<jupyter_start><jupyter_text>Linear Representations1. Homomorphism $\alpha: G \rightarrow GL(n,k)$ where n is the dimension of a vector space over field $k$. That is $n \times n $ matrices.2. A group action $G \times X \rightarrow X:(g,x) \mapsto x$ where  >1. $X$ is vector space and  >2. $(gh)x) = g(hx)$ >3. $ex=x$ >4. $g(u + v) = gu + gv$ >5. $g(ρv) = ρ(gv)$---A group action on a set of finite size is a subgroup of permutations, they may be represented as special matrices of size n for example:  $\begin{pmatrix}1 & 2 & 3\\ 2 & 1 & 3\end{pmatrix} \Leftrightarrow  \begin{pmatrix}0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1\end{pmatrix}$---A 1-dimensional representation is simply a homomorphism to multiplicative group of the field.$g \mapsto 1$ is the trivial representation.a 1-dimensional representation is constant on each conjugacy class for $S_n$: $g \mapsto \left \{\begin{matrix}1 & if \ g \ is \ even \\ -1 & if \ g \ is \ odd\end{matrix}\right.$for $D_4$:  $r^4=1, c^2=1, (rc)^2=1 \Rightarrow c=\pm1,r=\pm1$  there are 4 1-dimensional representations&nbsp;2-dimensional representations of $S_3$: $\langle s, t \vert s^3,t^2, st=ts^2  \rangle$ over $\mathbb{C}$  suppose $s$ has a eigenvector $v_1$ with eigenvalue $\lambda \ne 1, \ \lambda^3=1, \ \lambda = \exp(i2\pi/3)$  $s(tv_1) = ts^2v_1=\lambda^2(tv_1) \Rightarrow s \mapsto \begin{pmatrix}\lambda & 0 \\ 0 & \lambda^2\end{pmatrix}$ and the next eigenvector is $v_2==tv_1$  on the other hand $tv_1=v_2, tv_2=v_1 \Rightarrow t \mapsto \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$the other case is $s \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, then t will be free: $t \mapsto \begin{pmatrix} \lambda & 0 \\ 0 & \mu \end{pmatrix}$, $\lambda,\mu=\pm1$&nbsp;The representation $\alpha$ of $G$ in the vector space $V$ over $k$ is said tobe **simple (irreducible)** if no proper subspace of $V$ is stable under $G$.$dim \ \alpha \le \vert G \vert$A simple representation of a finite abelian group over $\mathbb{C}$ is necessarily 1-dimensional. (commutative matrices can be diagonalised simultaneously) **Arithmetic of Representations**$\alpha:g \mapsto A(g), \  \beta:g \mapsto B(g), \ \alpha+\beta: g \mapsto \begin{pmatrix} A(g) & 0 \\ 0 & B(g)\end{pmatrix}$. The representation space is  $V \oplus W$$\alpha:g \mapsto A(g), \  \beta:g \mapsto B(g), \ \alpha\beta: g \mapsto A(g) \otimes B(g)$. . The representation space is  $V \otimes W$> $\alpha+\beta=\beta+ \alpha$  > $\alpha+(\beta+\gamma)=(\alpha+\beta)+ \gamma$  > $\alpha\beta=\beta \alpha$  > $\alpha(\beta\gamma)=(\alpha\beta) \gamma$  > $\alpha(\beta+\gamma)=\alpha\beta+ \alpha\gamma$  Conjugacy: if $\alpha$ is the representation of $G$ in $V$, $\alpha^*$ is the representation of $G$ in dual space $V^*$  $\alpha:g \mapsto A(g), \ \alpha^* \mapsto (A(g)^{-1})^\prime$> $(\alpha^*)^*=\alpha$  > $(\alpha+\gamma)^*=\alpha^*+\beta^*$  > $(\alpha\beta)^*=\alpha^*\beta^*$  &nbsp;The represenation is said to be **semisimple** if it is expressible as a sum of simple representations:$\alpha = \sigma_1+\sigma_2+ ... + \sigma_r$ .Equivalently each stable subspace $U \subset V$ has at least one complementary stable subspace $W$, ie $V=U \oplus W$.---Let $Q$ be a positive (hermitian) form in $\mathbb{R}^n$ ($\mathbb{C}^n$). Avaraging it over group: $P(u,v) =\frac{1}{\vert G\vert} \sum_ {g \in G} Q(gu,gv)$$P(u,v)=0 \Rightarrow gu \perp gv \ for \ all \ g \in G$ and $u \in U \Rightarrow v \in U^{\bot}$---**Maschke’s Theorem**: Suppose α is a representation of the finitegroup $G$ over $k$ ($\vert G \vert \ne 0 \ in \ k$). Then α is semisimple.The expression for a semisimple representation $\alpha$ as a sum of simple parts $\alpha = \sigma_1+\sigma_2+ ... + \sigma_r$ is unique up to order.The character $\chi_\alpha:G \rightarrow k$ is defined as $\chi(g)=tr(\alpha(g))$> $\chi_{\alpha+\beta}(g)=\chi_\alpha(g)+\chi_\beta(g)$  > $\chi_{\alpha\beta}(g)=\chi_\alpha(g)\chi_\beta(g)$  > $\chi_{\alpha^*}(g)=\chi_{\alpha}(g^{-1})$  > $\chi_{1}(g)=1$  > $\chi_\alpha(e)=dim \alpha$  Two representations are equivalent if and only if their characters are equal:> $\alpha = \beta \Leftrightarrow \chi_\alpha(g)=\chi_\beta(g) \ for \ all \ g\in G$For representations of $G$ over $C$>$\chi (I) = n$  >$\vert \chi(g) \vert \le n$,  >$\vert \chi(g) \vert = n$ implies $g = e^{iθ}I$  >$\chi$ is constant on any given conjugacy class of $G$  >$\chi (g^{−1}) = \chi^*(g)$  >$\chi(g)$ is an algebraic number for all $g$.1. The number of simple representations of $G$ over $C$ is equal to the number of conjugacy classes in $G$2. The dimensions of the simple representations $σ_1, . . . , σ_s$ of $G$ over $C$ satisfy the relation  >$dim^2 σ_1 + · · · + dim^2 σ_s = \vert G \vert$.3. The dimension of each simple representation $σ_i$ divides the order of the group: $dim σ_i\> |\> \vert G \vert$.$\alpha$ is representation of $G$ and $\beta$ is representation of $H$, then $\alpha \times \beta$ (over $U \otimes V$) is representation of $G \times H$. and $\alpha \times \beta$ is simple iff $\alpha$ and $\beta$ are simple. Theorem. A representation of $G$ is irreducible if and only if $\sum_{g \in G} |\chi(g)|^2 = \vert G \vert$The characters are orthogonal, that is: $\sum_{i=1}^{r} r_i (\chi^p_i)^* \chi^q_i = |G|\delta_{pq}$ and $\sum_{p=1}^r (\chi^p_i)^* \chi^q_j = \frac{|G|}{r_i}\delta_{ij}$  where $p,q$ denote representations, $\chi^p_i$ is the character of $i$th conjugacy calss in $p$th irreducable representation, and $r_i$ is size of $i$th conjugacy class.Theorem. If $\rho$ is an arbitrary representation of $G$ with character $\chi$, and $\rho^p$ are theinequivalent irreducible representations of $G$ with characters $\chi^p$, then $\rho = \bigoplus_p c_p \rho^p$, where $c_p = \frac{1}{|G|} \sum_{i=1}^{r} r_i (\chi_i^p)^*\chi_i$The regular representation contains $dim \rho^p$ instances of each irreducible representation: $\chi_i^R = \sum (dim \rho^p)\chi_i^p$<jupyter_code># Z3 sympy exercises
import sympy.combinatorics.permutations as P
import sympy.combinatorics.named_groups as G
from sympy.matrices import eye
from sympy import *
init_printing(use_latex='mathjax')
P.Permutation.print_cyclic
c = G.CyclicGroup(3)
el = [e for e in c.elements]
el.sort(key=lambda x: c.coset_rank(x))
display(el)
m = eye(3)
regR = [m.permuteFwd(e) for e in el]
display(regR)
lamda = symbols('lamda')
poly=[factor(p.charpoly(lamda)) for p in regR]
display(poly)
(p1,d1) = regR[1].diagonalize()

