Title: Theory / Important / Math / Al gebra / Abstract Algebra - Dummit - Foote+14{ / 6 Representation Theory=
Tags: fromMath mathToDef examplesSwitchExample1 hasSeenConjDef exampleToQuestion mathToSee closed defToAlgebra property setAdditionMultiplication
On every note of this deck, unless a note says otherwise:
Chapter: 18
Section: 1

Name: A linear representation of \(G\)
Definition: A homomorphism from \(G\) into \(GL(V)\) for \(F\) a field and \(V\) a \(F\)-vector space
Page: 840
Variables: \((G,\cdot_G)\) a finite group,
Index: (1)
Kind: Def,

Name: The degree of \(\phi\)
Definition: The dimension of \(V\).
Page: 840
Variables: \((G,\cdot _g)\) a finite group,
\((F,+_F,\cdot _F)\) a field,
\((V,+_V)\) a \(F\)-vector space,
\(\phi\) a representation of \(G\) on \(V\) over \(F\)
Index: (1)
Kind: Def,

Name: A matrix representation of \(G\)
Definition: An homomorphism from \(G\) into \(GL_n(F)\).
Page: 840
Variables: \((G,\cdot _g)\) a finite group,
\((F,+_F,\cdot _F)\) a field,
\(n\in\mathbb Z^+\),
Index: (2)
Kind: Def,

Name: A faithful linear representation/matrix representation
Definition: An injective linear/matrix representation.
Page: 840, 844
Variables: \((G,\cdot _g)\) a finite group,
\((F,+_F,\cdot _F)\) a field,
Index: (3), (2), (3)
Kind: Def, Ex, Ex
Example: Regular representation

Smaller0: \(G\) is Abelian
Greater0: commutative
Page: 841
Chapter: 18
Section: 1
Variables: \((G,\cdot _g)\) a finite group,
\((F,+_F,\cdot _F)\) a field,
Smaller to greater: iff
Prefix Greater: \(FG\) is

Page: 842
Chapter: 18
Section: 1
Name: \(V\) as a left \(FG\)-module
Set: \(FG\)
Variables: \((G,\cdot _g)\) a finite group,
\((F,+_F,\cdot _F)\) a field,
\((V,+_V)\) a \(F\)-vector space,
\(\phi\) a representation of \(G\) on \(V\) over \(F\)
+: Componentwise addition
0: \(\sum_{g\in G}0_Rg\)
×: right action generated by \(g_i\cdot_{\text{module} }v=\phi(g_i)(v)\). Hence \(\left(\sum_{FG} \alpha_i g_i\right)\cdot_{\text{module} } v=\sum_{FG}\alpha_i\left(\phi(g_i)(v)\right)\)
Ring-base: \(F\)

Smaller0: \(v\cdot_{\text{vector space}}\alpha\)
Greater0: \(v\cdot_{FG\text{module}}(\alpha1_G)\)
Page: 843
Chapter: 18
Section: 1
Variables: \((G,\cdot _g)\) a finite group,
\((F,+_F,\cdot _F)\) a field,
\((V,+_V)\) a \(F\)-vector space,
\(\phi\) a representation of \(G\) on \(V\) over \(F\),
\(\alpha\in F\),
\(v\in V\)
Smaller to greater: =
Hide middle: yes

Name: The representation of \(G\) afforded by \(V\)
Definition: Let \((V,+_V,\cdot _{F,V})\) a \(F\)-vector space, defined by \(\cdot _{F,V}:(f,v)\mapsto f\cdot_{FG,V}v\).
The representation \(\phi:g\mapsto\left(v\mapsto g \cdot_{FG,V} v\right) \)
Page: 843
Variables: \((G,\cdot _g)\) a finite group,
\((F,+_F,\cdot _F)\) a field,
\((V,+_V,\cdot_{FG,V})\) a \(FG\)-module,

Name: A \(G\)-invariant subspace of \(V\)
Definition: A subspace \(U\subseteq V\) such that \(g\cdot_{FG,U}u\in U\) for each \(g\in G, u\in U\).
Page: 843
Variables: \((G,\cdot _g)\) a finite group,
\((F,+_F,\cdot _F)\) a field,
\((V,+_V,\cdot_{FG,V})\) a \(FG\)-module,
Name2: A \(G\)-stable subspace of \(V\)

Smaller0: \(U\) is a \(G\)-stable subspace of \(V\)
Greater0: \(U\) is a \(FG\)-submodule of \(V\)
Page: 843
Chapter: 18
Section: 1
Variables: \((G,\cdot _g)\) a finite group,
\((F,+_F,\cdot _F)\) a field,
\((V,+_V,\cdot_{FG,V})\) a \(FG\)-module,
\(U\subseteq V\)
Smaller to greater: iff

