Table algebras generated by elements of small degrees
by
Zvi Arad
Bar-Ilan University
Coauthors: Mikhail Muzychuk

The pair (A, B) is called a table algebra if A is a commutative associative C-algebra of dimension k with a distinguished basis

B = {1=b₁, b₂, ..., b_k} if:

I) ∀i, j, m b_ib_j = ∑_m=1^k l_ijmb_m where l_ijm ∈ R⁺∪{0} (nonnegative reals).
II) ∃-:A → A algebra automorphism s.t. B = [`(B)] and \Bar\Bar a = a ∀a ∈ A. (If b<sub>i</sub> = [`b]_i = b<sub>[`i]</sub>, then b<sub>i</sub> is called real).
III) ∀i, j l<sub>ij1</sub> ≠ 0 iff j = [`i].

Arad and Blau proved that there exists a unique algebra homomorphism f: A → C s.t. f(b_i) = f([`b]_i) ∈ R⁺, 1 ≤ ∀i ≤ n.
{f(b_i) | b_i ∈ B} are called the degree of (A, B).

The goal of our lecture is to give a survey of current research on algebras generated by a basis element of small degree n where n=2, 3, 4 and 5. Natural examples of table algebras are (Z[C(G)], Cl_a(G)) and (Ch(G), Irr(G)) where G is a finite group G.

Date received: June 29, 2000