...
 
Commits (2)
......@@ -99,12 +99,13 @@ InstallGlobalFunction( BlockDiagonalRepresentationFast, function(rho, args...)
# We don't know the basis that the new_rho(g) are written in, but
# since the representations are isomorphic, there is a basis
# change matrix A such that new_rho(g) = A^-1 * rho(g) * A
# change matrix A such that new_rho(g) = A^-1 * rho(g) * A for all g.
#
# This is an intertwining operator for rho and new_rho, or
# representation isomorphism.
# To calculate A, we use a random element of G
g := G.1;
basis_change := BasisChangeMatrixSimilar@(Image(new_rho, g), Image(rho, g));
# Note: this is where the heavy lifting of the function is
basis_change := LinearRepresentationIsomorphismSlow(new_rho, rho);
basis := TransposedMat(basis_change);
......
......@@ -10,8 +10,22 @@
#! there exists a linear map $A : V \to W$ such that for all $g \in
#! G$, $\tau(g)A = A\rho(g)$, this function returns one such $A$. $A$
#! is the isomorphism between the representations. If the
#! representations are not isomorphic, then fail is returned.
DeclareGlobalFunction( "RepresentationIsomorphism" );
#! representations are not isomorphic, then fail is returned. The
#! method used involves solving linear systems (size depending on the
#! degree of the reps), and depends on the size of $G$ as little as
#! possible.
DeclareGlobalFunction( "LinearRepresentationIsomorphism" );
#! @Arguments rho, tau
#! @Returns A matrix $A$ or fail
#! @Description The same as <Ref
#! Func="LinearRepresentationIsomorphism" />, but this function uses a
#! simpler method which involves summing over $G$. This is slow for
#! large $G$, but might be fast in the special case of a very large
#! group and very small degree representation.
DeclareGlobalFunction( "LinearRepresentationIsomorphismSlow" );
#! @Arguments rho, tau
......
......@@ -3,13 +3,13 @@
# Finds the fixed point space of the map A -> tau(gen)*A*rho(gen^-1)
FixedSpace@ := function(rho, tau, gen)
return fail;
end;
# Picks a random (nonzero) vector in the intersection of some vector
# spaces over C
RandomVectorIntersection@ := function(spaces)
return fail;
end;
# Wraps an n^2 long list into a n long list of n long lists
......@@ -17,8 +17,8 @@ WrapMatrix@ := function(vec, n)
return List([0..n-1], i -> vec{[1+n*i..n*(i+1)]});
end;
InstallGlobalFunction( RepresentationIsomorphism, function(rho, tau)
local G;
InstallGlobalFunction( LinearRepresentationIsomorphism, function(rho, tau)
local G, n, gens, fixed_spaces, A_cand;
if not AreRepsIsomorphic(rho, tau) then
return fail;
......@@ -46,6 +46,52 @@ InstallGlobalFunction( RepresentationIsomorphism, function(rho, tau)
return A_cand;
end );
# checks if it is the case that for all g in G, tau(g)*A = A*rho(g)
IsRepresentationIsomorphism@ := function(rho, tau, A)
local G, gens, n;
G := Source(rho);
n := DegreeOfRepresentation(rho);
# we only need to check the property for the generators
gens := GeneratorsOfGroup(G);
# need bijection and G-action preserving
return RankMat(A) = n and ForAll(gens, g -> Image(tau, g) * A = A * Image(rho, g));
end;
# calculates an isomorphism between rho and tau by summing over G
# (slow, but works)
# TODO: can I use a trick to sum over the generators instead of G?
InstallGlobalFunction( LinearRepresentationIsomorphismSlow, function(rho, tau, args...)
local G, n, candidate, tries;
G := Source(rho);
n := DegreeOfRepresentation(rho);
if not AreRepsIsomorphic(rho, tau) then
return fail;
fi;
# we just pick random invertible matrices and sum over the group
# until we actually get a representation isomorphism. This almost
# always happens, so we "should" get one almost always on the
# first time.
candidate := NullMat(n, n);
tries := 0;
repeat
candidate := RandomInvertibleMat(n);
candidate := Sum(G, g -> Image(tau, g) * candidate * Image(rho, g^-1));
tries := tries + 1;
until IsRepresentationIsomorphism@(rho, tau, candidate);
if Length(args) > 0 and args[1] = "print tries" then
Print(tries, " tries\n");
fi;
return candidate;
end );
# Tells you if two representations of the same group are isomorphic by
# examining characters
InstallGlobalFunction( AreRepsIsomorphic, function(rep1, rep2)
......
