Geometric representation theory of restricted Lie algebras

We modify the Hochschild -map to construct central extensions of a restricted Lie algebra. Such central extension gives rise to a goup scheme that leads to a geometric construction of unrestricted representations. For a classical semisimple Lie algebra, we construct equivariant line bundles whose global sections afford representations with a nilpotentp-character.Supported by NSF.Supported by NSF and EPSRC.     

Geometric representation theory for unitary groups of operator algebras

Geometric realizations for the restrictions of GNS representations to unitary groups of C^∗-algebras are constructed. These geometric realizations use an appropriate concept of reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic sections, a class of complex coadjoint orbits of the corresponding real Banach-Lie groups is described and some homogeneous holomorphic Hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed.     

Dissection of solutions in cooperative game theory using representation techniques

We compute a decomposition for the space of cooperative TU-games under the action of the symmetric group S _n . In particular we identify all irreducible subspaces that are relevant to the study of symmetric linear solutions – namely those that are isomorphic to the irreducible summands of . We then use such decomposition to derive, in a very economical way, some old and some new results for linear symmetric solutions.     

Geometric Gibbs theory

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space, which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.

     

Geometric representation of substitutions of Pisot type

We prove that a substitutive dynamical system of Pisot type contains a factor which is isomorphic to a minimal rotation on a torus. If the substitution is unimodular and satisfies a certain combinatorial condition, we prove that the dynamical system is measurably conjugate to an exchange of domains in a self-similar compact subset of the Euclidean space.

     

Spherical subcategories in representation theory

Mathematische Zeitschrift - We introduce a new invariant for triangulated categories: the poset of spherical subcategories ordered by inclusion. This yields several numerical invariants, like the...     

Ghosts in modular representation theory

A ghost over a finite p-group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis—the statement that ghosts between finite-dimensional G-representations factor through a projective—we define the ghost number of kG to be the smallest integer l such that the composite of any l ghosts between finite-dimensional G-representations factors through a projective. In this paper we study ghosts and the ghost numbers of p-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation k, holds for all p-groups. We then compute the ghost numbers of all cyclic p-groups and all abelian 2-groups with C₂ as a summand. We obtain bounds on the ghost numbers for abelian p-groups and for all 2-groups which have a cyclic subgroup of index 2. Using these bounds we determine the finite abelian groups which have ghost number at most 2. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory.     

Geometric composition in quilted Floer theory

We prove that Floer cohomology of cyclic Lagrangian correspondences is invariant under transverse and embedded composition under a general set of assumptions.     

Identities from representation theory

We give a new Jacobi–Trudi-type formula for characters of finite-dimensional irreducible representations in type $C_n$ using characters of the fundamental representations and non-intersecting lattice paths. We give equivalent determinant formulas for the decomposition multiplicities for tensor powers of the spin representation in type $B_n$ and the exterior representation in type $C_n$. This gives a combinatorial proof of an identity of Katz and equates such a multiplicity with the dimension of an irreducible representation in type $C_n$. By taking certain specializations, we obtain identities for $q$-Catalan triangle numbers, a slight modification of the $q,t$-Catalan number of Stump, $q$-triangle versions of Motzkin and Riordan numbers, and generalizations of Touchard’s identity. We use (spin) rigid tableaux and crystal base theory to show some formulas relating Catalan, Motzkin, and Riordan triangle numbers.     

Z 分次表示理论

箭图Hecke 代数的Z 分次表示理论是“代数群、量子群及Hecke 代数” 领域中当前最活跃的研究方向之一. 箭图Hecke 代数及其分圆版本产生于对量子群及其可积最高权表示的范畴化的研究, 它们与数学及数学物理的许多不同分支如Lie 代数、量子群、Kazhdan-Lusztig 理论、代数几何(箭图簇,反常层)、扭结理论、拓扑量子场论(TQFT) 等都有着紧密的联系与相互作用. 本文详细介绍了该方向的最新进展、前沿以及研究前景.     

Geometric representation of cubic graphs with four directions

We show that every connected cubic graph can be drawn in the plane with straight-line edges using only four distinct slopes and disconnected cubic graphs with five distinct slopes.