Cubillages of cyclic zonotopes

Abstract

АннотацияThis paper (written in Russian) presents a survey of new and earlier results on fine zonotopal tilings (briefly, cubillages) of cyclic zonotopes. The combinatorial theory of these objects is of interest in its own right and also has a connection to higher Bruhat orders, triangulations of cyclic polytopes, and Tamari-Stasheff posets applied in the study of Kadomtsev-Petviashvily equations, and etc.

Higher Bruhat orders of types B and C

“…In this section we give necessary definitions and recall basic facts on cubillages of cyclic zonotopes used in the paper. For details, see [3].…”

“…Geometrically, the inversion system Inv(Q) can be constructed as follows (for details see [3,Sect. 15]).…”

“…Flips. Next we recall facts about local transformations, or flips, in cubillages (for details, see [3,Sects. 8,12]).…”

“…Instead, we use the combinatorial representation in the form of inversions. For example, for n = 6, one can take the two-element system {1234, 3456} in Gr( [6], 4), which gives rise to a symmetric cubillage in Z (6,3). It is convenient to use the corresponding Ziegler's trellis.…”

“…Proof Let Q be a symmetric cubillage of the zonotope Z(n, d) different from the standard one. Then (see [3,Sect. 12]) it contains a capsid K with the anti-standard filling. If K is symmetric (K = K • ), we can make a simple lowering flip in Q and approach the standard cubillage (reducing the system Inv(Q) by one element).…”