Merge remote-tracking branch 'origin/master' into #1

README.md

@@ -3,10 +3,16 @@
 Automatical Generator of Conformal Bootstrap Equation
 
 For more information, see
-[autoboot: A generator of bootstrap equations with global symmetry](https://arxiv.org/abs/1903.10522).
+[autoboot: A generator of bootstrap equations with global symmetry](https://arxiv.org/abs/1903.10522),
+or [An Automated Generation of Bootstrap Equations for Numerical Study of Critical Phenomena](https://arxiv.org/abs/2006.04173).
 
 Some usages also can be checked by typing `?someSymbolName` (for example, `?getGroup`) in [Mathematica](http://reference.wolfram.com/language/tutorial/GettingInformationAboutWolframLanguageObjects.html).
 ``?somePackageName`*`` (for example, ``?ClebschGordan`*``) will give package-lebel information.
+These usages are also documented in [inv.md](doc/inv.md), [group.md](doc/group.md).
+
+- [Setup](#setup)
+- [Usage](#usage)
+- [Example](#example)
 
 ## Setup
 

doc/CustomGroup.md

@@ -6,6 +6,14 @@ You also attach labels of irrep-objects for all (unique) irreps.
 [`getO[3]`](https://github.com/selpoG/autoboot/tree/master/grouplie.m) is
 a good example for creating custom group-objects.
 
+- [`isrep`](#isrep)
+- [`minrep`](#minrep)
+- [`dim`](#dim)
+- [`dual`](#dual)
+- [`id`](#id)
+- [`prod`](#prod)
+- [`gG` and `gA`](#gg-and-ga)
+
 ## `isrep`
 
 `g[isrep[r]]` must give `True` or `False` (`True` if and only if `r` is a valid irrep-object for `g`).

doc/IrrepLabels.md

@@ -1,5 +1,17 @@
 # Labels for Irreps
 
+- [`G=group[g,i]`](#ggroupgi)
+- [`G=so[2]`](#gso2)
+- [`G=o[2]`](#go2)
+- [`G=so[3]`](#gso3)
+- [`G=o[3]`](#go3)
+- [`G=su[2]`](#gsu2)
+- [`G=su[4]`](#gsu4)
+- [`G=dih[N]` (`N`: even)](#gdihn-n-even)
+- [`G=dih[N]` (`N`: odd)](#gdihn-n-odd)
+- [`G=dic[N]`](#gdicn)
+- [`G=pGroup[G1,G2]`](#gpgroupg1g2)
+
 All labels of irreps are defined as follows:
 
