REPRESENTATION THEORY of FINITE GROUPS 1. Important Stuff

VIPUL NAIK

Abstract. The course Representation Theory of Finite Groups was taught by Senthamarai Kan- nan. Here, I give the list of important results proved in this course.

1. Important stuff taught in the course 1.1. Orthogonality results. Let G be a ﬁnite group. For this list, all representations are over Q.

• First orthogonality theorem (result stated): The characters of irreducible representations of G form an orthonormal basis for the space of class functions. This set of characters is denoted as X(G). Thus, the cardinality of X(G) is the number of conjugacy classes of G. • The multiplicity of each character in the regular representation equals its degree. Thus the sum of squares of degrees of irreducible representations is the cardinality of G. • Grand orthogonality theorem (result stated): If we choose the irreducible unitary representations, all the matrix entry functions for these irreducible unitary representations form an orthogonal basis for the space of functions on the group. This result is the • Second orthogonality theorem (result stated): Given any two distinct conjugacy classes, the inner product of those conjugacy classes with respect to irreducible representations is 0. • The characters of representations are precisely the N0-linear combinations of elements of X(G). This is termed R+(G). A class function that is a Z-linear combination of elements of X(G) is termed a virtual character(deﬁned). 1.2. Other results. • The product of two characters is a character. In fact, it is the character of the tensor product of the corresponding representations. Thus, R(G) is a ring. • If χ is a character, and k is relatively prime to the order of the group, then x 7→ χ(xk) is also a character. The operator that takes χ to this character is termed Ψk. • If χ(x) = χ(y), then χ(xk) = χ(yk) whenever k is relatively prime to the order of the group. • Given any conjugacy class, there is a character that takes the value 0 at that conjugacy class. This is proved by using norm-squared, and using a Cauchy-Scwarz inequality to exchange product and summation. 1.3. Induction and restriction. We deﬁned induction and restriction on characters. We have the following results: • Frobenius reciprocity G G G • IndH (f1.ResH f2) = IndH (f1).f2. • Mackey’s lemma and Mackey’s irreducibility criterion. This essentially gives a relation between the characters from two subgroups.

2. Artin’s theorem 2.1. Statement. Artin’s theorem states that th class functions induced from cyclic subgroups cover the whole group. Here’s a formulation of Artin’s theorem:

Theorem 1 (Artin’s theorem). Let X be a collection of subgroups of a group G. Then, the following are equivalent: • The union of conjugates of subgroups inside X is the whole of G. • FC (G) is generated by the images of Ind : FC (H) → FC (G) for H ∈ X.

c Vipul Naik, B.Sc. (Hons) Math and C.S., Chennai Mathematical Institute. 1 • Every element of R(G) is a Q-linear combination of elements induced from R(H) as H varies over members of X.

2.2. Proof idea of Artin’s theorem. The equivalence of the second and third statements essentially follows from the fact that if anything in R(G) is a combination of elements induced from R(H)s, then it must be a Q-linear combination. The hard part is showing that the ﬁrst two are equivalent. For this, we prove that surjectivity of the induction map is equivalent to injectivity of the restriction map. This comes from the following: If H is a subgroup of G, then a class function on G is zero restricted to H if and only if it is orthogonal to all class functions induced from H.

2.3. For cyclic subgroups. Artin’s theorem tells us, in particular, that the ring FC (G) is generated by the class functions induced from FC (H) where H varies over cyclic subgroups. For cyclic subgroups, we can actualyl say something stronger. We prove the following results: • Given a cyclic group C, the class function that sends a particular generator to |C| and every other element to 0 is in A ⊗ R(C). This follows from the second orthogonality theorem. • Thus, in particular, for a cyclic group, any class function that sends generators to multiples of |C| and other thigns to 0, lies inside A ⊗ R(C). • On a group, the class function that sends each element to |G| is the sum of the induced class functions from the θ class functions of cyclic subgroups. • By using the induction-restriction formula, we obtain that any class function with integer values divisible by |G| is in the sum of the induced parts from R(C) for cyclic subgroups C.

2.4. Why we cannot get Z-linear combinations. The main problem is that it may happen that the characters may always take values in a certain ideal. This is because the indices ofall the cyclic subgroups may not be relatively prime.

3. Brauer’s theorem 3.1. Statement. Quick deﬁnitions:

Definition. Let p be a prime. A finite group is said to be p-elementary(defined) if it is the direct product of a p-group and a cyclic group of order relatively prime to p.

