Setting the scene for Betti characters

Finite group actions on free resolutions and modules arise naturally in many interesting examples. Understanding these actions amounts to describing the terms of a free resolution or the graded components of a module as group representations which, in the non modular case, are completely determined by their characters. With this goal in mind, we introduce a Macaulay2 package for computing characters of finite groups on free resolutions and graded components of finitely generated graded modules over polynomial rings.Comment: 7 page

The continuum limit of the quark mass step scaling function in quenched lattice QCD

The renormalisation group running of the quark mass is determined non-perturbatively for a large range of scales, by computing the step scaling function in the Schroedinger Functional formalism of quenched lattice QCD both with and without O(a) improvement. A one-loop perturbative calculation of the discretisation effects has been carried out for both the Wilson and the Clover-improved actions and for a large number of lattice resolutions. The non-perturbative computation yields continuum results which are regularisation independent, thus providing convincing evidence for the uniqueness of the continuum limit. As a byproduct, the ratio of the renormalisation group invariant quark mass to the quark mass, renormalised at a hadronic scale, is obtained with very high accuracy.Comment: 23 pages, 3 figures; minor changes, references adde

Conjugates for Finding the Automorphism Group and Isomorphism of Design Resolutions

Consider a combinatorial design D with a full automorphism group G D. The automorphism group G of a design resolution R is a subgroup of G D. This subgroup maps each parallel class of R into a parallel class of R. Two resolutions R 1 and R 2 of D are isomorphic if some automorphism from G D maps each parallel class of R 1 to a parallel class of R 2. If G D is very big, the computation of the automorphism group of a resolution and the check for isomorphism of two resolutions might be difficult. Such problems often arise when resolutions of geometric designs (the designs of the points and t-dimensional subspaces of projective or affine spaces) are considered. For resolutions with given automorphisms these problems can be solved by using some of the conjugates of the predefined automorphisms. The method is explained in the present paper and an algorithm for construction of the necessary conjugates is presented. ACM Computing Classification System (1998): F.2.1, G.1.10, G.2.1

Resolutions of C^n/Z_n Orbifolds, their U(1) Bundles, and Applications to String Model Building

We describe blowups of C^n/Z_n orbifolds as complex line bundles over CP^{n-1}. We construct some gauge bundles on these resolutions. Apart from the standard embedding, we describe U(1) bundles and an SU(n-1) bundle. Both blowups and their gauge bundles are given explicitly. We investigate ten dimensional SO(32) super Yang-Mills theory coupled to supergravity on these backgrounds. The integrated Bianchi identity implies that there are only a finite number of U(1) bundle models. We describe how the orbifold gauge shift vector can be read off from the gauge background. In this way we can assert that in the blow down limit these models correspond to heterotic C^2/Z_2 and C^3/Z_3 orbifold models. (Only the Z_3 model with unbroken gauge group SO(32) cannot be reconstructed in blowup without torsion.) This is confirmed by computing the charged chiral spectra on the resolutions. The construction of these blowup models implies that the mismatch between type-I and heterotic models on T^6/Z_3 does not signal a complication of S-duality, but rather a problem of type-I model building itself: The standard type-I orbifold model building only allows for a single model on this orbifold, while the blowup models give five different models in blow down.Comment: 1+27 pages LaTeX, 2 figures, some typos correcte

Factorable Monoids : Resolutions and Homology via Discrete Morse Theory

We study groups and monoids that are equipped with an extra structure called factorability. A factorable group can be thought of as a group G together with the choice of a generating set S and a particularly well-behaved normal form map G → S*, where S* denotes the free group over S. This is related to the theory of complete rewriting systems, collapsing schemes and discrete Morse theory. Given a factorable monoid M, we construct new resolutions of Z over the monoid ring ZM. These resolutions are often considerably smaller than the bar resolution E*M. As an example, we show that a large class of generalized Thompson groups and monoids fits into the framework of factorability and compute their homology groups. In particular, we provide a purely combinatorial way of computing the homology of Thompson's group F

A software package for Mori dream spaces

Mori dream spaces form a large example class of algebraic varieties, comprising the well known toric varieties. We provide a first software package for the explicit treatment of Mori dream spaces and demonstrate its use by presenting basic sample computations. The software package is accompanied by a Cox ring database which delivers defining data for Cox rings and Mori dream spaces in a suitable format. As an application of the package, we determine the common Cox ring for the symplectic resolutions of a certain quotient singularity investigated by Bellamy/Schedler and Donten-Bury/Wi\'sniewski.Comment: 11 pages, minor changes, to appear in LMS Journal of Computation and Mathematic