THE REPRESENTATION RING OF SL 1. Introduction and ...

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THE REPRESENTATION RING OF SLd. AZI KHADIJA1 AND HINDA HAMRAOUI2∗. Abstract. Let k be a field, GLd be the algebraic linear group of order d over.
Gulf Journal of Mathematics Vol 4, Issue 4 (2016) 48-53

THE REPRESENTATION RING OF SLd AZI KHADIJA1 AND HINDA HAMRAOUI2∗ Abstract. Let k be a field, GLd be the algebraic linear group of order d over the field k and SLd be the subgroup of matrices with a determinant of 1. The aim of this paper is to describe the representation rings of those groups and their associated completed rings.

1. Introduction and preliminaries The representation theory group was created by Frobenius, Schur and developed by many others like Borel and Serre. The representation theory group has so many applications to the resolution of the heat equations, to theoretical physics, to algebraic geometry and algebraic topology. In [2] we will exploit our calculations of the representations rings of the linear group GLd and the special group SLd and the various morphisms between them to deduce morphisms between their associated completed rings. From [5], where Riou proves a motivic Atiyah isomorphism for linear group GLd which generalizes Atiyah isomorphism, for finite group [1], between the completed representation ring and the ring of the K-theory of the classifying space, we prove a motivic Atiyah isomorphism for the special group SLd . Definition 1.1. Let d be an integer. (1) The linear algebraic group GLd is the group of invertible d × d matrices together with the operation of ordinary matrix multiplication. Let id be the morphism: GLd −→ GLd+1 M 0 M 7−→ 0 1

(1.1)

Then the system {GLd , id } is direct. (2) We recall that the infinite linear group GL is the direct limit of the system {GLd , id }. Date: Accepted: Oct 24, 2016. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 46L55; Secondary 44B20. Key words and phrases. Representation, general linear group, special linear group, completed ring. 48

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2. Representation ring of the linear group Denote by Gdm the maximal torus of GLd and jd : Gdm → GLd the embedding. Let Rk (GLd ) , Rk (Gdm ) be their k-representation rings equipped with direct sum and tensor product of representations and jd∗ : Rk (GLd ) → Rk (Gdm ) be the restriction morphism. Recall X1 , .., Xd , X1−1 , .., Xd−1 the irreducible characters of Gdm . Thanks to [6], we have:   Lemma 2.1. The ring Rk (Gdm ) and the ring Z X1 , .., Xd , X1−1 , .., Xd−1 are isomorphic.   Rk (Gdm ) = Z X1 , .., Xd , X1−1 , .., Xd−1 (2.1) Let Sd be the symmetric group and σ a permutation of Sd : σ.(aii ) = bii where bii = aσ(i)σ(i) defines a natural action of Sd on the torus Gdm . From this follows an action of Sd on the k-representation ring Rk (Gdm ). Let (Rk (Gdm ))Sd be the sub-ring of Rk (Gdm ) invariant under Sd action. It follows from [6] that: Lemma 2.2. The morphism jd induces an isomorphism jd∗ : Rk (GLd ) → (Rk (Gdm ))Sd .

(2.2)

Let ti = Xi − 1 and si be the ith symmetric polynomial in the ti ’s. Since X1 × X2 × ... × Xd is invertible in Z [X1 , ..., Xd , , (X1 )−1 , ..., (Xd )−1 ] so is (1 + t1 ) × (1 + t2 ) × ... × (1 + td ) = 1 + s1 + s2 + ... + sd and we have: Proposition 2.3. The isomorphism jd∗ in the Lemma 2.2 becomes:   Rk (GLd ) ∼ = Z s1 , ..., sd , (1 + s1 + ... + sd )−1

(2.3)

Proof. We have Z [X1 , ..., Xd ] = Z [t1 , ..., td ]. Then : (Z [X1 , ..., Xd ])Sd = (Z [t1 , ..., td ])Sd = Z [s1 , ..., sd ] . On the other hand :     Z X1 , ..., Xd , (X1 )−1 , ..., (Xd )−1 = Z [X1 , ..., Xd ] (X1 )−1 , ..., (Xd )−1 Which implies:  S Rk (GLd ) = Z X1 , ..., Xd , X1−1 , ..., Xd−1 d  S = Z X1 , ..., Xd ][X1−1 , ..., Xd−1 d   = (Z[X1 , ..., Xd ])Sd X1−1 , ..., Xd−1   = Z[s1 , ..., sd ] X1−1 , ..., Xd−1   = Z s1 , ..., sd , (1 + s1 + ... + sd )−1  The ring Rk (GLd ) is augmented by d : s 7−→ dim(s), denote Id its augmentaˆ k (GLd ) its Id -adic completed ring. Recall the general following tion ideal and R result: Lemma 2.4. The augmentation ideal Id is generated by s1 , s2 , ..., sd . Id = hs1 , s2, ..., sd i

