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group.md

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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

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.