16,993 research outputs found
Pattern Avoidance and the Bruhat Order
The structure of order ideals in the Bruhat order for the symmetric group is
elucidated via permutation patterns. A method for determining non-isomorphic
principal order ideals is described and applied for small lengths. The
permutations with boolean principal order ideals are characterized. These form
an order ideal which is a simplicial poset, and its rank generating function is
computed. Moreover, the permutations whose principal order ideals have a form
related to boolean posets are also completely described. It is determined when
the set of permutations avoiding a particular set of patterns is an order
ideal, and the rank generating functions of these ideals are computed. Finally,
the Bruhat order in types B and D is studied, and the elements with boolean
principal order ideals are characterized and enumerated by length.Comment: 18 pages, 7 figure
Ore's theorem on subfactor planar algebras
This article proves that an irreducible subfactor planar algebra with a
distributive biprojection lattice admits a minimal 2-box projection generating
the identity biprojection. It is a generalization (conjectured in 2013) of a
theorem of Oystein Ore on distributive intervals of finite groups (1938), and a
corollary of a natural subfactor extension of a conjecture of Kenneth S. Brown
in algebraic combinatorics (2000). We deduce a link between combinatorics and
representations in finite group theory.Comment: 14 pages. It reproduces some preliminaries of arXiv:1702.02124 and
arXiv:1703.04486, for being self-containe
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