......@@ -15,7 +15,7 @@
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......@@ -83,38 +83,46 @@
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap3.html#X826D5ADF7FA87782">3.1-2 IsFiniteGroupPermutationRepresentation</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4.html#X7F968DF987DE4A6E">4 <span class="Heading">Computing decompositions of representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap4.html#X7C22F13E80A74438">4.1 <span class="Heading">Algorithms due to Serre</span></a>
<div class="ContChap"><a href="chap4.html#X7D9B253E794EF912">4 <span class="Heading">Isomorphisms between representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap4.html#X7AEE81C2809E0B98">4.1 <span class="Heading">Finding explicit isomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X8316BC7684A911AE">4.1-1 CanonicalDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X7DBB32527BDB15B3">4.1-2 IrreducibleDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X7FA264B77B938090">4.1-3 IrreducibleDecompositionCollected</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X7F0D3CFB7800149A">4.1-1 LinearRepresentationIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X841DE7D08491325F">4.1-2 LinearRepresentationIsomorphismSlow</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X86EB9DD586958473">4.1-3 AreRepsIsomorphic</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5.html#X81CAB9157959A34D">5 <span class="Heading">Paralellised functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap5.html#X79F2EFEC7C3EA80C">5.1 <span class="Heading">Decompositions</span></a>
<div class="ContChap"><a href="chap5.html#X7F968DF987DE4A6E">5 <span class="Heading">Computing decompositions of representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap5.html#X7C22F13E80A74438">5.1 <span class="Heading">Algorithms due to Serre</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X7CDC782E7D975A34">5.1-1 IrreducibleDecompositionCollectedParallel</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X8696CDB57D6ADF40">5.1-2 BlockDiagonalRepresentationParallel</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X8316BC7684A911AE">5.1-1 CanonicalDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X7DBB32527BDB15B3">5.1-2 IrreducibleDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X7FA264B77B938090">5.1-3 IrreducibleDecompositionCollected</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap6.html#X810FFB1C8035C8BE">6 <span class="Heading">Utility functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap6.html#X7C5563A37D566DA5">6.1 <span class="Heading">Miscellaneous</span></a>
<div class="ContChap"><a href="chap6.html#X81CAB9157959A34D">6 <span class="Heading">Paralellised functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap6.html#X79F2EFEC7C3EA80C">6.1 <span class="Heading">Decompositions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6.html#X8668A3897972C08E">6.1-1 BlockDiagonalMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6.html#X7D17785482F143B0">6.1-2 ComposeHomFunction</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6.html#X79D2F263840ED625">6.1-3 Replicate</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6.html#X7CDC782E7D975A34">6.1-1 IrreducibleDecompositionCollectedParallel</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6.html#X8696CDB57D6ADF40">6.1-2 BlockDiagonalRepresentationParallel</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap6.html#X7C3EDA5E7A24196C">6.2 <span class="Heading">Representation theoretic functions</span></a>
</div>
<div class="ContChap"><a href="chap7.html#X810FFB1C8035C8BE">7 <span class="Heading">Utility functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap7.html#X7C5563A37D566DA5">7.1 <span class="Heading">Miscellaneous</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7.html#X8668A3897972C08E">7.1-1 BlockDiagonalMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7.html#X7D17785482F143B0">7.1-2 ComposeHomFunction</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7.html#X79D2F263840ED625">7.1-3 Replicate</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap7.html#X7C3EDA5E7A24196C">7.2 <span class="Heading">Representation theoretic functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6.html#X841424DF824E258B">6.2-1 TensorProductRepLists</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6.html#X7E5D5434844A6C95">6.2-2 DirectSumRepList</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6.html#X86EB9DD586958473">6.2-3 AreRepsIsomorphic</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6.html#X85147CF97B912CC3">6.2-4 DegreeOfRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7.html#X841424DF824E258B">7.2-1 TensorProductRepLists</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7.html#X7E5D5434844A6C95">7.2-2 DirectSumRepList</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7.html#X85147CF97B912CC3">7.2-3 DegreeOfRepresentation</a></span>
</div></div>
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......@@ -124,7 +132,7 @@
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......@@ -18,7 +18,7 @@
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......@@ -86,38 +86,46 @@
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap3_mj.html#X826D5ADF7FA87782">3.1-2 IsFiniteGroupPermutationRepresentation</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4_mj.html#X7F968DF987DE4A6E">4 <span class="Heading">Computing decompositions of representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap4_mj.html#X7C22F13E80A74438">4.1 <span class="Heading">Algorithms due to Serre</span></a>
<div class="ContChap"><a href="chap4_mj.html#X7D9B253E794EF912">4 <span class="Heading">Isomorphisms between representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap4_mj.html#X7AEE81C2809E0B98">4.1 <span class="Heading">Finding explicit isomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4_mj.html#X8316BC7684A911AE">4.1-1 CanonicalDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4_mj.html#X7DBB32527BDB15B3">4.1-2 IrreducibleDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4_mj.html#X7FA264B77B938090">4.1-3 IrreducibleDecompositionCollected</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4_mj.html#X7F0D3CFB7800149A">4.1-1 LinearRepresentationIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4_mj.html#X841DE7D08491325F">4.1-2 LinearRepresentationIsomorphismSlow</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4_mj.html#X86EB9DD586958473">4.1-3 AreRepsIsomorphic</a></span>