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Proof. We know that Id is generated by Xi − 1. Since Xi − 1 is an element of the ideal generated by si , then Id ⊂ hs1 , s2, ..., sd i. Conversely, each symmetric polynomial si is expressed in terms of Xi − 1, so hs1 , s2, ..., sd i ⊂ Id .  Lemma 2.5. Let A be a commutative ring, I an ideal of A and a ∈ A becomes invertible in A/I. Denote by Aa = A[1/a] the local ring. Then for each n ∈ N the obvious map A/I n → Aa /(Aa .I)n is an isomorphism. Thus, the I-adic completion Aˆ of A and J-adic completion of A[1/a], where J is the ideal generated by the image of I in Aa , are the same. \ Aˆ = A[1/a]

(2.4)

Proof. See [5, Lemma III.91 on p.120].



Proposition 2.6. The Id -adic completion of the ring Rk (GLd ) is the ring of formal power series Z [[s1 , ..., sd ]]. ˆ k (GLd ) ∼ R (2.5) = Z [[s1 , s2 ..., sd ]] Proof. We deduce from the Proposition 2.3 that Rk (GLd ) is isomorphic to the ring Z [s1 , ..., sd , (1 + s1 + ... + sd )−1 ]. By I-adic completion where I = hs1 , ..., sd i we obtain : ˆ k (GLd ) ∼ R + s1 + ... + sd )−1 ] = Z [s1 , ..., sd , (1\ Since 1 + s1 + ... + sd is invertible through Rk (GLd ) → Rk (GLd )/Id we get: ∼ ˆ k (GLd ) ∼ R + s1 + ... + sd )−1 ] ∼ = Z [s1 , ..., sd , (1\ = Z [s\ 1 , ..., sd ] = Z [[s1 , ..., sd ]]  Proposition 2.7. Let i∗d be the restriction morphisms: Rk (GLd+1 ) −→ Rk (GLd ). We have i∗d (sd+1 ) = 0 and i∗d (si ) = si for each i ≤ d. Proof. Recall that the torus Gdm is the group of the diagonal matrices and Xi its characters. From each inclusion : id : Gdm −→ Gd+1 m M 7−→ id (M ) arises a morphism d i∗d : R(Gd+1 m ) −→ R(Gm ) ∗ Xi 7−→ id (Xi )

• For i = d + 1: We have i∗d (Xd+1 ) = Xd+1 ◦ id : M −→ 1. Then i∗d (td+1 ) = i∗d (Xd+1 − 1) = 0. Which implies i∗d (sd+1 ) = 0. • For i ≤ d: Let M = (aii )i≥1 be a diagonal matrix in Gdm . We have i∗d (Xi ) = Xi ◦ id : M −→ aii for each i ≤ d. Then i∗d (Xi ) = Xi for each i ≤ d. We obtain i∗d (ti ) = ti for each i ≤ d. Thus i∗d (si ) = si for each i ≤ d. Finally i∗d (sd+1 ) = 0 and i∗d (si ) = si . 

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bk (GLd+1 ) −→ R bk (GLd ) are Corollary 2.8. The transition morphisms ˆı∗d : R surjective for each d. 3. Representation ring of the special linear group Definition 3.1. Let d be an integer. (1) The special group SLd is the group of matrices with a determinant of 1. (2) Its maximal torus is denoted by Td = Gdm ∩ SLd . Proposition 3.2. The representation ring of Td is given by:   Rk (Td ) = Z X1 , ..., Xd , X1−1 , ..., Xd−1 /X1 × X2 ... × Xd = 1

(3.1)

