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

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

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

    My confusion is in the definition of W being a submodule (i.e. gw\in W and the requirement that it is a module in its own right corresponding to matrices A(g) i.e. 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 X be a reducible matrix representation with block form given by \left(\begin{array}{cc}A(g)&B(g)\\0&C(g)\end{array  }\right) where A,B,C are square matrices of the same size.

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

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

    My confusion is in the definition of W being a submodule (i.e. gw\in W and the requirement that it is a module in its own right corresponding to matrices A(g) i.e. 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 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 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 X(g)w_i \in W for all 1 \leq i \leq k and the last d-k coordinates of X(g)w_i is zero. Since W is a G-submodule of V and g(V/W)=\{gv + gW | v \in V\}=\{v' + W | v' \in V \}, it follows that V/W is a G-invariant subspace of V. Now V/W is an orthogonal complement of W, 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 V=\mathbb{C}^d; let W=\mathbb{C}\{e_1, e_2, \cdots, e_k\}, where e_i is the column vector with a 1 in the ith row and zeros elsewhere. Then the orthogonal complement of W, i.e., V/W = \mathbb{C}\{e_{k+1}, e_{k+2}, \cdots, e_d \}, where X(g)e_i \in (V/W) for  k+1 \leq i \leq d and all 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 \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 \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 \mathbb{Re} is an additive group of real numbers, then \rho: \mathbb{Re} \rightarrow GL_2(\mathbb{Re}) defined by 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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