PLANTEAMIENTO DEL PROBLEMA Ubicación del problema en un contexto
1 ARCOTEL: Agencia de Regulación y Control de las Telecomunicaciones
The following results will also be useful in later sections.
Lemma 1.7.18. [15, Theorem 29.7] and [8, Proposition 1.8.12]. Let F < E be
fields, and ρ, ρ0 :G → GLn(F) be representations. Then if ρ and ρ0 are equivalent
as representations overE, then they are equivalent overF.
Lemma 1.7.19. [14, Theorem 74.9] and [8, Proposition 1.8.13]. Let ρ : G → GLn(E)be an absolutely irreducible representation ofGwith corresponding character χ, withchar(E) =p >0. Let Fdenote the smallest subfield ofEcontaining the base
field Fp and the character values gχ for all g∈G. Then ρ is equivalent overE to a
representation with image in GLn(F).
Corollary 1.7.20. [8, Corollary 1.8.14] LetρandEbe as in Lemma 1.7.19. Ifρ is
equivalent toρφfor an automorphismφof E, then ρis equivalent to a representation
with image in GLn(K), where K is the fixed field ofφ.
1.8
Tensor products
1.8.1 Definitions
We provide a very brief introduction to tensor products of representations and matrices.
Definition 1.8.1.LetV andW be vector spaces over a fieldFwith bases{vi:i∈I}
and{wj :j ∈J}respectively. Then thetensor product ofV andW, denotedV⊗W,
is a vector space with basis{vi⊗wj :i∈I, j ∈J}satisfying the following relations,
fora, b∈I,c, d∈J,λ∈F:
• (va+vb)⊗wc= (va⊗wc) + (vb⊗wc).
• va⊗(wc+wd) = (va⊗wc) + (va⊗wd).
• λ(va⊗wc) = (λva)⊗wc=va⊗(λwc).
Remark 1.8.2. This construction can be extended in a straightforward way when
V andW are representations of a groupGto give a representationV ⊗W ofG; see [30, Chapter 4] for details.
Definition 1.8.3. Let F be a field, and A = (ai,j)a×b and B = (bk,l)c×d, with
ai,j, bk,l ∈F. Then we define the Kronecker product of A and B to be the ac×bd
block matrix, with blocks of sizec×d, where for 1≤r≤a, 1≤s≤b, the (r, s)-th block isar,sB.
The next results are direct from the definitions and will be used frequently without further reference.
Lemma 1.8.4. Let G be a group, and ρ1, ρ2 representations of G over the same
fieldF, with corresponding modulesV1 andV2 respectively. Then the moduleV1⊗V2
has action group given by {(gρ1)⊗(gρ2) :g∈G}.
Lemma 1.8.5. Let A and B be square matrices over the same field F. Then if A
is ann×n matrix andB is anm×m matrix then det(A⊗B) = det(A)mdet(B)n.
We now provide some results about tensor products of representations. Lemma 1.8.6. [8, Lemma 1.9.3] LetGbe a group, and ρ1, ρ2 representations of G
over the same field F, with corresponding modules V1 and V2 respectively. Then:
• For any field automorphism φof F, we have that (V1⊗V2)φ=V1φ⊗V
φ
2 . • (V1⊗V2)∗=V1∗⊗V2∗, where V∗ denotes the dual module of V.
• V1⊗V2∼=V2⊗V1.
Proposition 1.8.7. [8, Proposition 1.9.4] Let G be a group, andρ1, ρ2 representa-
tions ofGover the same fieldF, with corresponding modules V1 andV2 respectively.
Suppose thatGρ1 andGρ2 preserve nondegenerate bilinear formsB1 andB2 respec-
tively. Then:
(i) G(ρ1⊗ρ2) preserves the bilinear form B1⊗B2.
(ii) If B1 and B2 are both symmetric or both anti-symmetric, then B1 ⊗B2 is
symmetric. If one of B1 or B2 is symmetric and the other is anti-symmetric
thenB1⊗B2 is anti-symmetric.
(iii) Ifchar(F) = 2andB1 andB2 are alternating, thenB1⊗B2 is also alternating,
and G(ρ1⊗ρ2) also preserves a quadratic form Q. If F is finite then Q is of
plus type.
