httk property definitionsProperty definitions using the OPTIMADE property definition format, hosted by httk
This page documents an OPTIMADE Property Definition. See https://schemas.optimade.org/ for more information.
ID: https://schemas.httk.org/defs/v0.1/properties/pointgroups/character_table_complex
Definition name: character_table_complex
Property name: Complex character table
Description: Complex irreducible character table of the crystallographic point group.
Type: list
Rows correspond to complex irreducible representations and columns follow the order of conjugacy_classes.
Each character is stored as a dictionary with string fields re and im representing its real and imaginary components.
Components can be integers, rational fractions, or algebraic expressions such as sqrt(3)/2; they are not restricted to rational fraction strings.
For example, the C3 eigenvalue exp(2*pi*i/3) is represented by re: "-1/2" and im: "sqrt(3)/2", preserving its algebraic value without replacing the radical by a decimal approximation.
For a representation D and class representative g, the character is trace(D(g)); every member of a conjugacy class has the same character.
Each row's character list MUST have length n_conjugacy_classes, and its identity-class character MUST equal dimension.
The number of complex irreducible rows MUST equal n_conjugacy_classes, and the sum of their squared dimensions MUST equal order.
The row inner product is weighted by class size: sum_C size(C)*conj(chi_a(C))*chi_b(C)/order = delta_ab.
Conjugate one-dimensional rows use distinct _a and _b label suffixes even when their label_markup renderings coincide; join rows by label, not by markup.
The current generator places polynomial bases only in character_table_real; the optional basis fields retained here are not emitted.
In particular, the real two-dimensional span for a conjugate pair must not be interpreted as a basis for either single complex row.
Examples:
[{"label": "A", "dimension": 1, "characters": [{"re": "1", "im": "0"}], "frobenius_schur_indicator": 1, "label_markup": {"latex": "A", "unicode": "A"}}][{"label": "Ag", "dimension": 1, "characters": [{"re": "1", "im": "0"}, {"re": "1", "im": "0"}], "frobenius_schur_indicator": 1, "label_markup": {"latex": "A_{g}", "unicode": "Ag"}}, {"label": "Au", "dimension": 1, "characters": [{"re": "1", "im": "0"}, {"re": "-1", "im": "0"}], "frobenius_schur_indicator": 1, "label_markup": {"latex": "A_{u}", "unicode": "Au"}}][{"label": "A", "dimension": 1, "characters": [{"re": "1", "im": "0"}, {"re": "1", "im": "0"}, {"re": "1", "im": "0"}], "frobenius_schur_indicator": 1, "label_markup": {"latex": "A", "unicode": "A"}}, {"label": "E_a", "dimension": 1, "characters": [{"re": "1", "im": "0"}, {"re": "-1/2", "im": "sqrt(3)/2"}, {"re": "-1/2", "im": "-sqrt(3)/2"}], "frobenius_schur_indicator": 0, "label_markup": {"latex": "E", "unicode": "E"}}, {"label": "E_b", "dimension": 1, "characters": [{"re": "1", "im": "0"}, {"re": "-1/2", "im": "-sqrt(3)/2"}, {"re": "-1/2", "im": "sqrt(3)/2"}], "frobenius_schur_indicator": 0, "label_markup": {"latex": "E", "unicode": "E"}}]JSON definition:
{
"$id": "https://schemas.httk.org/defs/v0.1/properties/pointgroups/character_table_complex",
"$schema": "https://schemas.optimade.org/meta/v1.3/optimade/property_definition.json",
"title": "Complex character table",
"x-optimade-type": "list",
"x-optimade-definition": {
"kind": "property",
"version": "0.1.0",
"format": "1.3",
"name": "character_table_complex",
"label": "character_table_complex_pointgroups"
},
"type": [
"array",
"null"
],