</div></div>
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<div class="ContChap"><a href="chap5_mj.html#X81CAB9157959A34D">5 <span class="Heading">Paralellised functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap5_mj.html#X79F2EFEC7C3EA80C">5.1 <span class="Heading">Decompositions</span></a>
<div class="ContChap"><a href="chap5_mj.html#X7F968DF987DE4A6E">5 <span class="Heading">Computing decompositions of representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap5_mj.html#X7C22F13E80A74438">5.1 <span class="Heading">Algorithms due to Serre</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X7CDC782E7D975A34">5.1-1 IrreducibleDecompositionCollectedParallel</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X8696CDB57D6ADF40">5.1-2 BlockDiagonalRepresentationParallel</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X8316BC7684A911AE">5.1-1 CanonicalDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X7DBB32527BDB15B3">5.1-2 IrreducibleDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X7FA264B77B938090">5.1-3 IrreducibleDecompositionCollected</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap6_mj.html#X810FFB1C8035C8BE">6 <span class="Heading">Utility functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap6_mj.html#X7C5563A37D566DA5">6.1 <span class="Heading">Miscellaneous</span></a>
<div class="ContChap"><a href="chap6_mj.html#X81CAB9157959A34D">6 <span class="Heading">Paralellised functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap6_mj.html#X79F2EFEC7C3EA80C">6.1 <span class="Heading">Decompositions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8668A3897972C08E">6.1-1 BlockDiagonalMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7D17785482F143B0">6.1-2 ComposeHomFunction</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X79D2F263840ED625">6.1-3 Replicate</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7CDC782E7D975A34">6.1-1 IrreducibleDecompositionCollectedParallel</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8696CDB57D6ADF40">6.1-2 BlockDiagonalRepresentationParallel</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap6_mj.html#X7C3EDA5E7A24196C">6.2 <span class="Heading">Representation theoretic functions</span></a>
</div>
<div class="ContChap"><a href="chap7_mj.html#X810FFB1C8035C8BE">7 <span class="Heading">Utility functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap7_mj.html#X7C5563A37D566DA5">7.1 <span class="Heading">Miscellaneous</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7_mj.html#X8668A3897972C08E">7.1-1 BlockDiagonalMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7_mj.html#X7D17785482F143B0">7.1-2 ComposeHomFunction</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7_mj.html#X79D2F263840ED625">7.1-3 Replicate</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap7_mj.html#X7C3EDA5E7A24196C">7.2 <span class="Heading">Representation theoretic functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X841424DF824E258B">6.2-1 TensorProductRepLists</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7E5D5434844A6C95">6.2-2 DirectSumRepList</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X86EB9DD586958473">6.2-3 AreRepsIsomorphic</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X85147CF97B912CC3">6.2-4 DegreeOfRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7_mj.html#X841424DF824E258B">7.2-1 TensorProductRepLists</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7_mj.html#X7E5D5434844A6C95">7.2-2 DirectSumRepList</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7_mj.html#X85147CF97B912CC3">7.2-3 DegreeOfRepresentation</a></span>
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<div class="ChapSects"><a href="chap4.html#X7F968DF987DE4A6E">4 <span class="Heading">Computing decompositions of representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap4.html#X7C22F13E80A74438">4.1 <span class="Heading">Algorithms due to Serre</span></a>
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<div class="ChapSects"><a href="chap4.html#X7D9B253E794EF912">4 <span class="Heading">Isomorphisms between representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap4.html#X7AEE81C2809E0B98">4.1 <span class="Heading">Finding explicit isomorphisms</span></a>
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<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X8316BC7684A911AE">4.1-1 CanonicalDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X7DBB32527BDB15B3">4.1-2 IrreducibleDecomposition</a></span>
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<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap4.html#X7F0D3CFB7800149A">4.1-1 LinearRepresentationIsomorphism</a></span>
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<h3>4 <span class="Heading">Computing decompositions of representations</span></h3>
<h3>4 <span class="Heading">Isomorphisms between representations</span></h3>
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<h4>4.1 <span class="Heading">Algorithms due to Serre</span></h4>
<h4>4.1 <span class="Heading">Finding explicit isomorphisms</span></h4>
<p>These operations compute various decompositions of a representation <span class="Math">\rho : G \to GL(V)</span> where <span class="Math">G</span> is finite and <span class="Math">V</span> is a finite-dimensional <span class="Math">\mathbb{C}</span>-vector space. The terms used here are taken from Serre's Linear Representations of Finite Groups.</p>
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<h5>4.1-1 LinearRepresentationIsomorphism</h5>
<h5>4.1-1 CanonicalDecomposition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LinearRepresentationIsomorphism</code>( <var class="Arg">rho</var>, <var class="Arg">tau</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: A matrix <span class="Math">A</span> or fail</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CanonicalDecomposition</code>( <var class="Arg">rho</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Returns: List of vector spaces <span class="Math">V_i</span>, each <span class="Math">G</span>-invariant and a direct sum of isomorphic irreducibles. That is, for each <span class="Math">i</span>, <span class="Math">V_i \cong \oplus_j W_i</span> (as representations) where <span class="Math">W_i</span> is an irreducible <span class="Math">G</span>-invariant vector space.</p>
