Unitriangular matrix group:UT(4,p)

From Groupprops
Jump to: navigation, search
This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
View other such prime-parametrized groups

Definition

The cases p = 2 (see unitriangular matrix group:UT(4,2)) and p = 3 (see unitriangular matrix group:UT(4,3)) differ somewhat from the cases of other primes. This is noted at all places in the page where it becomes significant.

As a group of matrices

Given a prime p, the group UT(4,p) is defined as the unitriangular matrix group of degree four over the prime field \mathbb{F}_p. Explicitly, this is described as the following group under matrix multiplication:

\left \{ \begin{pmatrix}
1 & a_{12} & a_{13} & a_{14}\\
0 & 1 & a_{23} & a_{24} \\
0 & 0 & 1 & a_{34}\\
0 & 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{13},a_{14},a_{23},a_{24},a_{34} \in \mathbb{F}_p \right \}

The multiplication of matrices A = (a_{ij}) and B = (b_{ij}) gives the matrix C = (c_{ij}) where:

  • c_{12} = a_{12} + b_{12}
  • c_{13} = a_{13} + b_{13} + a_{12}b_{23}
  • c_{14} = a_{14} + b_{14} + a_{12}b_{24} + a_{13}b_{34}
  • c_{23} = a_{23} + b_{23}
  • c_{24} = a_{24} + b_{24} + a_{23}b_{34}
  • c_{34} = a_{34} + b_{34}

The identity element is the identity matrix, i.e., the matrix where all off-diagonal entries are zero and all diagonal entries are 1.

The inverse of a matrix A = (a_{ij}) is the matrix M = (m_{ij}) where:

  • m_{12} = -a_{12}
  • m_{13} = -a_{13} + a_{12}a_{23}
  • m_{14} = -a_{14} + a_{12}a_{24} + a_{13}a_{34} - a_{12}a_{23}a_{34}
  • m_{23} = -a_{23}
  • m_{24} = -a_{24} + a_{23}a_{34}
  • m_{34} = -a_{34}

In coordinate form

We may define the group as the set of ordered 6-tuples (a_{12},a_{13},a_{14},a_{23},a_{24}, a_{34}) over \mathbb{F}_p (the coordinates are allowed to repeat), with the multiplication law, identity element, and inverse operation given by:

 (a_{12},a_{13},a_{14},a_{23},a_{24},a_{34}) (b_{12},b_{13},b_{14},b_{23},b_{24},b_{34}) =

(a_{12} + b_{12},a_{13} + b_{13} + a_{12}b_{23},a_{14} + b_{14} + a_{12}b_{24} + a_{13}b_{34}, a_{23} + b_{23}, a_{24} + b_{24} + a_{23}b_{34},a_{34} + b_{34})

\mbox{Identity element} = (0,0,0,0,0,0)

(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})^{-1} = (-a_{12},-a_{13} + a_{12}a_{23}, -a_{14} + a_{12}a_{24} + a_{13}a_{34} - a_{12}a_{23}a_{34}, -a_{23}, -a_{34}, -a_{24} + a_{23}a_{34})

The matrix corresponding to the 6-tuple (a_{12},a_{13},a_{14},a_{23},a_{24}, a_{34}) is:

\begin{pmatrix}
1 & a_{12} & a_{13} & a_{14}\\
0 & 1 & a_{23} & a_{24} \\
0 & 0 & 1 & a_{34}\\
0 & 0 & 0 & 1 \\\end{pmatrix}

This definition clearly matches the earlier definition, based on the rules of matrix multiplication.

Families

These groups fall in the more general family UT(n,p) of unitriangular matrix groups. The unitriangular matrix group UT(n,p) can be described as the group of unipotent upper-triangular matrices in GL(n,p), which is also a p-Sylow subgroup of the general linear group GL(n,p). This further can be generalized to UT(n,q) where q is the power of a prime p. UT(n,q) is the p-Sylow subgroup of GL(n,q).

Element structure

Further information: element structure of unitriangular matrix group:UT(4,p)

Summary

Item Value
number of conjugacy classes 2p^3 + p^2 - 2p
equals number of irreducible representations. See number of irreducible representations equals number of conjugacy classes, linear representation theory of unitriangular matrix group of degree four over a finite field
order p^6
Follows from the general formula, order of UT(n,q), n = 4, q = p is q^{n(n-1)/2} = p^6
conjugacy class size statistics 1 (p times), p (p^2 - 1 times), p^2 (p^3 + p^2 - 2p times), p^3 (p^3 - p^2 - p + 1 times)
order statistics Case p = 2: order 1 (1 element), order 2 (27 elements), order 4 (36 elements)
Case p = 3: order 1 (1 element), order 3 (512 elements), order 9 (216 elements)
Case p \ge 5: order 1 (1 element), order p (p^6 - 1 elements)
exponent p^2 if p < 5
p if p \ge 5


Conjugacy class structure

For the right-most column for the type of matrix, we use the (row number, column number) notation for matrix entries. Explicitly, the matrix under consideration is:

\begin{pmatrix}
1 & a_{12} & a_{13} & a_{14} \\
0 &      1 & a_{23} & a_{24} \\
0 &      0 &      1 & a_{34} \\
0 &      0 &      0 &     1  \\
\end{pmatrix}

The subgroups mentioned in the table below are:

Subgroup Visual description Condition Order
center \begin{pmatrix}
1 &      0 &      0 & a_{14} \\
0 &      1 &      0 &     0 \\
0 &      0 &      1 &     0 \\
0 &      0 &      0 &     1  \\
\end{pmatrix} a_{12} = a_{13} = a_{23} = a_{24} = a_{34} = 0 p
derived subgroup \begin{pmatrix}
1 &      0 & a_{13} & a_{14} \\
0 &      1 &      0 & a_{24} \\
0 &      0 &      1 &     0 \\
0 &      0 &      0 &     1  \\
\end{pmatrix} a_{12} = a_{23} = a_{34} = 0 p^3
unique abelian subgroup of maximum order \begin{pmatrix}
1 &      0 & a_{13} & a_{14} \\
0 &      1 & a_{23} & a_{24} \\
0 &      0 &      1 &     0 \\
0 &      0 &      0 &     1  \\
\end{pmatrix} a_{12} = a_{34} = 0 p^4
Nature of conjugacy class Jordan block size decomposition Minimal polynomial Size of conjugacy class Number of such conjugacy classes Total number of elements Order of elements in each such conjugacy class Type of matrix (constraints on a_{ij}, i < j)
identity element 1 + 1 + 1 + 1 x - 1 1 1 1 1 all the a_{ij}, i < j are zero
non-identity element, but central (has Jordan blocks of size 1,1,2 respectively) 2 + 1 + 1 (x - 1)^2 1 p - 1 p - 1 p a_{14} \ne 0, all the others are zero
non-central but in derived subgroup, has Jordan blocks of size 1,1,2 2 + 1 + 1 (x - 1)^2 p 2(p - 1) 2p(p - 1) p a_{12} = a_{23} = a_{34} = 0
Among a_{13} and a_{24}, exactly one of them is nonzero.
a_{14} may be zero or nonzero
non-central but in derived subgroup, Jordan blocks of size 2,2 2 + 2 (x - 1)^2 p (p - 1)^2 p(p - 1)^2 p a_{12} = a_{23} = a_{34} = 0
Both a_{13} and a_{24} are nonzero.
a_{14} may be zero or nonzero
outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 1,1,2 2 + 1 + 1 (x - 1)^2 p^2 p - 1 p^2(p - 1) p a_{12} = a_{34} = a_{14} = 0
a_{23} is nonzero
a_{13} and a_{24} are arbitrary
outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 2,2 2 + 2 (x - 1)^2 p^2 (p - 1)^2 p^2(p - 1)^2 p a_{12} = a_{34} = 0
a_{23} and a_{14} are both nonzero
a_{13} and a_{24} are arbitrary
outside abelian subgroup of maximum order, Jordan blocks of size 1,1,2 2 + 1 + 1 (x - 1)^2 p^2 2(p - 1) 2p^2(p - 1) p Two subcases:
Case 1: a_{12} = a_{23} = a_{13} = 0, a_{34} nonzero, a_{14}, a_{24} arbitrary
Case 2: a_{23} = a_{24} = a_{34} = 0, a_{12} nonzero, a_{13}, a_{14} arbitrary
outside abelian subgroup of maximum order, Jordan blocks of size 2,2 2 + 2 (x - 1)^2 p^2 (p - 1)^2 p^2(p - 1)^2 p a_{12}, a_{34} both nonzero
a_{23} = 0
a_{14}, a_{24} arbitrary
a_{13} uniquely determined by other values
outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order q^4 3 + 1 (x - 1)^3 p^2 (p - 1)^2(p + 1) p^2(p - 1)^2(qp+ 1) p if p odd
4 if p = 2
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order q^3 3 + 1 (x - 1)^3 p^3 2(p - 1)^2 2p^3(p - 1)^2 p if p odd
4 if p = 2
Two subcases:
Case 1: a_{12}, a_{23} nonzero, a_{34} = 0, other entries arbitrary
Case 2: a_{23},a_{34} nonzero, a_{12} = 0, other entries arbitrary
Jordan block of size 4 4 (x - 1)^4 p^3 (p - 1)^3 p^3(p - 1)^3 p^2 if p < 5
p if p \ge 5
a_{12}, a_{23}, a_{34} nonzero
a_{13}, a_{14}, a_{24} arbitrary
Total (--) -- -- -- 2p^3 + p^2 - 2p p^6 -- --


Linear representation theory

Further information: linear representation theory of unitriangular matrix group:UT(4,p)

Summary

Item Value
number of conjugacy classes (equals number of irreducible representations over a splitting field) 2p^3 + p^2 - 2p. See number of irreducible representations equals number of conjugacy classes, element structure of unitriangular matrix group of degree four over a finite field
degrees of irreducible representations 1 (occurs p^3 times), p (occurs p^3 - p times), p^2 (occurs p^2 - p times)
sum of squares of degrees of irreducible representations p^6 (equals order of the group)
see sum of squares of degrees of irreducible representations equals order of group
lcm of degrees of irreducible representations p^2


GAP implementation

We assume p is assigned a prime number value beforehand.

Description Functions used
SylowSubgroup(GL(4,p),p) SylowSubgroup, GL
SylowSubgroup(SL(4,p),p) SylowSubgroup, SL
SylowSubgroup(PGL(4,p),p) SylowSubgroup, PGL
SylowSubgroup(PSL(4,p),p) SylowSubgroup, PSL