Linear representation theory of unitriangular matrix group:UT(3,p)

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This article gives specific information, namely, linear representation theory, about a family of groups, namely: unitriangular matrix group:UT(3,p). This article restricts attention to the case where the underlying ring is a finite prime field.
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This article describes the linear representation theory of the unitriangular matrix group of degree three over a finite field of size equal to a prime number p.

See also linear representation theory of unitriangular matrix group of degree three over a finite field.

Summary

Item Value
number of conjugacy classes (equals number of irreducible representations over a splitting field) p^2 + p - 1. See number of irreducible representations equals number of conjugacy classes, element structure of unitriangular matrix group of degree three over a finite field
degrees of irreducible representations over a splitting field (such as \overline{\mathbb{Q}} or \mathbb{C}) 1 (occurs p^2 times), p (occurs p - 1 times)
sum of squares of degrees of irreducible representations p^3 (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
condition for a field (characteristic not equal to p) to be a splitting field The polynomial x^p - 1 should split completely.
For a finite field of size q, this is equivalent to q \equiv 1 \pmod p.
field generated by character values, which in this case also coincides with the unique minimal splitting field (characteristic zero) Field \mathbb{Q}(\zeta) where \zeta is a primitive p^{th} root of unity. This is a degree p - 1 extension of the rationals.
unique minimal splitting field (characteristic c \ne 0,p) The field of size c^r where r is the order of c mod p.
degrees of irreducible representations over the rational numbers 1 (1 time), p - 1 (p + 1 times), p(p - 1) (1 time)
Orbits over a splitting field under the action of the automorphism group Case p = 2: Orbit sizes: 1 (degree 1 representation), 1 (degree 1 representation), 2 (degree 1 representations), 1 (degree 2 representation)
Case odd p: Orbit sizes: 1 (degree 1 representation), p^2 - 1 (degree 1 representations), p - 1 (degree p representations)
number: 4 (for p = 2), 3 (for odd p)
Orbits over a splitting field under the multiplicative action of one-dimensional representations Orbit sizes: p^2 (degree 1 representations), and p - 1 orbits of size 1 (degree p representations)


Particular cases

Prime number p p^3 (equals order of UT(3,p) Unitriangular matrix group UT(3,p) Degrees of irreducible representations (1 occurs p^2 times, p occurs p - 1 times) Number of irreducible representations (equals p^2 + p - 1) Linear representation theory page
2 8 dihedral group:D8 1 (occurs 4 times), 2 (occurs 1 time) 5 linear representation theory of dihedral group:D8
3 27 unitriangular matrix group:UT(3,3) 1 (occurs 9 times), 3 (occurs 2 times) 11 linear representation theory of unitriangular matrix group:UT(3,3)
5 125 unitriangular matrix group:UT(3,5) 1 (occurs 25 times), 5 (occurs 4 times) 29 linear representation theory of unitriangular matrix group:UT(3,5)
7 343 unitriangular matrix group:UT(3,7) 1 (occurs 49 times), 7 (occurs 6 times) 55 linear representation theory of unitriangular matrix group:UT(3,7)

Irreducible representations

Description of collection of representations Parameter for describing each representation How the representation is described Degree of each representation Number of representations Sum of squares of degrees
trivial representation -- trivial representation, sends everything to (1) 1 1 1
representation with kernel a maximal subgroup maximal subgroup, which is equivalent to specifying a one-dimensional subspace of a two-dimensional space over \mathbb{F}_p, and a nontrivial homomorphism from the quotient, identified with the additive group of \mathbb{F}_p, to \mathbb{C}^\ast. Equivalently, a linear functional on a two-dimensional vector space one-dimensional representation with kernel the maximal subgroup and homomorphism as described on the quotient. 1 p^2 - 1 p^2 - 1
representation induced from a nontrivial one-dimensional representation on an elementary abelian maximal subgroup that is 1 on a complement to the center in that subgroup. The original representation restricted to the center, which is effectively specifying a nontrivial character of the additive group mod p Induce the representation. p p - 1 p^3 - p^2
Total (3 rows) -- -- -- p^2 + p - 1 p^3

Character table

FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):
Orthogonality relations: Character orthogonality theorem | Column orthogonality theorem
Separation results (basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zero
Numerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integers
Character value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma

The character table is as follows. The characters of the one-dimensional representations are easy to see: they are 1 on their respective kernels, and take values the p^{th} roots of unity on the cosets of these kernels.

The p-dimensional representations have characters as follows: the character value on the center is p times the value of the corresponding one-dimensional character on the center. The character value outside the center is zero.

Kirillov orbit method for character computation

Further information: Kirillov orbit method for finite Lazard Lie group

The analysis is pretty similar both for the case p = 2 and for odd p, but there are a few minor differences, so the cases are presented separately.

Case of odd prime: orbit analysis

In this case, the Lazard Lie ring is the niltriangular matrix Lie ring:NT(3,p) (for more information, see Baer correspondence between NT(3,p) and UT(3,p)). Explicitly, the bijection is given by matrix exponentiation:

X \mapsto I + X + \frac{X^2}{2}

where the addition and multiplication are carried out as matrices. Note that higher powers of X do not appear because X^3 becomes zero.

The additive group of NT(3,p) is a \mathbb{F}_p-vector space, i.e., it is an elementary abelian group with exponent p. Thus, the characters of this all map to the group of p^{th} roots of unity. Modulo a choice of isomorphism between the additive group of \mathcal{F}_p and the multiplicative group of p^{th} roots of unity, the Pontryagin dual can be identified with the dual vector space to NT(3,p) as a \mathbb{F}_p-vector space. Since the dual vector space is easier to understand, we will carry out our analysis in these terms.