<p>Let <span class="Math">\rho : G \to GL(V)</span> and <span class="Math">\tau : G \to GL(W)</span>. If there exists a linear map <span class="Math">A : V \to W</span> such that for all <span class="Math">g \in G</span>, <span class="Math">\tau(g)A = A\rho(g)</span>, this function returns one such <span class="Math">A</span>. <span class="Math">A</span> is the isomorphism between the representations. If the representations are not isomorphic, then fail is returned. The method used involves solving linear systems (size depending on the degree of the reps), and depends on the size of <span class="Math">G</span> as little as possible.</p>
<p>Computes the canonical decomposition of <span class="Math">V</span> into <span class="Math">\oplus_i\;V_i</span> using the formulas for projections <span class="Math">V \to V_i</span> due to Serre.</p>
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<h5>4.1-2 LinearRepresentationIsomorphismSlow</h5>
<h5>4.1-2 IrreducibleDecomposition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LinearRepresentationIsomorphismSlow</code>( <var class="Arg">rho</var>, <var class="Arg">tau</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: A matrix <span class="Math">A</span> or fail</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IrreducibleDecomposition</code>( <var class="Arg">rho</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Returns: List of vector spaces <span class="Math">W_j</span> such that <span class="Math">V = \oplus_j W_j</span> and each <span class="Math">W_j</span> is an irreducible <span class="Math">G</span>-invariant vector space.</p>
<p>The same as <code class="func">LinearRepresentationIsomorphism</code> (<a href="chap4.html#X7F0D3CFB7800149A"><span class="RefLink">4.1-1</span></a>), but this function uses a simpler method which involves summing over <span class="Math">G</span>. This is slow for large <span class="Math">G</span>, but might be fast in the special case of a very large group and very small degree representation.</p>
<p>Computes the decomposition of <span class="Math">V</span> into irreducible subprepresentations.</p>
<p><a id="X86EB9DD586958473" name="X86EB9DD586958473"></a></p>
<p><a id="X7FA264B77B938090" name="X7FA264B77B938090"></a></p>
<h5>4.1-3 AreRepsIsomorphic</h5>
<h5>4.1-3 IrreducibleDecompositionCollected</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AreRepsIsomorphic</code>( <var class="Arg">rho</var>, <var class="Arg">tau</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: true if <var class="Arg">rho</var> and <var class="Arg">tau</var> are isomorphic as representations, false otherwise.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IrreducibleDecompositionCollected</code>( <var class="Arg">rho</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Returns: List of lists <span class="Math">V_i</span> of vector spaces <span class="Math">V_{ij}</span> such that <span class="Math">V = \oplus_i \oplus_j V_{ij}</span> and <span class="Math">V_{ik} \cong V_{il}</span> for all <span class="Math">i</span>, <span class="Math">k</span> and <span class="Math">l</span> (as representations).</p>
<p>Computes the decomposition of <span class="Math">V</span> into irreducible subrepresentations, grouping together the isomorphic subrepresentations.</p>
<p>Note that two representations are isomorphic iff they give similar matrices, also iff they have the same irreducible decomposition. We use characters to determine the latter: the first is expensive to check for large degree representations.</p>
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<div class="ChapSects"><a href="chap5.html#X81CAB9157959A34D">5 <span class="Heading">Paralellised functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap5.html#X79F2EFEC7C3EA80C">5.1 <span class="Heading">Decompositions</span></a>
<p><a id="X7F968DF987DE4A6E" name="X7F968DF987DE4A6E"></a></p>
<div class="ChapSects"><a href="chap5.html#X7F968DF987DE4A6E">5 <span class="Heading">Computing decompositions of representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap5.html#X7C22F13E80A74438">5.1 <span class="Heading">Algorithms due to Serre</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X7CDC782E7D975A34">5.1-1 IrreducibleDecompositionCollectedParallel</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X8696CDB57D6ADF40">5.1-2 BlockDiagonalRepresentationParallel</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X8316BC7684A911AE">5.1-1 CanonicalDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X7DBB32527BDB15B3">5.1-2 IrreducibleDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X7FA264B77B938090">5.1-3 IrreducibleDecompositionCollected</a></span>
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<h3>5 <span class="Heading">Paralellised functions</span></h3>
<h3>5 <span class="Heading">Computing decompositions of representations</span></h3>
<p><a id="X79F2EFEC7C3EA80C" name="X79F2EFEC7C3EA80C"></a></p>
<p><a id="X7C22F13E80A74438" name="X7C22F13E80A74438"></a></p>
<h4>5.1 <span class="Heading">Decompositions</span></h4>
<h4>5.1 <span class="Heading">Algorithms due to Serre</span></h4>
<p>These functions are copies of other functions in the package, but made to run in parallel. Usually, the trick is to rearrange the formulas so that we can run independent calculations per irrep in the list of (relevant) irreps of our group <span class="Math">G</span>.</p>
<p>These operations compute various decompositions of a representation <span class="Math">\rho : G \to GL(V)</span> where <span class="Math">G</span> is finite and <span class="Math">V</span> is a finite-dimensional <span class="Math">\mathbb{C}</span>-vector space. The terms used here are taken from Serre's Linear Representations of Finite Groups.</p>
<p><a id="X7CDC782E7D975A34" name="X7CDC782E7D975A34"></a></p>
<p><a id="X8316BC7684A911AE" name="X8316BC7684A911AE"></a></p>
<h5>5.1-1 IrreducibleDecompositionCollectedParallel</h5>
