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Full-Text Articles in Algebra
The Generalized Burnside And Representation Rings, Eric B. Kahn
The Generalized Burnside And Representation Rings, Eric B. Kahn
University of Kentucky Doctoral Dissertations
Making use of linear and homological algebra techniques we study the linearization map between the generalized Burnside and rational representation rings of a group G. For groups G and H, the generalized Burnside ring is the Grothendieck construction of the semiring of G × H-sets with a free H-action. The generalized representation ring is the Grothendieck construction of the semiring of rational G×H-modules that are free as rational H-modules. The canonical map between these two rings mapping the isomorphism class of a G-set X to the class of its permutation module …
Rees Products Of Posets And Inequalities, Tricia Muldoon Brown
Rees Products Of Posets And Inequalities, Tricia Muldoon Brown
University of Kentucky Doctoral Dissertations
In this dissertation we will look at properties of two different posets from different perspectives. The first poset is the Rees product of the face lattice of the n-cube with the chain. Specifically we study the Möbius function of this poset. Our proof techniques include straightforward enumeration and a bijection between a set of labeled augmented skew diagrams and barred signed permutations which label the maximal chains of this poset. Because the Rees product of this poset is Cohen-Macaulay, we find a basis for the top homology group and a representation of the top homology group over the symmetric …
Algebraic And Combinatorial Properties Of Certain Toric Ideals In Theory And Applications, Sonja Petrovic
Algebraic And Combinatorial Properties Of Certain Toric Ideals In Theory And Applications, Sonja Petrovic
University of Kentucky Doctoral Dissertations
This work focuses on commutative algebra, its combinatorial and computational aspects, and its interactions with statistics. The main objects of interest are projective varieties in Pn, algebraic properties of their coordinate rings, and the combinatorial invariants, such as Hilbert series and Gröbner fans, of their defining ideals. Specifically, the ideals in this work are all toric ideals, and they come in three flavors: they are defining ideals of a family of classical varieties called rational normal scrolls, cut ideals that can be associated to a graph, and phylogenetic ideals arising in a new and increasingly popular area of …