SymmetricPowers

In [3]:

from IPython.display import *

In [4]:

var('a:z')

V=Matrix(3,1,[u,v,w])
A=Matrix(3,3,[s,1,1,1,0,0,0,1,0])
B=A*V
A,B

Out[4]:

$$\left ( \left[\begin{matrix}s & 1 & 1\\1 & 0 & 0\\0 & 1 & 0\end{matrix}\right], \quad \left[\begin{matrix}s u + v + w\\u\\v\end{matrix}\right]\right )$$

In [5]:

N=3
mm=Poly((u+v+w)**N).monoms()
mm

Out[5]:

$$\left [ \left ( 3, \quad 0, \quad 0\right ), \quad \left ( 2, \quad 1, \quad 0\right ), \quad \left ( 2, \quad 0, \quad 1\right ), \quad \left ( 1, \quad 2, \quad 0\right ), \quad \left ( 1, \quad 1, \quad 1\right ), \quad \left ( 1, \quad 0, \quad 2\right ), \quad \left ( 0, \quad 3, \quad 0\right ), \quad \left ( 0, \quad 2, \quad 1\right ), \quad \left ( 0, \quad 1, \quad 2\right ), \quad \left ( 0, \quad 0, \quad 3\right )\right ]$$

In [6]:

dd=len(mm)
xx=eye(dd)
X=[]

for i in range(dd):
    F=x**mm[i][0]*y**mm[i][1]*z**mm[i][2]
    GG=F.subs(x,B[0]).subs(y,B[1]).subs(z,B[2]);FF=expand(GG)
    X.append([FF.coeff(u**mm[i][0]*v**mm[i][1]*w**mm[i][2]) for i in range(dd)])
 
#display(X)

In [7]:

MX=Matrix(1,dd,X[0])
for i in range(1,dd):
    XM=Matrix(1,dd,X[i])
    MX=MX.col_join(XM)
    

MX

Out[7]:

$$\left[\begin{matrix}s^{3} & 3 s^{2} & 3 s^{2} & 3 s & 6 s & 3 s & 1 & 3 & 3 & 1\\s^{2} & 2 s & 2 s & 1 & 2 & 1 & 0 & 0 & 0 & 0\\0 & s^{2} & 0 & 2 s & 2 s & 0 & 1 & 2 & 1 & 0\\s & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & s & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & s & 0 & 0 & 1 & 1 & 0 & 0\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\end{matrix}\right]$$

This is the Nth symmetric power of A. In this example, N=3.

In [8]:

IX=eye(A.shape[0])
delta=series(((IX-t*A).det())**(-1),t,0,10)
for i in range(10):
    display(delta.coeff(t,i))

$$1$$

$$s$$

$$s^{2} + 1$$

$$s^{3} + 2 s + 1$$

$$s^{4} + 3 s^{2} + 2 s + 1$$

$$s^{5} + 4 s^{3} + 3 s^{2} + 3 s + 2$$

$$s^{6} + 5 s^{4} + 4 s^{3} + 6 s^{2} + 6 s + 2$$

$$s^{7} + 6 s^{5} + 5 s^{4} + 10 s^{3} + 12 s^{2} + 7 s + 3$$

$$s^{8} + 7 s^{6} + 6 s^{5} + 15 s^{4} + 20 s^{3} + 16 s^{2} + 12 s + 4$$

$$s^{9} + 8 s^{7} + 7 s^{6} + 21 s^{5} + 30 s^{4} + 30 s^{3} + 30 s^{2} + 17 s + 5$$

These are the traces of the symmetric powers of A.

In [9]:

Res=series(trace((IX-t*A).inv()),t,0,10)
for i in range(10):
    display(Res.coeff(t,i))

$$3$$

$$s$$

$$s^{2} + 2$$

$$s^{3} + 3 s + 3$$

$$s^{4} + 4 s^{2} + 4 s + 2$$

$$s^{5} + 5 s^{3} + 5 s^{2} + 5 s + 5$$

$$s^{6} + 6 s^{4} + 6 s^{3} + 9 s^{2} + 12 s + 5$$

$$s^{7} + 7 s^{5} + 7 s^{4} + 14 s^{3} + 21 s^{2} + 14 s + 7$$

$$s^{8} + 8 s^{6} + 8 s^{5} + 20 s^{4} + 32 s^{3} + 28 s^{2} + 24 s + 10$$

$$s^{9} + 9 s^{7} + 9 s^{6} + 27 s^{5} + 45 s^{4} + 48 s^{3} + 54 s^{2} + 36 s + 12$$

These are the traces of the powers of A.

Symmetric Powers