We shall walk through the problem at little steps and day-by-day, not so much interest in the final solution per se , but with a higher aim to revise the modern algebra lessons along the way.

The French professors who designed this problem had done beautifully using all the concepts learned in the 2-year Classe Préparatoire (or Prépa, equivalent to Bachelor degree in Math & Science) – it is like an orchestra composer who pieces together all instruments to play a beautiful symphony – the catch is that the student must have a good grasp of all algebra topics.

SOLUTION

I – Operators on the functions with finite support

1. a. Prove that V is a vector subspace of .

Proof:
Recall the definitions:

= v.s.{}

Support = supp(ƒ) = {}

V = {f | supp(f) is a finite set}.

To prove V a vector subspace of ,
1) V must be non-empty?

V contains null function, so not empty subset of

2) closed under vector addition?

supp (f+g) supp(f) supp (g)

3) closed under scalar multiplication?

supp() = supp(f) for

Given ƒ ∈ , E(ƒ) ∈
by E(ƒ)(κ)= ƒ(κ+1), κ ∈

1. b. Show that E∈ L( ) and V is stable by E.

E by definition is an operator of shift, hence a linear transformation, thus
E∈ L( ).

——
[Solutions for XLC paper]:

Click to access Polytechnique_MP_A_XLC_corr_1.pdf

Math Paper A (XLC)
Duration: 4 hours

Use of calculator disallowed.

We propose to study the algebras of the remarkable endomorphisms of vector spaces of infinite dimension.

Preamble

An root of unity is called primitive if it generates the group of roots of unity.

In this problem, all vector spaces are over the base field of complex numbers field .

If ε is a vector space, the algebra of the endomorphisms of ε is denoted by L(ε), and the group of the automorphisms of ε is denoted by GL(ε).

denotes the identity mapping of ε.

If u ∈ L(ε), denotes the sub-algebra of L(ε) of the Polynomials in u.

denotes the vector space of the functions of to .

If ƒ is the function of to , supp(ƒ) denotes the set of κ ∈ such that ƒ(κ) ≠0
We call this set the support of ƒ.

Throughout the problem, V denotes the set of functions of to of which the support is a finite set.

I – Operators on the functions with finite support

1. a. Prove that V is a vector subspace of .
Given ƒ ∈ ,
we define E(ƒ) ∈
by
E(ƒ)(κ)= ƒ(κ+1), κ ∈

1. b. Show that E∈ L( ) and V is stable by E.

In the following, E denotes uniquely the endomophism of V induced.

2. Show that E ∈ GL(V).

3. For , we define by:

3.a. Prove that the family is the base of V.

3.b. Calculate E().
Let ,
we define the respective linear mappings
by:
and

4. Prove that

if and only if for all

In the remaining of Section I (but not in the following Sections), we asume the conditions in question 4 are verified.

5. Prove that

if and only if

6.a. Prove that for the vector space generated by has finite dimension.

6.b. Deduce that a vector subspace non-reduced to {0} of V, stable by H, contains at least one of the .

In the remaining of Section I (but not in the following Sections), we asume the conditions in question 5 are verified and

7.a. Prove that

7.b. Prove that E and F are not of finite order in the group GL(V).

7.c. Calculate the kernel of H and prove that

8. denotes the polynomials with complex number coefficients in one indeterminate X.

8.a. Prove that is isomorphic (as Algebra) to .

8.b. Prove that is isomorphic (as Algebra) to .

8.c. Prove that is isomorphic (as Algebra) to .

II – Interlude

In all the rest of the problem, we fix an odd interger and q a primitive root of unity.

9. Prove that is a primitive root of unity.

Let

10. Consider the element is :

10.a. Calculate .
Prove that

10.b. Let b root of a.
Calacute the eigenvectors of and the associated eigenvalues in function of b, q and .

Let’s define a linear mapping by
, and
define r and p respectively the residue and the quotient of the euclidian division of i by ℓ; ie:

11. Prove that is a projector of image .

III – Quantum Operators

12. Prove that
if and only if

In the following problem, we asume the conditions in question 12 are verified and

13. Prove that .

14. Prove that
if and only if

In the following problem, we asume the conditions in question 14 are verified.

15.a. Prove that are periodic over , of periods dividing .

15.b. Prove that the period of .

15.c. Prove that the period of is also equal to .

16. Let with being inverse of H.

16.a. Prove that
.

16.b. For , prove that is an eigenvector of C.

16.c. Deduce that C is a homothety of v of which we calculate the ratio of in function of .

16.d. Let’s fix . Prove that the mapping
is a bijection of onto .

16.e. Let’s fix . Prove that the mapping
is a surjection of onto but not a bijection.

IV – Modular Quantum Operators

Let like in the Section II. We say an element of L(V) is compatible with if

17.a. Prove that if is commutative with ,then is compatible with .

17.b. Prove that are compatible with .

Let the set of endomorphisms which are compatible with .

18. Prove that is a sub-algebra of L(V).

19. Prove that .

20.a. Show that there exists an unique morphism of algebras such that:

20.b. Prove that is contained in the kernel of if and only if the image of is a vector subspace of v generated by the vectors where is the euclidian division of .

21. Let’s study in this question .

21.a. Determine .

21.b. Deduce .

21.c. Calculate the dimension of the vector subspace of

21.d. Calculate the eigenvectors of

22. Let W a non zero sub-space of stable by .

22.a. Show that W contains at least one of the vectors .

22.b. What do you say if W is in addition stable by ?

23. Give a necessary and sufficient condition on in order for the operator to be nilpotent.

—End—

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