac{N^4 t^4}{24}-\frac{N^3 t^4}{4}+\frac{11 N^2 t^4}{24}+\frac{N^2 t^2}{2}-\frac{N t^4}{4}+\frac{N t^2}{2}$

$PL[s1,0,t]'=\frac{1}{8} N^4 \text{s1}^4 t^4+\frac{1}{6} N^4 \text{s1}^3 t^4+\frac{5}{24} N^4 \text{s1}^2 t^4+\frac{1}{8} N^4 \text{s1} t^4+\frac{1}{4} N^3 \text{s1}^4 t^4-\frac{1}{4} N^3 \text{s1}^2 t^4-\frac{1}{4} N^3 \text{s1} t^4-\frac{1}{8} N^2 \text{s1}^4 t^4-\frac{1}{6} N^2 \text{s1}^3 t^4+\frac{1}{2} N^2 \text{s1}^3 t^2-\frac{5}{24} N^2 \text{s1}^2 t^4+\frac{1}{2} N^2 \text{s1}^2 t^2-\frac{1}{8} N^2 \text{s1} t^4+\frac{1}{2} N^2 \text{s1} t^2-\frac{1}{4} N \text{s1}^4 t^4+\frac{1}{2} N \text{s1}^3 t^2+\frac{1}{4} N \text{s1}^2 t^4+\frac{1}{2} N \text{s1}^2 t^2+\frac{1}{4} N \text{s1} t^4+\frac{1}{2} N \text{s1} t^2$

$PL[s1,s2,t]'=\frac{3}{8} N^4 \text{s1}^2 \text{s2} t^4+\frac{1}{3} N^4 \text{s1} \text{s2} t^4+\frac{1}{4} N^3 \text{s1}^2 \text{s2} t^4+\frac{1}{2} N^2 \text{s1}^4 \text{s2}^4 t^2+7 N^2 \text{s1}^4 \text{s2}^3 t^2+6 N^2 \text{s1}^4 \text{s2}^2 t^2+\frac{5}{2} N^2 \text{s1}^4 \text{s2} t^2+7 N^2 \text{s1}^3 \text{s2}^4 t^2+9 N^2 \text{s1}^3 \text{s2}^3 t^2+\frac{13}{2} N^2 \text{s1}^3 \text{s2}^2 t^2+3 N^2 \text{s1}^3 \text{s2} t^2+6 N^2 \text{s1}^2 \text{s2}^4 t^2+\frac{13}{2} N^2 \text{s1}^2 \text{s2}^3 t^2+\frac{9}{2} N^2 \text{s1}^2 \text{s2}^2 t^2-\frac{3}{8} N^2 \text{s1}^2 \text{s2} t^4+\frac{5}{2} N^2 \text{s1}^2 \text{s2} t^2+\frac{5}{2} N^2 \text{s1} \text{s2}^4 t^2+3 N^2 \text{s1} \text{s2}^3 t^2+\frac{5}{2} N^2 \text{s1} \text{s2}^2 t^2-\frac{1}{3} N^2 \text{s1} \text{s2} t^4+\frac{3}{2} N^2 \text{s1} \text{s2} t^2+\frac{1}{2} N \text{s1}^4 \text{s2}^4 t^2+N \text{s1}^4 \text{s2}^3 t^2+\frac{1}{2} N \text{s1}^4 \text{s2} t^2+N \text{s1}^3 \text{s2}^4 t^2-N \text{s1}^3 \text{s2}^3 t^2+\frac{1}{2} N \text{s1}^3 \text{s2}^2 t^2-N \text{s1}^3 \text{s2} t^2+\frac{1}{2} N \text{s1}^2 \text{s2}^3 t^2+\frac{1}{2} N \text{s1}^2 \text{s2}^2 t^2-\frac{1}{4} N \text{s1}^2 \text{s2} t^4+\frac{1}{2} N \text{s1}^2 \text{s2} t^2+\frac{1}{2} N \text{s1} \text{s2}^4 t^2-N \text{s1} \text{s2}^3 t^2+\frac{1}{2} N \text{s1} \text{s2}^2 t^2-\frac{1}{2} N \text{s1} \text{s2} t^2$

