# Quiz:Linear representation theory of symmetric group:S3

For background, see linear representation theory of symmetric group:S3.

### Basic stuff

Up to equivalence, there are three irreducible representations of symmetric group:S3 in characteristic zero: the one-dimensional trivial representation, the one-dimensional sign representation (that sends every permutation to its sign), and the standard representation of symmetric group:S3, a two-dimensional representation.

1 Which of the irreducible representations is realized over the field of rational numbers?

 trivial representation only trivial representation and sign representation only all three representations

2 Which of the irreducible representations can be realized using orthogonal matrices (i.e., matrices in the orthogonal group for the standard dot product) over the field of rational numbers?

 trivial representation only trivial representation and sign representation only all three representations

3 The tensor product of the standard representation and the sign representation is a representation of the symmetric group of degree three. What representation is it?

 the standard representation the sum of the trivial and the sign representation the sum of two copies of the sign representation the sum of two copies of the trivial representation

4 The symmetric group of degree three can also be viewed as a dihedral group of degree three and order six, acting on a set of size three. The 3-cycles become rotations and the transpositions become reflections. This defines a two-dimensional representation over the real numbers. Which of these is the two-dimensional representation?

 the standard representation the sum of the trivial and the sign representation the sum of two copies of the sign representation the sum of two copies of the trivial representation