In this talk we dive into the representation theory of Lie algebras. Representation theory is an area of math that studies algebraic structures and the objects they “act on.” In addition, it is also a great tool that takes problems in abstract algebra and turns them into linear algebra problems. More specifically, we will be looking at what is called the current algebra $sl_2 \otimes \mathbb{C}[t]$ , where $sl_2$ is the space of complex $2\times 2$ matrices whose trace is zero, and $\mathbb{C}[t]$ is the space of polynomials with complex coefficients. The combinatorics involved in this talk are motivated by giving an explicit filtration (chain of submodules) of a well-studied family of modules, the level l Demazure modules, which are indexed by a natural number \ell and a ‘vector lambda.’ We know such a construction must exist by Naoi who proved the existence for every $\ell≥1$. In this talk we will construct maps that determines our explicit filtration. In defining these maps we have created a “game” that amounts to turning 1’s into 2’s. This “game” can also be played on a directed graph, turning the filtration question into one that investigates certain ways you can traverse this graph. We will investigate this graph and talk about some generalizations.
P-adic numbers after its introduction by Kurt Hensel more than a century ago, has been a mainstay in the field of number theory. An abstract concept by itself, in this talk we will look at how we can visualize p-adic numbers with a tree structure and look at some basic questions in p-adic analysis that can be answered with this visualization.
The zero divisor graph of a commutative ring $R$ is formed by taking the nonzero zero divisors of $R$ as the vertices and connecting two vertices exactly when the corresponding product of the two elements is zero. We will show that all 44 planar zero divisor graphs are subgraphs of planar graphs with a Hamiltonian cycle and that all 46 genus one zero divisor graphs are subgraphs of toroidal graphs with a Hamiltonian cycle.
A group with a cyclically symmetric presentation admits an automorphism of finite order called the shift. In this talk we look at cyclically presented groups which admit a certain decomposition, and relate the shift dynamics for the group to the components of the decomposition. Topological methods are used to identify fixed points for powers of the shift.
Given an arbitrary Coxeter system (W,S), Lusztig defined its asymptotic Hecke algebra J, an associative algebra closely related to the usual Hecke algebra and the category of Soergel bimodules for (W,S). The algebra J decomposes as a direct sum of subalgebras indexed by the 2-sided Kazhdan-Lusztig cells of W, and we present some results on the subalgebra $J_c$ corresponding to a particular cell c known as the subregular cell. We show that products in $J_c$ can be computed by repeated use of (variations of) the Clebsch-Gordan formula arising from the representation theory of $sl_2$, and we use this multiplication rule to obtain alternative descriptions of $J_c$ for Coxeter systems of certain types (such as all simply-laced ones).
We consider nth order linear recurrence relations of the form $S_k=a_{k-1}S_{k-1}+a_{k-2}S_{k-2}+\cdots+a_{k-n}S_{k-n}$ over the finite field $Z_p$, where $p$ is a prime not equal to 2. The results regarding the distribution of elements in the sequence $\{S_0,S_1, \dots \}$ are well known for second order linear recurrence relations, however, we expand some results using matrix groups, linear algebra and $G$-sets in the finite vector space $\left(Z_p\right)^k$. It is our hope that this alternate approach may provide a set of material or examples that could be utilized in undergraduate mathematics courses.
Cohen-Macaulay rings play a central role in commutative algebra and there are many connections between systems of parameters and the Cohen-Macaulay property. In a Cohen-Macaulay ring, every system of parameters is also a regular sequence (roughly speaking it behaves like a set of polynomial variables). A classical result due to Rees says that when working in a Cohen-Macaulay ring, a certain class of modules of homomorphisms defined by systems of parameters is always isomorphic to a certain free module of rank one. Recently, K. Bahmanpour and R. Naghipour showed that in a non-Cohen-Macaulay ring, the same class of modules of homomorphisms is sometimes decomposable as a direct sum, and therefore is not a free module of rank one. In this talk, we will present stronger theorems in the non-Cohen-Macaulay case, and present illustrative examples about the decompositions obtained.
I will give an overview of the representation theory of the queer Lie superalgebra q(n), focusing in particular on the representations of q(2) in its BGG category O.
Many partition results and $q$-series identities are classically derived through analytical techniques, though the results beg for a combinatoric interpretation. -- the standard examples being the Ramanujan congruences. Historically, the combinatorics were filled in by studying integer valued functions on the set of partitions, namely the rank and the crank functions.
We show how the ordinary partition statistics can be extended into the more general overpartition case.