Related papers: Invariant trilinear forms for SL(3,R)

Let $S$ be the sphere of dimension $n-1, n\geq 4$. Let $(\pi_{\lambda})_{\lambda\in \mathbb C}$ be the scalar principle series of representations of the conformal group $SO_0(1,n)$, realized on $\mathcal C^\infty(S)$. For $\boldsymbol…

We prove that the Lipshitz-Ozsv\'ath-Thurston correspondence between extended type D structures of knot complements and $\mathbb{F}[U, V]/(UV)$ knot Floer complexes can be arranged so that $\iota_K$-invariant splittings of knot Floer chain…

We extend the short presentation due to [Borel '53] of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its Schubert varieties. Our result is stated in a root-system uniform…

By the family index theory, we generalize some well-known $SL(2,Z)$ modular forms to the family case and obtain some new anomaly cancellation formulas for the determinant line bundle and index gerbes, and certain results about eta…

We prove a short, root-system uniform, combinatorial classification of Levi-spherical Schubert varieties for any generalized flag variety $G/B$ of finite Lie type. We apply this to the study of multiplicity-free decompositions of a Demazure…

A formula for the structure constants of the multiplication of Schubert classes is obtained in arXiv:1909.05283. In this note, we prove analogous formulae for the Chern--Schwartz--MacPherson (CSM) classes and Segre--Schwartz--MacPherson…

For several objects of interest in geometric complexity theory, namely for the determinant, the permanent, the product of variables, the power sum, the unit tensor, and the matrix multiplication tensor, we introduce and study a fundamental…

The works of Donaldson and Mark make the structure of the Seiberg-Witten invariant of 3-manifolds clear. It corresponds to certain torsion type invariants counting flow lines and closed orbits of a gradient flow of a circle-valued Morse map…

Generalizing Schubert cells in type A and a cell decomposition if Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of an oriented cycle admit an affine cell decomposition…

We investigate the breaking of SU(3) into its subgroups from the viewpoints of explicit and spontaneous breaking. A one-to-one link between these two approaches is given by the complex spherical harmonics, which form a complete set of…

We classify simple flops on smooth threefolds, or equivalently, Gorenstein threefold singularities with irreducible small resolution. There are only six families of such singularities, distinguished by Koll{\'a}r's {\em length} invariant.…

This monograph starts with an upper triangular matrix with integer entries and 1's on the diagonal. It develops from this a spectrum of structures, which appear in different contexts, in algebraic geometry, representation theory and the…

Let $G$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$, and $P_1,\dots, P_l$ be an arbitrary set of $l$ splitting parabolic subgroups of $G$. We determine all such sets with the property that $G$ acts with finitely many…

This is a survey of the author's series of three papers math.DG/0111324, math.DG/0111326, math.DG/0204343 using analysis to investigate special Lagrangian 3-folds (SL 3-folds) in C^3 invariant under the U(1)-action (z_1,z_2,z_3) -->…

We prove that the Schubert structure constants of the quantum $K$-theory ring of any minuscule flag variety or quadric hypersurface have signs that alternate with codimension. We also prove that the powers of the deformation parameter $q$…

The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this…

We construct a simple topological invariant of certain 3-manifolds, including quotients of the 3-sphere by finite groups, based on the fact that the tangent bundle of an orientable 3-manifold is trivialisable. This invariant is strong…

We consider a product of three copies of infinite symmetric group and its representations spherical with respect to the diagonal subgroup. We show that such representations generate functors from a certain category of simplicial…

We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag…

Inspired by the structure of $srs$ Laves networks in $\mathbb{R}^3$ that underpin the celebrated gyroid surface, we construct a Laves network of identical three-coordinated vertices on $S^3$ with double-twist. This network is a subset of…