Geometric contextuality from the Maclachlan-Martin Kleinian groups

There are contextual sets of multiple qubits whose commutation is parametrized thanks to the coset geometry $\mathcal{G}$ of a subgroup $H$ of the two-generator free group $G=\left\langle x,y\right\rangle$. One defines geometric contextuality from the discrepancy between the commutativity of cosets on $\mathcal{G}$ and that of quantum observables.It is shown in this paper that Kleinian subgroups $K=\left\langle f,g\right\rangle$ that are non-compact, arithmetic, and generated by two elliptic isometries $f$ and $g$ (the Martin-Maclachlan classification), are appropriate contextuality filters. Standard contextual geometries such as some thin generalized polygons (starting with Mermin's $3 \times 3$ grid) belong to this frame. The Bianchi groups $PSL(2,O\_d)$, $d \in \{1,3\}$ defined over the imaginary quadratic field $O\_d=\mathbb{Q}(\sqrt{-d})$ play a special role

The enumeration of fully commutative affine permutations

We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci--Del Lungo--Pergola--Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations.Comment: 18 pages; final versio

"Building" exact confidence nets

Confidence nets, that is, collections of confidence intervals that fill out the parameter space and whose exact parameter coverage can be computed, are familiar in nonparametric statistics. Here, the distributional assumptions are based on invariance under the action of a finite reflection group. Exact confidence nets are exhibited for a single parameter, based on the root system of the group. The main result is a formula for the generating function of the coverage interval probabilities. The proof makes use of the theory of "buildings" and the Chevalley factorization theorem for the length distribution on Cayley graphs of finite reflection groups.Comment: 20 pages. To appear in Bernoull

Group-theoretic Approach for Symbolic Tensor Manipulation: II. Dummy Indices

Computational Group Theory is applied to indexed objects (tensors, spinors, and so on) with dummy indices. There are two groups to consider: one describes the intrinsic symmetries of the object and the other describes the interchange of names of dummy indices. The problem of finding canonical forms for indexed objects with dummy indices reduces to finding double coset canonical representatives. Well known computational group algorithms are applied to index manipulation, which allow to address the simplification of expressions with hundreds of indices going further to what is needed in practical applications.Comment: 14 pages, 1 figure, LaTe