<h5>5.1-1 CanonicalDecomposition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IrreducibleDecompositionCollectedParallel</code>( <var class="Arg">rho</var>, <var class="Arg">num_jobs</var>[, <var class="Arg">irreps</var>] )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: List of lists <span class="Math">V_i</span> of vector spaces <span class="Math">V_{ij}</span> such that <span class="Math">V = \oplus_i \oplus_j V_{ij}</span> and <span class="Math">V_{ik} \cong V_{il}</span> for all <span class="Math">i</span>, <span class="Math">k</span> and <span class="Math">l</span> (as representations).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CanonicalDecomposition</code>( <var class="Arg">rho</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Returns: List of vector spaces <span class="Math">V_i</span>, each <span class="Math">G</span>-invariant and a direct sum of isomorphic irreducibles. That is, for each <span class="Math">i</span>, <span class="Math">V_i \cong \oplus_j W_i</span> (as representations) where <span class="Math">W_i</span> is an irreducible <span class="Math">G</span>-invariant vector space.</p>
<p>Computes the canonical decomposition of <span class="Math">V</span> into <span class="Math">\oplus_i\;V_i</span> using the formulas for projections <span class="Math">V \to V_i</span> due to Serre.</p>
<p><a id="X7DBB32527BDB15B3" name="X7DBB32527BDB15B3"></a></p>
<h5>5.1-2 IrreducibleDecomposition</h5>
<p>This function uses the same algorithm as <code class="func">IrreducibleDecompositionCollectedHybrid</code> (<span class="RefLink">???</span>), but runs the per-irrep calculation to break down the canonical summands in parallel. Uses <var class="Arg">irreps</var> as the list of relevant irreps, if given.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IrreducibleDecomposition</code>( <var class="Arg">rho</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Returns: List of vector spaces <span class="Math">W_j</span> such that <span class="Math">V = \oplus_j W_j</span> and each <span class="Math">W_j</span> is an irreducible <span class="Math">G</span>-invariant vector space.</p>
<p><a id="X8696CDB57D6ADF40" name="X8696CDB57D6ADF40"></a></p>
<p>Computes the decomposition of <span class="Math">V</span> into irreducible subprepresentations.</p>
<h5>5.1-2 BlockDiagonalRepresentationParallel</h5>
<p><a id="X7FA264B77B938090" name="X7FA264B77B938090"></a></p>
<h5>5.1-3 IrreducibleDecompositionCollected</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IrreducibleDecompositionCollected</code>( <var class="Arg">rho</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Returns: List of lists <span class="Math">V_i</span> of vector spaces <span class="Math">V_{ij}</span> such that <span class="Math">V = \oplus_i \oplus_j V_{ij}</span> and <span class="Math">V_{ik} \cong V_{il}</span> for all <span class="Math">i</span>, <span class="Math">k</span> and <span class="Math">l</span> (as representations).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; BlockDiagonalRepresentationParallel</code>( <var class="Arg">rho</var>, <var class="Arg">num_jobs</var>[, <var class="Arg">irreps</var>] )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: The same result as <code class="func">BlockDiagonalRepresentationFast</code> (<a href="chap1.html#X87AA8AD6818E9DCA"><span class="RefLink">1.1-3</span></a>).</p>
<p>Computes the decomposition of <span class="Math">V</span> into irreducible subrepresentations, grouping together the isomorphic subrepresentations.</p>
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<div class="ChapSects"><a href="chap5_mj.html#X81CAB9157959A34D">5 <span class="Heading">Paralellised functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap5_mj.html#X79F2EFEC7C3EA80C">5.1 <span class="Heading">Decompositions</span></a>
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<div class="ChapSects"><a href="chap5_mj.html#X7F968DF987DE4A6E">5 <span class="Heading">Computing decompositions of representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap5_mj.html#X7C22F13E80A74438">5.1 <span class="Heading">Algorithms due to Serre</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X7CDC782E7D975A34">5.1-1 IrreducibleDecompositionCollectedParallel</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X8696CDB57D6ADF40">5.1-2 BlockDiagonalRepresentationParallel</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X8316BC7684A911AE">5.1-1 CanonicalDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X7DBB32527BDB15B3">5.1-2 IrreducibleDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X7FA264B77B938090">5.1-3 IrreducibleDecompositionCollected</a></span>
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<h3>5 <span class="Heading">Paralellised functions</span></h3>
<h3>5 <span class="Heading">Computing decompositions of representations</span></h3>
<p><a id="X79F2EFEC7C3EA80C" name="X79F2EFEC7C3EA80C"></a></p>
<p><a id="X7C22F13E80A74438" name="X7C22F13E80A74438"></a></p>
<h4>5.1 <span class="Heading">Decompositions</span></h4>
<h4>5.1 <span class="Heading">Algorithms due to Serre</span></h4>
<p>These functions are copies of other functions in the package, but made to run in parallel. Usually, the trick is to rearrange the formulas so that we can run independent calculations per irrep in the list of (relevant) irreps of our group <span class="SimpleMath">\(G\)</span>.</p>
<p>These operations compute various decompositions of a representation <span class="SimpleMath">\(\rho : G \to GL(V)\)</span> where <span class="SimpleMath">\(G\)</span> is finite and <span class="SimpleMath">\(V\)</span> is a finite-dimensional <span class="SimpleMath">\(\mathbb{C}\)</span>-vector space. The terms used here are taken from Serre's Linear Representations of Finite Groups.</p>
<p><a id="X7CDC782E7D975A34" name="X7CDC782E7D975A34"></a></p>
<p><a id="X8316BC7684A911AE" name="X8316BC7684A911AE"></a></p>
<h5>5.1-1 IrreducibleDecompositionCollectedParallel</h5>
<h5>5.1-1 CanonicalDecomposition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IrreducibleDecompositionCollectedParallel</code>( <var class="Arg">rho</var>, <var class="Arg">num_jobs</var>[, <var class="Arg">irreps</var>] )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: List of lists <span class="SimpleMath">\(V_i\)</span> of vector spaces <span class="SimpleMath">\(V_{ij}\)</span> such that <span class="SimpleMath">\(V = \oplus_i \oplus_j V_{ij}\)</span> and <span class="SimpleMath">\(V_{ik} \cong V_{il}\)</span> for all <span class="SimpleMath">\(i\)</span>, <span class="SimpleMath">\(k\)</span> and <span class="SimpleMath">\(l\)</span> (as representations).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CanonicalDecomposition</code>( <var class="Arg">rho</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Returns: List of vector spaces <span class="SimpleMath">\(V_i\)</span>, each <span class="SimpleMath">\(G\)</span>-invariant and a direct sum of isomorphic irreducibles. That is, for each <span class="SimpleMath">\(i\)</span>, <span class="SimpleMath">\(V_i \cong \oplus_j W_i\)</span> (as representations) where <span class="SimpleMath">\(W_i\)</span> is an irreducible <span class="SimpleMath">\(G\)</span>-invariant vector space.</p>
