arXiv subjects

Christopher Manon

Publications and source records attributed to Christopher Manon.

42 records · Page 3Linked to original sources

arXiv2012

Coordinate rings for the moduli of $SL_2(\C)$ quasi-parabolic principal bundles on a curve and toric fiber products

We continue the program started in \cite{M1} to understand the combinatorial commutative algebra of the projective coordinate rings of the moduli stack $\mathcal{M}_{C, \vec{p}}(SL_2(\C))$ of quasi-parabolic $SL_2(\C)$ principal bundles on a generic marked projective curve. We find general bounds on the degrees of polynomials needed to present these algebras by studying their toric degenerations. In particular, we show that the square of any effective line bundle on this moduli stack yields a Koszul projective coordinate ring. This leads us to formalize the properties of the polytopes used in proving our results by constructing a category of polytopes with term-orders. We show that many of results on the projective coordinate rings of $\mathcal{M}_{C, \vec{p}}(SL_2(\C))$ follow from closure properties of this category with respect to fiber products.

math.AC↗

arXiv2011

Dissimilarity maps on trees and the representation theory of $GL_n(\C)$

We revisit the representation theory in type $A$used previously to establish that the dissimilarity vectors of phylogenetic trees are points on the tropical Grassmannian variety. We use a different version of this construction to show that the space of phylogenetic trees $K_n$ maps to the tropical varieties of every flag variety of $GL_n(\C).$ Using this map, we interpret the tropicalization of the semistandard tableaux basis of an irreducible representation of $GL_n(\C)$ as combinatorial invariants of phylogenetic trees.

math.AG↗

arXiv2011

Gorenstein Semigroup Algebras of Weighted Trees

We classify exactly when the toric algebras $\C[S_{\tree}(\br)]$ are Gorenstein. These algebras arise as toric deformations of algebras of invariants of the Cox-Nagata ring of the blow-up of $n-1$ points on $\mathbb{P}^{n-3}$, or equivalently algebras of the ring of global sections for the Plücker embedding of weight varieties of the Grassmanian $Gr_2(\C^n)$, and algebras of global sections for embeddings of moduli of weighted points on $\mathbb{P}^1$. As a corollary, we find exactly when these families of rings are Gorenstein as well.

math.AC↗

arXiv2010

Valuations from representation theory and tropical geometry

We recall the space of seminorms discussed by Payne in \cite{P} and define a slight modification, the space of graded valuations. After explaining how these spaces relate to tropical geometry, we describe examples of graded valuations which come from the representation theory of reductive groups.

math.CO↗

arXiv2010

The $m-$dissimilarity map and representation theory of $SL_m$

We give another proof that $m$-dissimilarity vectors of weighted trees are points on the tropical Grassmanian, as conjectured by Cools, and proved by Giraldo in response to a question of Sturmfels and Pachter. We accomplish this by relating $m$-dissimilarity vectors to the representation theory of $SL_m.$

math.AG↗

arXiv2008

The Toric Geometry of Triangulated Polygons in Euclidean Space

Speyer and Sturmfels [SpSt] associated Gröbner toric degenerations $\mathrm{Gr}_2(\C^n)^{\tree}$ of $\mathrm{Gr}_2(\C^n)$ to each trivalent tree $\tree$ with $n$ leaves. These degenerations induce toric degenerations $M_{\br}^{\tree}$ of $M_{\br}$, the space of $n$ ordered, weighted (by $\br$) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers as stratified symplectic spaces and describe the action of the compact part of the torus as "bendings of polygons." We prove the conjecture of Foth and Hu [FH] that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida [KY].

math.SG↗