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How to compute a unitary representation of finite group isomorphic to a given rep?

0

$begingroup$

Suppose I am given some representation of a finite group: $rho : G to textGL(n, mathbbC)$. I want to compute a unitary representation $tau$ which is isomorphic to $rho$.

I know about Weyl's trick, where you define a $G$-invariant inner product, $langle v,w rangle = frac1Gsum_g in G langle rho(g)v, rho(g)w rangle_0$ where $langle,rangle_0$ is the dot product (the usual inner product). Then $rho$ is unitary with respect to the new inner product (by relabelling the sum, easy argument).

But how does this help us compute the representation of $tau$ where the images are unitary matrices? By this I mean unitary according to the original inner product.

Here are my thoughts so far: Inner products on $mathbbC^n$ differ by a linear map, $langle v,w rangle = langle Av, w rangle_0$ for some $A in textGL(n, mathbbC)$, positive definite. Also $langle v,w rangle = v^* A^* w$. Maybe at this point I can do the Cholesky decomposition of $A$: $langle v,w rangle = v^* A^* w = v^* L^*L w = langle Lv, Lw rangle_0$. So the new inner product is just the dot product after doing the linear map $L$. And since this inner product is $G$-invariant, doesn't this imply $L$ is $G$-invariant also?

The Cholesky decomposition is problematic though, since in computer algebra systems (like GAP, SageMath), arbitrary square roots in cyclotomic fields (the closest thing to $mathbbC$) are not possible.

Any advice on my approach or better ideas are welcome.

representation-theory computer-algebra-systems computational-algebra

share|cite|improve this question

edited Apr 6 at 17:36

Alexander Konovalov

5,24222057

asked Apr 6 at 11:25

pizzarollpizzaroll

400110

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Once you have the new inner product, apply Gram-Schmidt to the old basis to get a basis which is orthonormal in the new inner product. Conjugating the representation by this change-of-basis matrix should give you what you want.
  $endgroup$
  – Joppy
  Apr 6 at 12:34
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am actually trying to avoid the Gram-Schmidt process. When working in exact cyclotomic fields, square rooting large real numbers means we have to go to a really big degree cyclotomic field which is really bad for computation time. You are right though, that is a method I didn't think of.
  $endgroup$
  – pizzaroll
  Apr 6 at 14:04
  

  
  
  
  

add a comment | 

0

$begingroup$

Suppose I am given some representation of a finite group: $rho : G to textGL(n, mathbbC)$. I want to compute a unitary representation $tau$ which is isomorphic to $rho$.

I know about Weyl's trick, where you define a $G$-invariant inner product, $langle v,w rangle = frac1Gsum_g in G langle rho(g)v, rho(g)w rangle_0$ where $langle,rangle_0$ is the dot product (the usual inner product). Then $rho$ is unitary with respect to the new inner product (by relabelling the sum, easy argument).

But how does this help us compute the representation of $tau$ where the images are unitary matrices? By this I mean unitary according to the original inner product.

Here are my thoughts so far: Inner products on $mathbbC^n$ differ by a linear map, $langle v,w rangle = langle Av, w rangle_0$ for some $A in textGL(n, mathbbC)$, positive definite. Also $langle v,w rangle = v^* A^* w$. Maybe at this point I can do the Cholesky decomposition of $A$: $langle v,w rangle = v^* A^* w = v^* L^*L w = langle Lv, Lw rangle_0$. So the new inner product is just the dot product after doing the linear map $L$. And since this inner product is $G$-invariant, doesn't this imply $L$ is $G$-invariant also?

The Cholesky decomposition is problematic though, since in computer algebra systems (like GAP, SageMath), arbitrary square roots in cyclotomic fields (the closest thing to $mathbbC$) are not possible.

Any advice on my approach or better ideas are welcome.

representation-theory computer-algebra-systems computational-algebra

share|cite|improve this question

edited Apr 6 at 17:36

Alexander Konovalov

5,24222057

asked Apr 6 at 11:25

pizzarollpizzaroll

400110

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Once you have the new inner product, apply Gram-Schmidt to the old basis to get a basis which is orthonormal in the new inner product. Conjugating the representation by this change-of-basis matrix should give you what you want.
  $endgroup$
  – Joppy
  Apr 6 at 12:34
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am actually trying to avoid the Gram-Schmidt process. When working in exact cyclotomic fields, square rooting large real numbers means we have to go to a really big degree cyclotomic field which is really bad for computation time. You are right though, that is a method I didn't think of.
  $endgroup$
  – pizzaroll
  Apr 6 at 14:04
  

  
  
  
  

add a comment | 

$begingroup$

Suppose I am given some representation of a finite group: $rho : G to textGL(n, mathbbC)$. I want to compute a unitary representation $tau$ which is isomorphic to $rho$.

I know about Weyl's trick, where you define a $G$-invariant inner product, $langle v,w rangle = frac1Gsum_g in G langle rho(g)v, rho(g)w rangle_0$ where $langle,rangle_0$ is the dot product (the usual inner product). Then $rho$ is unitary with respect to the new inner product (by relabelling the sum, easy argument).

But how does this help us compute the representation of $tau$ where the images are unitary matrices? By this I mean unitary according to the original inner product.

Here are my thoughts so far: Inner products on $mathbbC^n$ differ by a linear map, $langle v,w rangle = langle Av, w rangle_0$ for some $A in textGL(n, mathbbC)$, positive definite. Also $langle v,w rangle = v^* A^* w$. Maybe at this point I can do the Cholesky decomposition of $A$: $langle v,w rangle = v^* A^* w = v^* L^*L w = langle Lv, Lw rangle_0$. So the new inner product is just the dot product after doing the linear map $L$. And since this inner product is $G$-invariant, doesn't this imply $L$ is $G$-invariant also?

The Cholesky decomposition is problematic though, since in computer algebra systems (like GAP, SageMath), arbitrary square roots in cyclotomic fields (the closest thing to $mathbbC$) are not possible.

Any advice on my approach or better ideas are welcome.

representation-theory computer-algebra-systems computational-algebra

share|cite|improve this question

edited Apr 6 at 17:36

Alexander Konovalov

5,24222057

asked Apr 6 at 11:25

pizzarollpizzaroll

400110

$endgroup$

share|cite|improve this question

edited Apr 6 at 17:36

Alexander Konovalov

5,24222057

asked Apr 6 at 11:25

pizzarollpizzaroll

400110

share|cite|improve this question

edited Apr 6 at 17:36

Alexander Konovalov

5,24222057

asked Apr 6 at 11:25

pizzarollpizzaroll

400110

edited Apr 6 at 17:36

Alexander Konovalov

5,24222057

edited Apr 6 at 17:36

Alexander Konovalov

5,24222057

asked Apr 6 at 11:25

pizzarollpizzaroll

400110

asked Apr 6 at 11:25

pizzarollpizzaroll

400110