<p>Computes the canonical decomposition of <span class="SimpleMath">\(V\)</span> into <span class="SimpleMath">\(\oplus_i\;V_i\)</span> using the formulas for projections <span class="SimpleMath">\(V \to V_i\)</span> due to Serre.</p>
<p><a id="X7DBB32527BDB15B3" name="X7DBB32527BDB15B3"></a></p>
<h5>5.1-2 IrreducibleDecomposition</h5>
<p>This function uses the same algorithm as <code class="func">IrreducibleDecompositionCollectedHybrid</code> (<span class="RefLink">???</span>), but runs the per-irrep calculation to break down the canonical summands in parallel. Uses <var class="Arg">irreps</var> as the list of relevant irreps, if given.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IrreducibleDecomposition</code>( <var class="Arg">rho</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Returns: List of vector spaces <span class="SimpleMath">\(W_j\)</span> such that <span class="SimpleMath">\(V = \oplus_j W_j\)</span> and each <span class="SimpleMath">\(W_j\)</span> is an irreducible <span class="SimpleMath">\(G\)</span>-invariant vector space.</p>
<p><a id="X8696CDB57D6ADF40" name="X8696CDB57D6ADF40"></a></p>
<p>Computes the decomposition of <span class="SimpleMath">\(V\)</span> into irreducible subprepresentations.</p>
<h5>5.1-2 BlockDiagonalRepresentationParallel</h5>
<p><a id="X7FA264B77B938090" name="X7FA264B77B938090"></a></p>
<h5>5.1-3 IrreducibleDecompositionCollected</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IrreducibleDecompositionCollected</code>( <var class="Arg">rho</var> )</td><td class="tdright">(&nbsp;attribute&nbsp;)</td></tr></table></div>
<p>Returns: List of lists <span class="SimpleMath">\(V_i\)</span> of vector spaces <span class="SimpleMath">\(V_{ij}\)</span> such that <span class="SimpleMath">\(V = \oplus_i \oplus_j V_{ij}\)</span> and <span class="SimpleMath">\(V_{ik} \cong V_{il}\)</span> for all <span class="SimpleMath">\(i\)</span>, <span class="SimpleMath">\(k\)</span> and <span class="SimpleMath">\(l\)</span> (as representations).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; BlockDiagonalRepresentationParallel</code>( <var class="Arg">rho</var>, <var class="Arg">num_jobs</var>[, <var class="Arg">irreps</var>] )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: The same result as <code class="func">BlockDiagonalRepresentationFast</code> (<a href="chap1_mj.html#X87AA8AD6818E9DCA"><span class="RefLink">1.1-3</span></a>).</p>
<p>Computes the decomposition of <span class="SimpleMath">\(V\)</span> into irreducible subrepresentations, grouping together the isomorphic subrepresentations.</p>
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<div class="ChapSects"><a href="chap7.html#X810FFB1C8035C8BE">7 <span class="Heading">Utility functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap7.html#X7C5563A37D566DA5">7.1 <span class="Heading">Miscellaneous</span></a>
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<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7.html#X8668A3897972C08E">7.1-1 BlockDiagonalMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7.html#X7D17785482F143B0">7.1-2 ComposeHomFunction</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap7.html#X7C3EDA5E7A24196C">7.2 <span class="Heading">Representation theoretic functions</span></a>
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<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7.html#X841424DF824E258B">7.2-1 TensorProductRepLists</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7.html#X7E5D5434844A6C95">7.2-2 DirectSumRepList</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7.html#X85147CF97B912CC3">7.2-3 DegreeOfRepresentation</a></span>
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<h3>7 <span class="Heading">Utility functions</span></h3>
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<h4>7.1 <span class="Heading">Miscellaneous</span></h4>
<p><a id="X8668A3897972C08E" name="X8668A3897972C08E"></a></p>
<h5>7.1-1 BlockDiagonalMatrix</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; BlockDiagonalMatrix</code>( <var class="Arg">blocks</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: Matrix given by putting the given matrix <var class="Arg">blocks</var> on the diagonal</p>
<p><a id="X7D17785482F143B0" name="X7D17785482F143B0"></a></p>
<h5>7.1-2 ComposeHomFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ComposeHomFunction</code>( <var class="Arg">hom</var>, <var class="Arg">func</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: Homomorphism g given by g(x) = func(hom(x)).</p>
<p>This is mainly for convenience, since it handles all GAP accounting issues regarding the range, ByImages vs ByFunction, etc.</p>
<p><a id="X79D2F263840ED625" name="X79D2F263840ED625"></a></p>
<h5>7.1-3 Replicate</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Replicate</code>( <var class="Arg">elem</var>, <var class="Arg">n</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: List of <var class="Arg">n</var> copies of <var class="Arg">elem</var></p>
<p><a id="X7C3EDA5E7A24196C" name="X7C3EDA5E7A24196C"></a></p>
<h4>7.2 <span class="Heading">Representation theoretic functions</span></h4>
<p><a id="X841424DF824E258B" name="X841424DF824E258B"></a></p>
<h5>7.2-1 TensorProductRepLists</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; TensorProductRepLists</code>( <var class="Arg">list1</var>, <var class="Arg">list2</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: All possible tensor products given by <span class="Math">\rho \otimes \tau</span> where <span class="Math">\rho</span> is taken from <var class="Arg">list1</var> and <span class="Math">\tau</span> is taken from <var class="Arg">list2</var>.</p>
<p><a id="X7E5D5434844A6C95" name="X7E5D5434844A6C95"></a></p>
