Browsing by Author "Mohan S. Putcha, Committee Chair"

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    Fine Bruhat Intersections for Reductive Monoids
    (2004-07-20) Taylor, Dewey Terese; Mohan S. Putcha, Committee Chair
    Fine Bruhat intersections for reductive groups have been studied by several authors in connection with Kazhdan-Lusztig theory, canonical bases and Lie Theory. The purpose of this thesis is to study the analogous intersections for reductive monoids. We determine the conditions under which the following intersections BsB ∩ B⁻qB BsB ∩ B⁻qB⁻ are nonempty. First we study these intersections for the monoid Mn(K). This work gives rise to two new orderings, ≤1 and ≤2, on the monoid of partial permutation matrices. More precisely, ≤1 and ≤2 are orderings that exist within a particular J-class for a reductive monoid M. The J-classes of Mn(K) consist of matrices of the same rank. Combinatorial descriptions of the orderings are given and their relation to the Bruhat-Chevalley order is discussed. These results are then generalized to an arbitrary reductive monoid. We show that ≤1 is a partial order on R, the Renner monoid, and that ≤2 is in general not a partial order on elements of R, but rather on equivalence classes of elements in R. We describe the equivalence classes for the matrices and conclude with theorems for the partial permutation matrices of rank r < n .
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    Transformation Semigroups Over Groups
    (2008-03-25) Petersen, Richard Francis; Kailash Misra, Committee Member; Ronald Fulp, Committee Member; Tom Lada, Committee Member; Mohan S. Putcha, Committee Chair
    The semigroup analogue of the symmetric group, S_{n}, is the full transformation semigroup, T_{n}. T_{n} is the set of all mappings from the set {1,2,..n} to itself. This semigroup has been studied in great detail, especially in connection with automata theory. The wreath product of a group G by S_{n} has been studied for almost one hundred years. In this thesis, we study the wreath product of a group G by T_{n}. These wreath products are expressed as GwrS_{n} and GwrT_{n}, respectively. Many interesting theorems and properties for wreath products will be discussed. For example, the result of John Howie that every element in T_{n} − S_{n} can be expressed as a product of idempotents, is generalized to show that any element of GwrT_{n}- GwrS_{n} can be expressed as a product of idempotents. It will also be shown that GwrT_{n} is unit regular. Chapter five begins with a review of Green's relations for a moniod, M. Green's relations for T_{n} are also reviewed and R and L-classes for the wreath product GwrT_{n} are determined. Finally, in the last two chapters, the conjugacy class structures of GwrT_{n} are determined. Just as the conjugacy classes of GwrS_{n} are indexed by colored partitions, we show that the conjugacy classes of GwrT_{n} are indexed by certain colored directed graphs.