## Final Report Summary - HILBERT (Gröbner strata in multigraded Hilbert schemes)

Publishable Summary

Young diagrams, or equivalently, partitions, are ubiquitous in many branches of mathematics. They appear in representation theory, indexing Specht modules, Schubert varieties in the Grassmannian, Schur polynomials, Schubert polynomials etc. In the French notation, as finite standard sets Delta in N^2, they correspond to monomial ideals M_Delta in the polynomial ring in two variables. Thus they correspond to the torus fixed points on Hilbert schemes of points in the affine plane.

The present research project investigates both incarnations of Young diagrams, along with generalizations of them in the Hilbert-schematic setting. These generalizations, called Connect Four decompositions, or C4 decompositions, and C4 games, were introduced by the principal investigator (PI) of this project. They are finer invariants than d-dimensional standard sets, the straightforward generalization of standard sets from dimension two to arbitrary dimension.

A standard set Delta in N^d of a finite cardinality n leads to a number of different C4 games. This observation, along with an earlier theorem of the PI, led Bernd Sturmfels to conjecture that the set of C4 games of a given Delta be in bijection with the irreducible components of the Hilbert scheme of ideals I in the polynomial ring k[x_1,…,x_d] whose lexicographic Gröbner deformation equals M_Delta. The first major result of the present research project is the proof of Sturmfels’ conjecture for the Hilbert scheme of ideals with the properties listed above plus the constraint that ideal I be supported in n distinct points.

C4 being a combinatorial operation of finite objects, the problem of determining C4’s computational complexity arises. Together with coauthors Laurent Evain and Bjarke Hammersholt Roune, the PI showed that the generating complexity of computing all C4 decompositions of a given Delta is polynomial in the datum of Delta. Moreover, this problem was translated into a very natural problem in graph theory.

Together with coauthors Laurent Evain and Jenna Rajchgot, respectively, the PI enlarged three aspects of the existing theory. The first is the study of Białynicki-Birula schemes in the Hilbert scheme of points in affine d-space, joint with Laurent Evain. In the case d = 2, these schemes are affine cells, and have been prominent objects of interest for a long time. For larger d, however, they are more difficult to handle; the approach of choice here is the use of functorial techniques. The second aspect concerns the case d = 2, in which the PI showed a combinatorial duality between the Hilbert scheme of ideals supported in n distinct points and the Hilbert scheme of ideals supported at the origin to hold. The third aspect, joint with Jenna Rajchgot, studies subschemes of arbitrary Hilbert schemes and torus orbits therein. Two polyhedral complexes were constructed and shown to reflect certain geometric properties of the original scheme.

Finally, in joint work with Allen Knutson, the PI turned to Young diagrams in Schubert calculus. An exceedingly natural deformation of the ring of symmetric functions, implemented in torus-equivariant K-theory of Grassmannians, was constructed. The coefficients describing its multiplicative structure were determined in a positive way.

Young diagrams, or equivalently, partitions, are ubiquitous in many branches of mathematics. They appear in representation theory, indexing Specht modules, Schubert varieties in the Grassmannian, Schur polynomials, Schubert polynomials etc. In the French notation, as finite standard sets Delta in N^2, they correspond to monomial ideals M_Delta in the polynomial ring in two variables. Thus they correspond to the torus fixed points on Hilbert schemes of points in the affine plane.

The present research project investigates both incarnations of Young diagrams, along with generalizations of them in the Hilbert-schematic setting. These generalizations, called Connect Four decompositions, or C4 decompositions, and C4 games, were introduced by the principal investigator (PI) of this project. They are finer invariants than d-dimensional standard sets, the straightforward generalization of standard sets from dimension two to arbitrary dimension.

A standard set Delta in N^d of a finite cardinality n leads to a number of different C4 games. This observation, along with an earlier theorem of the PI, led Bernd Sturmfels to conjecture that the set of C4 games of a given Delta be in bijection with the irreducible components of the Hilbert scheme of ideals I in the polynomial ring k[x_1,…,x_d] whose lexicographic Gröbner deformation equals M_Delta. The first major result of the present research project is the proof of Sturmfels’ conjecture for the Hilbert scheme of ideals with the properties listed above plus the constraint that ideal I be supported in n distinct points.

C4 being a combinatorial operation of finite objects, the problem of determining C4’s computational complexity arises. Together with coauthors Laurent Evain and Bjarke Hammersholt Roune, the PI showed that the generating complexity of computing all C4 decompositions of a given Delta is polynomial in the datum of Delta. Moreover, this problem was translated into a very natural problem in graph theory.

Together with coauthors Laurent Evain and Jenna Rajchgot, respectively, the PI enlarged three aspects of the existing theory. The first is the study of Białynicki-Birula schemes in the Hilbert scheme of points in affine d-space, joint with Laurent Evain. In the case d = 2, these schemes are affine cells, and have been prominent objects of interest for a long time. For larger d, however, they are more difficult to handle; the approach of choice here is the use of functorial techniques. The second aspect concerns the case d = 2, in which the PI showed a combinatorial duality between the Hilbert scheme of ideals supported in n distinct points and the Hilbert scheme of ideals supported at the origin to hold. The third aspect, joint with Jenna Rajchgot, studies subschemes of arbitrary Hilbert schemes and torus orbits therein. Two polyhedral complexes were constructed and shown to reflect certain geometric properties of the original scheme.

Finally, in joint work with Allen Knutson, the PI turned to Young diagrams in Schubert calculus. An exceedingly natural deformation of the ring of symmetric functions, implemented in torus-equivariant K-theory of Grassmannians, was constructed. The coefficients describing its multiplicative structure were determined in a positive way.