Linear representation theory of unitriangular matrix group:UT(3,Zp^2)

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This article gives specific information, namely, linear representation theory, about a family of groups, namely: unitriangular matrix group of degree three.
View linear representation theory of group families | View other specific information about unitriangular matrix group of degree three

Let p be a prime number. This article describes the linear representation theory of the unitriangular matrix group of degree three over the ring \mathbb{Z}_{p^2} = \mathbb{Z}/p^2\mathbb{Z}.

Summary

Item Value
degrees of irreducible representations over a splitting field (such as \overline{\mathbb{Q}} or \mathbb{C}) 1 (occurs p^4 times), p (occurs p^3 - p^2 times), p^2 (occurs p^2 - p times)
number of conjugacy classes equals number of irreducible representations over a splitting field p^4 + p^3 - p
condition for a field (characteristic not p) to be a splitting field must contain a primitive (p^2)^{th} root of unity. For a finite field of size s, this is equivalent to s \equiv 1 \pmod{p^2}.
minimal splitting field (characteristic zero) \mathbb{Q}(\zeta) where \zeta is a primitive (p^2)^{th} root of unity. This is a degree p(p-1) extension of the rationals.
Coincides with the field generated by character values
unique minimal splitting field (characteristic c \ne 0,p) The field of size c^r where r is the order of c mod p.


Particular cases

p p^2 Ring \mathbb{Z}_{p^2} = \mathbb{Z}/p^2\mathbb{Z} Group UT(3,\mathbb{Z}_{p^2}) Linear representation theory
2 4 ring:Z4 unitriangular matrix group:UT(3,Z4) linear representation theory of unitriangular matrix group:UT(3,Z4)
3 9 ring:Z9 unitriangular matrix group:UT(3,Z9) [[linear representation th

Kirillov orbit method for character computation

Case of odd prime: orbit analysis

Further information: Kirillov orbit method for finite Lazard Lie group

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 orbits of functionals (equals p - 1 times or p(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 or p(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) of index p in NT(3,\mathbb{Z}/p^2\mathbb{Z}). In our case, this is equivalent to the requirement that the kernel contain the center and contain all the p-multiples. 1 1 p + 1 p^2 - 1 p + 1 p^2 - 1
nonzero functional whose kernel is not a subring but has index p. In other words, the kernel contains the subgroup of p-multiples but not the center. p^2 p 1 p - 1 p^2 p^3 - p^2
nonzero functional whose kernel is a subring of index p^2 that contains the center (and hence the subring is also an ideal), and the quotient is cyclic of order p^2. 1 1 p(p+1) p^2(p^2 - 1) p(p + 1) p^4 - p^2
nonzero functional whose kernel is a subring (not an ideal) of index p^2 that contains the p-multiples in the center but not the whole center. The quotient is cyclic of order p^2. p^2 p p^2 - 1 (p - 1)^2(p + 1) p(p^2 - 1) p^2(p-1)^2(p + 1) = p^5 - p^4 - p^3 + p^2
nonzero functional whose kernel is a subring (not an ideal) of index p^2 that intersects the center trivially p^4 p^2 1 p(p - 1) p^4 p^6 - p^5
Total (6 rows) -- -- -- p^4 + p^3 - p (equals total number of irreducible representations, which equals number of conjugacy classes) p^4 + p^3 + 2p^2 + p + 2 (equal number of additive subgroups of Lazard Lie ring with cyclic quotient group) p^6 (equals order of the whole group)

Case of odd prime: character computation

Zero functional case

Item Value
Size of orbit 1
Description of degenerate subring whole ring NT(3,\mathbb{Z}_{p^2})
Description of stabilizer of any member of the orbit whole group UT(3,\mathbb{Z}_{p^2})
index of the stabilizer equals size of the orbit, which is 1.
Description of polarizing subring whole ring NT(3,\mathbb{Z}_{p^2}) 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 of prime index

