What algebraic structure do the representations of a fixed finite group carry? On isomorphism classes of representations there are two operations, and . Both are commutative and associative up to isomorphism, and distributes over : . (The isomorphisms of Propositions 9.8.8 and 9.9.3 that express these laws are given by formulas such as , so they commute with the actions of .) This is almost a ring; what is missing is additive inverses, and we add them formally.
Definition 14.5.1 (Representation ring). Let be the free abelian group on the set of isomorphism classes of representations of ; its elements are finite formal sums with . The representation ring is the quotient of by the subgroup generated by all elements . Its multiplication is defined on generators by and extended bilinearly.
The multiplication is well defined on , which is free on the classes , and it descends to because is an ideal: using , and similarly for products on the other side. It is commutative, associative and distributive because is, and its identity is the class of the trivial representation, since . In we have , and now differences also make sense.
Proposition 14.5.2 (The character map). The map induces an injective ring homomorphism where is a ring under pointwise addition and multiplication. The classes of the irreducible representations form a basis of the abelian group , so , where is the number of conjugacy classes of .
Proof. The homomorphism sending each generator to vanishes on because , so it induces an additive map on . It is multiplicative because , and . By complete reducibility, every representation satisfies in with , so the classes generate as an abelian group. If , then applying gives , so all because the are linearly independent. Hence the form a basis, and is injective. Finally, is the number of conjugacy classes by Theorem 13.6.2. ∎
General elements of are called virtual representations. The actual representations, those with all , form a cone inside , that is, a subset closed under addition. The image of is the lattice of virtual characters .
Example 14.5.3 (The representation ring of ). By Example 13.3.6, , and comparing characters gives and . So in , which has basis and identity ,
What, then, is an arbitrary class function the character of? Allowing complex coefficients, let be the complex vector space with basis , with the multiplication of extended bilinearly. The map extends to a -linear map , with , which is still multiplicative.
Corollary 14.5.4 (Class functions as virtual characters). The map is an isomorphism of -algebras.
Proof. It is a multiplicative linear map that sends the basis of to the basis of (Theorem 13.6.2). ∎
Note. The theorems of Artin and Brauer (Theorems 15.3.4 and 15.3.5) describe the lattice of virtual characters inside in terms of characters induced from subgroups of .
Exercise 14.5.5 (Character Tables and the Representation Ring).
- Using the character table of , show that .
- Let be the standard representation of . Show that by computing from the formula and justify the formula.
- Verify column orthogonality (Proposition 13.6.5) for the character table of .
- Using the character table of , decompose and into irreducibles.
- Let be the irreducible representations of , and write in . Show that the structure constants are nonnegative integers, that , and that .
- Let be the standard representation of , . Show that .
Show solution
(1) The character of is , in the order of the table. With the class sizes , So , and the dimensions agree: . Alternatively, , and .
(2) If has eigenvalues on with eigenvectors , the vectors , , form a basis of of eigenvectors with eigenvalues (Proposition 9.10.19). Write and . Expanding, so , which is the formula. Next we need . Cubing the representatives: , , , , and . So, in the order of the table, Entry by entry, which is . So .
(3) Write and , so that and ; all entries are real. Using the rows , the sums for are using for either class of -cycles. These are , , and . For : and for the two classes of -cycles, .
(4) The character of is . The odd classes are those of , and , and vanishes on all three, so and .
The character of is . With the class sizes , where we have omitted the last two classes, on which vanishes. So , of dimension .
(5) We have , and by Corollary 13.5.6, with , a nonnegative integer. Since the form a basis of , , which is symmetric in and .
(6) Since , the space is -dimensional, and acts on it by (Theorem 9.11.3). The permutation representation is , so the permutation matrix of is conjugate to a block-diagonal matrix with blocks and the matrix of , and equals the determinant of the permutation matrix, which is (Proposition 5.4.24). So acts on as multiplication by , and .