## Agregation 2014 – Mathematiques Generales – Parts 4-6

This is the second part of the Mathematiques Generales French Agregation written exam 2014. For the complete notation list and the first three parts look at this post.

**Part 4 – Reduced form of permutations**

For we denote the set of pairs such that . We call the set of inversions of a permutation the set

and we denote the cardinal of .

**1.** For which permutations is the number maximum?

For we denote the transposition which changes and .

**4.2** (a) Let . Prove that

and that is obtained from by adding or removing an element of .

(b) Find explicitly in function of the element of which makes it differ from .

Let . We call word a finite sequence of elements of . We say that is the length of and that the elements are the letters of . The case of a void word is authorized.

A writing of a permutation is a word such that . We make the convention that the permutation which corresponds to the void word is the identity.

**4.3** (a) Prove that the group is generated by .

(b) Let . Prove that has a writing of length and that every writing of is of length at least .

A writing of a permutation is called reduced if its length is .

**4.4** Find a reduce writing of each of the elements of .

**4.5** Let and be a non-reduced writing of . Prove that ther exist two integers such that is a writing of .

The result of question 4.5 can be interpreted like this: if a word is a non-reduced writing of a permutation , then we can obtain a shorter writing for by ommiting from two letters chosen conveniently. Repeating this operation as long as it takes, we arrive at extracting from a reduced writing of .

**4.6** Consider two reduced writings of a permutation different from identity. Let be the first letter of and the first letter of .

(a) Suppose that . Prove that has a reduced writing starting with .

(b) Suppose that . Prove that has a reduced writing starting with .

Given two words of the same length we write if we are in one of the following situations:

– There exists a word and elements and such that ,

– There exists a word and elements and such that ,

We write if there is a finite sequence of words such that

(The case which corresponds at is authorized.)

**4.7** (a) Let and be two words of the same length. Prove that if then .

(b) Let and be two reduced writings of the same permutation. Prove that .

**Part 5 – Bruhat Decomposition**

In this part and is a field. Let be the vector space with its standard base . We identify and .

We call standard flag the flag where and for is the vector space spanned by .

We denote the subgroup of formed of the matrices which are invertible and upper triangular.

To every permutation there corresponds a permutation matrix defined in Part 3. It is a matrix with coefficients in that belongs to . We denote the set of matrices of the form witn . This is a subset of .

**5.1** For this question we take or . Denote the permutation defined by , for . Prove that is dense in .

(Indication: use Theorem D)

For and we denote the matrix with coefficients in with on the diagonal and in position and zero everywhere else.

**5.2** Prove that for every .

We use the notations from Part 3.

**5.3** Let be the standard flag.

(a) Prove that is the stabilizer of .

(b) Prove that for every the pair of flags is in position .

(c) Prove that for every we have .

(d) Prove that is the disjoint union of , i.e. and that the are pairwise disjoint.

Given two subsets of we denote . We use the notations of Part 4.

**5.4** Let such that .

(a) Prove that for every the product belongs to .

(Indication: use 3.7 (b))

(b) Deduce that .

From question 5.4 we deduce that for every permutation and every reduced writing of we have

The end of this part has the objective of finding the closures of . To give a meaning to this probleme we place ourselves in the cases or .

Given two permutations and we write if for every we have .

**5.5** (a) Prove that is an order relation on .

(b) Prove that the identity permutation is the smallest element of for the order .

(c) Does the set endowed with the order relation has a largest element?

**5.6** Let such that and let . Prove that:

(a) If then .

(b) If then .

**5.7** Let be a vector space of finite dimension and be two linear subspaces of and a positive integer. Prove that

is a closed subset of .

**5.8** Let . Prove that the following four statements are equivalent.

(i) .

(ii) For every reduced writing of there exists a strictly increasing sequence of indices such that is a reduced writing of .

(iii) There exists a reduced writing of and a strictly increasing sequence of indices such that is a reduced writing of .

(iv) The set is included in the closure of in .

**5.9** Prove that for every the closure of in is given by

**Part 6 – Unitriangular totally positive matrices**

As noted in the end of the second part, we study here the set of inferior unitriangular matrices. In the following all matrices are considered of size , where .

Like in Part 5 we denote for and the matrix with on the diagonal and in position with zeros elswhere. We can easily verify that is TP if and only if . Therefore the product of such matrices is TP.

Using the notations of Part 4 each word defines an application defined by

If is the void word, then the domain of definition of contains only one element and we define to be the identity matrix on that element.

**6.1** In this question we study the case .

(a) On the example of the word prove that the image of is a sub-manifold of .

(b) Let be the six words found in question 4.4. Verify that the images of the applications are pairwise disjoint and that their union is the set of inferior unitriangular TP matrices.

The phenomenons observed in question 6.1 happen for every . In the following we will prove some partial results related to this.

**6.2** Let and .

(a) Suppose that . Verify that .

(b) Suppose that . Prove that there existe a unique element such that

Is the application from to itself bijective?

The question 6.2 implies that and have the same image if , with the notation introduced in part 4. The question 4.7(b) shows that if we have then the image of is constant as runs through the set of reduced writings of . We denote that image . The question 5.2 and the identity from 5.4(b) show that . Therefore by question 5.3 the sets are pairwise disjoint.

**6.3** (a) Let , , and . Prove that belongs to if or to if the opposite holds.

(b) Let , and a reduced writing of . Prove that the application is bijective.

**6.4** (a) Prove that for every the closure of in is

(b) Prove that stable under multiplication and is closed in .

We can prove that is the set of inferior unitriangular TP matrices and that if is a reduced writing of a permutation, then the application is an extension which implies that are sub-manifolds of .