Schur's Lemma (2)

Here's the follow up post to Schur's Lemma, in case you were anxiously awaiting it.

The corollary says that if α=β, f has to be a multiple of the identity function (or unit matrix).



For the proof one looks at an eigenvalue λ_i. And subtracts the following on both sides:



This changes to:



Now we apply the Schur's Lemma from last time and find out that has to be either invertible or the 0 function. But it cannot be invertible because λ_i is an eigenvalue (actually the only eigenvalue). And we have what we wanted.

Quantum cryptography

A pretty cool book I just read is Simon Singh's Code Book about cryptography. As my blog title suggests I'll mostly talk about the quantum part. Just a short summary.

Today's standard is the RSA cipher. Its speciality is that you need a different key for enciphering and deciphering. In other words: everyone can a encipher a message intended for you, but only you can read it. This is possible because exponentiating and subsequent modulo taking is in principle bijective (with an appropriate domain) but it is difficult to find the inverse function. At the heart of this method are two (large) prime numbers p and q. Their product N is made public and can be used for encrypting. For decrypting you need p and q. If p and q are large enough (and well chosen so that a few other mathematical tricks don't work), it is in principle impossible to deduce p and q out of N and only the person that initially multiplied p and q will be able to read the message.

Apparently quantum computers might be spefically suited for the task of factorising N. A regular computer would check numbers one at a time. The input for a quantum computer is a superposition of all the numbers to be checked. The computation can be carried out on this whole set. If done correctly the measurement at the end will lead the correct result with a high probability.

The way to escape from every decryption even quantum computers is quantum cryptography. It relies either on the uncertainty principle or on quantum entanglement. According to modern understanding of science it is in principle unbreakable. Unless of course some physicist eventually finds some of those hidden variables, how you can hop back and forth between parallel universes or some more realistic way of life beyond quantum theory.

P.S. There may be some quantum computers around already and people that know every dirty bit of secret about you.

Simplified molecular input line entry specification

The whole name is almost as catchy as "SMILES". I used to think it was a strange way of representing molecules for computers. But it actually seems like a more straight forward way than IUPAC nomenclature. It's also shorter, and there's a possibility to have unique names. (It's more difficult to pronounce though.)

You can try out SMILES strings at this page it's kind of fun. How to do it is described on wikipedia for example.

Ethane is just CC.

Add double and triple bonds like this:
C#CC=C for butenyne.

Add a branch in parentheses:
CC(C)CCC for 2-Methyl-n-pentane

If you want a ring add a number after the two atoms to be joined together:
C1C(C)CCC1 for Methyl-cyclo-pentane

Add a pyridyl group to the C next to the methyl group (aromatic atoms are written in lower case, and you have to include a second ring closure)
C1(c2ncccc2)C(C)CCC1 for (2-Pyridyl-)-2-methyl-c-pentane

You can add an extra oxirane ring:
C1(c2ncccc2)C(C)CCC13OC3

You can mess with stereochemistry (using @ and @@)
C1(c2ncccc2)[C@@H](C)CC[C@]13OC3

If you still haven't had enough, you can add a double bond in E configuration to the pyridyl ring:
C1(c2nc(/C=C(Cl)\C)ccc2)[C@@H](C)CC[C@]13OC3

or Z configuration
C1(c2nc(/C=C(Cl)/C)ccc2)[C@@H](C)CC[C@]13OC3

Go SMILES!

Schur's Lemma

Schur's Lemma is a nice piece of group theory apparently needed to prove the orghogonality relations which are in turn another nice piece of group theory. I did not find the whole proof anywhere, so I will show it here. The nice thing is: Once you put things into correct mathematical notation, you're almost done.

We have a group G whose elements we call R. And we may think of the elements as symmetry operations.

We have two representations α and β of this group. So what is a representation of a group? In most cases you may think of it as a vector space V or its basis, and let the group operations work on it. To be precise you have to define how the group operations work on this vector space. Then a representation becomes a mapping (a homomorphism) between the group G and the group of invertible linear transformations on V (instead of linear transformations you may also think of matrices).



Typically β=α but for this prove it can be different and may work on a different vector space W



For Schur's Lemma we also need that α is irreducible. That means every subspace U of V that is invariant with all the group operations contains either just the null vector or is the whole V.



Finally we have another linear transformation (or matrix) f



which has the following property



In other words the two mappings are the same for all the group operations.

Then Schur's Lemma says that f has to be either the 0 mapping or it has to be invertible. In the second case are the dimensions of the representations the same, dim(V)=dim(W):



If you have read this far, you may be wondering why that is.

Proof:

First we can assume that



without loss of generality because we could just relabel things if they aren't.

You have to consider the kernel of f. First we make the following observation



In other words:



But because α is irreducible, we know that ker(f), a subspace of V, has to look like:



In the second case obviously because everything is mapped onto the null vector. In the first case f is injective and it has to be bijective because W has at most the same dimension as V (it turns out that dim(V)=dim(W)).

That's what we wanted to show.

The applications of this are also kind of interesting. Maybe later.