Complex Representations of Finite Monoids
Mohan S. Putcha · Proceedings of the London Mathematical Society · 1996
In this paper we study the complex representations of arbitrary finite monoids M. Let θ be an irreducible character of a maximal subgroup (or Schützenberger group) of M. Related to the Schützenberger representations of M. we construct left and right induced characters θ− and θ− of the unit group G of M. We show that the sum of suitable intertwining numbers (θ−, θ−) is equal to the number of conjugacy classes of M. The semigroup induced character θ ¯ of G is shown to be a summand of θ+ ∩ θ−with equality occurring in many natural examples. We apply our results to the full transformation semigroup, determining explicitly all the irreducible characters. We also study the algebra of invariants of the complex monoid algebra. We use this to show that for the multiplicative monoid of triangular matrices over a finite field, all the irreducible complex representations restrict to irreducible representations of the unit group.