SymmetrizedDependentComponents

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SymmetrizedDependentComponents[comp,sym]

gives the list of components that are equivalent to the component comp by the symmetry sym.

Details

The component comp must be given as a list of positive integers.

Examples

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Basic Examples (1)Summary of the most common use cases

Components of a depth-3 array related by symmetry to component {1,3,5}:

In[1]:=1

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-ix48h7

Out[1]=1

Components vanishing by symmetry are also related to other components:

In[2]:=2

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-gu91vj

Out[2]=2

Scope (2)Survey of the scope of standard use cases

This is an array with symmetry:

In[1]:=1

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-lrvv65

Out[1]=1

These are the dependent components associated to component {1,1,2}:

In[2]:=2

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-gyyzdf

Out[2]=2

The respective values coincide by symmetry:

In[3]:=3

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-39qugz

Out[3]=3

In an array with no symmetry, all components are independent:

In[1]:=1

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-dvdfi0

Out[1]=1

Properties & Relations (4)Properties of the function, and connections to other functions

Using Symmetric, SymmetrizedDependentComponents is essentially equivalent to Permutations:

In[1]:=1

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-teleit

Out[1]=1

SymmetrizedDependentComponents allows permuting only some elements:

In[2]:=2

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-vjhn3v

Out[2]=2

SymmetrizedDependentComponents is an orbit computation under Permute action with the group associated to the symmetry permutations:

In[1]:=1

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-2jhdes

In[2]:=2

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-0vvh6r

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-bmq246

Out[3]=3

Take a symmetry for a depth-4 array:

In[1]:=1

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-2stpkk

There are 55 independent components in dimension 5:

In[2]:=2

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-4mdeqz

Out[2]=2

Compute the respective dependent components and flatten the result:

In[3]:=3

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-fyzeh4

Out[3]=3

The remaining components are all zero by symmetry:

In[4]:=4

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-z4b041

Out[4]=4

The relationship of the values of the dependent components to each other depends on the phases of the symmetry generators. For antisymmetry, the signs of the values alternate:

In[1]:=1

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-v06777

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-ox2km7

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-zy4my6

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-62w0br

Out[4]=4

Complex phases are also possible:

In[5]:=5

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-m3l560

Out[5]=5

In[6]:=6

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-z3m7r3

Out[6]=6

In[7]:=7

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-yylm91

Out[7]=7

In[8]:=8

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-2fb56b

Out[8]=8

Neat Examples (1)Surprising or curious use cases

Plot random orbits of components of an array with symmetric blocks:

In[1]:=1

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-x2uxgv

In[2]:=2

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-ffibpo

Take an array of depth 6 having 2 symmetric blocks of 3 levels:

In[3]:=3

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-gd9ppf

Out[3]=3

Random orbits of the array, after flattening it to a matrix:

In[4]:=4

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https://wolfram.com/xid/0bsvv2e84zk9k84poz-84wba9

Out[4]=4

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