<h5>7.2-2 DirectSumRepList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DirectSumRepList</code>( <var class="Arg">list</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: Direct sum of the list of representations <var class="Arg">list</var></p>
<p><a id="X85147CF97B912CC3" name="X85147CF97B912CC3"></a></p>
<h5>7.2-3 DegreeOfRepresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DegreeOfRepresentation</code>( <var class="Arg">rho</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: Degree of the representation <var class="Arg">rho</var>. That is, <span class="Math">\mbox{Tr}(\rho(e_G))</span>, where <span class="Math">e_G</span> is the identity of the group <span class="Math">G</span> that <var class="Arg">rho</var> has as domain.</p>
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<div class="ChapSects"><a href="chap7_mj.html#X810FFB1C8035C8BE">7 <span class="Heading">Utility functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap7_mj.html#X7C5563A37D566DA5">7.1 <span class="Heading">Miscellaneous</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7_mj.html#X8668A3897972C08E">7.1-1 BlockDiagonalMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7_mj.html#X7D17785482F143B0">7.1-2 ComposeHomFunction</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7_mj.html#X79D2F263840ED625">7.1-3 Replicate</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap7_mj.html#X7C3EDA5E7A24196C">7.2 <span class="Heading">Representation theoretic functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7_mj.html#X841424DF824E258B">7.2-1 TensorProductRepLists</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap7_mj.html#X7E5D5434844A6C95">7.2-2 DirectSumRepList</a></span>
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<h3>7 <span class="Heading">Utility functions</span></h3>
<p><a id="X7C5563A37D566DA5" name="X7C5563A37D566DA5"></a></p>
<h4>7.1 <span class="Heading">Miscellaneous</span></h4>
<p><a id="X8668A3897972C08E" name="X8668A3897972C08E"></a></p>
<h5>7.1-1 BlockDiagonalMatrix</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; BlockDiagonalMatrix</code>( <var class="Arg">blocks</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: Matrix given by putting the given matrix <var class="Arg">blocks</var> on the diagonal</p>
<p><a id="X7D17785482F143B0" name="X7D17785482F143B0"></a></p>
<h5>7.1-2 ComposeHomFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ComposeHomFunction</code>( <var class="Arg">hom</var>, <var class="Arg">func</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: Homomorphism g given by g(x) = func(hom(x)).</p>
<p>This is mainly for convenience, since it handles all GAP accounting issues regarding the range, ByImages vs ByFunction, etc.</p>
<p><a id="X79D2F263840ED625" name="X79D2F263840ED625"></a></p>
<h5>7.1-3 Replicate</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Replicate</code>( <var class="Arg">elem</var>, <var class="Arg">n</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: List of <var class="Arg">n</var> copies of <var class="Arg">elem</var></p>
<p><a id="X7C3EDA5E7A24196C" name="X7C3EDA5E7A24196C"></a></p>
<h4>7.2 <span class="Heading">Representation theoretic functions</span></h4>
<p><a id="X841424DF824E258B" name="X841424DF824E258B"></a></p>
<h5>7.2-1 TensorProductRepLists</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; TensorProductRepLists</code>( <var class="Arg">list1</var>, <var class="Arg">list2</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: All possible tensor products given by <span class="SimpleMath">\(\rho \otimes \tau\)</span> where <span class="SimpleMath">\(\rho\)</span> is taken from <var class="Arg">list1</var> and <span class="SimpleMath">\(\tau\)</span> is taken from <var class="Arg">list2</var>.</p>
<p><a id="X7E5D5434844A6C95" name="X7E5D5434844A6C95"></a></p>
<h5>7.2-2 DirectSumRepList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DirectSumRepList</code>( <var class="Arg">list</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: Direct sum of the list of representations <var class="Arg">list</var></p>
<p><a id="X85147CF97B912CC3" name="X85147CF97B912CC3"></a></p>
<h5>7.2-3 DegreeOfRepresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; DegreeOfRepresentation</code>( <var class="Arg">rho</var> )</td><td class="tdright">(&nbsp;function&nbsp;)</td></tr></table></div>
<p>Returns: Degree of the representation <var class="Arg">rho</var>. That is, <span class="SimpleMath">\(\mbox{Tr}(\rho(e_G))\)</span>, where <span class="SimpleMath">\(e_G\)</span> is the identity of the group <span class="SimpleMath">\(G\)</span> that <var class="Arg">rho</var> has as domain.</p>
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<h3>Index</h3>
<code class="func">AreRepsIsomorphic</code> <a href="chap6.html#X86EB9DD586958473">6.2-3</a> <br />
<code class="func">AreRepsIsomorphic</code> <a href="chap4.html#X86EB9DD586958473">4.1-3</a> <br />
<code class="func">BlockDiagonalBasis</code>, for IsGroupHomomorphism <a href="chap1.html#X796E97C0829CAC27">1.1-1</a> <br />
<code class="func">BlockDiagonalMatrix</code> <a href="chap6.html#X8668A3897972C08E">6.1-1</a> <br />
<code class="func">BlockDiagonalMatrix</code> <a href="chap7.html#X8668A3897972C08E">7.1-1</a> <br />
<code class="func">BlockDiagonalRepresentation</code>, for IsGroupHomomorphism <a href="chap1.html#X85D284E386A4257F">1.1-2</a> <br />
<code class="func">BlockDiagonalRepresentationFast</code> <a href="chap1.html#X87AA8AD6818E9DCA">1.1-3</a> <br />
<code class="func">BlockDiagonalRepresentationParallel</code> <a href="chap5.html#X8696CDB57D6ADF40">5.1-2</a> <br />
<code class="func">CanonicalDecomposition</code>, for IsGroupHomomorphism <a href="chap4.html#X8316BC7684A911AE">4.1-1</a> <br />
<code class="func">BlockDiagonalRepresentationParallel</code> <a href="chap6.html#X8696CDB57D6ADF40">6.1-2</a> <br />
<code class="func">CanonicalDecomposition</code>, for IsGroupHomomorphism <a href="chap5.html#X8316BC7684A911AE">5.1-1</a> <br />
<code class="func">ClassSumCentralizer</code> <a href="chap2.html#X87E5BAEB82DC00C3">2.2-2</a> <br />
<code class="func">ComposeHomFunction</code> <a href="chap6.html#X7D17785482F143B0">6.1-2</a> <br />
<code class="func">DegreeOfRepresentation</code> <a href="chap6.html#X85147CF97B912CC3">6.2-4</a> <br />
<code class="func">DirectSumRepList</code> <a href="chap6.html#X7E5D5434844A6C95">6.2-2</a> <br />
