Quiz:Linear representation theory of symmetric group:S3

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

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Item	Value
Degrees of irreducible representations over a splitting field (and in particular over $\mathbb{C}$)	1,1,2 maximum: 2, lcm: 2, number: 3 sum of squares: 6, quasirandom degree: 1
Schur index values of irreducible representations	1,1,1
Smallest ring of realization for all irreducible representations (characteristic zero)	$\mathbb{Z}$
Minimal splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero)	$\mathbb{Q}$ (hence, it is a rational representation group)
Condition for being a splitting field for this group	Any field of characteristic not two or three is a splitting field. In particular, $\mathbb{Q}$ and $\R$ are splitting fields.
Minimal splitting field in characteristic $p \ne 0,2,3$	The prime field $\mathbb{F}_p$
Smallest size splitting field	field:F5, i.e., the field of five elements.

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Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2 or 3, except in the last two columns, where we consider what happens in characteristic 2 and characteristic 3.

Name of representation type	Number of representations of this type	Degree	Schur index	Criterion for field	Kernel	Quotient by kernel (on which it descends to a faithful representation)	Characteristic 2	Characteristic 3
trivial	1	1	1	any	whole group	trivial group	works	works
sign	1	1	1	any	A3 in S3	cyclic group:Z2	works, same as trivial	works
standard (two-dimensional irreducible)	1	2	1	any	trivial subgroup, i.e., it is faithful	symmetric group:S3	works	indecomposable but not irreducible

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Representation/Conjugacy class representative	$()$ (identity element) -- size 1	$(1,2,3)$ (3-cycle) -- size 2	$(1,2)$ (2-transposition) -- size 3
Trivial representation	1	1	1
Sign representation	1	1	-1
Standard representation	2	-1	0

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