1,249 research outputs found
Lattice congruences, fans and Hopf algebras
We give a unified explanation of the geometric and algebraic properties of
two well-known maps, one from permutations to triangulations, and another from
permutations to subsets. Furthermore we give a broad generalization of the
maps. Specifically, for any lattice congruence of the weak order on a Coxeter
group we construct a complete fan of convex cones with strong properties
relative to the corresponding lattice quotient of the weak order. We show that
if a family of lattice congruences on the symmetric groups satisfies certain
compatibility conditions then the family defines a sub Hopf algebra of the
Malvenuto-Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has
a basis which is described by a type of pattern-avoidance. Applying these
results, we build the Malvenuto-Reutenauer algebra as the limit of an infinite
sequence of smaller algebras, where the second algebra in the sequence is the
Hopf algebra of non-commutative symmetric functions. We also associate both a
fan and a Hopf algebra to a set of permutations which appears to be
equinumerous with the Baxter permutations.Comment: 34 pages, 1 figur
Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals
We prove a theorem unifying three results from combinatorial homological and
commutative algebra, characterizing the Koszul property for incidence algebras
of posets and affine semigroup rings, and characterizing linear resolutions of
squarefree monomial ideals. The characterization in the graded setting is via
the Cohen-Macaulay property of certain posets or simplicial complexes, and in
the more general nongraded setting, via the sequential Cohen-Macaulay property.Comment: 31 pages, 1 figure. Minor changes from previous version. To appear in
Advances in Mathematic
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