\contentsline {section}{\tocsection {}{1}{Introduction}}{1}
\contentsline {section}{\tocsection {}{2}{Combinatorics}}{6}
\contentsline {subsection}{\tocsubsection {}{2.1}{Final Elements in The Weyl Group}}{6}
\contentsline {subsection}{\tocsubsection {}{2.2}{Final Types and Young Diagrams}}{8}
\contentsline {subsection}{\tocsubsection {}{2.3}{Canonical Types}}{12}
\contentsline {subsection}{\tocsubsection {}{2.4}{Admissible Elements}}{15}
\contentsline {subsection}{\tocsubsection {}{2.5}{Shuffles}}{16}
\contentsline {section}{\tocsection {}{3}{The Flag Space}}{17}
\contentsline {subsection}{\tocsubsection {}{3.1}{The Flag space of the Hodge bundle}}{17}
\contentsline {subsection}{\tocsubsection {}{3.2}{The canonical flag of an abelian variety}}{19}
\contentsline {section}{\tocsection {}{4}{Strata on the Flag Space}}{20}
\contentsline {subsection}{\tocsubsection {}{4.1}{The Stratification}}{20}
\contentsline {subsection}{\tocsubsection {}{4.2}{Some Important Strata}}{21}
\contentsline {subsection}{\tocsubsection {}{4.3}{Shuffling flags}}{25}
\contentsline {subsection}{\tocsubsection {}{4.4}{The E-O strata on ${{\@mathcal A}}_g \otimes {\@mathbb F}_p$}}{28}
\contentsline {section}{\tocsection {}{5}{Extension to the boundary}}{28}
\contentsline {section}{\tocsection {}{6}{Existence of boundary components}}{30}
\contentsline {section}{\tocsection {}{7}{Superspecial fibres}}{31}
\contentsline {section}{\tocsection {}{8}{Local structure of strata}}{34}
\contentsline {subsection}{\tocsubsection {}{8.1}{Stratified Spaces}}{34}
\contentsline {subsection}{\tocsubsection {}{8.2}{Height $1$-Maps}}{34}
\contentsline {section}{\tocsection {}{9}{Punctual flag spaces}}{37}
\contentsline {section}{\tocsection {}{10}{Pieri formulas}}{41}
\contentsline {section}{\tocsection {}{11}{Irreducibility properties}}{45}
\contentsline {section}{\tocsection {}{12}{The Cycle Classes}}{48}
\contentsline {subsection}{\tocsubsection {}{12.1}{Fulton's formulas}}{48}
\contentsline {subsection}{\tocsubsection {}{12.2}{The $p$-rank strata}}{50}
\contentsline {subsection}{\tocsubsection {}{12.3}{The $a$-number Strata}}{52}
\contentsline {subsection}{\tocsubsection {}{12.4}{Positivity of tautological classes}}{54}
\contentsline {section}{\tocsection {}{13}{Tautological rings}}{54}
\contentsline {section}{\tocsection {}{14}{Comparison with ${\@mathcal S}(g,p)$}}{57}
\contentsline {section}{\tocsection {}{15}{Appendix $g=3,4$}}{58}
\contentsline {subsection}{\tocsubsection {}{15.1}{Admissible Strata for $g=3$}}{58}
\contentsline {subsection}{\tocsubsection {}{15.2}{E-O Cycle Classes for $g=3$}}{58}
\contentsline {subsection}{\tocsubsection {}{15.3}{E-O Cycle Classes for $g=4$}}{59}
\contentsline {section}{\tocsection {}{}{References}}{59}