I just glanced at the book Representation Theory: A First Course by William

Fulton and in the book there is the symbol [math]\mathfrak{S}_{\lambda}[/math]. What does this symbol mean?

 

[math]\mathfrak{S}_{d}[/math] means symmetric group but what does [math]\mathfrak{S}_{\lambda}[/math] mean?

The symmetric group on a set [math]X \in Set[/math] is denoted at [math]\mathfrak{S}_{X}[/math]. It consists of the collection of bijections [math]X \rightarrow X[/math] with the group operation being composition of functions.

 

The symmetric group of degree [math]d[/math] is the symmetric group on the set [math]\{ 1,2, \cdots , d \}[/math]. It is denoted [math]\mathfrak{S}_{d}[/math].

 

So, [math]\mathfrak{S}_{\lambda}[/math] refers explicitly to the set [math]\lambda \in Set[/math] or [math]\lambda \in \mathbb{N}[/math] and the set is a subset of the natural numbers.