Name: The trivial representation
Definition: The representation \((V,\phi)\) with \(V\) a \(1\)-dimensional vector space and \(\phi(g)=ID\).
Page: 844
Variables: \((G,\cdot _g)\) a finite group,
\((F,+_F,\cdot _F)\) a field,
Index: (1)
Kind: Ex
Property: Not faithful representation if \(|G|>1\)

Name: The regular representation
Definition: The representation of \(G\) on \(FG\), \(g\mapsto (g'\mapsto gg')\).
Page: 844
Variables: \((G,\cdot _g)\) a finite group,
\((F,+_F,\cdot _F)\) a field,
Index: (2)
Kind: Ex
Typ: Faithful

Q: The matrix of [...] over basis \(G\) is [...]
A: \(g\) in the regular representation; a \(|G|\times |G|\)-matrix \(M\) with \(M_{g',g''}\) being 1 iff \(g\cdot g''=g'\) and \(0\) otherwise
Page: 844
Chapter: 18
Section: 1
Variables: \((G,\cdot _g)\) a finite group,
\((F,+_F,\cdot _F)\) a field,
\(g\in G\)
Index: (2)
Kind: Ex

Name: Representations
Page: 844
Section: 1
Chapter: 18
Index: (4)
Kind: Ex
Under: precomposition by a morphism of group

Name: Representation of \(D_{2n}\)
Page: 845, 848
Variables: \(n\in\mathbb Z^+\)
Index: (6), (8)
Kind: Ex, Ex
Example: \(r\mapsto \left(\begin{array}{ll}\cos 2\pi/n&-\sin 2\pi/n\\\sin 2\pi/n&\cos 2\pi/n\end{array}\right)\), \(s\mapsto \left(\begin{array}{ll}0&1\\1&0\end{array}\right)\).
irreducible (assuming \(n\ge 3\))

Name: Representation of \(Q_8\) over \(\mathbb C\)
Page: 845, 848
Variables: \(n\in\mathbb Z^+\)
Index: (7), (4)
Kind: Ex
Example: \(i\mapsto \left(\begin{array}{ll}i&0\\0&-i\\\end{array}\right)\), \(j\mapsto \left(\begin{array}{ll}0&-1\\1&0\end{array}\right)\).
Property: Faithfull, irreducible

Question: Example of representation of quaternions, irreduccible over real and not over complex
Answer: \(i\mapsto \left(\begin{array}{llll}&-1&&\\1&&&\\&&&-1\\&&1&\\\end{array}\right)\)
and
\(j\mapsto \left(\begin{array}{llll}&&-1&\\&&&1\\1&&&\\&-1&&\\\end{array}\right)\)
Page: 845
Chapter: 18
Section: 1
Index: 9
Kind: Example

Question: Isomorphism from \(F[x]/(x^n-1)\) to \(F\langle\mathbb Z/n\mathbb Z\rangle\) ?
Answer: \(\sum a_kx^k\mapsto\sum a_kk\) with \(a_k\in F\)
Page: 842
Chapter: 18
Section: 1
Index: (1)
Kind: Ex

Question: A representation of \(G\) is
Answer: determined by a matrix \(M_s\) for each \(s\in S\). Such that for each \(a_1\dots a_n=a'_1\dots a'_{n'}\in R\), \(M_{a_1}\dots M_{a_n}=M_{a'_1}\dots M_{a'_{n'} }\)
Page: 845
Chapter: 18
Section: 1
Variables: \(S\) a set of letter,
\(\mathcal R\) a set of equality of words
Let \(G=\langle S\mid \mathcal R\rangle\) a presentation of a group
Index: (6)
Kind: Ex

Name: \(\phi_1\) and \(\phi_2\) are equivalent representations of \(G\)
Definition: The representations of \(G\) whose \(FG\)-modules afford the \(\phi_i\) are isomorphic
Page: 846
Variables: \(G\) a finite group,
\(F\) a field,
\(\phi_i:G\to V_i\) for \(i\) being 1 or 2,
Kind: Def
Definition2: \(\phi_1(g)= P\phi_2(g) P^{-1}\) for \(P\) an invertible matrix, for each \(g\in G\)
Name2: \(\phi_1\) and \(\phi_2\) are similar representations of \(G\)

Name: Inequivalent representations of \(G\)
Definition: The relation on representations of \(G\) which are not equivalent.
Page: 846
Variables: \(G\) a finite group,
\(F\) a field,
Kind: Def

