Supercharacters and pattern subgroups in the upper triangular groups

Let $U_n(q)$ denote the upper triangular group of degree $n$ over the finite field ${\text{F}}_q$ with $q$ elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters $\text{Irr}(U_n(q))\mathrm{.}$ In this paper we present a correspondence between supercharacters and pattern subgroups of the form $U_k(q)\cap {}^w{U}_k(q)$ where $w$ is a monomial matrix in $G{L}_k(q)$ for some $k<n\mathrm{.}$