Statement of Brauer’s theorem: Let G be a finite group. Then, R(G) is in the Z-span of characters induced from elementary subgroups. 3.2. First step of proof. Let A be the ring generated by the |G|th roots of unity. The first step is in showing that if χ is any character, then χ(x) = χ(xs) where xs is the p-semisimple (or p-regular) part of x. 3.3. Next step of proof. We proceed as follows: • For a p-regular element g, let C be the cyclic subgroup generated by g. Then, consider the element of R(C) that associates |C| to g and 0 to other elements. Let P be a Sylow subgroup of the centralizer of g. Then C × P is an elementary p-subgroup. Extend the element of A ⊗ R(C) to an element of A ⊗ R(C × P ) by composing with the projection. Now induce to an element of A ⊗ R(G). This character is zero on all elements whose semisimple parts are not conjugate to x, and is not 0 modulo p on all elements whose semisimple part is conjugate to x. We crucially use the fact that the index of P in the centralizer is relatively priem to p. • Add such elements up for all semisimple conjugacy classes. The sum we get is a character ψ that is not zero modulo p for any element. Let N = φ(pr). Take l(ψN − 1). Since ψ ∈ A ⊗ R(G), so is ψN . Also, since ψN − 1 takes values divisible by pr for every r, the product takes values divisible by |G| and is hence actually in R(G). Thus, taking lψN − l(ψN − 1) we obtain that l is in Vp(G). • We now put the things together for each prime p. 2 4. Rationality questions 4.1. Basic results. • Every representation of a group is realizable over the field containing the |G|th roots of unity. • The characters over a field K containing Q form an otrhonormal basis for the space of functions of ΓK -conjugacy classes, that is, the classes by taking the equivalence relation generated by conjugation and powering by things allowed in the Galois group of K over Q. • If RK (G) denotes the Z-span of characetrs of representations realizable over K and RK (G) denotes the Z-span of characters of representations such that the character has values fully inside K. Then, RK (G) has finite index in RK (G). This finite index is termed the Schur index. ALl this is closely related to the fact that Ψk are all operators from R(G) to R(G). Why is this important: roughly because it shows the following: Let K be a field extension of Q and σ be a Galois conjugation of K over Q. Suppose the character of a representation has values in K. Then σ applied to th characetr values also gives the character of a representation. Another interesting consequence is the following: If χ is the character of a representation, and χ(x) = χ(y), then χ(xk) = χ(yk).

5. Other results related to Brauer’s theorem 5.1. A theorem of Frobenius.

Theorem 2 (Frobenius’ thoerem). If f is a class function on G with values on A, and g = |G|, the function g/(g, n)Ψn(f) belongs to A ⊗ R(G).

ALternative formulations are: n • For each conjugacy class c, let fc be the indicator function for c. Then, the function g/(g, n)Ψ (fc) belongs to A ⊗ R(G). P • For each conjugacy class c of G, and each character χ of G, the expression 1/(g, n) xn∈c χ(x) ∈ A. The proof of Frobenius’ theorem arises by ﬁrst reducing it to the case fo elementary subgroups (using Brauer’s theorem) and then reducing from elementary subgroups to p-groups (by using the fact that it is a direct product) and then usnig the fact that every character of a p-group.

5.2. Converse to Brauer’s theorem.

Lemma 1. Let p be a prime number, and x a p-regular element. Let C×P be an associated elementary subgroup. Suppose H does not contain any conjugate of this associated elementary subgroup. Then any class function on H with values in A induced to a class function on G which takes a value congruent to 0 modulo pA.

From this lemma, we get the following result:

Theorem 3 (Green’s converse to Brauer’s theorem). Let X be a collection of subgroups with the property that R(G) is generated by the images of R(H) for H ∈ X. Then every elementary subgroup is contained in some conjugate of an element of X.

5.3. Generalizations of Brauer’s theorem. Brauer’s theorem does not hold in its usual form when we work over RK (G) where RK (G) is the ring of ring of characters of representations realizable over K. What we need to do is enlarge the class of “elementary” subgroups. 3 Deﬁnition. A subgroup H of G is said to be ΓK − p elementary if its is the semidirect product of a −1 t p-group P and a cyclic group C such that yxy is x for a suitable t ∈ ΓK .

6. Results proved in class/examinations 6.1. Results proved in the first half of the course. • In the examination: Let H ≤ G be finite group. If evry irreducible character of G restricts to an irreducible character of H then H is normal in G and the number of conjguacy classes in G is [ G : H ] times the number of conjugacy classes in H. Coversely, if the number of conjugacy classes in G is [ G : H ] times the number of conjugacy classes in H, then H is normal and every irreducible character of G restricts to an irreducible character fo H. • A slight generalization: Let H ≤ G be a finite group. Then every isotypical character of G restricts to an isotypical character of H if and only if H is normal and conjugacy-closed in G. • In the examination: Something on irreducible characters of G × G, obvious later on :) 6.2. Results proved on permutation representations. (1) A permutation representation is transitive if and only if the only eigenspace with eigenvalue 1 is that generated by the sum of basis elements. (2) A permutation representation is 2-transitive if and only if it is the direct sum of the one- dimensional representation given by sum of basis elements, and a linearly irreducible representation. 6.3. Some exercises that were done. (1) Every element of R(G) that is orthogonal to the regular character is a Z-linear combination of G elements of the form IndE(α − 1) where E is an elementary subgroup and α is a character of degree 1. (2) Any N0-linear combination of monomial characters is monomial. Thus, in fact, any element of R(G).

7. Student seminars: results (1) The degrees of irreducible representations divide the order of the group. (2) The degrees of irreducible representations divide the index of the center. (3) The square of the degree of an irreducible representation is less than or equal to the index of the center. (4) The very important result: if A is a normal subgroup of G, and χ is an irreducible character of G, then either the restriction of χ to A is isotypical, or there is a subgroup H containing A such that χ is induced from a character on H. (5) In particular, if A is an Abelian normal subgroup not contained inside the center, χ is induced from a character of a proper subgroup containing A. (6) The characters of irreducible representations divide the index of any normal Abelian subgroup. (7) Any character of a supersolvable group is monomial. Thus, any character of a group is monomi- ally generated.