Name: A matrix intertwining \(\phi_1\) and \(\phi_2\)
Definition: A matrix \(P\) such that \(\phi_1(g)= P\phi_2(g) P^{-1}\) for each \(g\in G\)
Page: 847
Variables: \(G\) a finite group,
\(F\) a field,
\(\phi_i:G\to V_i\) for \(i\) being 1 or 2, equivalent representation
Kind: Def

Smaller0: \(\phi_1\) and \(\phi_2\) are equivalent group representation
Greater0: the dimension of \(V_1\) and \(V_2\) are equal
Page: 847
Chapter: 18
Section: 1
Variables: \(G\) a finite group,
\(F\) a field,
\(\phi_1:G\to V_1, \phi_2:G\to V_2\)
Kind: Def

Name: An irreducible \(R\)-module
Definition: A non-zero \(R\)-module whose only submodules are 0 and itself
Page: 847, 848
Variables: \(R\) a ring
Index: (1), (3)
Kind: Def, Ex
Example: The augmentation ideal of \(FS_n\), assuming \(n>1\) and \(F\)'s characteristic is not \(n\).
Name2: A simple \(R\)-module

Name: A reducible \(R\)-module
Definition: A non-irreducible \(R\)-module
Page: 847
Variables: \(R\) a ring,
Index: (1)
Kind: Def
Definition2: A non-zero \(R\)-module which have a non zero proper submodule

Name: An indecomposable \(R\)-module
Definition: A \(R\)-module which is not the direct sum of two \(R\)-submodule
Page: 847
Variables: \(R\) a ring,
Index: (2)
Kind: Def

Name: A decomposable \(R\)-module
Definition: A \(R\)-module which is not indecomposable
Page: 847
Variables: \(R\) a ring,
Index: (2)
Kind: Def

Name: A completely reducible \(R\)-module
Definition: A direct sum of irreducible submodules
Page: 847
Variables: \(R\) a ring,
Index: (3)
Kind: Def

Name: A constituent of \(M\)
Definition: such that \(N\) is proper and
Page: 847
Variables: \(R\) a ring,
\(M\) a \(R\)-module
Index: (5)
Kind: Def
Definition2: \(M=N\oplus N'\) for some \(R\)-submodule \(N'\).
Conjdef: An \(R\)-submodule \(N\) of \(M\)
Assuming: \(M\) a completely reducible \(R\)-module

Name: An irreducible representation of \(G\)
Definition: A representation over a field \(F\) whose \(FG\)-module is irreducible.
Page: 847
Variables: \(G\) a group
Index: (1)
Kind: Def
Definition2: A nonzero \(F\)-vector space with no non-trivial proper \(G\)-invariant subspaces.
Example: A representation of \(G\) of dimension 1
Name2: A simple representation of \(G\)

Name: A reducible representation of \(G\)
Definition: A representation over a field \(F\) whose \(FG\)-module is reducible.
Page: 847, 848
Variables: \(G\) a group
Index: (1), (2) (3)
Kind: Def, ex
Example: The regular representation (assuming that \(|G|>1\)) has, as non-trivial submodule, the trace ideal and the representation ideal
Example2: The \(FS_n\)-module \((F^n,+,\circ)\) for some field \(F\),\(n>1\) and \(\circ\) the action \(\sigma\mapsto(e_i\mapsto e_{\sigma i})\)

Name: An indecomposable representation of \(G\)
Definition: A representation over a field \(F\) whose \(FG\)-module is indecomposable.
Page: 847
Variables: \(G\) a group
Index: (2)
Kind: Def

Name: A decomposable representation of \(G\)
Definition: A representation over a field \(F\) whose \(FG\)-module is decomposable
Page: 847
Variables: \(G\) a group
Index: (2)
Kind: Def

Name: A completely reducible representation of \(G\)
Definition: A representation over a field \(F\) whose \(FG\)-module is completely reducible
Page: 847
Variables: \(G\) a group
Index: (3)
Kind: Def

Smaller0: \(V\) is reducible, with some non-trivial submodule \(U\)
Greater0: of the form \(\left(\begin{array}{ll}\phi_1(g)&\psi(g)\\0&\phi_2(g)\end{array}\right)\) with \(\phi_1(g)=\phi(g)_{\mid U}\) for some submodule \(U\), and \(\phi_2(g)\) is the representation of \(G\) on \(V/U\)
Page: 848
Chapter: 18
Section: 1
Variables: \(F\) a field,
\(G\) a group
\((V,+_V,\phi)\) an \(FG\)-module
Smaller to greater: iff
Prefix Greater: each \(\phi(g)\) is