"description": "Complex irreducible character table of the crystallographic point group.\n\nRows correspond to complex irreducible representations and columns follow the order of `conjugacy_classes`.\nEach character is stored as a dictionary with string fields `re` and `im` representing its real and imaginary components.\nComponents can be integers, rational fractions, or algebraic expressions such as `sqrt(3)/2`; they are not restricted to rational fraction strings.\nFor example, the C3 eigenvalue `exp(2*pi*i/3)` is represented by `re: \"-1/2\"` and `im: \"sqrt(3)/2\"`, preserving its algebraic value without replacing the radical by a decimal approximation.\n\nFor a representation D and class representative g, the character is `trace(D(g))`; every member of a conjugacy class has the same character.\nEach row's character list MUST have length `n_conjugacy_classes`, and its identity-class character MUST equal `dimension`.\nThe number of complex irreducible rows MUST equal `n_conjugacy_classes`, and the sum of their squared dimensions MUST equal `order`.\nThe row inner product is weighted by class size: `sum_C size(C)*conj(chi_a(C))*chi_b(C)/order = delta_ab`.\nConjugate one-dimensional rows use distinct `_a` and `_b` label suffixes even when their `label_markup` renderings coincide; join rows by `label`, not by markup.\nThe current generator places polynomial bases only in `character_table_real`; the optional basis fields retained here are not emitted.\nIn particular, the real two-dimensional span for a conjugate pair must not be interpreted as a basis for either single complex row.",
"x-optimade-unit": "inapplicable",
"items": {
"x-optimade-type": "dictionary",
"type": [
"object",
"null"
],
"description": "One row of a point-group character table.",
"properties": {
"label": {
"x-optimade-type": "string",
"type": [
"string",
"null"
],
"description": "Irreducible-representation label.",
"x-optimade-unit": "inapplicable"
},
"label_markup": {
"$id": "https://schemas.httk.org/defs/v0.1/properties/core/string_markups",
"title": "String markups",
"x-optimade-type": "dictionary",
"x-optimade-definition": {
"kind": "property",
"version": "0.1.0",
"format": "1.3",
"name": "string_markups",
"label": "string_markups_core"
},
"x-optimade-unit": "inapplicable",
"type": [
"object",
"null"
],
"description": "Strings with alternate markup and/or encoding for display rendering.\n\nThe object is intended for display-oriented variants only, a sibling property should be used for canonical plain string value.\n\n**Requirements/Conventions**:\n\n- It MUST be a dictionary with the following keys:\n\n - **html**: OPTIONAL; String.\n HTML rendering of the sibling string, using inline HTML elements where needed for typographic structure such as subscripts, superscripts, overlines, fractions, and line breaks.\n\n - **latex**: OPTIONAL; String.\n LaTeX rendering of the sibling string, suitable for use with a LaTeX or MathJax-like renderer.\n\n - **unicode**: OPTIONAL; String.\n Unicode rendering of the sibling string, using Unicode code points for display features where practical.",
"properties": {
"html": {
"x-optimade-type": "string",
"x-optimade-unit": "inapplicable",
"type": [
"string",
"null"
],
"description": "HTML rendering of the sibling string."
},
"latex": {
"x-optimade-type": "string",
"x-optimade-unit": "inapplicable",
"type": [
"string",
"null"
],
"description": "LaTeX rendering of the sibling string."
},
"unicode": {
"x-optimade-type": "string",
"x-optimade-unit": "inapplicable",
"type": [
"string",
"null"
],
"description": "Unicode rendering of the sibling string."
}
},
"examples": [
{
"html": "<i>P</i> 2<sub>1</sub>/<i>c</i>",
"latex": "\\mathit{P}\\,2_{1}/c",
"unicode": "P2\u2081/c"
}
]
},
"dimension": {
"x-optimade-type": "integer",
"type": [
"integer",
"null"
],
"description": "Dimension of the irreducible representation.",
"x-optimade-unit": "inapplicable"
},
"characters": {
"x-optimade-type": "list",
"type": [
"array",
"null"
],
"description": "Characters for the conjugacy classes in table order.",
"items": {
"x-optimade-type": "dictionary",
"type": [
"object",
"null"
],
"description": "One complex character value represented as exact real and imaginary parts.",
"x-optimade-unit": "inapplicable",
"properties": {
"re": {
"x-optimade-type": "string",
"type": [
"string",
"null"
],
