Linear representation theory of groups of prime-cube order

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of prime-cube order.
View linear representation theory of group families | View other specific information about groups of prime-cube order

The discussion here is specifically intended to cater to odd primes $p$ and excludes the case $p = 2$, which is somewhat different. For more on $p = 2$, see linear representation theory of groups of order 8.

Group	GAP ID second part	Linear representation theory page
cyclic group of prime-cube order	1	linear representation theory of cyclic group of prime-cube order, linear representation theory of cyclic groups
direct product of cyclic group of prime-square order and cyclic group of prime order	2	linear representation theory of direct product of cyclic group of prime-square order and cyclic group of prime order, linear representation theory of finite abelian groups
unitriangular matrix group:UT(3,p)	3	linear representation theory of unitriangular matrix group:UT(3,p)
semidirect product of cyclic group of prime-square order and cyclic group of prime order	4	linear representation theory of semidirect product of cyclic group of prime-square order and cyclic group of prime order
elementary abelian group of prime-cube order	5	linear representation theory of elementary abelian group of prime-cube order, linear representation theory of elementary abelian groups

Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

Due to the fact that nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order, for groups of order $p^3$ for a fixed prime $p$, the nilpotency class determines the degrees of irreducible representations. (This also holds for $p = 2$, although we are not discussing that case.

Group	GAP ID second part	Degrees as list	Number of irreps of degree 1 (= order of abelianization)	Number of irreps of degree $p$	Total number of irreps (= number of conjugacy classes)
cyclic group of prime-cube order	1	1 ($p^3$ times)	$p^3$	0	$p^3$
direct product of cyclic group of prime-square order and cyclic group of prime order	2	1 ($p^3$ times)	$p^3$	0	$p^3$
unitriangular matrix group:UT(3,p)	3	1 ($p^2$ times), $p$ ($p - 1$ times)	$p^2$	$p - 1$	$p^2 + p - 1$
semidirect product of cyclic group of prime-square order and cyclic group of prime order	4	1 ($p^2$ times), $p$ ($p - 1$ times)	$p^2$	$p - 1$	$p^2 + p - 1$
elementary abelian group of prime-cube order	5	1 ($p^3$ times)	$p^3$	0	$p^3$

Splitting field

Two important notes:

• Odd-order and ambivalent implies trivial, so none of these groups are ambivalent groups, i.e., none of them have all their character real-valued. Therefore, none of them are rational groups or rational-representation groups. This is in sharp contrast to the $p = 2$, where three of the five groups of order $p^3$ are rational groups and two of them are rational-representation groups.
• Odd-order p-group implies every irreducible representation has Schur index one: All the irreducible representations have Schur index one, so they can be realized over the field generated by the character values. Note that this fact requires both the odd order and the fact that it is a group of prime power order. For $p = 2$, there is an example -- faithful irreducible representation of quaternion group -- that has Schur index two. Also, there is a group of order 63 that has an irreducible representation of Schur index three.

With these facts in mind, we present the list of minimal splitting fields. We denote by $\zeta_n$ a primitive $n^{th}$ root of unity. Note that the degree of the extension $\mathbb{Q}(\zeta_n)$ over $\mathbb{Q}$ is the Euler totient function $\varphi(n)$.

Group	GAP ID second part	Nilpotency class	minimal splitting field = field generated by character values (because it's an odd-order p-group)	degree of extension over $\mathbb{Q}$	minimal sufficiently large field	degree of extension over $\mathbb{Q}$	Note
cyclic group of prime-cube order	1	1	$\mathbb{Q}(\zeta_{p^3})$	$p^2(p - 1)$	$\mathbb{Q}(\zeta_{p^3})$	$p^2(p - 1)$	both same, because abelian
direct product of cyclic group of prime-square order and cyclic group of prime order	2	1	$\mathbb{Q}(\zeta_{p^2})$	$p(p - 1)$	$\mathbb{Q}(\zeta_{p^2})$	$p(p - 1)$	both same, because abelian
unitriangular matrix group:UT(3,p)	3	2	$\mathbb{Q}(\zeta_p)$	$p - 1$	$\mathbb{Q}(\zeta_p)$	$p - 1$
semidirect product of cyclic group of prime-square order and cyclic group of prime order	4	2	$\mathbb{Q}(\zeta_p)$	$p - 1$	$\mathbb{Q}(\zeta_{p^2})$	$p(p - 1)$	splitting not implies sufficiently large
elementary abelian group of prime-cube order	5	1	$\mathbb{Q}(\zeta_p)$	$p - 1$	$\mathbb{Q}(\zeta_p)$	$p - 1$	all same, because abelian