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Routines to determine quivers suppporting graded twisted Calabi-Yau algebras

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Here is a description of the relevant files for our Maple computations.

Files: cypack.mw / cypack.m ListSpdeg3.mw ListSpdeg4.mw

Descriptions.

cypack.mw / cypack.m This is the primary set of functions.

ListSpdeg3.mw / ListSpdeg4.mw Runs the search procedure for degree 3/4 superpotentials and produces all "good" matrices. Many of these are not actually good but Maple cannot judge them.

IsStronglyConnected(M) - checks whether a matrix M is strongly connected. This is implemented in mforms.mw to generate generic forms of potential matrices.

IsNormal(M) - checks whether a matrix M is normal.

CYpoly(M,d) - returns the matrix polynomial associated to a quiver supporting a CY3 algebra with a superpotential of degree d.

mpoly(M,d) - returns the determinant of CYpoly(M,d).

CY(M,d) - computes the roots of mpoly(M,d).

hseries(M,n,d) - returns a list of the first n terms of the matrix-valued Hilbert series associated to a quiver supporting a CY3 algebra with a superpotential of degree d.

thilb(M,n,d) - returns a list of the first n terms in hseries(M,n,d) after summing all entries in each term.

ph(M,n,d) - prints hseries(M,n,d). Helpful if you want the output as opposed to the list.

hpoly(M,n,d) - returns hseries(M,n,d) written as a polynomial in t up to degree n.

IsGood(M,d) - checks whether all roots of mpoly(M,d) lie on the unit circle.

first(M,d) - evaluates the derivative of mpoly(M,d) at t=1.

GK(M,d) - computes the GK dimension of a vacualgebra A living on the quiver with adjacency matrix M with superpotential of degree d, assuming such a thing exists.

mutate(M,v) - computes the adjacency matrix of a the quiver with adjacency matrix M after mutation at vertex v.

the next set of functions are used to construct generic forms for matrices in our classification.

mfind(u,v,w,gam,d) - first grabs all generic matrix forms with diagonal entries u,v,w and gamma=gam. For each possible form, the matrix polynomial is computed and compared to potentially valid products of cyclotomic polynomials. If a solution is found for indeterminates in generic forms then this solution is returned.

Routines to determine quivers suppporting graded twisted Calabi-Yau algebras

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