Schur’s Lemma

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.

Theorem:

Let V,W be representations of a group G. If f:V \mapsto W is a G-linear map, then the following are true assuming that V and W are not isomorphic:

1)If both V,W are irreducible, then f is either an isomorphism or the zero map.

2)If only V is irreducible, then f is either injective or the zero map.

3)If only W is irreducible, then f is either surjective or the zero map


If V,W are isomorphic as representations, then f is a scalar multiplication map.

Proof:

Let \rho_{V},\rho_{W} be the respective representations. Here, V,W are vector spaces over an algebraically closed field.

We tackle the first part of the theorem.

Consider the kernel of f, say Ker(f), which is set of all v \in V such that f(v)=0. Since f is a G-linear map, f(g.v)=g.f(v)=0 implies that if v \in Ker(f), then g.v \in Ker(f). Hence, Ker(f) is stable under the action of G which makes it a sub-representation. Since V is irreducible, this means that either Ker(f) must be zero or f must be the zero map. This proves that f is injective.

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Slick proof of Hilbert’s Nullstellensatz

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.


 

Lemma

If F is an infinite field and f \in F[x_{1},x_{2},\cdots ,x_{n}] is a non-constant polynomial and n \geq 2 whose total degree is d, then there exists a_{1},a_{2},\cdots ,a_{n-1} \in F such that the coefficient of x_{n}^{d} in

f(x_{1}+a_{1}x_{n},x_{2}+a_{2}x_{n},\cdots ,x_{n-1}+a_{n-1}x_{n},x_{n})

is non-zero.

Proof:

Let f_{d} represent the homogenous component of f of degree d.So, the coefficient of x_{d}^{n} in f(x_{1}+a_{1}x_{n},x_{2}+a_{2}x_{n},\cdots,x_{n-1}+a_{n-1}x_{n},x_{n}) is f_{d}(a_{1},\cdots,a_{n-1},1). As a polynomial in F[x_{1},\cdots,x_{n-1}], there is some point (a_{1},\cdots,a_{n-1} \in F^{n-1} where f_{d}(a_{1},\cdots,a_{n-1},1) doesn’t vanish. Choose such a point to establish the a_{i} and this guarantees a non-zero coefficient of x_{n}^{d}.

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Coxeter Groups(intro)

Reflection groups

In order to understand the intuition underlying the theory of Coxeter groups(and Weyl groups in particular groups in particular), you can go through the theory of reflection groups which I’ll only superficially treat preceding my exposition of Coxetr systems and the associated representation theory.

Consider some Euclidean space V and a vector \alpha \in V. A reflection associated with \alpha is a linear operator which sends \alpha to -\alpha and point-wise fixes the orthogonal hyperplane/subspace H_{\alpha}.

If the reflection is s_{\alpha}, then it can represented as:

s_{\alpha}(x)=x-\frac{2(x,\alpha)}{(\alpha,\alpha)}\alpha=x-2 proj_{\alpha}(x)

Clearly s_{\alpha} is an orthogonal transformation and the set of all reflections in V can be recognized as the subgroup of O(V)(orthogonal group of V) consisting of elements of order 2.

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