(p2,d2) = regR[2].diagonalize()
#display(d1)

p1_=simplify(p1**-1)
display([simplify(p1_*regR[1]*p1),simplify(p1_*regR[2]*p1)])


w = exp(I*2*pi/3)
display(w,w.expand(complex=true))
m = Matrix([[0,-1],[1,-1]])
display([m,m.inv(),m*m])<jupyter_output><empty_output><jupyter_text>3 representations over $C$: >$\{1,1,1\}$ $\{1,w,w^2\}$ $\{1,w^2,w\}$not trivial representation over $N$: $\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}$, $\begin{bmatrix}0 & -1\\ 1 & -1\end{bmatrix}$,$\begin{bmatrix}-1 & 1\\ -1 & 0\end{bmatrix}$not trivial orthogonal representation over $R$: $\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}$, $\begin{bmatrix}-\frac{1}{2} & -\frac{\sqrt 3}{2}\\ \frac{\sqrt 3}{2} & -\frac{1}{2}\end{bmatrix}$,$\begin{bmatrix}-\frac{1}{2} & \frac{\sqrt 3}{2}\\ -\frac{\sqrt 3}{2} & -\frac{1}{2}\end{bmatrix}$  Pauli matrices$\{±I,±iI,±X,±iX,±Y,±iY,±Z,±iZ\}$One of Generator Sets: $\langle iI,X,Y \rangle$Center is $\{±I,±iI\}$<jupyter_code>e, X, Y, Z = symbols('e X Y Z')

t = [e,X,Y,Z]
pauliSymbols = [e,X,Y,Z]
pauliSymbols.extend([-l for l in t])
pauliSymbols.extend([I*l for l in t])
pauliSymbols.extend([-I*l for l in t])

display([(e,pauliSymbols[e]) for e in range(16)])
from sympy.combinatorics.perm_groups import PermutationGroup
e8 = P.Permutation([8,9,10,11, 12,13,14,15, 4,5,6,7,  0,1,2,3])
e1 = P.Permutation([1,0,11,14, 5,4,15,10,   9,8,7,2,  13,12,3,6])
e2 = P.Permutation([2,15,0,9,  6,11,4,13,   10,3,8,5, 14,7,12,1])
display(e8)
display(e1)
display(e2)

p = PermutationGroup(e8,e1,e2)
display(p.degree)
p.center()
from sympy.combinatorics.free_groups import free_group
from sympy.combinatorics.fp_groups import FpGroup
from sympy.combinatorics.fp_groups import low_index_subgroups
F, i, x, y = free_group("i, x y")
G = FpGroup(F, [i**4, x**2, i*x*i**-1*x**-1, y**2, i*y*i**-1*y**-1, (x*y)**4, i*i*x*y*x**-1*y**-1])
print("Order is:",G.order())
l = low_index_subgroups(G,16)
print("SubGroup Count is:",len(l))
#for t in l:
#    print(len(t.table))<jupyter_output>Order is: 16
SubGroup Count is: 20
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