Lemma of Schur

In Mathematical and more precisely in Linear algebra, the lemma of Schur is a technical lemma used particularly in the theory of the Représentation of the groups.

It was shown in 1907 by Issai Schur (1875 - 1941) within the framework of its work on the theory of the Représentations of a group finished,

This lemma is at the base of the analysis of a Caractère of a representation of a group finished. it allows, for example to characterize the finished abelian groups.

Context

Motivation

The lemma of Schur represents one of the bases of the theory of the Représentations of a group finished and analysis of the algebra of the semi-simple modules. A representation of a group is the data of a morphism of a group G in the whole of the Automorphisme S of a vector Space. This approach initiated by Frobenius (1849 - 1917) in an article of 1896 proves to be profitable. Three years later, Heinrich Maschke (1853 1908) shows that any representation is direct Somme of irreducible representations.

The lemma of Schur is an essential technical lemma for the demonstration of a major result: not only the irreducible representations are identified by their character, but in more the characters of these representations orthogonal between them is all. This approach produces major results for the theory of the finished groups. It finally allowed the classification of the simple groups, but also the demonstration of results as a conjecture of William Burnside (1852 - 1927) stipulating that any finished group of an odd nature is resolvable. This result is at the origin of the Médaille Fields of Thompson born in 1932 .

If this lemma is also used in other contexts, that of the representation is nevertheless most important.

Representations of a finished group

See also: Theory of the representations of a group finished

Let us point out the definition of a representation and fix the notations for the remainder of the article. G indicates here a Groupe finished of order G . Its neutral element is noted 1, and if S and T is two elements of G the internal law of composition of the group on S and T is noted St . E indicates a vector space on a body noted K .

* a representation of the group G is the data of a vector space E of Dimension finished noted N and of a morphism of group ρ of G towards the linear group GL ( E ). A representation is noted ( E , ρ) or sometimes and wrongly E .

I.e. the ρ application is with value in the space of the bijective linear applications and preserves the law of the group, which is equivalent to:

\ rho_1=Id \ quad and \ quad \ forall S, T \ in G \ quad \ rho (S) \ circ \ rho (T) = \ rho (St)

Theorem of Maschke

Lemma

Statement

Are E and F two K vector spaces and Φ a Linear application nonnull of E in F.

* (1) If there exists an irreducible part U of L ( E ) such as:

\ forall U \ in U \ quad \ exists v \ in \ mathcal L (F) \ quad/\ quad \ phi \ circ u=v \ circ \ phi

Alors Φ is injective.

* (2) If there exists an irreducible part V of L ( F ) such as:

$\ forall v \ in V \ quad \ exists U \ in \ mathcal L (E) \ quad/\ quad \ phi \ circ u=v \ circ \ phi$

Alors

\ quad \ phi

is surjective .

Demonstration

(1) If Φ is not injective, its core NR is nonnull (and nonequal to E since Φ is nonnull). One has then:

\ forall U \ in U \ quad \ exists v \ in \ mathcal L (F) \ quad \ phi \ circ U (NR) =v \ circ \ phi (NR) = \ {0 \}.

It results from it that U ( NR ) is included in NR , which is contradictory with the assumption of irreducibility of U .
(2) The following properties are checked:

\ forall v \ in V \ quad \ exists U \ in \ mathcal L (E) \ quad v \ circ \ phi (E) = \ phi \ circ U (E) \ subset \ phi (E)

The image of Φ is thus stable by all the elements of V . It results from it by irreducibility of V (and the fact that Φ is nonnull) that the image of Φ is equal to F .

Corollaries

Corollary 1

* Is E a vector space of Dimension finished on a body K algebraically closed and U an irreducible part of L ( E ). If a endomorphism Φ of E commutates with any element of U, then Φ is a Homothétie.

Indeed, one can write, if Id indicates the identity application:

\ quad \ forall \ lambda \ in K \ quad \ forall U \ in U \ quad (\ phi - \ lambda. Id) \ circ u=u \ circ (\ phi \ lambda. Id)

One from of deduced by application from the lemma from Schur that Φ - λ. Id is an automorphism or is null. That is to say λ^* an eigenvalue of Φ, then Φ - λ^*. Id is the null application, which shows the corollary.