1.8.2 Symmetric powers
We provide a very brief definition of the symmetric power of a module; we refer the reader to [8, Section 5.2] for more details.
Definition 1.8.8. LetG be a group, and let V be an n-dimensional module with basise1, . . . , en. Then thesymmetric power Sk(V) ofV is obtained by quotienting
the tensor powerV⊗k by the moduleK :=h(v1⊗ · · · ⊗vk)−(v1σ−1 ⊗ · · · ⊗vkσ−1) :
Remark 1.8.9. It follows from the definition that Sk(V) is generated by elements of the formei1⊗ · · · ⊗eik+K with 1≤i1 ≤. . .≤ik≤n, and hence has dimension
n+k−1 k .
1.9
Number Theory
1.9.1 Legendre symbolsIn this section, we introduce some of the notions from number theory that we will require for the purposes of this thesis. This material is standard; see for instance [45] for a more thorough introduction to the material.
Definition 1.9.1. Letpbe an odd prime, anda∈Z. Thenais aquadratic residue modulop(or asquare modulop) if there exists an integerxsuch thatx2≡a mod p.
Ifa6≡0 mod p then the Legendre symbol is given by
a p := 1 ifais a square modulop,
−1 ifais not a square modulo p.
Ifp|athenap= 0.
Now suppose that q is a positive odd number, so that we can write q =
p1. . . ps for pi odd primes (not necessarily distinct). Then the Jacobi symbol is
given byaq= s Q i=1 a pi .
Note that the Jacobi symbol is an extension of the Legendre symbol; in particular, whenq is a prime the Legendre and Jacobi symbols agree.
We collect together a number of useful results involving the Legendre and Jacobi symbols.
Lemma 1.9.2. [45, Theorems 3.1, 3.4, 3.6 and 3.8] Let p, q be distinct, odd and positive integers, anda, b∈Z.
(i) ab p = a p b p . (ii) pqa=ap aq.
(iii) If p and q are coprime, then
p2 q = p q2 = 1.
(iv) If p and q are coprime, then pq pq= (−1)p−21
q−1 2 .
Using Lemma 1.9.2, it is straightforward to determine, given a specific in- teger a, the primes p such that a is a square modulo p. In particular, we collect the following standard results in the below lemma, that we will use from now on without further reference.
Lemma 1.9.3. Let p be an odd prime. Then:
(i) −1 p = 1 if p≡1 mod 4, −1 if p≡3 mod 4. (ii) 2 p = 1 if p≡ ±1 mod 8, −1 if p≡ ±3 mod 8. (iii) 3 p = 1 if p≡ ±1 mod 12, −1 if p≡ ±5 mod 12. (iv) 5 p = 1 if p≡ ±1 mod 5, −1 if p≡ ±2 mod 5.
We will also be interested in an extension of this notion to algebraic irra- tionalities.
1.9.2 Algebraic irrationalities
Definition 1.9.4. An algebraic irrationality is α ∈C\Q such that there exists a polynomial f ∈Q[x] such thatf(α) = 0. If f can be chosen to be irreducible and
have degree 2, then we say thatα is aquadratic irrationality.
It is a standard result that for an algebraic irrationality α there exists a unique irreducible monic polynomial f ∈ Q[x] such that f(α) = 0; such an f is called theminimal polynomial of α.
Definition 1.9.5. For algebraic irrationalities α1, . . . , αn, we define the number
field Q(α1, . . . , αn) to be the smallest subfield of Ccontaining Qand α1, . . . , αn.
The notion of a number field will be useful when performing various com- putations involving algebraic irrationalities in characteristic 0. We require little knowledge of number fields beyond the definition; for more information see [45, Section 9.3].
Definition 1.9.6. The minimal field of realisation of an algebraic irrationality α
in characteristic p is the smallest field F of characteristic p such that the minimal
polynomialf of α has a root inF.
We use the notation of theAtlas[12] to describe the algebraic irrationalities
we will need in the course of this thesis. We will not generally explicitly define these irrationalities (see [12] or [8, Section 4.2] for more details); for our purposes the information contained in the table below will suffice.
The columns in the table below are as follows:
• ‘Irrat’ gives the name of the algebraic irrationalityα inAtlas notation.
• ‘Real’ denotes whetherα∈Rorα∈C\R.
• ‘Degree’ denotes the degree of the minimal polynomial ofα.
• ‘Min poly’ denotes the minimal polynomial of α. We can compute these in
Magma.
• ‘p-modular reduction’ denotes the degree of the minimal field extension ofFp
containingα, which typically involves some congruence on the primep. Many of the entries in this table are direct from [8, Table 4.2 and Table 4.3] or [48, Table 2.2.1]. The only additional computation we require is for the irrationalitiesb35,d13,r5,y15 andy36.
• b35= −1+
√ −35
2 . The minimal polynomial ofb35isx2+x+ 9, b35always exists overFp2, and exists overFp if and only if
√
−35 does, which we can determine from the Legendre symbol when p is odd and via a direct calculation when
p= 2.
• yn = zn+zn−1 where zn denotes a primitive n-th root of unity. [8, Lemma
4.2.1] shows that forp-n,yn∈Fpe if and only if pe≡ ±1 mod n, and so our congruences follow from this (or a direct computation for the finite number of cases wherep|n).
• r5 = √
5, and congruences follow directly from the Legendre symbol.
• d13=z+z3+z9wherezis a primitive 13-th root of unity. ThusQ(d13)⊂Q(z),
and since the existence ofz overFq (whereq is a power of p) depends on the
value of p mod 13, hence so does the existence of d13 over Fq. Thus we can
Table 1.1: Table of algebraic irrationalities
Irrat Real Degree Min poly p-modular reduction
z3 No 2 x2+x+ 1 Deg 1:p≡0,1 (3) Deg 2:p≡2 (3) b5 Yes 2 x2+x−1 Deg 1:p≡0,1,4 (5) Deg 2:p≡2,3 (5) b7 No 2 x2+x+ 2 Deg 1:p≡0,1,2,4 (7) Deg 2:p≡3,5,6 (7) b11 No 2 x2+x+ 3 Deg 1:p≡0,1,2,3,4,5,9 (11) Deg 2:p≡2,6,7,8,10 (11) b31 No 2 x2+x+ 8 Deg 1:p≡0,1,2,4,5,7,8,9,10,14,16,18,19,20,25,28 (31) Deg 2:p≡3,6,11,12,13,15,17,21,22,23,24,26,27,29,30 (31) b35 No 2 x2+x+ 9 Deg 1:p≡1,3,4,5,7,9,11,12,13,16,17,27,29,33 (35) Deg 2:p≡2,6,8,18,19,22,23,24,26,31,32,34 (35) i No 2 x2+ 1 Deg 1:p≡1,2 (4) Deg 2:p≡3 (4) i2 No 2 x2+ 2 Deg 1:p≡1,2,3 (8) Deg 2:p≡5,7 (8) i5 No 2 x2+ 5 Deg 1:p≡1,2,3,5,7,9 (20) Deg 2:p≡11,13,17,19 (20) r3 Yes 2 x2−3 Deg 1:p≡1,2,3,11 (12) Deg 2:p≡5,7 (12) r5 Yes 2 x2−5 Deg 1:p≡0,1,4,(5) Deg 2:p≡2,3 (5) r6 Yes 2 x2−6 Deg 1:p≡1,2,3,5,19,23 (24) Deg 2:p≡7,11,13,17 (24) y7 Yes 3 x3+x2−2x−1 Deg 1:p≡0,1,6 (7) Deg 2:p≡2,3,4,5 (7) y9 Yes 3 x3−3x+ 1 Deg 1:p≡1,3,8 (9) Deg 3:p≡2,4,5,7 (9) d13 No 4 x4+x3+ 2x2−4x+ 3 Deg 1:p≡0,1,3,9 (13) Deg 2:p≡4,10,12 (13) Deg 4:p≡2,5,6,7,8,11 (13) y15 Yes 4 x4−x3−4x2+ 4x+ 1 Deg 1:p≡1,14 (15) Deg 2:p≡4,11 (15) Deg 4:p≡2,3,5,7,8,13 (15) y36 Yes 6 x6−6x4+ 9x2−3 Deg 1:p≡1,3,35 (36) Deg 2:p≡17,19 (36) Deg 3:p≡2,11,13,23,25 (36) Deg 6:p≡5,7,29,31 (36)