"description": "Real component serialized as an integer or rational string for the generated crystallographic tables.",
"x-optimade-unit": "inapplicable"
},
"im": {
"x-optimade-type": "string",
"type": [
"string",
"null"
],
"description": "Imaginary component serialized as a symbolic real-number string, including signed radicals such as `-sqrt(3)/2`; `0` denotes a real character.",
"x-optimade-unit": "inapplicable"
}
}
},
"x-optimade-unit": "inapplicable"
},
"frobenius_schur_indicator": {
"x-optimade-type": "integer",
"enum": [
0,
1,
null
],
"type": [
"integer",
"null"
],
"description": "The complex-irrep indicator `nu = sum_g chi(g^2)/order`.\nThe emitted value 1 means the irrep admits a real realization; 0 means it is of complex type and has a distinct complex-conjugate partner.\nThe crystallographic point groups covered here have no quaternionic-type irreps (whose indicator would be -1).",
"x-optimade-unit": "inapplicable"
},
"basis_linear": {
"x-optimade-type": "list",
"type": [
"array",
"null"
],
"description": "Linear basis polynomials spanning the isotypic component of this representation in the space of linear functions.\nPolynomials use an orthonormal Cartesian frame (x along a, z along c for hexagonal axes) with exact rational coefficients, e.g. `x`, `x+y`.\nThe listed polynomials are linearly independent; repeated copies of a representation are not separated into conventional multiplets.",
"items": {
"x-optimade-type": "string",
"type": [
"string",
"null"
],
"description": "One basis polynomial in Cartesian x, y, z with rational coefficients.",
"x-optimade-unit": "inapplicable"
},
"x-optimade-unit": "inapplicable"
},
"basis_rotation": {
"x-optimade-type": "list",
"type": [
"array",
"null"
],
"description": "Axial-vector (rotation) basis polynomials spanning the isotypic component of this representation, in the same Cartesian frame as `basis_linear`, e.g. `Rz`, `Rx+Ry`.",
"items": {
"x-optimade-type": "string",
"type": [
"string",
"null"
],
"description": "One basis polynomial in Rx, Ry, Rz with rational coefficients.",
"x-optimade-unit": "inapplicable"
},
"x-optimade-unit": "inapplicable"
},
"basis_quadratic": {
"x-optimade-type": "list",
"type": [
"array",
"null"
],
"description": "Quadratic basis polynomials spanning the isotypic component of this representation in the space of homogeneous quadratic functions, in the same Cartesian frame as `basis_linear`, e.g. `x^2+y^2`, `x^2-y^2`, `xz`.",
"items": {
"x-optimade-type": "string",
"type": [
"string",
"null"
],
"description": "One quadratic basis polynomial in x^2, y^2, z^2, xy, xz, yz with rational coefficients.",
"x-optimade-unit": "inapplicable"
},
"x-optimade-unit": "inapplicable"
}
},
"x-optimade-unit": "inapplicable"
},
"examples": [
[
{
"label": "A",
"dimension": 1,
"characters": [
{
"re": "1",
"im": "0"
}
],
"frobenius_schur_indicator": 1,
"label_markup": {
"latex": "A",
"unicode": "A"
}
}
],
[
{
"label": "Ag",
"dimension": 1,
"characters": [
{
"re": "1",
"im": "0"
},
{
"re": "1",
"im": "0"
}
],
"frobenius_schur_indicator": 1,
"label_markup": {
"latex": "A_{g}",
"unicode": "Ag"
}
},
{
"label": "Au",
"dimension": 1,
"characters": [
{
"re": "1",
"im": "0"
},
{
"re": "-1",
"im": "0"
}
],
"frobenius_schur_indicator": 1,
"label_markup": {
"latex": "A_{u}",
"unicode": "Au"
}
}
],
[
{
"label": "A",
"dimension": 1,
"characters": [
{
"re": "1",
"im": "0"
},
{
"re": "1",
"im": "0"
},
{
"re": "1",
"im": "0"
}
],
"frobenius_schur_indicator": 1,
"label_markup": {
"latex": "A",
"unicode": "A"
}
},
{
"label": "E_a",
"dimension": 1,
"characters": [
{
"re": "1",
"im": "0"
},
{
"re": "-1/2",
"im": "sqrt(3)/2"
},
{
"re": "-1/2",
"im": "-sqrt(3)/2"
}
],
"frobenius_schur_indicator": 0,
"label_markup": {
"latex": "E",
"unicode": "E"
}
},
{
"label": "E_b",
"dimension": 1,
"characters": [
{
"re": "1",
"im": "0"
},
{
"re": "-1/2",
"im": "-sqrt(3)/2"
},
{
"re": "-1/2",
"im": "sqrt(3)/2"
}
],
"frobenius_schur_indicator": 0,
"label_markup": {
"latex": "E",
"unicode": "E"
}
}
]
]
}