17 research outputs found
Combinatorial Hopf algebra structure on packed square matrices
We construct a new bigraded Hopf algebra whose bases are indexed by square
matrices with entries in the alphabet , , without
null rows or columns. This Hopf algebra generalizes the one of permutations of
Malvenuto and Reutenauer, the one of -colored permutations of Novelli and
Thibon, and the one of uniform block permutations of Aguiar and Orellana. We
study the algebraic structure of our Hopf algebra and show, by exhibiting
multiplicative bases, that it is free. We moreover show that it is self-dual
and admits a bidendriform bialgebra structure. Besides, as a Hopf subalgebra,
we obtain a new one indexed by alternating sign matrices. We study some of its
properties and algebraic quotients defined through alternating sign matrices
statistics.Comment: 35 page
Selected Topics in Gravity, Field Theory and Quantum Mechanics
Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories