Notes in MATH4307 - Topics in Algebra and Number Theory

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Published	10/24/2024	Let \(G\) be a group and \(X\) be a set. Recall the definition of a group action of \(G\) on \(X\) (denoted&nb…
Published	10/24/2024	How can I write the two conditions for a group action using nicer notation?
Published	10/24/2024	How can I view a group action \(G \curvearrowright X\) as a homomorphism \(G \to \mathrm{Sym}(X)\)?
Published	10/24/2024	What is a \(G\)-set?
Published	10/24/2024	What is a \(G\)-module?
Published	10/24/2024	What is a representation?
Published	10/24/2024	Let \(G\) be a group. Recall the definition of the group algebra of \(G\), denoted \(\mathbb{C} [G]\)
Published	10/24/2024	Recall the form of elements on \(\mathbb{C}[G]\)
Published	10/24/2024	Recall addition and multiplication on \(\mathbb{C}[G]\)
Published	10/24/2024	Let \(G\) be a group. Recall the left-regular representation of \(G\)
Published	10/24/2024	Recall the definition of the symmetric group \(S_n\)
Published	10/24/2024	Recall the definition for a group action to be faithful
Published	10/24/2024	Recall the definition for a group action to be free
Published	10/24/2024	Recall the definition of the stabilizer group \(G_x\)
Published	10/24/2024	Recall the orbit of \(x\), denoted \(G \cdot x \)
Published	10/24/2024	Recall the definition for a group action to be transitive
Published	10/24/2024	Recall the Orbit-Stabilizer theorem
Published	10/24/2024	Recall Burnside's Lemma
Published	10/24/2024	What does \(X^g\) mean for some \(G \curvearrowright X\) and \(g \in G\)?
Published	10/24/2024	What does \(X/G\) mean for \(G \curvearrowright X\)?
Published	10/24/2024	What does it mean for a homomorphism between \(G\)-modules, \(\varphi:V\to W\), to be \(G\) linear?
Published	10/24/2024	How can I represent \(\rho : G \to GL(V)\) as a matrix if I have \(\{b_1,\dots,b_n\}\) as a basis for \(V\)?
Published	10/24/2024	When is a rep \(\rho\) faithful?
Published	10/24/2024	Two reps \(\rho : G \to GL(V)\) and \(\tau: G \to GL(W)\) are equivalent if there exists an {{c1::isomorphism of vector spaces}}&n…
Published	10/24/2024	When are two matrix reps \(\rho: G \to GL(n, \mathbb{K})\), \(\tau: G \to GL(m, \mathbb{K})\) equivalent?
Published	10/24/2024	How can I write \(\|G\|\) in terms of the conjugacy classes and the centralisers?
Published	10/24/2024	What is the defining rep/permutation rep of \(S_n\)?
Published	10/24/2024	Let \(V\) be a \(G\)-module. Recall the definition of a submodule of \(V\).
Published	10/24/2024	Let \(V\) be a \(G\)-module. Recall the definition of a trivial submodule of \(V\).
Published	10/24/2024	Let \(V\) be a \(G\)-module. Recall the definition of a proper submodule of \(V\).
Published	10/24/2024	Let \(V\) be a \(G\)-module. Recall the definition of \(V\) being reducible.
Published	10/24/2024	Recall the definition of simple/irreducible module
Published	10/24/2024	Let \(V\) be a \(G\)-module. Recall the definition of a \(G\)-invariant inner product on \(V\)
Published	10/24/2024	Let \(W\) be a submodule of \(V\) that has an inner product. Define \(W^\perp\).
Published	10/24/2024	If \(W\) is a submodule of \(V\), what can we say about \(W^\perp\)?
Published	10/24/2024	Recall the definition of a \(G\)-module \(V\) to be semisimple/completely reducible.
Published	10/24/2024	Recall Maschke's theorem
Published	10/24/2024	Recall part 1 of Schur's lemma
Published	10/24/2024	Let \(V,W\) be \(G\)-modules such that \(V\) is irreducible. Then \({{c1::\mathrm{Hom}_G(V,W) = 0}} \iff\){{c2::there do…
Published	10/24/2024	Recall part 2 of Schur's lemma (identity map)
Published	10/24/2024	Let \(V,W\) be finite dimensional \(G\)-modules over \(\mathbb{C}\) where \(\|G\| < \infty\). Define the multiplicity o…
Published	10/24/2024	Let \(V,W\) (where \(W \neq 0\)) be finite dimensional \(G\)-modules over \(\mathbb{C}\) such that \(V\) is ir…
Published	10/24/2024	Let \(\rho\) be a finite dimensional rep over \(\mathbb{C}\) of \(G, \|G\|< \infty\). Define the character.
Published	10/24/2024	Why is the characteristic a well-defined function? How can I show that?
Published	10/24/2024	\(\chi_\rho (1) = {{c1::\dim(V)}}\)
Published	10/24/2024	\({{c1::\chi_{\rho}(g^{-1})}} = {{c2:: \overline{\chi_\rho (g)} }}\)
Published	10/24/2024	Recall the inner product on the function space of a group
Published	10/24/2024	Let \(\rho, \tau\) be irreducible finite dimensional representations over \(\mathbb{C}\) of \(G\), \(\|G\| <\infty\), w…
Published	10/24/2024	Let \(\mathbb{C}[G] \cong V = \bigoplus^r_{i=1} V^{(i) \oplus m_i}\). Then,1) \(m_i = {{c1:: \dim(V^{(i)}) }}\) so that \(\|G\| = {{…
Published	10/24/2024	Recall the trivial representation
Published	10/24/2024	Recall the signed representation of \(S_n\) on \(\mathbb{C}\). Why is it a rep?
Published	10/24/2024	Let \(V (=\mathbb{C}^n)\) be the defining rep of \(S_n\). Give an example of a non-trivial, proper submodule.
Published	10/24/2024	Let \(V\) be the left-regular rep of \(S_n\). Provide two examples of proper, non-trivial submodules.
Published	10/24/2024	Recall the definition of the restricted representation.
Published	10/24/2024	Let \(H \leq G\) such that \([G:H]=k\) and let \(\rho :H \to GL(V)\) be an \(n\)-dimensional rep of \(H\). The…
Published	10/24/2024	Given a partition \(\lambda \vdash n\), what is the order of \(YT(\lambda)\)?
Published	10/24/2024	Are Young Tableux strict?
Published	10/24/2024	Recall \(SYT(\lambda)\) (standard young tableux)
Published	10/24/2024	Recall the Young subgroup
Published	10/24/2024	How does the young subgroup act on elements of \(YT(\lambda)\)?
Published	10/24/2024	When are two young tableux equivalent?
Published	10/24/2024	Recall the set of young tabloids of shape \(\lambda\)
Published	10/24/2024	Recall the young permutation module for \(\lambda \vdash n\)
Published	10/24/2024	Recall the size of \(T(\lambda)\)
Published	10/24/2024	What is the young permutation module \(M_\lambda\) equivalent to?
Published	10/24/2024	Let \(t \in YT(\lambda)\). What is a row/column stabiliser of \(\lambda\)?
Published	10/24/2024	Recall the form of the polytabloid labelled by \(t \in YT(\lambda)\)
Published	10/24/2024	\({{c1:: e_{\sigma \cdot t} }}= {{c2:: \sigma \cdot e_ t }}\)
Published	10/24/2024	Recall the definition of a specht module
Published	10/24/2024	Recall the procedure for determining the dominance relation on tabloids
Published	10/24/2024	If \(t \in SYT(\lambda)\), then \(\text{max}(\text{Pos}(e_t )) = {{c1::[t]}}\)
Published	10/24/2024	The \(e_t\) indexed by \(t \in SYT(\lambda)\) are {{c1::linearly independent}}
Published	10/24/2024	If \(\sigma \in C(t)\), then \(e_{\sigma \cdot t} {{c1::= \text{sgn}(\sigma) e_t }}\)
Published	10/24/2024	Recall the Garnir element
Published	10/24/2024	Recall \(S_{A,B}\)
Published	10/24/2024	Give a basis for the Specht module \(S^\lambda\)
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