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 be a prime number. This article describes the linear representation theory of the unitriangular matrix group of degree three over the ring .
Summary
Item | Value |
---|---|
degrees of irreducible representations over a splitting field (such as or ) | 1 (occurs times), (occurs times), (occurs times) |
number of conjugacy classes equals number of irreducible representations over a splitting field | |
condition for a field (characteristic not ) to be a splitting field | must contain a primitive root of unity. For a finite field of size , this is equivalent to . |
minimal splitting field (characteristic zero) | where is a primitive root of unity. This is a degree extension of the rationals. Coincides with the field generated by character values |
unique minimal splitting field (characteristic ) | The field of size where is the order of mod . |
Particular cases
Ring | Group | 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 times or 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 or 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 in . In our case, this is equivalent to the requirement that the kernel contain the center and contain all the -multiples. | 1 | 1 | ||||
nonzero functional whose kernel is not a subring but has index . In other words, the kernel contains the subgroup of -multiples but not the center. | 1 | |||||
nonzero functional whose kernel is a subring of index that contains the center (and hence the subring is also an ideal), and the quotient is cyclic of order . | 1 | 1 | ||||
nonzero functional whose kernel is a subring (not an ideal) of index that contains the -multiples in the center but not the whole center. The quotient is cyclic of order . | ||||||
nonzero functional whose kernel is a subring (not an ideal) of index that intersects the center trivially | 1 | |||||
Total (6 rows) | -- | -- | -- | (equals total number of irreducible representations, which equals number of conjugacy classes) | (equal number of additive subgroups of Lazard Lie ring with cyclic quotient group) | (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 |
Description of stabilizer of any member of the orbit | whole group index of the stabilizer equals size of the orbit, which is 1. |
Description of polarizing subring | whole ring 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 |
Description of stabilizer of any member of the orbit | whole group index of the stabilizer equals size of the orbit, which is 1. |
Description of polarizing subring | whole ring is the only polarizing subring. It has index 1. |
Induced representation, i.e., the irreducible representation of corresponding to this coadjoint orbit | The pullback to (via the bijection between <nath>NT(3,\mathbb{Z}_{p^2})</math> and ) 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 under the bijection between and . |
Number of such irreducible representations = number of such coadjoint orbits | (these are essentially the linear functionals on the quotient by the subgroup generated by the center and the -multiples, which is a two-dimensional vector space over ) |
Nonzero functional whose kernel has prime index and is not an ideal/subring
Item | Value |
---|---|
Size of orbit | |
Description of degenerate subring | subring generated by the center of and the subring comprising the -multiples. |
Description of stabilizer of any member of the orbit | subring generated by the center of and the subgroup generated by the powers. index of the stabilizer equals size of the orbit, which is . |
Description of polarizing subring | Any subring of order (which must therefore also be an ideal) is a polarizing subring for any representative of the orbit. Each polarizing subring has index . |
Induced representation, which is the irreducible representation of corresponding to this coadjoint orbit. | Induce from any subgroup of order any one-dimensional character that is nontrivial on the center (an order subgroup of it) and is trivial on the set of powers (an order subgroup). |
Degree of irreducible representation obtained = index of polarizing subring = square root of size of orbit | |
Description of character of the irreducible representation | The character restricted to the join of the center and the subgroup of powers is times the restriction of the original character of . Outside the join, the character value is zero. |
Number of such irreducible representations = number of such coadjoint orbits |
Nonzero functional whose kernel is an ideal of prime-square index
Item | Value |
---|---|
Size of orbit | 1 |
Description of degenerate subring | whole ring |
Description of stabilizer of any member of the orbit | whole group index of the stabilizer equals size of the orbit, which is 1. |
Description of polarizing subring | whole ring is the only polarizing subring. It has index 1. |
Induced representation, i.e., the irreducible representation of corresponding to this coadjoint orbit | The pullback to (via the bijection between and ) 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 under the bijection between and . |
Number of such irreducible representations = number of such coadjoint orbits | (there are ideals of index with cyclic quotient group, and for each such ideal, there are 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 | |
Description of degenerate subring | subring generated by the center of and the subring comprising the -multiples. |
Description of stabilizer of any member of the orbit | subgroup generated by the center of and the subgroup generated by the powers. index of the stabilizer equals size of the orbit, which is . |
Description of polarizing subring | Any subring of order (which must therefore also be an ideal) is a polarizing subring for any representative of the orbit. Each polarizing subring has index . |
Induced representation, which is the irreducible representation of corresponding to this coadjoint orbit. | Induce from any subgroup of order any one-dimensional character that is nontrivial on the set of powers in the center. |
Degree of irreducible representation obtained = index of polarizing subring = square root of size of orbit | |
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 |
Nonzero functional whose kernel has prime-square index and intersects the center trivially
Item | Value |
---|---|
Size of orbit | |
Description of degenerate subring | center of |
Description of stabilizer of any member of the orbit | subgroup generated by the center of and the subgroup generated by the powers. index of the stabilizer equals size of the orbit, which is . |
Description of polarizing subring | Some sort of subring of order . Each polarizing subring has index . |
Induced representation, which is the irreducible representation of 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 | |
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 |
Case
Further information: Kirillov orbit method for finite inner-Lazard Lie group
Essentially, the calculations above also work for the case , even though we don't strictly have the Lazard correspondence.