Georgi Lie Algebras Chapter 1 Solutions

==Problem 1==
Find the multiplication table for group with 3 elements and prove that it is unique.

==Solution==
Clear that if $e,a$ are the elements of the group, then only other extra element allowed is - $b=a^{-1}$. The multiplication table is as follows: $$\left(\begin{array}{ccccccc}& & e& & a& & b\\ e& & e& & a& & b\\ a& & a& & b& & e\\ b& & b& & e& & a\end{array}\right)$$

==Problem 2==
Find all essentially different possible multiplication tables for groups with four elements.

==Solution==
One is Klein Group 4. If $a,b,c,e$ are four elements. $$\left(\begin{array}{ccccccccc}& & e& & a& & b& & c\\ e& & e& & a& & b& & c\\ a& & a& & e& & c& & b\\ b& & b& & c& & e& & a\\ c& & c& & b& & a& & e\end{array}\right)$$ Other is Symmetric Group of 4 elements with the following table. $$\left(\begin{array}{cccccccc}& & e& & a& & b& c\\ e& & e& & a& & b& & c\\ a& & a& & e& & c& & b\\ b& & b& & c& & e& & a\\ c& & c& & b& & a& & e\end{array}\right)$$

==Problem 3==
Show that the representation $(1.135)$ is reducible.

==Solution==
Schur's lemma in special case states that if $A$ is a complex matrix of order $n$ that commutes with all matrices from $G(Here S_n)$, then A is scalar and the representation is reducible. Standard representation consists of single $1$ in each row and column. Take all $n\times n$ representation of the permutation group $S_n$. Apply the first projection formula, $$\varphi =\frac{1}{|G|}\sum _{i=1}^{n}g$$ where $G=S_n$ and $g\in End(V)=n\times n$ matrix of each group element. Clear that the Projection $\varphi$ consists of a an $n\times n$ matrix with $1$ in every row and column. Also it is trivial to show that this matrix $\varphi$ commutes with all group elements $g$ (in matrix form). Thus $\varphi$ is a scalar matrix and the representation is reducible.

==Problem 4==
Suppose $D_1$ and $D_2$ are equivalent irreducible representations of finite group of $G$ such that $D_2(g)=S{D}_1(g)S^{-1}$ $\mathrm{\forall }g\in G$, what can you say about an Operator $A$ that satisfies $A{D}_1(g)=D_2(g)A$?

==Solution==
We are given that both $D_1,D_2$ are equivalent irreds. Consider a G-Module homomorphism $A:D_1\to D_2$ given by $A{D}_1(g)=D_2(g)A$ for all $g\in G$. Then, $$\begin{array}{r}A{D}_1(g)=D_2(g)A\\ =S{D}_1(g)S^{-1}A\\ \text{ Or }{S}^{-1}A{D}_1(g)=D_1(g)S^{-1}A\end{array}$$ Now, this can be considered as G-module homomorphism $S^{-1}A:D_1\to D_1$. As, per Schur's lemma, the operator $S^{-1}A=\lambda I$. Therefore, $$\begin{array}{r}{S}^{-1}A=\lambda I\\ A=\lambda S\end{array}$$

==Problem 1E==
Find the Group of all discrete rotations that leave a regular tetrahedron invariant by labelling $4$ vertices and considering the rotations noas permutatof the four vertices. This defines four dimensional representation of a group. Find conjugacy classes and the character of the irreducible representations of the group.

==Solution==
Discrete rotations that leave the Tetrhedron invariant include,

• Identity
• Rotation around an axis thru vertex perpendicular to opposite plane by
$\pm {120}^o$. There are $4$ such axis and $2$ per axis giving a total of 8. If the Tetrahedron is labelled by $1,2,3,4$, the permutations are $(1,3,2),(1,2,3),(1,4,3),(2,4,3),(1,2,4),(2,3,4),(1,3,4),(1,4,2)$.
• Three (3) ${180}^o$ rotations that map edges to opposite edges.These permutations are \$(1,3)(2,4),(1,4)(2,3),(1,2)(3,4)$.

Total number of elements that leave Tetrahedron invariant under discrete rotation are $1+8+3=12$. This is isomorphic to Alternating Group of 4 elements. The conjugacy classes are $(),(1,2)(3,4),(1,2,3),(1,2,4)$. Using Sage Math or by using the character table computations, Irreducible characters: $$\left(\begin{array}{rrrrr}Conj->& 1& (123)& (132)& (12)(34)\\ U& 1& 1& 1& 1\\ U^{\prime }& 1& \omega & {\omega }^2& 1\\ U^{{}^{″}}& 1& {\omega }^2& \omega & 1\\ V& 3& 0& 0& -1\end{array}\right)$$

==Problem 1F==


==Solution==

• Each of the 4 blocks can move in $x,y$ directions. Then there are $8$ degrees of freedom of $x,y$ coordinates. Call this $D_2$.
• 4 Blocks arranged in a square has symmetries has Dihedral Group 4 symmetry
• \includegraphics[scale=5.0]{PermutationsColor.jpg}
• Dihedral group has $8$ elements.