3 research outputs found
Root and weight semigroup rings for signed posets
We consider a pair of semigroups associated to a signed poset, called the
root semigroup and the weight semigroup, and their semigroup rings,
and , respectively.
Theorem 4.1.5 gives generators for the toric ideal of affine semigroup rings
associated to signed posets and, more generally, oriented signed graphs. These
are the subrings of Laurent polynomials generated by monomials of the form
. This result appears to be new
and generalizes work of Boussicault, F\'eray, Lascoux and Reiner, of Gitler,
Reyes, and Villarreal, and of Villarreal. Theorem 4.2.12 shows that strongly
planar signed posets have rings ,
which are complete intersections, with
Corollary 4.2.20 showing how to compute in this case. Theorem 5.2.3
gives a Gr\"obner basis for the toric ideal of in type B,
generalizing Proposition 6.4 of F\'eray and Reiner. Theorems 5.3.10 and 5.3.1
give two characterizations (via forbidden subposets versus via inductive
constructions) of the situation where this Gr\"obner basis gives a complete
intersection presentation for its initial ideal, generalizing Theorems 10.5 and
10.6 of F\'eray and Reiner.Comment: 170 pages; 63 figures; PhD Dissertation, University of Minnesota,
August 201
On a Subposet of the Tamari Lattice
We discuss some properties of a subposet of the Tamari lattice introduced by Pallo (1986), which we call the comb poset. We show that three binary functions that are not well-behaved in the Tamari lattice are remarkably well-behaved within an interval of the comb poset: rotation distance, meets and joins, and the common parse words function for a pair of trees. We relate this poset to a partial order on the symmetric group studied by Edelman (1989)
On a Subposet of the Tamari Lattice
We explore some of the properties of a subposet of the Tamari lattice
introduced by Pallo, which we call the comb poset. We show that three binary
functions that are not well-behaved in the Tamari lattice are remarkably
well-behaved within an interval of the comb poset: rotation distance, meets and
joins, and the common parse words function for a pair of trees. We relate this
poset to a partial order on the symmetric group studied by Edelman.Comment: 21 page