Irreducible constituents of minimal degree in supercharacters of the finite unitriangular groups

Let q be a prime power and U the group of lower unitriangular matrices of order n for some natural number n. We give a lower bound for the degrees of irreducible constituents of André-Yan supercharacters and classify the supercharacters having constituents whose degree assume this lower bound. Moreover we show that the number of distinct irreducible characters of U meeting this condition is a polynomial in (q−1) with nonnegative integral coefficients and exhibit monomial sources for those.