 ## `G=group[g,i]`

doc/group.md

@@ -0,0 +1,212 @@
+# `group.m`
+
+Supported groups are direct products of finite groups and some Lie groups.
+Supported finite groups are those whose irreps was calculated by `GAP` (in `sgd/`),
+and any dihedral and quartenion groups.
+Supported Lie groups are `su[2]`, `su[4]`, `so[2]`, `o[2]`, `so[3]`, `o[3]`.
+We support only compact groups, so we can assume any finite dimensional irrep can be unitarized.
+
+This package imports `groupd.m` and `grouplie.m`.
+
+- [Get Groups](#get-groups)
+- [Group Data](#group-data)
+- [Irrep-Objects](#irrep-objects)
+- [`groupd.m`](#groupdm)
+- [`grouplie.m`](#groupliem)
+
+---
+
+## Get Groups
+
+### `getGroup`
+
+`getGroup[g,i]` loads data from `sgd/sg.g.i.m` and returns group-object `group[g,i]`.
+`g` is the order of the finite group, `i` is the number of the group assined by `GAP`.
+
+### `product`
+
+`product[g1,g2]` returns group-object `pGroup[g1,g2]`
+which represents direct product of two group-object `g1`, `g2`.
+
+### `group`
+
+`group[g,i]` is a group-object whose order is `g` and whose number assigned by `GAP` is `i`.
+Before using this value, you have to call `getGroup[g,i]` to get proper group-object.
+
+### `pGroup`
+
+`pGroup[g1,g2]` is a group-object which is a direct product of `g1`, `g2`.
+Before using this value, you have to call `product[g1,g2]` to get proper group-object.
+
+### `setGroup`
+
+`setGroup[G]` loads `inv.m` with global symmetry `G`.
+This action clears all values calculated by `inv.m` previously.
+
+### `available`
+
+`available[g,i]` gives whether `group[g,i]` are supported or not.
+
+### `setPrecision` (only in `ngroup.md`, not in `group.md`)
+
+After calling `setPrecision[prec]`, all calculation in this package will be done in precision `prec`
+and any number less than `1/10^(prec-10)` will be choped.
+
+It is assumed that `prec` is sufficiently bigger than `10` and `setPrecision` is called just once just after loading this package.
+
+---
+
+## Group Data
+
+A group-object `g` has attributes `ncg`, `ct`, `id`, `dim`, `prod`, `dual`, `isrep`, `gG`, `gA`, `minrep`.
+You can evaluate attributes in putting it in `g[...]`.
+For example, `g[dim[r]]` gives the dimension of irrep `r`.
+
+### `ncg`
+
+`ncg` is the number of conjugacy classes, which is also the number of inequivalent irreps.
+This is not defined for Lie groups.
+
+### `ct`
+
+`ct` is the character table. This is not defined for Lie groups.
+
+### `id`
+
+`id` is the trivial representation.
+
+### `dim`
+
+`dim[r]` is the dimension of irrep `r`.
+
+### `prod`
+
+`prod[r,s]` gives a list of all irreps arising
+in irreducible decomposition of direct product representation of `r` and `s`.
+`prod[r,s]` may not be duplicate-free.
+
+### `dual`
+
+`dual[r]` gives dual representation of irrep `r`.
+
+### `isrep`
+
+`isrep[r]` gives whether `r` is recognised as a irrep-object of the group-object or not.
+
+### `gG`
+
+`gG` is a list of all generator-objects of finite group part of the group-object.
+
+### `gA`
+
+`gA` is a list of all generator-objects of Lie algebra part of the group-object.
+
+### `minrep`
+
+`minrep[r,s]` gives `r` if `r < s` else `s`. `r` and `s` are irrep-objects.
+
+---
+
+## Irrep-Objects
+
+We need all irreps to be sorted in some linear order.
+All irrep-objects of `G=group[g,i]` are `rep[1]`, `rep[2]`, ..., `rep[n]` (`n=G[ncg]`).
+all irrep-objects of `pGroup[g1,g2]` are `rep[r1,r2]`,
+where `r1` is a irrep-object of `g1` and `r2` is a irrep-object of `g2`.
+`minrep` compares irreps in lexical order.
+
+### `rep`
+
+`rep[n]` is `n`-th irrep-object (`n` is assined by `GAP` and corresponds to the index of `ct`).
+This is recognised only by `group[g,i]`.
+
+`rep[r1,r2]` is natural irrep-object of `pGroup[g1,g2]`
+where `r1` is irrep-object of `g1`, `r2` is irrep-object of `g2`.
+This is recognised only by `pGroup[g1,g2]`.
+
+### `v`
+
+`v[n]` is spin-`n` irrep-object.
+This is recognised only by `dih[n]`, `dic[n]`, `su[2]`, `so[3]`, `o[2]` and `so[2]`.
+
+`v[n,s]` is spin-`n` irrep-object with sign `s`. This is recognised only by `o[3]`.
+
+### `i`
+
+`i[a]` is one-dimensional irrep-object with sign `a`.
+This is recognised only by `dih[n]` (`n`: odd) and `o[2]`.
+
+`i[a,b]` is one-dimensional irrep-object with sign `a,b`.
+This is recognised only by `dih[n]` (`n`:even), `dic[n]`.
+
+---
+
+## `groupd.m`
+
+If `n` is even, all irrep-objects of `dih[n]` are `i[1,1], i[1,-1], i[-1,1], i[-1,-1], v[1], ..., v[n/2-1]`.
+If `n` is odd, all irrep-objects of `dih[n]` are `i[1], i[-1] v[1], ..., v[(n-1)/2]`.
+All irrep-objects of `dic[n]` are `i[1,1], i[1,-1], i[-1,1], i[-1,-1], v[1], ..., v[n-1]`.
+
+### `getDihedral`
+
+`getDihedral[n]` returns group-object `dih[n]` which represents the dihedral group of order `2n`.
+
+### `getDicyclic`
+
+`getDicyclic[n]` returns group-object `dic[n]` which represents the dicyclic group of order `4n`.
+
+### `dih`
+
+`dih[n]` is a group-object which is the dihedral group of order `2n`.
+Before using this value, you have to call `getDihedral[n]` to get proper group-object.
+
+### `dic`
+
+`dic[n]` is a group-object which is the dicyclic group of order `4n`.
+Before using this value, you have to call `getDicyclic[n]` to get proper group-object.
+
+---
+
+## `grouplie.m`
+
+All irrep-objects of `G=su[2]` are `v[0], v[1/2], v[1], v[3/2], ...`.
+
+All irrep-objects of `G=su[4]` are `v[n, m, l]` (`n,m,l=0,1,2,...` and `n >= m >= l`).
+
+All irrep-objects of `G=o[3]` are `v[0,1], v[0,-1], v[1,1], v[1,-1], v[2,1], v[2,-1], v[3,1], v[3,-1], ...`.
+
+All irrep-objects of `G=so[3]` are `v[0], v[1], v[2], v[3], ...`.
+
+All irrep-objects of `G=o[2]` are `i[1], i[-1], v[1], v[2], v[3], ...`.
+
+All irrep-objects of `G=so[2]` are `v[x]` (`x \in \mathbb{R}`).
+
+### `getSU`
+
+`getSU[n]` returns group-object `su[n]` which represents the special unitary group of rank `n`.
+`n` must be `2,4`.
+
+### `getO`
+
+`getO[n]` returns group-object `o[n]` which represents the orthogonal group of rank `n`.
+`n` must be `2,3`.
+
+### `getSO`
+
+`getSO[n]` returns group-object `su[n]` which represents the special orthogonal group of rank `n`.
+`n` must be `2,3`.
+
+### `su`
+
+`su[n]` is a group-object which is the special unitary group of rank `n`.
+Before using this value, you have to call `getSU[n]` to get proper group-object.
+
+### `o`
+
+`o[n]` is a group-object which is the orthogonal group of rank `n`.
+Before using this value, you have to call `getO[n]` to get proper group-object.
+
+### `so`
+
+`so[n]` is a group-object which is the special orthogonal group of rank `n`.
+Before using this value, you have to call `getSO[n]` to get proper group-object.