This is a standard result that any beginner in the study of the representation theory would be aware of. I merely present restate the theorem a little(essentially I don’t) and prove it.
Let be representations of a group . If is a map, then the following are true assuming that and are not isomorphic:
1)If both are irreducible, then is either an isomorphism or the zero map.
2)If only is irreducible, then is either injective or the zero map.
3)If only is irreducible, then is either surjective or the zero map
If are isomorphic as representations, then is a scalar multiplication map.
Let be the respective representations. Here, are vector spaces over an algebraically closed field.
We tackle the first part of the theorem.
Consider the kernel of , say , which is set of all such that . Since is a G-linear map, implies that if , then . Hence, is stable under the action of which makes it a sub-representation. Since is irreducible, this means that either must be zero or must be the zero map. This proves that is injective.
Continue reading “Schur’s Lemma”
It is a well known fact that there are multiple proofs for the Nullstellensatz which do not use Noether’s normalization lemma. Serge Lang’s proof(in his book) and Zariski’s proof both fall under this category. In fact, Daniel Allcock of UT Austin posted a proof which essentially builds from the edifice of Zariski’s proof(see that here). He claims that it is the simplest proof for the Nullstellensatz and frankly this is quite true considering the proof uses nothing more than basic field theory, is only one page long and just requires some familiarity with transcendence bases, and the transcendence degree. Yet in the true spirit of simplicity, T. Tao has presented a proof which uses nothing more than the extended Euclidean algorithm and “high school algebra” albeit with a lengthy case-by-case analysis(see the proof here).
Most of these proofs(except Tao’s) go about proving the ‘Weak’ Nullstellensatz and obtain the ‘Strong’ Nullstelensatz through the famous Rabinowitsch trick.
But a few days, I found something truly magnificent, a proof by Enrique Arrondo in the American Mathematical Monthly which proves the Nullstellensatz using a weaker version of Noether normalization and techniques similar to that of Tao, Ritrabata Munshi. The proof is essentially a simplification of a proof by R. Munshi.
Here, I present a special case of the normalization lemma.
If is an infinite field and is a non-constant polynomial and whose total degree is , then there exists such that the coefficient of in
Let represent the homogenous component of of degree .So, the coefficient of in is . As a polynomial in , there is some point where doesn’t vanish. Choose such a point to establish the and this guarantees a non-zero coefficient of .
Continue reading “Slick proof of Hilbert’s Nullstellensatz”
This is a part of a series of posts which will deal with different types of counterexamples in topology and algebra.
Continue reading “Counterexamples in Topology and Algebra-I”