Q1) Is every finite dimentional representation of a finite group completely reducible?
A representation is completely reducible if you work over any fields F such that the characteristic of F does not divide the order of the finite group G (indeed, this is exactly the statement of Maschke's Theorem). As a non-example, it was explained in the lecture notes why there is a 2-dimensional representation of C_5 over F_5 which is neither irreducible nor completely reducible (you cannot find an invariant complementary subspace in this case).
Q2) To show whether a representation is irreducible or not can I use characters whenever possible or should I try to find the G-invariant subspaces first?
To show irreduciblility, character theory should be the easiest way to do so. But please be aware that : 1. Character Theory works only for complex representations(i.e. the underlying vector space is over C) 2. Sometimes you will be asked to write down the explicit invariant subspaces instead of just determining whether the rep is irreducible or not.
Q3) Is there character theory over R?
No. In particular, Schur's Lemma fails.
Q4) What is the standard representation of S_n and how do we pick a basis?
The standard representation S_n is {a_1e_1+a_2e_2+...+a_ne_n | a_1+a_2+...+a_n=0} and G acts on the indices of the e_i's in the obvious way. This representation is (n-1) dimensional. To pick a basis, we need (n-1) linearly independent elements. For instance, in the case n=4, if v=ae_1+be_2+ce_3+de_4 belongs to the standard rep, then a+b+c+d=0 , equivalently, a=-b-c-d. Hence, you can rewrite v as
v=b(e_2-e_1)+c(e_3-e_1)+d(e_4-e_1). In this way, you immediately see that {e_i-e_1 : i=2,3,4} forms a basis. Of course, there are many ways to choose a basis (by writing b in terms of a,c,d instead, for example). Nevetheless, by the power of Linear algebra, this choice does not affect the trace hence you get the same character!
Q5) How do we determine if there exists G-equivariant homomorphisms between two representations V and W?
In Prop 16.1 of lecture notes, you have proved that dim(Hom_G(V,W))= inner product of the characters of V and W. Therefore, it sufficies to compute the inner product of the characters of V and W : if you get zero, then there are none; otherwise, they exist.
Q6) How did we determine that S_4/A_4={ eA_4, (12)A_4 } ?
Firstly one has to recall some fundamental properties of (left) cosets:
Let G be a group and H be its subgroup.
1. Any two left cosets have the same number of elements.
2. The left cosets partition a group G.
3. Lagrange Thm : Number of Left cosets equal |G|/|H|.
There are 24 elements in S_4 and half (i.e. 12) of them are even permutations , so the size of A_4 is 12.
To calculate the cosets explicitly, firstly A_4=eA_4 itself forms a left coset, now pick any element that is not in A_4, for example (12) (you can choose anyone you like), we form another left coset : (12)A_4. By Lagrange Thm, we expect 24/12=2 lefts cosets of A_4 in S_4. So we are done.
The process of computing all the left cosets have been discussed in Tutorial 7 in Abstract Algebra and Tutorial 2 in Groups and Symmetries respectively. If you want more (worked) examples, you may have a look on my Group and Sym tutorial notes here.
Q7) What does a faithful representation mean if we think of representations as vector spaces with G-linear action (instead of a homomorphism)?
If you like thinking in "vector-space terms", a representation V is faithful if and only if the only element in G that fixes everything in V is the identity element.
For example, the (left) regular representation C[G] is faithful because an element g in G satisfies g·e_x=e_x for all x if and only if g=e.
Q8) In the tutorial it was mentioned : A complex representation V is irreducible if and only if the inner product of the character of V with itself equals to 1. We have seen the proof for (<=) in the lecture. How about the converse?
Here's a quick proof : recall that in general, dim(Hom_G(V,W))= inner product of the characters of V and W. Hence, if V is irreducible, then
inner product of the characters of V with itself =dim(Hom_G(V,V)) which equals to 1 by Schur's Lemma.
Q9) How do you show a representation is a subrepresentation of another?
Over C, you can simply compute the inner product of the characters that they correspond to.
In general, to relate two representations (no matter it's to show one is a subrep. of another , or to show that they are isomorphic), one needs a G-equivariant map. Conceptually such a map is "difficult to exist".
1) Suppose such a function f exists. By the definiton of G-equivariance g．f(v)= f(g．v), of course it's more convenient to work in matrix term: \rho_1(g)Av=A\rho_2(g)v here A is the matrix represnting the linear map f.
2) Now exercise your knowledge in Linear Algebra that what sort of restriction this relation gives on f.
For instance, in 2015 A1(v) \rho_2(g) is a scalar, this shows that Av has to be an eigenvector w.r.t. the matrix \rho_1(g). This is the motivation behind my solution (and my choice of the function f). This is also possible that the above relation unwinds into some nonsense and you conclude that such function cannot exist (see, for example, 2017B6 last part).
Q10) Does the order matter when we compute the inner product of two characters?
No. (Indeed, it's a good exercise to think about why. Hint : for any character \chi, one has \chi(g^{-1})=complex conjugate of \chi(g)).
Q11) We know that if a group G is abelian, then all its irreducible representations are 1-dimensional. Is the converse true?
Yes. But apparently this uses deeper result(s) from the course:
Suppose all the irreducible rep are 1-dimensional. By the formula that \sum dim(V_i)^2=|G|, we have that |G|= number of irreducible rep. Now by one of the "miracles", number of irreducible rep=number of conjugacy classes of G. Thus we have |G|=number of conjugacy classes of G. This implies that every element of G forms a conjugacy class by itself and hence G is abelian.
Q12) If W is a subrepresentation of a permutation representation V, is W necessary isomorphic to a permutation representation?
No. Take V=C[{1,2,3}] and W=V_{stand}, the standard rep. of S_3. Then W is not isomorphic to a permutation rep. (There are lots of ways to see this, for example, the character of standard rep. can take negative values whereas that of a permutation rep. cannot; or you know that V_{stand} is an irreducible rep whereas a permutation rep is never irreducible (unless it's one-dimensional))