Proof. We have Td = {A = (aii )i=1,..,d with aii ∈ k ∗ such that a11 ×a22 ×...×ann = 1}, then Td is isomorphic to (k ∗ )d /a11 × a22 × ... × ann = 1. The irreductible representations of Td are the restrictions of the irreductible representations GLd1 to Td . Since a1 × a2 × ... × an = 1 in Td then X1 × X2 ... × Xd = 1 in Rk (Td ). As a result Rk (Td ) = Z X1 , ..., Xd , X1−1 , ..., Xd−1 /X1 × X2 ... × Xd = 1.  Proposition 3.3. Let ti = Xi − 1. Denote si the ith symmetric polynomial on ti . Hence, Rk (SLd ) = Z [s1 , ..., sd ] /s1 + ... + sd = 0 (3.2) Proof. According to [6] we have Rk (Td )Sd = Rk (SLd ) and it follows from the Proposition 3.2 that   Rk (Td ) = Z X1 , ..., Xd , X1−1 , ..., Xd−1 /X1 × X2 ... × Xd = 1 Then   S Rk (SLd ) = Z X1 , ..., Xd , X1−1 , ..., Xd−1 /X1 × X2 ... × Xd = 1 d  S = Z X1 , ..., Xd , X1−1 , ..., Xd−1 d /X1 × X2 ... × Xd = 1 Using the Lemma 2.2 and the Proposition 2.3 we get  S   Z X1 , ..., Xd , X1−1 , ..., Xd−1 d = Z s1 , s2 , ..., sd , (1 + s1 + ... + sd )−1 We have X1 ×X2 ×...×Xd = (t1 +1)(t2 +1)...(td +1) = 1+s1 +...+sd which implies s1 +...+sd = 0. Then Rk (SLd ) = Z [s1 , ..., sd , (1 + s1 + .. + sd )−1 ] /s1 +...+sd = 0. Finally Rk (SLd ) = Z [s1 , ..., sd ] /s1 + .. + sd = 0.  Proposition 3.4. The completed ring of Rk (SLd ) is bk (SLd ) = Z [[s1 , ..., sd ]] /s1 + ... + sd = 0 R 4. Links between the previous representation rings Consider the following embeddings: • id : GLd ,→ GLd+1 • jd : SLd ,→ SLd+1 • kd : SLd ,→ GLd

(3.3)

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We denote by sd the group morphism bellow: GLd −→  SLd+1  A 0 A 7−→ 0 det(A)−1 We have sd ◦ kd = jd . representation rings.

(4.1)

Each of them induces a morphism between the k-

Proposition 4.1. The morphism s∗d : R(SLd+1 ) −→ R(GLd ) σ −→ s∗d (σ) is an isomorphism. be the maximal torus of GLd+1 and Sd+1 the symmetric group. Proof. Let Gd+1 m The following diagram is commutative: Sd R(GLd ) ∼ (R(Gd+1 = m )) ↑s∗d ↑t∗d d+1 ∼ R(SLd+1 ) = R(Gm ∩ SLd+1 )Sd+1 Where td is the restriction of sd to Gdm   −1 −1 −1   ∗ Z X1 , X2 , ..., Xd+1 , X1 , X2 , ...., Xd+1 td : −→ Z X1 , X2 , ..., Xd , X1−1 , X2−1 , ...., Xd−1 X1 X2 ....Xd+1 = 1   diag(ajj ) ∗ td (Xi ) (diag(ajj )) = Xi π(ajj )−1 We have • For 1 ≤ i ≤ d t∗d (Xi ) = Xi • For i = d + 1 t∗d (Xd+1 ) = X1−1 X2−1 ....Xd−1 So :   −1 −1 −1 −1   Z X , X , ..., X , X , X , X , ..., X , X 1 2 d d+1 1 2 d d+1 −→ Z X1 , X2 , ..., Xd , X1−1 , X2−1 , ..., Xd−1 t∗d : X1 X2 ...Xd Xd+1 = 1 From the diagram that s∗d is an isomorphism.  Acknowledgement. The authors would like to thank Tom De Liso, Pierre Vogel, Christophe Deninger, Joel Riou, Perre Pascual and Fabien Morel for the interesting discussions. References 1. M.F. Atiyah, Characters and cohomology of finite groups, Publications Math´ematiques de l’I.H.E.S. Soc. 38 (1961), 23-64. 2. K.Azi and H.Hamraoui Atiyah Motivic theorem for the special group SLd communication au colloque international de Taza/Maroc. 3. H. Hamraoui and B. Kahn, Analogue orthogonal et symplectique d’un th´eor`eme de Rector, Preprint, (2002). 4. D.L. Rector, Modular characters and K-theory with coefficients in a finite field , Journal of Pure and Applied Algebra, 4 (1974), 137-158.

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5. J. Riou, Op´erateurs sur la K-th´eorie alg´ebrique et r´egulateurs via la th´eorie homotopique des sch´emas. Th`ese de Doctorat (2006) 6. J.P. Serre, Groupe de Grothendieck des sch´emas en groupes r´eductifs d´eploy´es. Publications Math´ematiques de l’IHES. 34 (1968), 37-52. 1

CRESC, EGE, Rabat, Morocco. E-mail address: azi [email protected] 2

Department of Mathematics, University Hassan II, Casablanca, Morocco E-mail address: [email protected]