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Harm Derksen

Publications and source records attributed to Harm Derksen.

68 records · Page 4Linked to original sources
arXiv2008

Symmetric and Quasi-Symmetric Functions associated to Polymatroids

To every subspace arrangement X we will associate symmetric functions P[X] and H[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant H[X] specializes to the Tutte polynomial T[X]. Billera, Jia and Reiner recently introduced a quasi-symmetric function F[X] (for matroids) which behaves valuatively with respect to matroid base polytope decompositions. We will define a quasi-symmetric function G[X] for polymatroids which has this property as well. Moreover, G[X] specializes to P[X], H[X], T[X] and F[X].

math.CO↗
arXiv2007

The Combinatorics of Quiver Representations

We give a description of faces of all codimensions for the cones of weights of rings of semi-invariants of quivers. For a triple flag quiver and faces of codimension 1 this reduces to the result of Knutson-Tao-Woodward on the facets of the Klyachko cone. We give new applications to Littlewood-Richardson coefficients, including a product formula for LR-coefficients corresponding to triples of partitions lying on a wall of the Klyachko cone. We systematically review and develop the necessary methods (exceptional and Schur sequences, orthogonal categories, semi-stable decompositions, GIT quotients for quivers). In the Appendix we include a version of Belkale's geometric proof of Fulton's conjecture that works for arbitrary quivers.

math.RT↗
arXiv2007

Computing invariants of algebraic group actions in arbitrary characteristic

Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring K[X]^G in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes K[X]^G in terms of a so-called colon-operation. From this, generators of K[X]^G can be obtained in finite time if it is finitely generated. Under the additional hypothesis that K[X] is factorial, we present an algorithm that finds a quasi-affine variety whose coordinate ring is K[X]^G. Along the way, we develop some techniques for dealing with non-finitely generated algebras. In particular, we introduce the finite generation locus ideal.

math.AC↗
arXiv2005

On the number of subrepresentations of a general quiver representation

It is well-known that the intersection multiplicities of Schubert classes in the Grassmanian are Littlewood-Richardson coefficients. We generalize this statement in the context of quiver representations. Here the intersection multiplicity of Schubert classes is replaced by the number of subrepresentations of a general quiver reprsentation, and the Littlewood-Richardson coefficients are replaced by the the dimension of a certain space of semiinvariants.

math.AG↗
arXiv2005

A Skolem-Mahler-Lech Theorem in Positive Characteristic and Finite Automata

Lech proved in 1953 that the set of zeroes of a linear recurrence sequence in a field of characteristic 0 is the union of a finite set and finitely many infinite arithmetic progressions. This result is known as the Skolem-Mahler-Lech theorem. Lech gave a counterexample to a similar statement in positive characteristic. We will present some more pathological examples. We will state and prove a correct analog of the Skolem-Mahler-Lech theorem in positive characteristic. The zeroes of a recurrence sequence in positive characteristic can be described using finite automata.

math.NT↗
arXiv2005

Hilbert series of subspace arrangements

The vanishing ideal I of a subspace arrangement is an intersection of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of a product J of the linear ideals without any assumptions on the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. The results can be applied to Generalized Principal Component Analysis (GPCA), a tool that is useful for computer vision and image processing.

math.AC↗
arXiv2005

Multimagic Squares

In this paper we give the first method for constructing n-multimagic squares (and hypercubes) for any n. We give an explicit formula in the case of squares and an effective existence proof in the higher dimensional case. Finally we prove that n-multimagic squares do not exist for certain orders.

math.CO↗
arXiv2003

Degree Bounds for Syzygies of Invariants

Suppose that G is a linearly reductive group. We study the minimal free resolution of the invariant ring. If G is a finite linearly reductive group, then the ring of invariants is generated in degree at most |G|, the group order. We prove that the ideal of relations of a minimal set of generators of the invariant ring is generated in degree at most 2|G|.

math.AC↗
arXiv2003

Error correcting codes and B_h-sequences

We construct error correcting nonlinear binary codes using a construction of Bose and Chowla in additive number theory. Our method extends a construction of Graham and Sloane for constant weight codes. The new codes improve 1028 of the 7168 best known h-error correcting codes of wordlength at most 512 and h at most 14. We give assymptotical comparisons to shortened BCH codes. Tables of new lower bounds for "A(n,d)" are included.

math.CO↗
arXiv2003

Castelnuovo-Mumford regularity by approximation

The Castelnuovo-Mumford regularity of a module gives a rough measure of its complexity. We bound the regularity of a module given a system of approximating modules whose regularities are known. Such approximations can arise naturally for modules constructed by inductive combinatorial means. We apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups.

math.AC↗
arXiv2003

Universal denominators of Hilbert series

The denominator of the Hilbert series of a finitely generated R-module M does not always divide the denominator of the Hilbert series of R. For this reason, we define the universal denominator. The universal denominator of a module M is the least common multiple of the denominators of the Hilbert series of all submodules of M. The universal denominator behaves nicely with respect to short exact sequences and tensor products. It also has interesting geometric interpretations. Formulas are given for the universal denominator for rings of invariants. Dixmier gave a conjectural formula for the denominator of the Hilbert series of invariants of binary forms. We show that the universal denominator is actually equal to Dixmier's formula in that case.

math.AC↗