Abstract combinatorial commutation properties for separating well-foundedness of unions of relations can be applied to generic path orderings used in termination proofs.

Using the standard Coxeter presentation for the symmetric group Sn, two reduced expressions for the same group element are said to be commutation equivalent if we can obtain one expression from the other by applying a finite sequence of commutations. The resulting equivalence classes of reduced expressions are called commutation classes. How many commutation classes are there for the longest el...

Umbral calculus’ deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of Lie algebras on a lattice. Related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus. This observat...

Recent interest in discrete, classical integrable systems has focused on their connection to quantum integrable systems via the Bethe equations. In this note, solutions to the rational nested Bethe ansatz (RNBA) equations are constructed using the “completed Calogero-Moser phase space” of matrices which satisfy a finite dimensional analogue of the canonical commutation relationship. A key featu...