Title: Theory / Unimportant / Math / Al gebra / Group / Representation theory-Steinberg
Tags: closed fromMath examplesSwitchExample1 toBinaryRelation

Name: A representation of \(G\)
Definition: An homomorphism \(G\to GL(V)\) with \(V\) a finite dimensional vector space
Page: 13, 14
Chapter: 3
Section: 1
Variables: \(G\) a group
Kind: Def 1, Ex 1-6
Example: Trivial representation
Example2: [$]\defFun{\mathbb Z/n\mathbb Z}{\mathbb C^*}{m}{e^{2\pi i\frac mn}}[/$]
Applied to2: \(\mathbb Z/n\mathbb Z\)

Definition: \(\phi(g)\)
Page: 13
Chapter: 3
Section: 1
Variables: \(G\) a group
\(V\) a finite dimensional vector space
\(\phi:G\to GL(V)\) a representation of \(G\)
\(g\in G\)
Notation: \(\phi_g\)

Name: The degree of the representation
Definition: The dimension of \(V\)
Page: 13
Chapter: 3
Section: 1
Variables: \(G\) a group
\(V\) a finite dimensional vector space
\(\phi:G\to GL(V)\) a representation of \(G\)

Name: The trivial representation of \(G\)
Definition: [$]\defFun{G}{\mathbb C^*}{g}{1}[/$]
Page: 13
Chapter: 3
Section: 1
Variables: \(G\) a group

Page: 14
Definition: there exists \(T:V_1\to V_2\) invertible such that \(\phi^{(2)}_{,g}=T\phi^{(1)}_{,g}T^{-1}\) for all \(g\)
Chapter: 3
Section: 1
Variables: \(G\) a group
\(F\) a field
\(V_1,V_2\) finite-dimensional \(F\) vector space
\(\phi^{(1)}:G\to GL(V_1)\),\(\phi^{(2)}:G\to GL(V_2)\)
Notation: \(\phi^{(1)}\sim\phi^{(2)}\)
Name: \(\phi^{(1)}\) and \(\phi^{(2)}\) are equivalent representation
Kind: Def
Index: 7
Example: for \(G=\mathbb Z/n\mathbb Z\), \(V_1=V_2=\mathbb C^2\).
\(\phi^{(1)}_{,m}=\left[\begin{array}{ll}\cos\left(\frac{2\pi m}{n}\right)&-\sin\left(\frac{2\pi m}{n}\right)\\\sin\left(\frac{2\pi m}{n}\right)&\cos\left(\frac{2\pi m}{n}\right)\end{array}\right]\) and \(\phi^{(2)}_{,m}=\left[\begin{array}{ll}e^{2\pi i\frac mn}&0\\0&e^{-2\pi i\frac mn}\end{array}\right]\), it suffices to take \(T=\left[\begin{array}{ll}i&-i\\1&1\end{array}\right]\) \(T=\frac1{2i}\left[\begin{array}{ll}1&i\\-1&i\end{array}\right]\)
Assuming: the dimensions of \(V_1\) and of \(V_2\) are equal

Name: The standard representation of \(S_n\)
Definition: [$]\defFun{S_n}{GL_n(F)}{\sigma}{e_{\sigma(i)}}[/$] where \(e_i\) is the \(i\)-th unit vector of the standard basis
Page: 15
Chapter: 3
Section: 1
Variables: \(F\) a field
\(n\in\mathbb N\)
Index: 9
Kind: Ex

Name: A \(G\)-invariant subspace of \(V\)
Definition: A subspace \(W\le V\) such that, for all \(g\in G\), \(w\in W\), \(\phi_gw\in W\)
Page: 15
Chapter: 3
Section: 1
Variables: \(G\) a group
\(V\) a finite dimensional vector space
\(\phi:G\to GL(V)\) a representation of \(G\)
Index: 10
Kind: Def

Name: The direct sum of the representations \(\phi^{(1)}\) and \(\phi^{(2)}\)
Definition: [$]\defFun{G}{GL(V_1\oplus V_2)}{g}{\defFun{V_1\oplus V_2}{V_1\oplus V_2}{(v_1,v_2)}{(\phi^{(1)}_{g},\phi^{(2)}_{g})}}[/$]
Page: 15
Chapter: 3
Section: 1
Variables: \(G\) a group
\(F\) a field
\(V_1,V_1\) finite-dimensional \(F\) vector space
\(\phi^{(1)}:G\to GL(V_1)\),\(\phi^{(2)}:G\to GL(V_2)\)
Notation: \(\phi^{(1)}\oplus\phi^{(2)}\)
Index: 11
Kind: Def

Question: The representation of \(\phi^{(1)}\oplus\phi^{(2)}\) is
Answer: [$]\defFun{G}{GL_{n_1+n_2}}{g}{\left[\begin{array}{ll}\phi^{(1)}_{g}&0\\0&\phi_{g}^{(2)}\end{array}\right]}[/$]
Page: 16
Chapter: 3
Section: 1
Variables: \(G\) a group
\(F\) a field
\(V_1,V_1\) finite-dimensional \(F\) vector space
\(\phi^{(1)}:G\to GL_{n_1}(F)\),\(\phi^{(2)}:G\to GL_{n_2}(F)\)
Index: 11
Kind: Def

