Classification of smooth affine spherical varieties - Knop, Friedrich; Van Steirteghem, Bart (Transformation Groups, 2006)
Let G be a complex reductive group. A normal G-variety X is called spherical if a Borel subgroup of G has a dense orbit in X. Of particular interest are spherical varieties which are smooth and affine since they form local models for multiplicity free Hamiltonian K-manifolds, K a maximal compact subgroup of G. In this paper, we classify all smooth affine spherical varieties up to coverings, central tori, and C-x-fibration

COMBINATORIAL CHARACTERIZATION OF THE WEIGHT MONOIDS OF SMOOTH AFFINE SPHERICAL VARIETIES - Pezzini, Guido; van Steirteghem, Bart (Transactions of the American Mathematical Society, 2019)
Let G be a connected reductive group, and let X be a smooth affine spherical G-variety, both defined over the complex numbers. A well-known theorem of I. Losev's says that X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper, we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties.
CRIS-Team WoS Importer:2019-08-27

Equivariant degenerations of spherical modules for groups of type A - Papadakis, Stavros Argyrios; Van Steirteghem, Bart (Annales de l'Institut Fourier, 2012)
V. Alexeev and M. Brion introduced, for a given a complex reductive group, a moduli scheme of affine spherical varieties with prescribed weight monoid. We provide new examples of this moduli scheme by proving that it is an affine space when the given group is of type A and the prescribed weight monoid is that of a spherical module. © Association des Annales de l'institut Fourier, 2012, tous droits réservés.
CRIS-Team Scopus Importer:2022-11-17
Keywords: Equivariant degeneration; Invariant Hilbert scheme; Spherical module; Spherical variety

Equivariant Degenerations of Spherical Modules: Part II - Papadakis, Stavros Argyrios; Van Steirteghem, Bart (Algebras and Representation Theory, 2016)
We determine, under a certain assumption, the Alexeev–Brion moduli scheme M of affine spherical G-varieties with a prescribed weight monoid. In Papadakis and Van Steirteghem (Ann. Inst. Fourier (Grenoble). 62(5) 1765–1809 19) we showed that if G is a connected complex reductive group of type A and is the weight monoid of a spherical G-module, then M is an affine space. Here we prove that this remains true without any restriction on the type of G.
CRIS-Team Scopus Importer:2022-11-17
Keywords: Equivariant degeneration; Invariant Hilbert scheme; Spherical module; Spherical variety

Momentum polytopes of projective spherical varieties and related Kahler geometry - Cupit-Foutou, Stephanie; Pezzini, Guido; van Steirteghem, Bart (Selecta Mathematica-New Series, 2020)
We apply the combinatorial theory of spherical varieties to characterize the momentum polytopes of polarized projective spherical varieties. This enables us to derive a classification of these varieties, without specifying the open orbit, as well as a classification of all Fano spherical varieties. In the setting of multiplicity free compact and connected Hamiltonian manifolds, we obtain a necessary and sufficient condition involving momentum polytopes for such manifolds to be Kahler and classify the invariant compatible complex structures of a given Kahler multiplicity free compact and connected Hamiltonian manifold.
CRIS-Team WoS Importer:2020-04-21

Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit - Iarrobino, Anthony; Khatami, Leila; Van Steirteghem, Bart; Zhao, Rui (Linear Algebra and Its Applications, 2018)
The Jordan type of a nilpotent matrix is the partition giving the sizes of its Jordan blocks. We study pairs of partitions (P,Q), where Q=Q(P) is the Jordan type of a generic nilpotent matrix A commuting with a nilpotent matrix B of Jordan type P. T. Košir and P. Oblak have shown that Q has parts that differ pairwise by at least two. Such partitions, which are also known as “super distinct” or “Rogers–Ramanujan” are exactly those that are stable or “self-large” in the sense that Q(Q)=Q. In 2012 P. Oblak formulated a conjecture concerning the cardinality of Q−1(Q) when Q has two parts, and proved some special cases. R. Zhao refined this to posit that the partitions in Q−1(Q) for Q=(u,u−r) with u>r>1 could be arranged in an (r−1)×(u−r) table T(Q) where the entry in the k-th row and ℓ-th column has k+ℓ parts. We prove this Table Theorem, and then generalize the statement to propose a Box Conjecture for the set of partitions Q−1(Q) for an arbitrary partition Q whose parts differ pairwise by at least two.
CRIS-Team Scopus Importer:2022-11-17
Keywords: Commuting nilpotent matrices; Generic commuting orbit; Jordan type; Nilpotent orbit; Partition

On Some Families of Smooth Affine Spherical Varieties of Full Rank - Paulus, Kay; Pezzini, Guido; van Steirteghem, Bart (Acta Mathematica Sinica-English Series, 2018)
Let G be a complex connected reductive group. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify: (a) all such varieties for G = SL(2) × ℂ × and (b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and Knop’s classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of Woodward’s result that every reflective Delzant polytope is the moment polytope of such a manifold.
CRIS-Team Scopus Importer:2022-11-17
Keywords: Affine spherical variety; moment polytope; multiplicity free Hamiltonian manifold; weight monoid

The moduli scheme of affine spherical varieties with a free weight monoid - Bravi, Paolo; Van Steirteghem, Bart (International Mathematics Research Notices, 2016)
We study Alexeev and Brion's moduli scheme MΓ of affine spherical varieties with weight monoid Γ under the assumption that Γ is free. We describe the tangent space to MΓ at its "most degenerate point" in terms of the combinatorial invariants of spherical varieties and deduce that the irreducible components of MΓ, equipped with their reduced induced scheme structure, are affine spaces.
CRIS-Team Scopus Importer:2022-11-17