arXiv subjects

Dominic Searles

Publications and source records attributed to Dominic Searles.

21 records · Page 2Linked to original sources

arXiv2013

Root-theoretic Young diagrams and Schubert calculus II

We continue the study of root-theoretic Young diagrams (RYDs) from [Searles-Yong '13]. We provide an RYD formula for the $GL_n$ Belkale-Kumar product, after [Knutson-Purbhoo '11], and we give a translation of the indexing set of [Buch-Kresch-Tamvakis '09] for Schubert varieties of non-maximal isotropic Grassmannians into RYDs. We then use this translation to prove that the RYD formulas of [Searles-Yong '13] for Schubert calculus of the classical (co)adjoint varieties agree with the Pieri rules of [Buch-Kresch-Tamvakis '09], which were needed in the proofs of the (co)adjoint formulas.

math.CO↗

arXiv2013

Root-theoretic Young diagrams and Schubert calculus: planarity and the adjoint varieties

We study root-theoretic Young diagrams to investigate the existence of a Lie-type uniform and nonnegative combinatorial rule for Schubert calculus. We provide formulas for (co)adjoint varieties of classical Lie type. This is a simplest case after the (co)minuscule family (where a rule has been proved by H.Thomas and the second author using work of R.Proctor). Our results build on earlier Pieri-type rules of P.Pragacz-J.Ratajski and of A.Buch-A.Kresch-H.Tamvakis. Specifically, our formula for OG(2,2n) is the first complete rule for a case where diagrams are non-planar. Yet the formulas possess both uniform and non-uniform features. Using these classical type rules, as well as results of P.-E.Chaput-N.Perrin in the exceptional types, we suggest a connection between polytopality of the set of nonzero Schubert structure constants and planarity of the diagrams. This is an addition to work of A.Klyachko and A.Knutson-T.Tao on the Grassmannian and of K.Purbhoo-F.Sottile on cominuscule varieties, where the diagrams are always planar.

math.CO↗

arXiv2011

On the Number of Facets of Polytopes Representing Comparative Probability Orders

Fine and Gill (1973) introduced the geometric representation for those comparative probability orders on n atoms that have an underlying probability measure. In this representation every such comparative probability order is represented by a region of a certain hyperplane arrangement. Maclagan (1999) asked how many facets a polytope, which is the closure of such a region, might have. We prove that the maximal number of facets is at least F_{n+1}, where F_n is the nth Fibonacci number. We conjecture that this lower bound is sharp. Our proof is combinatorial and makes use of the concept of flippable pairs introduced by Maclagan. We also obtain an upper bound which is not too far from the lower bound.

math.CO↗