For $SO(5)$

$PL[s1,s2,0]=2 \text{s1}^6 \text{s2}^6+2 \text{s1}^6 \text{s2}^5+2 \text{s1}^6 \text{s2}^4+\text{s1}^6 \text{s2}^3+\text{s1}^6 \text{s2}^2+2 \text{s1}^5 \text{s2}^6+2 \text{s1}^5 \text{s2}^5+2 \text{s1}^5 \text{s2}^4+\text{s1}^5 \text{s2}^3+\text{s1}^5 \text{s2}^2+\text{s1}^5+2 \text{s1}^4 \text{s2}^6+2 \text{s1}^4 \text{s2}^5+3 \text{s1}^4 \text{s2}^4+2 \text{s1}^4 \text{s2}^3+2 \text{s1}^4 \text{s2}^2+\text{s1}^4 \text{s2}+\text{s1}^4+\text{s1}^3 \text{s2}^6+\text{s1}^3 \text{s2}^5+2 \text{s1}^3 \text{s2}^4+2 \text{s1}^3 \text{s2}^3+2 \text{s1}^3 \text{s2}^2+\text{s1}^3 \text{s2}+\text{s1}^3+\text{s1}^2 \text{s2}^6+\text{s1}^2 \text{s2}^5+2 \text{s1}^2 \text{s2}^4+2 \text{s1}^2 \text{s2}^3+2 \text{s1}^2 \text{s2}^2+\text{s1}^2 \text{s2}+\text{s1}^2+\text{s1} \text{s2}^4+\text{s1} \text{s2}^3+\text{s1} \text{s2}^2+\text{s1} \text{s2}+\text{s1}+\text{s2}^5+\text{s2}^4+\text{s2}^3+\text{s2}^2+\text{s2}$

$PL[0,0,t]=\frac{N^5 t^5}{120}-\frac{N^4 t^5}{12}+\frac{7 N^3 t^5}{24}-\frac{5 N^2 t^5}{12}+\frac{N^2 t^2}{2}+\frac{N t^5}{5}+\frac{N t^2}{2}$

$PL[s,0,t]=\frac{1}{6} N^5 s^6 t^5+\frac{11}{60} N^5 s^5 t^5+\frac{1}{6} N^5 s^4 t^5+\frac{1}{8} N^5 s^3 t^5+\frac{3}{40} N^5 s^2 t^5+\frac{1}{30} N^5 s t^5+\frac{1}{6} N^4 s^6 t^5-\frac{1}{12} N^4 s^6 t^4-\frac{1}{12} N^4 s^5 t^4-\frac{1}{6} N^4 s^4 t^5-\frac{1}{12} N^4 s^4 t^4-\frac{1}{4} N^4 s^3 t^5-\frac{1}{4} N^4 s^2 t^5-\frac{1}{6} N^4 s t^5-\frac{1}{6} N^3 s^6 t^5-\frac{7}{12} N^3 s^5 t^5-\frac{1}{6} N^3 s^4 t^5-\frac{1}{8} N^3 s^3 t^5+\frac{1}{8} N^3 s^2 t^5+\frac{1}{6} N^3 s t^5-\frac{1}{6} N^2 s^6 t^5+\frac{1}{12} N^2 s^6 t^4+\frac{1}{12} N^2 s^5 t^4+\frac{1}{6} N^2 s^4 t^5+\frac{1}{12} N^2 s^4 t^4+\frac{1}{2} N^2 s^4 t^2+\frac{1}{4} N^2 s^3 t^5+\frac{1}{2} N^2 s^3 t^2+\frac{1}{4} N^2 s^2 t^5+\frac{1}{2} N^2 s^2 t^2+\frac{1}{6} N^2 s t^5+\frac{1}{2} N^2 s t^2+\frac{2}{5} N s^5 t^5+\frac{1}{2} N s^4 t^2+\frac{1}{2} N s^3 t^2-\frac{1}{5} N s^2 t^5+\frac{1}{2} N s^2 t^2-\frac{1}{5} N s t^5+\frac{1}{2} N s t^2$

