Abstract Nonsense

Crushing one theorem at a time

Matrix Functions Form an (almost) Orthonormal Basis (Pt. II)


Point of post: This is a continuation of this post.

Matrix Entry Functions Span the Group Algebra

We now show that \Lambda=\text{span }\mathcal{D}=\mathcal{A}(G). To do this we prove a result similar to that of the Stone-Weierstrass theorem for finite sets. Namely:

Theorem: Let X be some finite set and \mathscr{A} a subalgebra of \mathbb{C}^X which separates points, then \mathscr{A}=\mathbb{C}^X. (here the algebra structure is the usual pointwise scalar multiplication, pointwise addition, and pointwise product of functions).

Proof: Note that \left\{\delta_x\right\}_{x\in X} is a basis for X where \delta_x(y)=\delta_{x,y}. Note though, for each y\in X we are guaranteed some f_{x,y}\in\mathscr{A} such that f(x)=1 and f(y)=0. It’s evident though that

 

\displaystyle \delta_x=\prod_{y\ne x}f_{x,y}

 

and so in particular since \mathscr{A} is a subalgebra we have that \delta_x\in\mathscr{A} for every x\in X and since \mathscr{A} is, in particular, a subspace we have that \mathscr{A}\supseteq\text{span}\{\delta_x\}_{x\in X}=\mathbb{C}^X. The conclusion follows. \blacksquare

 

From this we pretty easily derive the desired result. Namely:

Theorem: \Lambda=\mathcal{A}(G).

Proof: We’ve proven in previous posts that \Lambda is a subalgebra of \mathbb{C}^G=\mathcal{A}(G) (this equality is as a set, since the group algebra has a different multiplicative structure than pointwise multiplication) which separates points, and thus the conclusion follows by the previous theorem. \blacksquare

Corollary: \mathcal{D} is a basis for \mathcal{A}(G) and thus in particular \#\left(\widehat{G}\right)<\infty and

\displaystyle \sum_{\alpha\in\widehat{G}}d_\alpha^2=\#\left(\mathcal{D}\right)=\dim\mathcal{A}(G)=|G|

 

Note though that since we also have the trivial irrep \tau:G\to\mathbb{C}:g\mapsto 1 that the above implies that, for example, any non-trivial irrep \rho has the quality that \deg\rho\leqslant \sqrt{|G|-1}. We shall use this to derive many, many more interesting result as the theory progesses.

References:

1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

2. Serre, Jean Pierre. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977. Print

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February 23, 2011 - Posted by | Algebra, Representation Theory | , , , ,

9 Comments »

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  7. Hey Alex,

    Nice blog!
    I’m starting one here:
    http://dmitrigekhtman.wordpress.com/
    but I’m kind of inexperienced.

    Would you mind sending me the code for this post?
    I’d like to figure out how to use WordPress’s LaTeX features more effectively.
    My e-mail address is dmitrgekh@gmail.com.

    Thanks,
    Dmitri

    Comment by Dmitri Gekhtman | July 4, 2011 | Reply

    • Hello Dmitri!

      Thank you very much for your support! I’m glad to see that you will be a new memeber to the fantastic wordpress.com blog community! I am having a bit of e-mail trouble at the moment, so I hope copy and pasting the code at the bottom of this response will suffice.

      Anyways, tell me a little bit about yourself. What’s your mathematical background, interests, etc.?

      Best,
      Alex

      Point of post: This is a continuation of this post.

      Matrix Entry Functions Span the Group Algebra

      We now show that \Lambda=\text{span }\mathcal{D}=\mathcal{A}(G). To do this we prove a result similar to that of the Stone-Weierstrass theorem for finite sets. Namely:

      Theorem: Let X be some finite set and \mathscr{A} a subalgebra of \mathbb{C}^X which separates points, then \mathscr{A}=\mathbb{C}^X. (here the algebra structure is the usual pointwise scalar multiplication, pointwise addition, and pointwise product of functions).

