I am trying to understand the first paragraph (Section 9.1) of Chapter 9 in Serre's Linear Representations of Finite Groups.
Let me reproduce it here.
Let $G$ be a finite group and let $\chi_1, \ldots, \chi_h$ be its distinct irreducible characters. A class function on $G$ is a character if and only if it is a linear combination of the $\chi_i$'s with non-negative integer coefficients. We will denote by $R^+ (G)$ the set of these functions, and by $R (G)$ the group generated by $R^+ (G)$, i.e., the set of differences of two characters. We have $$ R (G) = \mathbb{Z} \chi_1 \oplus \ldots \oplus \mathbb{Z} \chi_h $$.
My questions:
- What is $h$? Is it the number of characters of all irreducible representations of all elements in the group? For example, is $h = 5! \times 7$ for $G = S_5$?
- If $G = S_n$, can $$ R (G) = \mathbb{Z} \chi_1 \oplus \ldots \oplus \mathbb{Z} \chi_h $$ be written as $$ R (G) = \mathbb{Z} \chi_1 + \ldots + \mathbb{Z} \chi_h $$?
Thanks in advance!
In general, if $G$ is a finite group, the number $h$ of irreducible characters of $G$ is equal to the number of conjugacy classes of $G$. This essentially follows from the fact that the dimension of the space of class functions on $G$ is easily seen to be equal to the number of conjugacy classes, combined with the fact that the irreducible characters $\chi_1, \dots, \chi_h$ form an orthonormal basis for this space (you can find these facts in section 2.5 of Serre's book for example).
Also, to answer your second question, it is known that the conjugacy classes of the symmetric group $\mathbb{S}_n$ are in one to one correspondence with the partitions of $n$. Hence by the above fact, the number of irreducible characters of $\mathbb{S}_n$ is equal to $p(n)$, where $p(n)$ is the partition function.
Thus, since $p(5) = 7$, then $h = 7$ for $G = \mathbb{S}_5$.