Leading Coefficients of Kazhdan--Lusztig Polynomials in Type DD

Kazhdan--Lusztig polynomials arise in the context of Hecke algebras associated to Coxeter groups. The computation of these polynomials is very difficult for examples of even moderate rank. In type $A$ it is known that the leading coefficient, $\mu(x,w)$ of a Kazhdan--Lusztig polynomial $P_{x,w}$ is either 0 or 1 when $x$ is fully commutative and $w$ is arbitrary. In type $D$ Coxeter groups there are certain "bad" elements that make $\mu$-value computation difficult. The Robinson--Schensted correspondence between the symmetric group and pairs of standard Young tableaux gives rise to a way to compute cells of Coxeter groups of type $A$. A lesser known correspondence exists for signed permutations and pairs of so-called domino tableaux, which allows us to compute cells in Coxeter groups of types $B$ and $D$. I will use this correspondence in type $D$ to compute $\mu$-values involving bad elements. I will conclude by showing that $\mu(x,w)$ is 0 or 1 when $x$ is fully commutative in type $D$.Comment: Author's Ph.D. Thesis (2013) directed by R.M. Green at the University of Colorado Boulder. 68 page

QuESTlink -- Mathematica embiggened by a hardware-optimised quantum emulator

We introduce QuESTlink, pronounced "quest link", an open-source Mathematica package which efficiently emulates quantum computers. By integrating with the Quantum Exact Simulation Toolkit (QuEST), QuESTlink offers a high-level, expressive and usable interface to a high-performance, hardware-accelerated emulator. Requiring no installation, QuESTlink streamlines the powerful analysis capabilities of Mathematica into the study of quantum systems, even utilising remote multicore and GPU hardware. We demonstrate the use of QuESTlink to concisely and efficiently simulate several quantum algorithms, and present some comparative benchmarking against core QuEST.Comment: 11 pages, 5 figures; added new facilities and remote benchmarkin