The dual vector space of NT(3,p) is a three-dimensional vector space, comprising the linear functionals on NT(3,p). Any nonzero element of this vector space is a linear functional with kernel a subspace of codimension one, which in this case means an additive subgroup of index p. Further, for each subspace of codimension one (i.e., each additive subgroup of index p) there are p - 1 possible linear functionals to \mathbb{F}_p (and, once we have fixed the isomorphism to the p^{th} roots of unity, there are p - 1 characters with that subgroup as kernel).

The action of UT(3,p) by conjugation cannot send any linear functional to a different linear functional with the same kernel, because the group is a p-group and the set of linear functionals with a given kernel has size p - 1. Moreover, there is another action, namely post-multiplication by scalars, that commutes with the UT(3,p)-action. Thus, the nature of orbits of all the different linear functionals with the same kernel is the same.

If the kernel is an ideal, then it is also invariant under the conjugation action of the Lazard Lie group, so the orbit size is 1, and we thus get just a one-dimensional character of the group, coinciding with the corresponding character of the additive group. If the kernel is not an ideal, then it is not invariant under the conjugation action of the Lazard Lie group, so the orbit size is bigger than 1, and the corresponding irreducible representation of the group is not one-dimensional.

Further, note that the kernel of a linear functional is an additive subgroup, but it may or may not be a subring of the Lie ring. It is a subring if and only if it is an ideal, which in our case happens if and only if it is a subgroup of index p that contains the center of NT(3,p).

With this in mind, we can classify the orbits.

Type of linear functional (and hence, type of character of additive group) Size of orbit Degree of corresponding irreducible representation (equals square root of size of orbit) Number of orbits of kernels Number of corresponding irreducible representations = number of orbits of functionals (equals p - 1 times number of orbits of kernels, except in the zero functional case) Total number of kernels Total number of such linear functionals (or characters of the additive group) (equals p - 1 times number of orbits of kernels, except in the zero functional case)
zero functional 1 1 1 1 1 1
nonzero functional whose kernel is a subring (and hence an ideal) in NT(3,p). In our case, this is equivalent to the requirement that the kernel contain the center of NT(3,p), which is a subring of order p. 1 1 p + 1 p^2 - 1 p + 1 p^2 - 1
nonzero functional whose kernel is not a subring p^2 p 1 p - 1 p^2 p^3 - p^2
Total -- -- -- p^2 + p - 1 -- p^3

Case of odd prime: character computation

We can use the explicit construction of the induced representation for the Kirillov orbit method, as well as the corresponding character formula, to compute the characters of the irreducible representations of UT(3,p).

Zero functional case

Item Value
Size of orbit 1
Description of degenerate subring whole ring NT(3,p)
Description of stabilizer of any member of the orbit whole group UT(3,p)
index of the stabilizer equals size of the orbit, which is 1.
Description of polarizing subring whole ring NT(3,p) is the only polarizing subring. It has index 1.
Induced representation Trivial representation of the whole group.
Description of character 1 on all group elements
Number of such orbits, and hence number of irreducible representations 1

Nonzero functional whose kernel is an ideal

Item Value
Size of orbit 1
Description of degenerate subring whole ring NT(3,p)
Description of stabilizer of any member of the orbit whole group UT(3,p)
index of the stabilizer equals size of the orbit, which is 1.
Description of polarizing subring whole ring NT(3,p) is the only polarizing subring. It has index 1.
Induced representation, i.e., the irreducible representation of UT(3,p) corresponding to this coadjoint orbit The pullback to UT(3,p) (via the bijection between <nath>NT(3,p)</math> and UT(3,p)) of the sole one-dimensional character in the coadjoint orbit.
Degree of irreducible representation obtained = index of polarizing subring = square root of size of orbit 1
Description of character of the irreducible representation The pullback of the original character of NT(3,p) under the bijection between NT(3,p) and UT(3,p).
Number of such irreducible representations = number of such coadjoint orbits p^2 - 1 (these are essentially the linear functionals on the quotient by the center, which is a two-dimensional vector space over \mathbb{F}_p)

Nonzero functional whose kernel is not an ideal/subring

Item Value
Size of orbit p^2
Description of degenerate subring center of NT(3,p)
Description of stabilizer of any member of the orbit center of UT(3,p)
index of the stabilizer equals size of the orbit, which is p^2.
Description of polarizing subring Any subring of order p^2 (which must therefore also be an ideal) is a polarizing subring for any representative of the orbit. Each polarizing subring has index p.
Induced representation, which is the irreducible representation of UT(3,p) corresponding to this coadjoint orbit. Induce from any subgroup of order p^2 any one-dimensional character that is nontrivial on the center (an order p subgroup of it).
Degree of irreducible representation obtained = index of polarizing subring = square root of size of orbit p
Description of character of the irreducible representation The character restricted to the center is p times the restriction of the original character of NT(3,p). Outside the center, the character value is zero.
Number of such irreducible representations = number of such coadjoint orbits p - 1

Case of the prime two

Further information: Kirillov orbit method for finite inner-Lazard Lie group

The procedure outlined above works for the prime two, with the following caveat: the unitriangular matrix group UT(3,2), which is isomorphic to dihedral group:D8, is not a Lazard Lie group, so the niltriangular matrix Lie ring NT(3,2) is not its Lazard Lie ring. However, we can still consider the action and use the variant of the Kirillov orbit method for inner-Lazard Lie groups.