The upper unitriangular group scheme U_n and its coordinate ring

Let $k$ be a field and let $n\ge 1$. Let ${\mathrm{G}\mathrm{L}}_n$ be the general linear group scheme over $k$ with its coordinate ring $k[x_{ij},{\det }^{-1}]$ (The general linear group scheme and its coordinate ring) and let $T_n$, $D_n$, $U_n$ be the closed subschemes of ${\mathrm{G}\mathrm{L}}_n$ defined by the following equations on the matrix entries, cut out as quotients of the coordinate ring of ${\mathrm{G}\mathrm{L}}_n$ (Closed immersions into affine schemes are quotient spectra):

• $T_n$: the upper triangular matrices, defined by $x_{ij}=0$ for $i>j$;
• $D_n$: the diagonal matrices, defined by $x_{ij}=0$ for $i\mathrm{\ne }j$;
• $U_n$: the upper unitriangular matrices, defined by $x_{ij}=0$ for $i>j$ and $x_{ii}=1$.

These are closed subgroup schemes of ${\mathrm{G}\mathrm{L}}_n$ (Morphisms and closed subgroup schemes of group schemes), because for each of them the defining equations are stable under matrix multiplication, inverse and identity; equivalently, by the valued-point criterion (Closed subgroup schemes are detected on all algebra-valued points), for every commutative unital $k$-algebra $R$ the $R$-points are the corresponding subgroups $T_n(R),D_n(R),U_n(R)$ of ${\mathrm{G}\mathrm{L}}_n(R)$, described by the same equations. In particular $U_n(R)=\{({\alpha }_{ij})\in {\mathrm{G}\mathrm{L}}_n(R):{\alpha }_{ij}=0(i>j),{\alpha }_{ii}=1\},$ the group of upper unitriangular matrices, and $T_n=D_n⋉U_n$ is the group of invertible upper triangular matrices, the semidirect product for the conjugation action of $D_n$ on $U_n$ (Upper triangular, lower triangular and diagonal square matrices over a commutative ring).

The coordinate ring of $U_n$ is $O(U_n)=k[X_{ij}\mid 1\le i<j\le n]$ with comultiplication $\mathrm{\Delta }(X_{ij})=X_{ij}\otimes 1+1\otimes X_{ij}+{\sum }_{i<l<j}{X}_{il}\otimes X_{lj},$ counit $\epsilon (X_{ij})=0$ and antipode determined by the inverse of a unitriangular matrix; the displayed formula is the matrix multiplication formula restricted to unitriangular matrices and visibly preserves the polynomial ring, so $O(U_n)$ is a polynomial algebra on the $\left(\genfrac{}{}{0px}{}{n}{2}\right)$ entries strictly above the diagonal (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis).