Irreducible Constituents of Monomial Characters
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.
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