The Unapologetic Mathematician

Mathematics for the interested outsider

Modules

With the group algebra in hand, we now define a “G-module” to be a module for the group algebra of G. That is, it’s a (finite-dimensional) vector space V and a bilinear map A:\mathbb{C}[G]\times V\to V. This map must satisfy A(\mathbf{e},v)=v and A(\mathbf{g},A(\mathbf{h},v))=A(\mathbf{gh},v).

This is really the same thing as a representation, since we may as well pick a basis \{e_i\} for V and write V=\mathbb{C}^d. Then for any g\in G we can write

\displaystyle A(\mathbf{g},e_i)=\sum\limits_{j=1}^dm_i^je_j

That is, A(\mathbf{g},\underbar{\hphantom{X}}) is a linear map from V to itself, with its matrix entries given by m_i^j. We define this matrix to be \rho(g), which must be a representation because of the conditions on A above.

Conversely, if we have a matrix representation \rho:G\to GL_d, we can define a module map for \mathbb{C}^d as

\displaystyle A(\mathbf{g},v)=\rho(g)v

where we apply the matrix \rho(g) to the column vector v. This must satisfy the above conditions, since they reflect the fact that \rho is a representation.

In fact, to define A, all we really need to do is to define it for the basis elements \mathbf{g}\in\mathbb{C}[G]. Then linearity will take care of the rest of the group algebra. That is, we can just as well say that a G-module is a vector space V and a function A:G\times V\to V satisfying the following three conditions:

  • A is linear in V: A(g,cv+dw)=cA(g,v)+dA(g,w).
  • A preserves the identity: A(e,v)=v.
  • A preserves the group operation: A(g,A(h,v))=A(gh,v).

The difference between the representation viewpoint and the G-module viewpoint is that representations emphasize the group elements and their actions, while G-modules emphasize the representing space V. This viewpoint will be extremely helpful when we want to consider a representation as a thing in and of itself. It’s easier to do this when we think of it as a vector space equipped with the extra structure of a G-action.

September 15, 2010 Posted by | Algebra, Group theory, Representation Theory | 11 Comments

   

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