We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations that are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics.

We continue in this paper our study of conjugacy classes of a reductive monoid $M$. The main theorems establish a strong connection with the Bruhat-Renner decomposition of $M$. We use our results to decompose the variety $M_{\nil}$ of nilpotent elements of $M$ into irreducible components. We also identify a class of nilpotent elements that we call standard and prove that the number of conjugacy classes of standard nilpotent elements is always finite.

We study the coordinate rings~$k[\Cmubar\cap\hbox{\Frakvii t}]$ of scheme-theoretic intersections of nilpotent orbit closures with the diagonal matrices. Here $\mu'$ gives the Jordan block structure of the nilpotent matrix. de Concini and Procesi~\cite{deConcini&Procesi} proved a conjecture of Kraft~\cite{Kraft} that these rings are isomorphic to the cohomology rings of the varieties constructed by Springer~\cite{Springer76,Springer78}. The famous $q$-Kostka polynomial~$\Klmt(q)$ is the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by~$\lambda$ in the ring $k[\Cmubar\cap\hbox{\Frakvii t}]$. \LS~\cite{L&S:Plaxique,Lascoux} gave combinatorially a decomposition of~$\Klmt(q)$ as a sum of ``atomic'' polynomials with non-negative integer coefficients, and Lascoux proposed a corresponding decomposition in the cohomology model. Our work provides a geometric interpretation of the atomic decomposition. The Frobenius-splitting results of Mehta and van der Kallen~\cite{Mehta&vanderKallen} imply a direct-sum decomposition of the ideals of nilpotent orbit closures, arising from the inclusions of the corresponding sets. We carry out the restriction to the diagonal using a recent theorem of Broer~\cite{Broer}. This gives a direct-sum decomposition of the ideals yielding the $k[\Cmubar\cap \hbox{\Frakvii t}]$, and a new proof of the atomic decomposition of the $q$-Kostka polynomials.