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Thread: Confusion regarding G-modules

  1. #1
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    Confusion regarding G-modules

    The following question illustrates the confusion I have (from The Symmetric Group by Bruce Sagan):

    Let $\displaystyle X$ be a reducible matrix representation with block form given by $\displaystyle \left(\begin{array}{cc}A(g)&B(g)\\0&C(g)\end{array }\right)$ where $\displaystyle A,B,C$ are square matrices of the same size.

    Let $\displaystyle V$ be a module for $\displaystyle X$ with submodule $\displaystyle W$ corresponding to $\displaystyle A$. Consider the quotient vector space $\displaystyle V/W=\{\mathbf{v}+W\mid \mathbf{v}\in W\}$.

    Show that $\displaystyle V/W$ is a $\displaystyle G$-module with corresponding matrices $\displaystyle C(g)$. Furthermore, show that we have $\displaystyle V\cong W\oplus (V/W)$.

    My confusion is in the definition of $\displaystyle W$ being a submodule (i.e. $\displaystyle gw\in W$ and the requirement that it is a module in its own right corresponding to matrices $\displaystyle A(g)$ i.e. $\displaystyle gw=A(g)w$.

    How does one do this question? I have trouble getting started and confusion with all the definitions. Any help would be greatly appreciated!

    Many thanks,

    Atticus
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  2. #2
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    Quote Originally Posted by AtticusRyan View Post
    The following question illustrates the confusion I have (from The Symmetric Group by Bruce Sagan):

    Let $\displaystyle X$ be a reducible matrix representation with block form given by $\displaystyle \left(\begin{array}{cc}A(g)&B(g)\\0&C(g)\end{array }\right)$ where $\displaystyle A,B,C$ are square matrices of the same size.

    Let $\displaystyle V$ be a module for $\displaystyle X$ with submodule $\displaystyle W$ corresponding to $\displaystyle A$. Consider the quotient vector space $\displaystyle V/W=\{\mathbf{v}+W\mid \mathbf{v}\in W\}$ (v in V?).

    Show that $\displaystyle V/W$ is a $\displaystyle G$-module with corresponding matrices $\displaystyle C(g)$. Furthermore, show that we have $\displaystyle V\cong W\oplus (V/W)$.

    My confusion is in the definition of $\displaystyle W$ being a submodule (i.e. $\displaystyle gw\in W$ and the requirement that it is a module in its own right corresponding to matrices $\displaystyle A(g)$ i.e. $\displaystyle gw=A(g)w$.

    How does one do this question? I have trouble getting started and confusion with all the definitions. Any help would be greatly appreciated!

    Many thanks,

    Atticus
    Let V be a G-module of dimension d with basis $\displaystyle B = \{w_1, w_2, \cdots , w_k, v_{k+1}, v_{k+2}, \cdots , v_d \}$, where a submodule W of dimension k has a basis $\displaystyle B' = \{w_1, w_2, \cdots , w_k\}$. Since the lower left corner of X(g) is zero and A corresponds to W, we see that $\displaystyle X(g)w_i \in W $ for all $\displaystyle 1 \leq i \leq k$ and the last d-k coordinates of $\displaystyle X(g)w_i$ is zero. Since W is a G-submodule of V and $\displaystyle g(V/W)=\{gv + gW | v \in V\}=\{v' + W | v' \in V \}$, it follows that $\displaystyle V/W$ is a G-invariant subspace of V. Now V/W is an orthogonal complement of W, $\displaystyle V \cong W \oplus (V/W)$. If W corresponds to A, then the orthogonal complement of W, i.e. , V/W, should correspond to C.
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  3. #3
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    Why should the orthogonal complement correspond to C? And does the matrix B come into it at all? If not, why is it mentioned, i.e. how is this case different from a block-diagonal case?
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  4. #4
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    Quote Originally Posted by AtticusRyan View Post
    Why should the orthogonal complement correspond to C? And does the matrix B come into it at all? If not, why is it mentioned, i.e. how is this case different from a block-diagonal case?
    Let $\displaystyle V=\mathbb{C}^d$; let $\displaystyle W=\mathbb{C}\{e_1, e_2, \cdots, e_k\}$, where $\displaystyle e_i$ is the column vector with a 1 in the ith row and zeros elsewhere. Then the orthogonal complement of W, i.e., $\displaystyle V/W = \mathbb{C}\{e_{k+1}, e_{k+2}, \cdots, e_d \}$, where $\displaystyle X(g)e_i \in (V/W)$ for $\displaystyle k+1 \leq i \leq d $ and all $\displaystyle g \in G$. This accounts for the matrix C(g) in X(g) and zero elsewhere.

    If X is a matrix representation with a block form $\displaystyle \left(\begin{array}{cc}A(g)&B(g)\\0&C(g)\end{array }\right)$, then X is a reducible representation. If X is a matrix representation with a block form $\displaystyle \left(\begin{array}{cc}A(g)&0\\0&C(g)\end{array}\r ight)$, then X is a decomposable representation. Note that there is an indecomposable representation but is reducible. For example, if $\displaystyle \mathbb{Re}$ is an additive group of real numbers, then $\displaystyle \rho: \mathbb{Re} \rightarrow GL_2(\mathbb{Re})$ defined by $\displaystyle a \mapsto \left(\begin{array}{cc}1&a\\0&1\end{array}\right)$.

    Since V is a decomposable G-module, there is a decomposable matrix representation of X such that B(g) is a zero matrix by changing a basis of V.
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