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Let $q$ be a prime power and $U$ the group of lower unitriangular matrices oforder $n$ for some natural number $n$. We give a lower bound for the degrees ofirreducible constituents of Andr\'{e}-Yan supercharacters and classify thesupercharacters 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 integralcoefficients and exhibit monomial sources for those.

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