Smaller0: \(V\) is decomposable, with \(V=U_1\oplus U_2\)
Greater0: each \(\phi(g)\) is of the form \(\left(\begin{array}{ll}\phi_1(g)&0\\0&\phi_2(g)\end{array}\right)\) with \(\phi_i(g)=\phi(g)_{\mid U_i}\) for some submodule \(U_i\)
Page: 848
Chapter: 18
Section: 1
Variables: \(F\) a field,
\(G\) a group
\((V,+_V,\phi)\) an \(FG\)-module
Smaller to greater: iff

Name: Example of \(FG\)-submodule of \(M\)
Page: 846
Variables: \(n\in\mathbb Z^+\),
\(F\) a field
Let \(M=(F^n,+,\circ)\) an \(n\)-dimensional \(F S_n\)-module, with representation \(\sigma\mapsto(e_i\mapsto e_{\sigma i})\)
Index: 10
Kind: Ex
Example: The trace submodule.
Example2: The augmentation ideal.
Extra: augmentation ideal is defined in Rings part

Name: The trace ideal of \(FG\)
Definition: The set of vectors \(\alpha\sum_{g\in G}g\) for some \(\alpha\in F\)
Page: 846
Variables: \(F\) a field
\(G\) a group
Index: 11
Kind: Ex
Typ: A submodule of dimension 1.
Denoted by: \(N\)

Smaller0: \(n\ge3\)
Greater0: is the singleton composed of the trace submodule
Page: 846
Chapter: 18
Section: 1
Variables: \(n\in\mathbb Z^+\),
\(F\) a field
\(M=(F^n,+,\circ)\) an \(n\)-dimensional \(F S_n\)-module, with representation \(\sigma\mapsto(e_i\mapsto e_{\sigma i})\)
Index: 10
Kind: Ex
Prefix Greater: the set of submodules of \(M\) of dimension 1 is

Smaller0: \(F\)'s characteristic is not \(n\)
Greater0: the augmentation ideal of \(FS_n\)
Page: 848
Chapter: 18
Section: 1
Variables: \(n>1\),
\(F\) a field
Index: 3
Kind: Ex
Greater1: irreduccible
Connect Greater0: is
Extra: Merge this and next ?

Smaller0: \(F\langle\mathbb Z/n\mathbb Z\rangle\)
Greater0: \(F[x]/(x^n-1)\)
Page: 842
Chapter: 18
Section: 1
Variables: \((F,+_F,\cdot _F)\) a field,
Index: (1)
Kind: Ex
Smaller to greater: is isomorphic to
Prefix Smaller: the group ring

Smaller0: \(F\)'s characteristic does not divide [$]\abs{G}[/$]
Greater0: it is an homomorphism and a projection
Page: 849, 852
Chapter: 18
Section: 1
Variables: \(G\) a finite group
\(F\) a field
\(V\) a \(FG\)-module
\(U\subseteq V\) a submodule
Kind: Theo
Prefix Greater: there exists \(P:V\to U\) such that
Name: Maschke 's Theorem
Greater1: a direct summand
Connect Greater0: and every submodule is
Construction: Let \(W_0\) a sub-vector space (not necessarily a module) such that \(U\oplus W_0\).
Let \(\pi_0\) the projection to \(U\)
Let [$]\pi=\defFun{V}{U}{v}{\frac 1{\abs{ G}}\sum_{g\in G}g(\pi_0g^{-1}\cdot v)}[/$], a \(FG\)-module homomorphism
Take \(W=\text{ker}\pi\)
Then \(U\oplus W\) as a vector space,

Smaller0: \(F\)'s characteristic does not divide [$]\abs{G}[/$]
Greater0: there is a basis of \(V\) such that for each \(g\in G\) the matrix of \(\phi(g)\) with respect to this basis is block is diagonal \(\left(\begin{array}{llll}\phi_1(g)&&&\\&\phi_2(g)&&\\&&\dots&\\&&&\phi_m(g)\end{array}\right)\) with \(\phi_i\) an irreducible matrix representation of \(G\)
Page: 851
Chapter: 18
Section: 1
Variables: \(G\) a finite group
\(F\) a field
\(\phi:G\to GL(V)\) a representation
Index: 2
Kind: Cor
Smaller1: \(\phi\)'s degree is finite
Construction: take \(W\subseteq V\) such that \(U\oplus V\) as a vector space,
\(\pi_0\) the projection to \(U\)
take [$]\pi=\frac 1{\abs{ G}}\sum_{g\in G}g\pi_0g^{-1}[/$]. Take \(W=\text{ker}\pi\)

Name: A projection onto \(U\)
Definition: \(\pi(u)=u\) for all \(u\in U\)
Page: 849
Variables: \(X\) a set,
\(U\subseteq X\)
Kind: Theo
Definition2: \(\pi\) is idempotent
Conjdef: A set \(X\to U\) such that