In the case of the representation of a finished group of order G , then any automorphism of the image has for Polynôme canceler P = X ^g - 1. Consequently, if K contains the Corps of decomposition of P the corollary still applies.

Corollary 2

* All irreducible representation in a space of size finished of an abelian group G on a body algebraically closed is of dimension 1 .

Indeed, the morphism of the representation is ρ. Whatever the element S of G ρ_s commutates with all the endomorphisms of the representation. According to corollary 1 ρ_s is a homothety. If the dimension of the space of representation were strictly higher than 1, each subspace of dimension 1 being thus invariant one would end in a contradiction.

Case of the finished groups

Corollary 3

* Is ( E , ρ¹) and ( F , ρ²) two representations of G irreducible on a body K of characteristic either null or first with G the order of the group and containing the body of decomposition of the polynomial X ^g - 1 and ψ a linear application of E in F , one defines the linear application φ E in F by:

\ varphi = \ frac {1} {G} \ sum_ {S \ in G} \ rho_s^2 \ circ \ psi \ circ (\ rho_s^1) ^ {- 1}

(1) If the representations are not isomorphous, then φ is null.

(2) If the body K is of null characteristic and algebraically closed and if E is equal to F φ is homothety of report/ratio 1 N . Tr (ψ), where N indicates degree of the representations.

Vérifions initially that φ checks the following property:

\ forall T \ in G \ quad \ varphi \ circ \ rho_t^1= \ rho_t^2 \ circ \ varphi \ quad or \; still \ quad \ varphi = \ rho_t^2 \ circ \ varphi \ circ (\ rho_t^1) ^ {- 1} \;

Let us notice first of all that the application of G in G which with S associates ts is a permutation, if T is an element of G . One from of deduced that:

\ forall T \ in G \ quad \ rho_t^2 \ circ \ varphi \ circ (\ rho_t^1) ^ {- 1} = \ frac {1} {G} \ sum_ {S \ in G} \ rho_t^2 \ circ \ rho_s^2 \ circ \ psi \ circ (\ rho_s^1) ^ {- 1} \ circ (\ rho_t^1) ^ {- 1} = \ frac {1} {G} \ sum_ {S \ in G} \ rho_ {ts} ^2 \ circ \ psi \ circ (\ rho_ {ts} ^1) ^ {- 1} = \ varphi

(1) As the representations is not isomorphous φ cannot be at the same time injective and surjective. The lemma of Schur shows that, as φ is not an automorphism, φ is the null application.
(2) If E is equal to F , assumptions of the corollary 1 are checked what shows that φ is a homothety. In this case, the two representations are identical and the expression defining φ is the average of G applications all similar to ψ and thus having same the trace as ψ. The traces of φ and ψ are thus equal. As for any homothety, φ is of report/ratio 1/n.Tr (φ), like the traces of φ and of ψ are equal, we showed that the report/ratio of homothety is equal to 1/n.Tr (φ).

Corollary 4

It is a fourth corollary which is used in the theory of the characters. It corresponds to the translation in term of matrix of the preceding corollary. Let us use the following notations, either has and B two representations in matric form of a group finished G of order G on the same body K of characteristic or null or first with dimension of has and G and such as the Polynôme X ^g - 1 or divided. Respective dimensions of E and F are noted N and m . The image of an element S of G by has (resp. B ) is noted has _ij ( S ) (resp B _ij ( S ))

There is then the following corollary with the assumptions of the preceding corollary:

* (1) If the representations R ¹ and R ² are not isomorphous, then:

$\ forall I, J \ in \; \ forall K, L \ in \ quad \ sum_ {S \ in G} a_ {ij} (S) .b_ {kl} (s^ {- 1}) =0$

* (2) If the two representations are isomorphous, then:

\ forall I, J, K, L \ in \ quad \ frac {1} {G} \ sum_ {S \ in G} a_ {ij} (S) .b_ {kl} (s^ {- 1}) = \ frac {1} {N} \ delta_ {it} \ delta_ {jk}

Where δ_ij indicates the Symbole of Kronecker.

