Presenting Schur superalgebras

We provide a presentation of the Schur superalgebra and its quantum analogue which generalizes the work of Doty and Giaquinto for Schur algebras. Our results include a basis for these algebras and a presentation using weight idempotents in the spirit of Lusztig's modified quantum groups.Comment: 28 pages, to appear in the Pacific Journal of Mathematic

Complexity of Modules over Lie Superalgebras

In this dissertation, a fair amount of work is dedicated to computing the complexity of modules over a classical Lie superalgebra \fg=\fg_{\0}\oplus \fg_{\1} over the complex numbers \C. We will consider the category \F of finite dimensional \fg-supermodules which are completely reducible as \fg_{\0}-modules. Every module M\in \F admits a minimal projective resolution whose terms have dimensions which increase at a polynomial rate of growth. This rate of growth is called the \emph{complexity} of $M$. In \cite{BKN1} the authors compute the complexity of the simple and the Kac modules over the general linear Lie superalgebra \gl(m|n) of type $A$. A natural continuation to their work is computing the complexity of the same family of modules over the ortho-symplectic Lie superalgebra \osp(2|2n) of type $C$. The two Lie superalgebras are both of \emph{Type I}, thus the Kac modules in the two cases are constructed by the same induction functor. This similarity will result in similar computations. In fact, our geometric interpretation of the complexity agrees with theirs. The complexity is not a categorical invariant. However, we compute a categorical invariant called the $z$-complexity, introduced in \cite{BKN1}, and we interpret this invariant geometrically in terms of a specific detecting subsuperalgebra. In addition, we compute the complexity and the $z$-complexity of the simple modules over the \emph{Type II} Lie superalgebras \osp(3|2), \dd, $G(3)$, and $F(4)$

Special Issue Call for Papers: Creativity in Mathematics

The Journal of Humanistic Mathematics is pleased to announce a call for papers for a special issue on Creativity in Mathematics. Please send your abstract submissions via email to the guest editors by March 1, 2019. Initial submission of complete manuscripts is due August 1, 2019. The issue is currently scheduled to appear in July 2020

Inquiry as an Entry Point to Equity in the Classroom

Although many policy documents include equity as part of mathematics education standards and principles, researchers continue to explore means by which equity might be supported in classrooms and at the institutional level. Teaching practices that include opportunities for students to engage in active learning have been proposed to address equity. In this paper, through aligning some characteristics of inquiry put forth by Cook, Murphy and Fukawa-Connelly with Gutiérrez\u27s dimensions of equity, we theoretically explore the ways in which active learning teaching practices that focus on inquiry could support equity in the classroom

Exploring Bounds for the Frobenius Number

Let G be a set of three natural numbers, G = {a, b, c}, such that gcd(a, b, c) = 1. The Frobenius number of G is the largest integer that cannot be written as a non-negative linear combination of elements of G. In this article, we present some experimental results on the Frobenius number

Formative Assessment of Creativity in Undergraduate Mathematics: Using a Creativity-in-Progress Rubric (CPR) on Proving

“The project described in this chapter introduces an assessment framework for mathematical creativity in undergraduate mathematics teaching and learning. One outcome of this project is a formative assessment tool, the Creativity-in-Progress Rubric (CPR) on proving, that can be implemented in an introductory proof course. Using multiple methodological tools on a case study, we demonstrate how implementing the CPR on proving can help researchers and educators to observe and assess a student’s development of mathematical creativity in proving. We claim if mathematicians who regularly engage in proving value creativity, then there should be some explicit discussion of mathematical creativity in proving early in a young mathematician’s career. In this chapter, we also outline suggestions on how to introduce mathematical creativity in the undergraduate classroom.” – p. 2