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

Nonzero functional whose kernel has prime index and is not an ideal/subring

Item Value
Size of orbit p^2
Description of degenerate subring subring generated by the center of NT(3,\mathbb{Z}_{p^2}) and the subring comprising the p-multiples.
Description of stabilizer of any member of the orbit subring generated by the center of NT(3,\mathbb{Z}_{p^2}) and the subgroup generated by the p^{th} powers.
index of the stabilizer equals size of the orbit, which is p^2.
Description of polarizing subring Any subring of order p^5 (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,\mathbb{Z}_{p^2}) corresponding to this coadjoint orbit. Induce from any subgroup of order p^5 any one-dimensional character that is nontrivial on the center (an order p^2 subgroup of it) and is trivial on the set of p^{th} powers (an order p^3 subgroup).
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 join of the center and the subgroup of p^{th} powers is p times the restriction of the original character of NT(3,\mathbb{Z}_{p^2}). Outside the join, the character value is zero.
Number of such irreducible representations = number of such coadjoint orbits p - 1

Nonzero functional whose kernel is an ideal of prime-square index

Item Value
Size of orbit 1
Description of degenerate subring whole ring NT(3,\mathbb{Z}_{p^2})
Description of stabilizer of any member of the orbit whole group UT(3,\mathbb{Z}_{p^2})
index of the stabilizer equals size of the orbit, which is 1.
Description of polarizing subring whole ring NT(3,\mathbb{Z}_{p^2}) is the only polarizing subring. It has index 1.
Induced representation, i.e., the irreducible representation of UT(3,\mathbb{Z}_{p^2}) corresponding to this coadjoint orbit The pullback to UT(3,\mathbb{Z}_{p^2}) (via the bijection between NT(3,\mathbb{Z}_{p^2}) and UT(3,\mathbb{Z}_{p^2})) 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,\mathbb{Z}_{p^2}) under the bijection between NT(3,\mathbb{Z}_{p^2}) and UT(3,\mathbb{Z}_{p^2}).
Number of such irreducible representations = number of such coadjoint orbits p^4 - p^2 (there are p(p+1) ideals of index p^2 with cyclic quotient group, and for each such ideal, there are p(p-1) characters with that ideal as kernel).

Nonzero functional whose kernel has prime-square index and intersects the center nontrivially but does not contain it

Item Value
Size of orbit p^2
Description of degenerate subring subring generated by the center of NT(3,\mathbb{Z}_{p^2}) and the subring comprising the p-multiples.
Description of stabilizer of any member of the orbit subgroup generated by the center of UT(3,\mathbb{Z}_{p^2}) and the subgroup generated by the p^{th} powers.
index of the stabilizer equals size of the orbit, which is p^2.
Description of polarizing subring Any subring of order p^5 (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,\mathbb{Z}_{p^2}) corresponding to this coadjoint orbit. Induce from any subgroup of order p^5 any one-dimensional character that is nontrivial on the set of p^{th} powers in the center.
Degree of irreducible representation obtained = index of polarizing subring = square root of size of orbit p
Description of character of the irreducible representation PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Number of such irreducible representations = number of such coadjoint orbits (p - 1)^2(p +1)

Nonzero functional whose kernel has prime-square index and intersects the center trivially

Item Value
Size of orbit p^4
Description of degenerate subring center of NT(3,\mathbb{Z}_{p^2})
Description of stabilizer of any member of the orbit subgroup generated by the center of UT(3,\mathbb{Z}_{p^2}) and the subgroup generated by the p^{th} powers.
index of the stabilizer equals size of the orbit, which is p^2.
Description of polarizing subring Some sort of subring of order p^4. Each polarizing subring has index p^2.
Induced representation, which is the irreducible representation of UT(3,\mathbb{Z}_{p^2}) corresponding to this coadjoint orbit. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Degree of irreducible representation obtained = index of polarizing subring = square root of size of orbit p^2
Description of character of the irreducible representation PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Number of such irreducible representations = number of such coadjoint orbits p(p - 1)

Case p = 2

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

Essentially, the calculations above also work for the case p = 2, even though we don't strictly have the Lazard correspondence.