Let us show the proposal (1):

If C a matrix of dimension m X N of coefficients ( C _jk), the translation of the point (1) of the preceding corollary shows that:

\ sum_ {S \ in G} A_s.C.B_ {s^ {- 1}} = 0 \ quad thus \ quad \ forall I \ in \; \ forall L \ in \ quad \ sum_ {jk} \ sum_ {S \ in G} a_ {ij} (S) .c_ {jk} .b_ {kl} (s^ {- 1}) = \ sum_ {jk} \ left (\ sum_ {S \ in G} a_ {ij} (S) .b_ {kl} (s^ {- 1}) \ right) .c_ {jk} =0

This equality is true for any matrix C , therefore for any value of C _jk, which shows the proposal (1) .

Let us show the proposal (2):

With the same notations (now m is equal to N ), one obtains according to the point (2) of the preceding corollary:

\ frac {1} {G} \ sum_ {S \ in G} A_s.C.B_ {s^ {- 1}} = \ frac {1} {N} Tr (C).Id \ quad thus \ quad \ forall I, J \ in \ quad \ frac {1} {G} \ sum_ {jk} \ sum_ {S \ in G} a_ {ij} (S) .c_ {jk} .b_ {kl} (s^ {- 1}) = \ frac {1} {N} \ sum_k c_ {kk}. \ delta_ {it}

One from of deduced:

\ forall I, J, K, L \ in \ quad \ frac {1} {G} \ sum_ {S \ in G} a_ {ij} (S). b_ {kl} (s^ {- 1}) = \ frac {1} {N} \ delta_ {it} \ delta_ {jk}

And the proposal (2) is shown.

Applications

Character

See also: Character of a representation of a group finished

It is the first historical application of the lemma. Are G a finished group of order G , ( E , ρ¹) and ( F , ρ²) two representations of G irreducible. It is supposed here that the body K is that of the complex numbers. One notes χ₁ and χ₂ the characters of the two representations. The characters are elements of the vector space noted C of the applications of G in the body of complexes of dimension. Its dimension is equal to G . One provides C with the square Produit < | > according to:
$\ forall X, there \ in C \ quad = \ frac {1} {G} \ sum_ {G \ in G} x_s.y_s^*$
If indicates a complex number there, ^* indicates its Conjugué here there.

* irreducible characters of a finished group, if the body has train orthonormal family C .

Indeed, it is a direct consequence of corollary 4. The associated article shows that the trace of the reverse of S equal to is combined trace of S , if S is an element of G . By using the notations of the preceding paragraph, one obtains:

< \ chi_1|\ chi_2>= \ frac {1} {G} \ sum_ {G \ in G} \ left (\ sum_ {i=1} ^n a_ {II} (S) \ right). \ left (\ sum_ {j=1} ^n b_ {jj} (S) \ right) ^*= \ frac {1} {G} \ sum_ {G \ in G} \ left (\ sum_ {i=1} ^n a_ {II} (S) \ right). \ left (\ sum_ {i=1} ^n b_ {jj} (s^ {- 1}) \ right) = \ frac {1} {G} \ sum_ {ij} \ left (\ sum_ {G \ in G} a_ {II} (S) b_ {jj} (s^ {- 1}) \ right)

If the two representations are not isomorphous, then the proposal (1) of the corollary makes it possible to conclude with orthogonality. If the two representations are isomorphous, according to the proposal (2) one obtains:

< \ chi_1|\ chi_2>= \ frac {1} {N} \ sum_ {ij} \ delta_ {ij}. \delta_{ij}=1

What shows the proposal. This result is one of the bases of the theory of the caractères.
The remark on the algebraic fence at the beginning of the paragraph on the finished groups makes it possible to extend this result to all the commutative body of null characteristic.