Semi-primary Lattices and Tableau Algorithms

We develop the theory of semi-primary lattices, a class of modular lattices, including abelian subgroup lattices and invariant subspace lattices, in which an integer partition is assigned to every element and every interval. Flags in these lattices give rise to chains of partitions, which may be encoded as tableaux. In certain of these lattices, Steinberg and van Leeuwen respectively have shown that relative positions and cotypes, which describe configurations of elements in flags, are generically computed by the well known Robinson-Schensted and evacuation algorithms on standard tableaux. We explore extensions of this to semi-primary lattices: we consider the nongeneric configurations, leading to nondeterministic variations of the Robinson-Schensted and evacuation tableau games, and consider exact and asymptotic enumeration of the number of ways to achieve certain configurations. We also introduce other configuration questions leading to new tableau games, and develop a number of deterministic and nondeterministic tableau operators that can be combined to describe the generic and degenerate configurations of flags undergoing various transformations. We also look at similar problems in the class of modular lattices whose complemented intervals have height at most 2, such as Stanley's Fibonacci lattice Z(r). Here the generic relative position is related to Fomin's analogue of the Robinson-Schensted correspondence in Z(1). Thesis Supervisor: Richard P. Stanley Title: Professor of Applied Mathematics