## Magma.maths.usyd.edu.au

Irreducible Constituents of Monomial Characters
HTTP://SCIENZE-COMO.UNINSUBRIA.IT/PREVITALI
H:= a subgroup of finite index, say n, of a group G;
T := a right transversal of H in G, thus G =
We assume that G be a subgroup of Sym(n);
T × T becomes G-set via (s, t) · g := (s · g, t · g);
The G-orbits on T × T are called orbitals;
X := (T × T )//G a set of representatives of (H, H)-cosets;
define bijections between orbitals, suborbits and (H, H)-cosets;
F := Q(ζ ), where ζ is a primitive -th root of 1 ∈ C;
t∈T W ⊗ t is the F G-module affording the monomial representa-
where s, t ∈ T , g ∈ G, is the associated monomial matrix;

**Definition**: The orbital (1, x) · G is µ-central if [H ∩ H x, x−1] ≤ ker µ.

**Theorem**: (P. 2005) EndG(V ) =

Λ F cΛ, where Λ varies in the family of
all µ-central orbitals, and c = cΛ is a matrix such that:
2. if Λ = (1, x) · G, x ∈ X, then c(1,x)·g = ρ1x(g),
where ρst(g) := µ(tg(t · g)−1(s · g)g−1s−1), s, t ∈ T , g ∈ G.

If µ = 1H, the trivial character of H, then V becomes the permutationmodule P affording the permutation character (1H)G.

a = aΛ is the adjacency matrix of the orbital Λ, that is, ast = 1 iff (s, t) ∈Λ, ast = 0 otherwise.

**Corollary**: (Higman , Bannai-ˆIto, Michler-Weller) EndG(P ) =

Reorder orbitals so that µ-central occur first and set ci := cΛ ;
We call the structure constants pkij with respect to the basis (c1, . . . , cr) ofC := EndG(V ) the generalized intersection numbers

**Theorem**: pkij may be efficiently obtained as a sum of µ-values depending

on the G-structure of T × T . Moreover, pki1 = δik and pk1j = δjk. In

particular, c1 is the identity matrix and the first row of ci is the i-th standard

vector.

**Corollary**: When µ = 1H, pkij is an intersection number and equals

First reduction: σ : cj −→ (pkij) is the right regular representation forC = EndG(V ).

σ reduces the size of matrices from n = |G : H| to r, the number ofµ-central orbitals.

**Example**: For G = PGL2(73), P ∈ Syl73(G), H = NG(P ), n = 2628

and r = 36.

Using the special shape of σ(ci) we obtain heuristically a set of generatorsfor σ(C) (as an algebra) in log2(r) steps.

Z0 := Z(σ(C)), the center of σ(C), can be efficiently obtained solving alinear system with a small number of equations.

Second reduction: Let τ : Z0 → (F )t be the right regular representationfor Z0, where t = dimF (Z0).

**Definition**: We say A is a one-generator algebra over a field E if A = E[a]

for some a ∈ A.

**Theorem**: (Chillag 1995 P. 2005) Let A be a commutative, semisimple,

finite-dimensional E-algebra, E a separable field. If |E| > dimE(A), thenA is a one-generator algebra.

**Corollary **Let Z = τ (Z0), then Z = F [z], for some z.

z is obtained using a probabilistic approach.

**Theorem**: Let F be an infinite field, Z a semisimple, finite dimensional,

commutative algebra over F , z1, . . . , zt an F -basis for Z. Then z =

ti=1 aizi satisfies Z = F [z] unless (a1, . . . , at) ∈ Zt lies in the union of
ij ≤ Et, where E is a splitting field for Z.

**Theorem**: Let Z = τ (Z(σ(C))) ≤ (F )t be generated by z and E =

Q(ζe), where |ζe| = Exp(G). Then

(a) z admits distinct eigenvalues λ1, . . . , λt in E∗, where t = dimF (Z).

(b) Let Li(x) be the Lagrange polynomials relative to λ1, . . . , λt, thenLi(z) are the central primitive idempotents of Z.

(c) Let fi = (χi, µG) be the multiplicity of χi in µG. Then f2
where ei = Li(τ −1(z)) is a primitive central idempotent for σ(C).

rj=1 aijσ(cj), where cj are the µ-adjacency matrices. Then
aij is the (1, j)-entry of ei. In particular, aij ∈ E.

**Definition**: Given a µ-central orbital Λj and g ∈ G we define the extended

Gollan-Ostermann number

where u ∈ T satisfies xj · hug = 1 · u, for some h ∈ H.

**Theorem**: Let ei = Li(σ−1τ −1(z)) = σ−1(ei), then the ei’s are the

pairwise orthogonal primitive central idempotents for EM (G). Moreover,

ei =

tj=1 aijcj for some aij ∈ E. Let pj(g) be the extended Gollan-
Ostermann numbers. If χi ∈ Irr(G|µG) corresponds to ei, then
i = (χi, µG)2 = rank(ei). In particular, di = χi(1) = nai1.

**Corollary**: When µ = 1H we obtain an algorithm by Michler and Weller

(2002).

**Corollary**: When G is finite and H = 1 we obtain an algorithm due to

Frobenius and Burnside.

Unfortunately arithmetic in the cyclotomic field E = Q(ζe) might be ex-pensive if e = Exp(G) is big;
Resort to a modular `a la Dixon approach;
p a prime congruent to 1 (mod e) and p > max(2n, t);
L := Fp and εe ∈ L∗ such that |εe| = e;
Build homorphism θ from Z[ζe] into L via
Set ML(g) := θ(M(g)), where we extend θ to matrices and M is themonomial representation;
Using a theorem of Brauer and Nesbitt we may express the modular re-
duction θ(χi(g)) as in the cyclotomic case;
Knowing the power maps in G we may lift these modular values uniquelyinto E.

Source: http://magma.maths.usyd.edu.au/Magma2006/talks/Previtali-IrrConstMonCharsSlideBerlin.pdf

SUMMARY OF BENEFITS Connecticut General Life Insurance Co. Long Island University – Buy-Up 2 Preferred Provider Organization Copay Plan Annual deductibles and maximums In-network Out-of-network Lifetime maximum Pre-Existing Condition Limitation (PCL) Coinsurance Maximum reimbursable charge • Determined based on the lesser of: • the health care profession

TERRA TREATMENT - the treatment you can trust! The Significance of Sutherlandia - PRO Sutherlandia has application in the treatment of all conditions that are associated with impaired immune function, stress or general debility of the body. It strengthens the body's natural immune response, supports the healing system and accelerates the recovery process. The full botanical name of Suther