<code class="func">IrreducibleDecomposition</code>, for IsGroupHomomorphism <a href="chap4.html#X7DBB32527BDB15B3">4.1-2</a> <br />
<code class="func">IrreducibleDecompositionCollected</code>, for IsGroupHomomorphism <a href="chap4.html#X7FA264B77B938090">4.1-3</a> <br />
<code class="func">IrreducibleDecompositionCollectedParallel</code> <a href="chap5.html#X7CDC782E7D975A34">5.1-1</a> <br />
<code class="func">ComposeHomFunction</code> <a href="chap7.html#X7D17785482F143B0">7.1-2</a> <br />
<code class="func">DegreeOfRepresentation</code> <a href="chap7.html#X85147CF97B912CC3">7.2-3</a> <br />
<code class="func">DirectSumRepList</code> <a href="chap7.html#X7E5D5434844A6C95">7.2-2</a> <br />
<code class="func">IrreducibleDecomposition</code>, for IsGroupHomomorphism <a href="chap5.html#X7DBB32527BDB15B3">5.1-2</a> <br />
<code class="func">IrreducibleDecompositionCollected</code>, for IsGroupHomomorphism <a href="chap5.html#X7FA264B77B938090">5.1-3</a> <br />
<code class="func">IrreducibleDecompositionCollectedParallel</code> <a href="chap6.html#X7CDC782E7D975A34">6.1-1</a> <br />
<code class="func">IsFiniteGroupLinearRepresentation</code>, for IsGroupHomomorphism <a href="chap3.html#X8631A1417C3C1D88">3.1-1</a> <br />
<code class="func">IsFiniteGroupPermutationRepresentation</code>, for IsGroupHomomorphism <a href="chap3.html#X826D5ADF7FA87782">3.1-2</a> <br />
<code class="func">Replicate</code> <a href="chap6.html#X79D2F263840ED625">6.1-3</a> <br />
<code class="func">LinearRepresentationIsomorphism</code> <a href="chap4.html#X7F0D3CFB7800149A">4.1-1</a> <br />
<code class="func">LinearRepresentationIsomorphismSlow</code> <a href="chap4.html#X841DE7D08491325F">4.1-2</a> <br />
<code class="func">Replicate</code> <a href="chap7.html#X79D2F263840ED625">7.1-3</a> <br />
<code class="func">RepresentationCentralizer</code> <a href="chap2.html#X7DEC18DE7E169D9D">2.2-1</a> <br />
<code class="func">RepresentationCentralizerBlocks</code> <a href="chap2.html#X798AD78F858A73FF">2.1-1</a> <br />
<code class="func">TensorProductRepLists</code> <a href="chap6.html#X841424DF824E258B">6.2-1</a> <br />
<code class="func">TensorProductRepLists</code> <a href="chap7.html#X841424DF824E258B">7.2-1</a> <br />
<p> </p>
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<div class="index">
<h3>Index</h3>
<code class="func">AreRepsIsomorphic</code> <a href="chap6_mj.html#X86EB9DD586958473">6.2-3</a> <br />
<code class="func">AreRepsIsomorphic</code> <a href="chap4_mj.html#X86EB9DD586958473">4.1-3</a> <br />
<code class="func">BlockDiagonalBasis</code>, for IsGroupHomomorphism <a href="chap1_mj.html#X796E97C0829CAC27">1.1-1</a> <br />
<code class="func">BlockDiagonalMatrix</code> <a href="chap6_mj.html#X8668A3897972C08E">6.1-1</a> <br />
<code class="func">BlockDiagonalMatrix</code> <a href="chap7_mj.html#X8668A3897972C08E">7.1-1</a> <br />
<code class="func">BlockDiagonalRepresentation</code>, for IsGroupHomomorphism <a href="chap1_mj.html#X85D284E386A4257F">1.1-2</a> <br />
<code class="func">BlockDiagonalRepresentationFast</code> <a href="chap1_mj.html#X87AA8AD6818E9DCA">1.1-3</a> <br />
<code class="func">BlockDiagonalRepresentationParallel</code> <a href="chap5_mj.html#X8696CDB57D6ADF40">5.1-2</a> <br />
<code class="func">CanonicalDecomposition</code>, for IsGroupHomomorphism <a href="chap4_mj.html#X8316BC7684A911AE">4.1-1</a> <br />
<code class="func">BlockDiagonalRepresentationParallel</code> <a href="chap6_mj.html#X8696CDB57D6ADF40">6.1-2</a> <br />
<code class="func">CanonicalDecomposition</code>, for IsGroupHomomorphism <a href="chap5_mj.html#X8316BC7684A911AE">5.1-1</a> <br />
<code class="func">ClassSumCentralizer</code> <a href="chap2_mj.html#X87E5BAEB82DC00C3">2.2-2</a> <br />
<code class="func">ComposeHomFunction</code> <a href="chap6_mj.html#X7D17785482F143B0">6.1-2</a> <br />
<code class="func">DegreeOfRepresentation</code> <a href="chap6_mj.html#X85147CF97B912CC3">6.2-4</a> <br />
<code class="func">DirectSumRepList</code> <a href="chap6_mj.html#X7E5D5434844A6C95">6.2-2</a> <br />
<code class="func">IrreducibleDecomposition</code>, for IsGroupHomomorphism <a href="chap4_mj.html#X7DBB32527BDB15B3">4.1-2</a> <br />
<code class="func">IrreducibleDecompositionCollected</code>, for IsGroupHomomorphism <a href="chap4_mj.html#X7FA264B77B938090">4.1-3</a> <br />
<code class="func">IrreducibleDecompositionCollectedParallel</code> <a href="chap5_mj.html#X7CDC782E7D975A34">5.1-1</a> <br />
<code class="func">ComposeHomFunction</code> <a href="chap7_mj.html#X7D17785482F143B0">7.1-2</a> <br />
<code class="func">DegreeOfRepresentation</code> <a href="chap7_mj.html#X85147CF97B912CC3">7.2-3</a> <br />
<code class="func">DirectSumRepList</code> <a href="chap7_mj.html#X7E5D5434844A6C95">7.2-2</a> <br />
<code class="func">IrreducibleDecomposition</code>, for IsGroupHomomorphism <a href="chap5_mj.html#X7DBB32527BDB15B3">5.1-2</a> <br />
<code class="func">IrreducibleDecompositionCollected</code>, for IsGroupHomomorphism <a href="chap5_mj.html#X7FA264B77B938090">5.1-3</a> <br />
<code class="func">IrreducibleDecompositionCollectedParallel</code> <a href="chap6_mj.html#X7CDC782E7D975A34">6.1-1</a> <br />
<code class="func">IsFiniteGroupLinearRepresentation</code>, for IsGroupHomomorphism <a href="chap3_mj.html#X8631A1417C3C1D88">3.1-1</a> <br />
<code class="func">IsFiniteGroupPermutationRepresentation</code>, for IsGroupHomomorphism <a href="chap3_mj.html#X826D5ADF7FA87782">3.1-2</a> <br />
<code class="func">Replicate</code> <a href="chap6_mj.html#X79D2F263840ED625">6.1-3</a> <br />
<code class="func">LinearRepresentationIsomorphism</code> <a href="chap4_mj.html#X7F0D3CFB7800149A">4.1-1</a> <br />
<code class="func">LinearRepresentationIsomorphismSlow</code> <a href="chap4_mj.html#X841DE7D08491325F">4.1-2</a> <br />
<code class="func">Replicate</code> <a href="chap7_mj.html#X79D2F263840ED625">7.1-3</a> <br />
<code class="func">RepresentationCentralizer</code> <a href="chap2_mj.html#X7DEC18DE7E169D9D">2.2-1</a> <br />
<code class="func">RepresentationCentralizerBlocks</code> <a href="chap2_mj.html#X798AD78F858A73FF">2.1-1</a> <br />
<code class="func">TensorProductRepLists</code> <a href="chap6_mj.html#X841424DF824E258B">6.2-1</a> <br />
<code class="func">TensorProductRepLists</code> <a href="chap7_mj.html#X841424DF824E258B">7.2-1</a> <br />
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