Definition: [$]\defFun{G}{GL(W)}{g}{\defFun{W}{W}{w}{\phi_g(w)}}[/$]
Page: 16
Chapter: 3
Section: 1
Variables: \(G\) a group
\(F\) a field
\(V\) finite-dimensional \(F\) vector space
\(\phi:G\to GL(V)\),
\(W\le V\)
Notation: \(\phi_{\mid W}\)
Index: 11
Kind: Def
Assuming: \(W\) is \(G\)-invariant

Name: A subrepresentation of \(\phi\)
Definition: Some \(\phi_{\mid W}\) for some \(W\subseteq V\) which is \(G\)-invariant
Page: 16
Chapter: 3
Section: 1
Variables: \(G\) a group
\(F\) a field
\(V\) finite-dimensional \(F\) vector space
\(\phi:G\to GL(V)\),
Index: 11
Kind: Def

Smaller0: \(\phi\)
Greater0: \(\phi_{\mid W_1}\oplus\phi_{\mid W_2}\)
Page: 16
Chapter: 3
Section: 1
Variables: \(G\) a group
\(F\) a field
\(V\) finite-dimensional \(F\) vector space
\(\phi:G\to GL(V)\),
\(W_1,W_2\le V\)
Index: 11
Kind: Def
Smaller to greater: \(\sim\)
Assuming: \(W_1,W_2\) invariants, \(V=W_1\oplus W_2\)

Name: An irreducible representation
Definition: A representation of \(\phi:G\to V\) such that its only \(G\)-invariant subspaces are \(\{0\}\) and \(V\)
Page: 17
Chapter: 3
Section: 1
Variables: \(G\) a group
Index: 15
Kind: Def
Example: Any representation of degree 1

Question: The representations of the trivial groups in \(GL_n(F)\) is
Answer: exactly the constant function to the diagonal unit matrix
Page: 17
Chapter: 3
Section: 1
Variables: \(F\) a field
\(n\in\mathbb N\)

Question: the \(G\)-invariant subspace of \(\phi:1\to V\) are
Answer: all the subspaces
Page: 17
Chapter: 3
Section: 1
Variables: \(F\) a field
\(V\) a finite-dimensional \(F\)-vector space
\(1\) the trivial group
\(\phi:1\to V\) a representation

Smaller0: \(\phi\)'s degree is 3
Greater0: there is no common eigenvector to all \(\phi_g\) with \(g\in G\)
Page: 17
Chapter: 3
Section: 1
Variables: \(F\) a field
\(V\) a finite-dimensional \(F\)-vector space
\(G\) a group
\(\phi:1\to V\) a representation
Index: 19
Kind: Prop
Smaller1: \(G\) is finite)
Greater1: \(\phi\) is irreducible
Connect Greater0: iff
Prefix Smaller: (
Connect Smaller0: and
Smaller2: \(\phi\)'s degree is 2
Connect Smaller1: or

Name: A completely reducible representation
Definition: A representation of the form \(\phi^{(1)}\oplus\dots\oplus\phi^{(n)}\) with each \(\phi^{(i)}\) irreducible
Page: 19
Chapter: 3
Section: 1
Variables: \(F\) a field
\(V\) a finite-dimensional \(F\)-vector space
\(G\) a group
Index: 21
Kind: Def

Name: A decomposable representation of \(G\)
Definition: A representation of the form \(\phi^{(1)}\oplus\phi^{(2)}\) with each \(\phi^{(i)}\) non-zero
Page: 19
Chapter: 3
Section: 1
Variables: \(F\) a field
\(V\) a finite-dimensional \(F\)-vector space
\(G\) a group
Index: 22
Kind: Def

Name: An indecomposable representation of \(G\)
Definition: A representation of \(G\) which is not decomposable
Page: 19
Chapter: 3
Section: 1
Variables: \(F\) a field
\(V\) a finite-dimensional \(F\)-vector space
\(G\) a group
Index: 22
Kind: Def

Name: The set of decomposable representation of \(G\)
Page: 19
Section: 1
Chapter: 3
Index: 23
Variables: \(F\) a field
\(V\) a finite-dimensional \(F\)-vector space
\(G\) a group
Kind: Lem
Under: taking an equivalent representation

Name: The set of irreducible representations of \(G\)
Page: 19
Section: 1
Chapter: 3
Index: 23
Variables: \(F\) a field
\(V\) a finite-dimensional \(F\)-vector space
\(G\) a group
Kind: Lem
Under: taking an equivalent representation

Name: The set of reducible representations of \(G\)
Page: 19
Section: 1
Chapter: 3
Index: 23
Variables: \(F\) a field
\(V\) a finite-dimensional \(F\)-vector space
\(G\) a group
Kind: Lem
Under: taking an equivalent representation

Name: The set of completely reducible representations of \(G\)
Page: 19
Section: 1
Chapter: 3
Index: 25
Variables: \(F\) a field
\(V\) a finite-dimensional \(F\)-vector space
\(G\) a group
Kind: Lem
Under: taking an equivalent representation