Structure of the rational monoid algebra for Boolean matrices of order 3

Abstract

We use the computer algebra system Maple to study the 512-dimensional associative algebra QB3, the rational monoid algebra of 3×3 Boolean matrices. Using the LLL algorithm for lattice basis reduction, we obtain a basis for the radical in bijection with the 42 non-regular elements of B3. The center of the 470-dimensional semisimple quotient has dimension 14; we use a splitting algorithm to find a basis of orthogonal primitive idempotents. We show that the semisimple quotient is the direct sum of simple two-sided ideals isomorphic to matrix algebras Md(Q) for d=1,1,1,2,3,3,3,3,6,6,7,9,9,12. We construct the irreducible representations of B3 over Q by calculating the representation matrices for a minimal set of generators.