$PL[s1,s2,t]=\frac{67}{6} N^3 \text{s2}^3 t^3 \text{s1}^6+\frac{15}{2} N^2 \text{s2}^3 t^3 \text{s1}^6+\frac{4}{3} N \text{s2}^3 t^3 \text{s1}^6+\frac{20}{3} N^3 \text{s2}^2 t^3 \text{s1}^6+2 N^2 \text{s2}^2 t^3 \text{s1}^6+\frac{1}{3} N \text{s2}^2 t^3 \text{s1}^6+\frac{11}{6} N^3 \text{s2} t^3 \text{s1}^6+\frac{1}{2} N^2 \text{s2} t^3 \text{s1}^6-\frac{1}{3} N \text{s2} t^3 \text{s1}^6+\frac{23}{2} N^2 \text{s2}^5 t^2 \text{s1}^6-\frac{1}{2} N \text{s2}^5 t^2 \text{s1}^6+17 N^2 \text{s2}^4 t^2 \text{s1}^6-2 N \text{s2}^4 t^2 \text{s1}^6+\frac{23}{2} N^2 \text{s2}^3 t^2 \text{s1}^6-\frac{3}{2} N \text{s2}^3 t^2 \text{s1}^6+\frac{9}{2} N^2 \text{s2}^2 t^2 \text{s1}^6-\frac{1}{2} N \text{s2}^2 t^2 \text{s1}^6+N^2 \text{s2} t^2 \text{s1}^6+32 N \text{s2}^6 t \text{s1}^6+31 N \text{s2}^5 t \text{s1}^6+21 N \text{s2}^4 t \text{s1}^6+9 N \text{s2}^3 t \text{s1}^6+3 N \text{s2}^2 t \text{s1}^6+15 N^3 \text{s2}^4 t^3 \text{s1}^5+11 N^2 \text{s2}^4 t^3 \text{s1}^5+2 N \text{s2}^4 t^3 \text{s1}^5+\frac{43}{3} N^3 \text{s2}^3 t^3 \text{s1}^5+4 N^2 \text{s2}^3 t^3 \text{s1}^5-\frac{1}{3} N \text{s2}^3 t^3 \text{s1}^5+\frac{41}{6} N^3 \text{s2}^2 t^3 \text{s1}^5+\frac{1}{2} N^2 \text{s2}^2 t^3 \text{s1}^5-\frac{1}{3} N \text{s2}^2 t^3 \text{s1}^5+\frac{11}{6} N^3 \text{s2} t^3 \text{s1}^5-\frac{1}{2} N^2 \text{s2} t^3 \text{s1}^5-\frac{1}{3} N \text{s2} t^3 \text{s1}^5+\frac{23}{2} N^2 \text{s2}^6 t^2 \text{s1}^5-\frac{1}{2} N \text{s2}^6 t^2 \text{s1}^5+22 N^2 \text{s2}^5 t^2 \text{s1}^5-3 N \text{s2}^5 t^2 \text{s1}^5+\frac{39}{2} N^2 \text{s2}^4 t^2 \text{s1}^5-\frac{5}{2} N \text{s2}^4 t^2 \text{s1}^5+\frac{25}{2} N^2 \text{s2}^3 t^2 \text{s1}^5-\frac{5}{2} N \text{s2}^3 t^2 \text{s1}^5+\frac{11}{2} N^2 \text{s2}^2 t^2 \text{s1}^5-\frac{1}{2} N \text{s2}^2 t^2 \text{s1}^5+\frac{3}{2} N^2 \text{s2} t^2 \text{s1}^5-\frac{1}{2} N \text{s2} t^2 \text{s1}^5+31 N \text{s2}^6 t \text{s1}^5+25 N \text{s2}^5 t \text{s1}^5+16 N \text{s2}^4 t \text{s1}^5+7 N \text{s2}^3 t \text{s1}^5+2 N \text{s2}^2 t \text{s1}^5+\frac{4}{15} N^5 \text{s2} t^5 \text{s1}^4+\frac{5}{6} N^4 \text{s2} t^5 \text{s1}^4-\frac{2}{3} N^3 \text{s2} t^5 \text{s1}^4-\frac{5}{6} N^2 \text{s2} t^5 \text{s1}^4+\frac{2}{5} N \text{s2} t^5 \text{s1}^4+\frac{53}{3} N^3 \text{s2}^4 t^3 \text{s1}^4+6 N^2 \text{s2}^4 t^3 \text{s1}^4+\frac{1}{3} N \text{s2}^4 t^3 \text{s1}^4+\frac{73}{6} N^3 \text{s2}^3 t^3 \text{s1}^4+\frac{3}{2} N^2 \text{s2}^3 t^3 \text{s1}^4-\frac{2}{3} N \text{s2}^3 t^3 \text{s1}^4+\frac{16}{3} N^3 \text{s2}^2 t^3 \text{s1}^4-\frac{1}{3} N \text{s2}^2 t^3 \text{s1}^4+\frac{3}{2} N^3 \text{s2} t^3 \text{s1}^4-\frac{1}{2} N^2 \text{s2} t^3 \text{s1}^4+17 N^2 \text{s2}^6 t^2 \text{s1}^4-2 N \text{s2}^6 t^2 \text{s1}^4+\frac{39}{2} N^2 \text{s2}^5 t^2 \text{s1}^4-\frac{5}{2} N \text{s2}^5 t^2 \text{s1}^4+\frac{33}{2} N^2 \text{s2}^4 t^2 \text{s1}^4-\frac{3}{2} N \text{s2}^4 t^2 \text{s1}^4+11 N^2 \text{s2}^3 t^2 \text{s1}^4-N \text{s2}^3 t^2 \text{s1}^4+\frac{11}{2} N^2 \text{s2}^2 t^2 \text{s1}^4+\frac{1}{2} N \text{s2}^2 t^2 \text{s1}^4+2 N^2 \text{s2} t^2 \text{s1}^4+21 N \text{s2}^6 t \text{s1}^4+16 N \text{s2}^5 t \text{s1}^4+11 N \text{s2}^4 t \text{s1}^4+5 N \text{s2}^3 t \text{s1}^4+2 N \text{s2}^2 t \text{s1}^4+\frac{13}{120} N^5 \text{s2}^2 t^5 \text{s1}^3+\frac{25}{12} N^4 \text{s2}^2 t^5 \text{s1}^3-\frac{41}{24} N^3 \text{s2}^2 t^5 \text{s1}^3-\frac{13}{12} N^2 \text{s2}^2 t^5 \text{s1}^3+\frac{3}{5} N \text{s2}^2 t^5 \text{s1}^3+\frac{7}{20} N^5 \text{s2} t^5 \text{s1}^3+\frac{1}{6} N^4 \text{s2} t^5 \text{s1}^3-\frac{3}{4} N^3 \text{s2} t^5 \text{s1}^3-\frac{1}{6} N^2 \text{s2} t^5 \text{s1}^3+\frac{2}{5} N \text{s2} t^5 \text{s1}^3+\frac{73}{6} N^3 \text{s2}^4 t^3 \text{s1}^3+\frac{3}{2} N^2 \text{s2}^4 t^3 \text{s1}^3-\frac{2}{3} N \text{s2}^4 t^3 \text{s1}^3+\frac{23}{3} N^3 \text{s2}^3 t^3 \text{s1}^3-N^2 \text{s2}^3 t^3 \text{s1}^3-\frac{2}{3} N \text{s2}^3 t^3 \text{s1}^3+\frac{10}{3} N^3 \text{s2}^2 t^3 \text{s1}^3-N^2 \text{s2}^2 t^3 \text{s1}^3-\frac{1}{3} N \text{s2}^2 t^3 \text{s1}^3+N^3 \text{s2} t^3 \text{s1}^3-N^2 \text{s2} t^3 \text{s1}^3+\frac{23}{2} N^2 \text{s2}^6 t^2 \text{s1}^3-\frac{3}{2} N \text{s2}^6 t^2 \text{s1}^3+\frac{25}{2} N^2 \text{s2}^5 t^2 \text{s1}^3-\frac{5}{2} N \text{s2}^5 t^2 \text{s1}^3+11 N^2 \text{s2}^4 t^2 \text{s1}^3-N \text{s2}^4 t^2 \text{s1}^3+8 N^2 \text{s2}^3 t^2 \text{s1}^3-N \text{s2}^3 t^2 \text{s1}^3+\frac{9}{2} N^2 \text{s2}^2 t^2 \text{s1}^3+\frac{1}{2} N \text{s2}^2 t^2 \text{s1}^3+2 N^2 \text{s2} t^2 \text{s1}^3+9 N \text{s2}^6 t \text{s1}^3+7 N \text{s2}^5 t \text{s1}^3+5 N \text{s2}^4 t \text{s1}^3+2 N \text{s2}^3 t \text{s1}^3+N \text{s2}^2 t \text{s1}^3+\frac{13}{120} N^5 \text{s2}^3 t^5 \text{s1}^2+\frac{25}{12} N^4 \text{s2}^3 t^5 \text{s1}^2-\frac{41}{24} N^3 \text{s2}^3 t^5 \text{s1}^2-\frac{13}{12} N^2 \text{s2}^3 t^5 \text{s1}^2+\frac{3}{5} N \text{s2}^3 t^5 \text{s1}^2+\frac{17}{40} N^5 \text{s2}^2 t^5 \text{s1}^2+\frac{1}{4} N^4 \text{s2}^2 t^5 \text{s1}^2-\frac{9}{8} N^3 \text{s2}^2 t^5 \text{s1}^2+\frac{1}{4} N^2 \text{s2}^2 t^5 \text{s1}^2+\frac{1}{5} N \text{s2}^2 t^5 \text{s1}^2+\frac{31}{120} N^5 \text{s2} t^5 \text{s1}^2-\frac{1}{4} N^4 \text{s2} t^5 \text{s1}^2-\frac{11}{24} N^3 \text{s2} t^5 \text{s1}^2+\frac{1}{4} N^2 \text{s2} t^5 \text{s1}^2+\frac{1}{5} N \text{s2} t^5 \text{s1}^2+\frac{16}{3} N^3 \text{s2}^4 t^3 \text{s1}^2-\frac{1}{3} N \text{s2}^4 t^3 \text{s1}^2+\frac{10}{3} N^3 \text{s2}^3 t^3 \text{s1}^2-N^2 \text{s2}^3 t^3 \text{s1}^2-\frac{1}{3} N \text{s2}^3 t^3 \text{s1}^2+\frac{3}{2} N^3 \text{s2}^2 t^3 \text{s1}^2-\frac{1}{2} N^2 \text{s2}^2 t^3 \text{s1}^2+\frac{1}{2} N^3 \text{s2} t^3 \text{s1}^2-\frac{1}{2} N^2 \text{s2} t^3 \text{s1}^2+\frac{9}{2} N^2 \text{s2}^6 t^2 \text{s1}^2-\frac{1}{2} N \text{s2}^6 t^2 \text{s1}^2+\frac{11}{2} N^2 \text{s2}^5 t^2 \text{s1}^2-\frac{1}{2} N \text{s2}^5 t^2 \text{s1}^2+\frac{11}{2} N^2 \text{s2}^4 t^2 \text{s1}^2+\frac{1}{2} N \text{s2}^4 t^2 \text{s1}^2+\frac{9}{2} N^2 \text{s2}^3 t^2 \text{s1}^2+\frac{1}{2} N \text{s2}^3 t^2 \text{s1}^2+3 N^2 \text{s2}^2 t^2 \text{s1}^2+N \text{s2}^2 t^2 \text{s1}^2+\frac{3}{2} N^2 \text{s2} t^2 \text{s1}^2+\frac{1}{2} N \text{s2} t^2 \text{s1}^2+3 N \text{s2}^6 t \text{s1}^2+2 N \text{s2}^5 t \text{s1}^2+2 N \text{s2}^4 t \text{s1}^2+N \text{s2}^3 t \text{s1}^2+N \text{s2}^2 t \text{s1}^2+\frac{7}{20} N^5 \text{s2}^3 t^5 \text{s1}+\frac{1}{6} N^4 \text{s2}^3 t^5 \text{s1}-\frac{3}{4} N^3 \text{s2}^3 t^5 \text{s1}-\frac{1}{6} N^2 \text{s2}^3 t^5 \text{s1}+\frac{2}{5} N \text{s2}^3 t^5 \text{s1}+\frac{1}{8} N^5 \text{s2} t^5 \text{s1}-\frac{1}{4} N^4 \text{s2} t^5 \text{s1}-\frac{1}{8} N^3 \text{s2} t^5 \text{s1}+\frac{1}{4} N^2 \text{s2} t^5 \text{s1}+\frac{3}{2} N^3 \text{s2}^4 t^3 \text{s1}-\frac{1}{2} N^2 \text{s2}^4 t^3 \text{s1}+N^3 \text{s2}^3 t^3 \text{s1}-N^2 \text{s2}^3 t^3 \text{s1}+\frac{1}{2} N^3 \text{s2}^2 t^3 \text{s1}-\frac{1}{2} N^2 \text{s2}^2 t^3 \text{s1}+\frac{1}{6} N^3 \text{s2} t^3 \text{s1}-\frac{1}{2} N^2 \text{s2} t^3 \text{s1}+\frac{1}{3} N \text{s2} t^3 \text{s1}+N^2 \text{s2}^6 t^2 \text{s1}+\frac{3}{2} N^2 \text{s2}^5 t^2 \text{s1}-\frac{1}{2} N \text{s2}^5 t^2 \text{s1}+2 N^2 \text{s2}^4 t^2 \text{s1}+2 N^2 \text{s2}^3 t^2 \text{s1}+\frac{3}{2} N^2 \text{s2}^2 t^2 \text{s1}+\frac{1}{2} N \text{s2}^2 t^2 \text{s1}+N^2 \text{s2} t^2 \text{s1}$