      Proof: Note that \left\{\delta_x\right\}_{x\in X} is a basis for X where \delta_x(y)=\delta_{x,y}. Note though, for each y\in X we are guaranteed some f_{x,y}\in\mathscr{A} such that f(x)=1 and f(y)=0. It’s evident though that

      \displaystyle \delta_x=\prod_{y\ne x}f_{x,y}

      and so in particular since \mathscr{A} is a subalgebra we have that \delta_x\in\mathscr{A} for every x\in X and since \mathscr{A} is, in particular, a subspace we have that \mathscr{A}\supseteq\text{span}\{\delta_x\}_{x\in X}=\mathbb{C}^X. The conclusion follows. \blacksquare

      From this we pretty easily derive the desired result. Namely:

      Theorem: \Lambda=\mathcal{A}(G).

      Proof: We’ve proven in previous posts that \Lambda is a subalgebra of \mathbb{C}^G=\mathcal{A}(G) (this equality is as a set, since the group algebra has a different multiplicative structure than pointwise multiplication) which separates points, and thus the conclusion follows by the previous theorem. \blacksquare

      Corollary: \mathcal{D} is a basis for \mathcal{A}(G) and thus in particular \#\left(\widehat{G}\right)<\infty and

      \displaystyle \sum_{\alpha\in\widehat{G}}d_\alpha^2=\#\left(\mathcal{D}\right)=\dim\mathcal{A}(G)=|G|

      Note though that since we also have the trivial irrep \tau:G\to\mathbb{C}:g\mapsto 1 that the above implies that, for example, any non-trivial irrep \rho has the quality that \deg\rho\leqslant \sqrt{|G|-1}. We shall use this to derive many, many more interesting result as the theory progesses.

      References:

      1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

      2. Serre, Jean Pierre. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977. Print

      Comment by Alex Youcis | July 7, 2011 | Reply

  8. Hmm, that didn’t work so well, did it? Haha. I’ll erase the $$ stuff. I assume you know you have to encase your LaTeX in $latex(space)(LaTeX)$ where obviously (space) means you need to put a space and (LaTeX) is whatever you want. I hope this is what you wanted. If you were more concerned with formatting (spaces, bolding, italicizing, etc.) or if you have any more questions, just let me know! Anyways:

    Point of post: This is a continuation of this post.

    Matrix Entry Functions Span the Group Algebra

    We now show that \Lambda=\text{span }\mathcal{D}=\mathcal{A}(G). To do this we prove a result similar to that of the Stone-Weierstrass theorem for finite sets. Namely:

    Theorem: Let X be some finite set and \mathscr{A} a subalgebra of \mathbb{C}^X which separates points, then \mathscr{A}=\mathbb{C}^X. (here the algebra structure is the usual pointwise scalar multiplication, pointwise addition, and pointwise product of functions).

    Proof: Note that \left\{\delta_x\right\}_{x\in X} is a basis for X where \delta_x(y)=\delta_{x,y}. Note though, for each y\in X we are guaranteed some f_{x,y}\in\mathscr{A} such that f(x)=1 and f(y)=0. It’s evident though that

    \text{ }

    \displaystyle \delta_x=\prod_{y\ne x}f_{x,y}

    \text{ }

    and so in particular since \mathscr{A} is a subalgebra we have that \delta_x\in\mathscr{A} for every x\in X and since \mathscr{A} is, in particular, a subspace we have that \mathscr{A}\supseteq\text{span}\{\delta_x\}_{x\in X}=\mathbb{C}^X. The conclusion follows. \blacksquare

    \text{ }

    From this we pretty easily derive the desired result. Namely:

    Theorem: \Lambda=\mathcal{A}(G).

    Proof: We’ve proven in previous posts that \Lambda is a subalgebra of \mathbb{C}^G=\mathcal{A}(G) (this equality is as a set, since the group algebra has a different multiplicative structure than pointwise multiplication) which separates points, and thus the conclusion follows by the previous theorem. \blacksquare

    Corollary: \mathcal{D} is a basis for \mathcal{A}(G) and thus in particular \#\left(\widehat{G}\right)<\infty and

    \displaystyle \sum_{\alpha\in\widehat{G}}d_\alpha^2=\#\left(\mathcal{D}\right)=\dim\mathcal{A}(G)=|G|

    Note though that since we also have the trivial irrep \tau:G\to\mathbb{C}:g\mapsto 1 that the above implies that, for example, any non-trivial irrep \rho has the quality that \deg\rho\leqslant \sqrt{|G|-1}. We shall use this to derive many, many more interesting result as the theory progesses.

    References:

    1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

    2. Serre, Jean Pierre. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977. Print

    Comment by Alex Youcis | July 7, 2011 | Reply


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