Perforated Tableaux: A Combinatorial Model for Crystal Graphs in Type An−1A_{n-1}

We present a combinatorial model, called \emph{perforated tableaux}, to study $A_{n-1}$ crystals, unifying several previously studied combinatorial models. We identify nodes in the $k$-fold tensor product of the standard crystal with length $k$ words in $[n]= \{ 1, \ldots n\}$. We model this crystal with perforated tableaux (ptableaux) with simpler crystal operators with which we can identify highest weights visually without computation (for all crystals directly, without reference to a canonical model of semistandard Young tableaux (SSYT)). We generalize the tensor products in the Littlewood-Richardson rule to all of $[n]^{\otimes k}$, and not just the irreducible crystals whose reading words come from SSYT. We relate evacuation (Lusztig involution) to products of ptableaux crystal operators, and find a combinatorial algorithm to compute commutators of highest weight ptableaux

Honeycombs from Hermitian Matrix Pairs

Knutson and Tao's work on the Horn conjectures used combinatorial invariants called hives and honeycombs to relate spectra of sums of Hermitian matrices to Littlewood-Richardson coefficients and problems in representation theory, but these relationships remained implicit. Here, let $M$ and $N$ be two $n ×n$ Hermitian matrices. We will show how to determine a hive $\mathcal{H}(M, N)={H_ijk}$ using linear algebra constructions from this matrix pair. With this construction, one may also define an explicit Littlewood-Richardson filling (enumerated by the Littlewood-Richardson coefficient $c_μν ^λ$ associated to the matrix pair). We then relate rotations of orthonormal bases of eigenvectors of $M$ and $N$ to deformations of honeycombs (and hives), which we interpret in terms of the structure of crystal graphs and Littelmann's path operators. We find that the crystal structure is determined \emphmore simply from the perspective of rotations than that of path operators

Honeycombs from Hermitian Matrix Pairs

Knutson and Tao's work on the Horn conjectures used combinatorial invariants called hives and honeycombs to relate spectra of sums of Hermitian matrices to Littlewood-Richardson coefficients and problems in representation theory, but these relationships remained implicit. Here, let $M$ and $N$ be two $n ×n$ Hermitian matrices. We will show how to determine a hive $\mathcal{H}(M, N)={H_ijk}$ using linear algebra constructions from this matrix pair. With this construction, one may also define an explicit Littlewood-Richardson filling (enumerated by the Littlewood-Richardson coefficient $c_μν ^λ$ associated to the matrix pair). We then relate rotations of orthonormal bases of eigenvectors of $M$ and $N$ to deformations of honeycombs (and hives), which we interpret in terms of the structure of crystal graphs and Littelmann's path operators. We find that the crystal structure is determined \emphmore simply from the perspective of rotations than that of path operators

Ethics in the classroom: a reflection on integrating ethical discussions in an introductory course in computer programming

In this paper, we describe our recent approaches to introducing students in a beginning computer science class to the study of ethical issues related to computer science and technology. This consists of three components: lectures on ethics and technology, in-class discussion of ethical scenarios, and a reflective paper on a topic related to ethics or the impact of technology on society. We give both student reactions to these aspects, and instructor perspective on the difficulties and benefits in exposing students to these ideas