For $SU(4)$ ,

Carten matrix is $\begin{pmatrix}2&-1&0\\ -1& 2& -1\\ 0& -1& 2\end{pmatrix}$

Weight diagram of symmetric 2 tensor is

$\chi_{symm}=z_1^2+z_2+z_1^3z_3/z_2^2+z_1^2z_2/z_3^2+z_1^4z_3^2/z_2^4+z_1z_2^3z_3+1/(z_1^2z_2^2)+z_2^2z_3^2+z_1^2z_2^2/z_3^4+z_1^3/(z_2z_3)$

$\chi_{cnsy}=z_3^2+z_2+z_3^3z_1/z_2^2+z_3^2z_2/z_1^2+z_3^4z_1^2/z_2^4+z_3z_2^3z_1+1/(z_3^2z_2^2)+z_2^2z_1^2+z_3^2z_2^2/z_1^4+z_3^3/(z_2z_1)$

Maximum rank

rank 4 is possible.

For $SU(5)$ ,

But strange number appear..

For $F4$ ,

[Georgi] 27.5 Anomalies

There is a peculiar constrain on unified theories that follows from the structure of quantum field theory, the mathematical language in which all these theoreis are formulated. The constraint is that if the creation operators for all the right-handed spin 1/2 particles transform according to a representation generated by matrices $T^R_a$ , then $T^R_a$ must satisfy $\begin{align}\mbox{Tr}\left( \{T^R_a,T^R_b\}T^R_c\right)=0.\end{align}$

You can show that this symmetric trace of three generators vanishes for all simple Lie algebras except $SU(N)$ for $N\ge 3$ (and $SO(6)$ which is equivalent to $SU(4)$ ). In $SU(N)$ , suppose that $T^D_a$ generate the representation $D$ of $SU(N)$ . Then define the invariant tensor $d^{abc}$ as follows: $\begin{align}\mbox{Tr}\left( \{T^{D^1}_a,T^{D^1}_b\}T^{D^1}_c\right)\equiv d^{abc}\end{align}$ for the defining representation $D^1$ . Then, for any representation, you can show that $\begin{align}\mbox{Tr}\left( \{T^D_a,T^D_b\}T^D_c\right)=A(D)d^{abc},\end{align}$ where $A(D)$ is an integer, which is calle the anomaly of the representation $D$ . Thus (23) is the statement that the creation operators for the right handed particles transform according to an anomaly free representation of the unifying group.

You can easily derive the following preperties of $A(D)$ $\begin{align}A(\bar{D})=-A(D)\\ A(D_1\oplus D_2)=A(D_1)+A(D_2)\\ A(D_1\otimes D_2)=\mbox{dim}(D_1)A(D_2)+\mbox{dim}(D_2)A(D_1)\end{align}$