Mathematics & Statistics
Permanent URI for this collection
Browse
Browsing Mathematics & Statistics by Subject "Algebraic combinatorics"
Now showing 1 - 2 of 2
Results Per Page
Sort Options
Item Open Access Algebra Structure On Set Partitions(2025-04-10) Solomon, Yohana; Zabrocki, MikeThe partition algebra is an algebra with a basis of set partitions diagrams. Its subalgebra includes diagram algebras such as the uniform block permutations and the group algebra of the symmetric group. We connect the Hopf algebra of uniform block permutations to the diagram algebra known as the party algebra. This is done by describing a new basis of the partition algebra and looking at the relationship to the basis given for the Hopf algebra of uniform block permutations. The product and coproduct of the Hopf algebra of uniform block permutations are the generalization of the product of the Malvenuto-Reutenauer Hopf algebra of permutations. We connect the product of the uniform block permutations with the bases of the partition algebra. The centralizer algebra has an internal product and we define an external product on the partition algebra. This algebra contains the algebra of uniform block permutations and the algebra of permutations.Item Open Access Polarization Operators in Superspace(2024-07-18) Chan, Kelvin Tian Yi; Bergeron, NantelThe classical coinvariant rings and its variants are quotient rings with rich connections to combinatorics, symmetric function theory and geometry. Studies of a generalization of the classical coinvariant rings known as the diagonal harmonics have fruitfully produced many interesting discoveries in combinatorics including the q, t-Catalan numbers and the Shuffle Theorem. The super coinvariant rings are a direct generalization of the classical coinvariant rings to one set of commuting variables and one set of anticommuting variables. N. Bergeron, Li, Machacek, Sulzgruber, and Zabrocki conjectured in 2018 that the super coinvariant rings are representation theoretic models for the Delta Conjecture at t = 0. In this dissertation, we explore the super coinvariant rings using algebraic and combina- torial methods. In particular, we study the alternating component of the super harmonics and discover a novel basis using polarization operators. We use polarization equivalence to establish a triangularity relation between the new basis and a known basis due to two groups of researchers Bergeron, Li, Machacek, Sulzgruber, and Zabrocki and Swanson and Wallach. Furthermore, we prove a folklore result on the cocharge statistics of standard Young tableaux and propose a basis for every irreducible representation appearing in the super harmonics.