| Description |
For certain quadratic algebras Q, we introduce the concept of
Q-tableaux, which are certain combinatorial objects drawn on the
square lattice. These tableaux are equivalent to the notion of a
planar automaton. Planar automata is a new concept (not to be confused
with cellular automata) which formalizes the idea of recognizing
certain "planar figures" drawn on a 2D lattice. Two quadratic algebras
well known in physics are good examples of planar automata: the most
simple Weyl-Heisenberg algebra defined by the commutation relation
UD=DU+Id (creation-annihilation operators in quantum mechanics) and
the so-called PASEP algebra defined by the relation DE=ED+E+D, in the
physics of dynamical systems far from equilibrium. The associated
Q-tableaux are respectively rook placements, permutations and the
so-called alternating, tree-like and permutation tableaux. Other
examples include non-crossing configurations of paths, tiling, plane
partitions and alternating sign matrices.
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