Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2007.02173

On Jordan classes for Vinberg's theta-groups

Giovanna Carnovale·Francesco Esposito·Andrea Santi
View original source ↗Read source document ↗Export metadata ↓

Abstract

Popov has recently introduced an analogue of Jordan classes (packets, or decomposition classes) for the action of a theta-group (G_0,V), showing that they are finitely-many, locally-closed, irreducible unions of G_0-orbits of constant dimension partitioning V. We carry out a local study of their closures showing that Jordan classes are smooth and that their closure is a union of Jordan classes. We parametrize Jordan classes and G_0-orbits in a given class in terms of the action of subgroups of Vinberg's little Weyl group, and include several examples and counterexamples underlying the differences with the symmetric case and the critical issues arising in the theta-situation.

Explore related subjects

math.RTmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Giovanna Carnovale, Francesco Esposito, Andrea Santi. 2021-07-07. On Jordan classes for Vinberg's theta-groups. https://doi.org/10.1007/s00031-021-09675-8

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On singular supports of Lusztig's perverse sheaves

We prove a conjecture of Lusztig on a microlocal characterization of his perverse sheaves. For any finite quiver without loops, an equivariant simple perverse sheaf on the variety of quiver representations is a Lusztig's perverse sheaf if and only if its singular support is contained in Lusztig's Lagrangian variety, that is, the variety of nilpotent representations of the preprojective algebra of the quiver.

math.RT↗
arXiv2026

Skein algebras and quantized Coulomb branches

To a compact oriented surface of genus at most one with boundary, we associate a quantized $K$-theoretic Coulomb branch in the sense of Braverman, Finkelberg, and Nakajima. In the case where the surface is a three- or four-holed sphere or a one-holed torus, we describe a relationship between this quantized Coulomb branch and the Kauffman bracket skein algebra of the surface. We formulate a general conjecture relating these algebras.

math.RT↗
arXiv2026

Quiver presentations for band algebras are defined over the integers

A band is a semigroup in which each element is idempotent. In recent years, there has been a lot of activity on the representation theory of the subclass of left regular bands due to connections to Markov chains associated to hyperplane arrangements, oriented matroids, matroids and CAT(0) cube complexes. We prove here that the integral semigroup algebra of a band is isomorphic to the integral path algebra of a quiver modulo an admissible ideal. This leads to a uniform bound quiver presentation for band algebras over all fields. Also, we answer a question of Margolis, Saliola and Steinberg by proving that the integral semigroup algebra of a CW left regular band is isomorphic to the quotient of the integral path algebra of the Hasse diagram of its support semilattice modulo the ideal generated by the sum of all paths of length two. This includes, for example, hyperplane face semigroup algebras.

math.RT↗