49 research outputs found
Real representations of C2-graded groups : the linear and Hermitian theories
No full textWe study linear and hermitian representations of finite C2-graded groups. We prove that the category of linear representations is equivalent to a category of antilinear representations as an infinity-category. We also prove that the category hermitian representations, as an ∞-category, is equivalent to a category of usual representations
Real representations of C2-graded groups : the antilinear theory
No full textWe use the structure of finite-dimensional graded algebras to develop the theory of antilinear representations of finite C2-graded groups. A finite C2-graded group is a finite group with a subgroup of index 2. In this theory the subgroup acts linearly, while the other coset acts antilinearly. We introduce antilinear blocks, whose structure is a crucial component of the theory. Among other things, we study characters and Frobenius-Schur indicators. As an example, we describe the antilinear representations of the C2-graded group An≤Sn
Brauer's 14th Problem and Dyson's Tenfold Way
No full textWe consider Brauer's 14th Problem in the context of "Real" structures on
finite groups and their antilinear representations. The problem is to count the
number of characters of different type using "group theory". While Brauer's
original problem deals only with the three types (real, complex and
quaternionic), here we consider the ten types coming from Dyson's tenfold way
Subregular representations of Sln and simple singularities of type An-1. Part II
No full textThe aim of this paper is to show that the structures on K-theory
used to formulate Lusztig's conjecture for subregular nilpotent sln-representations
are, in fact, natural in the McKay correspondence. The main result is a
categorification of these structures. The no-cycle algebra plays the special role
of a bridge between complex geometry and representation theory in positive
characteristic
Kac-Moody groups and their representations
No full textIn this expository paper we review some recent results about representations of Kac-Moody groups. We sketch the construction of these groups. If practical, we present the ideas behind the proofs of theorems. At the end we pose open questions
Duality for Hopf Algebroids
No full textAbstractOur goal is twofold. First, we formulate a duality between commutative bialgebroids and cocommutative bialgebras over a ring extension. Second, we show that for a certain action groupoid J, the Hopf algebroid of functions on the Frobenius kernel J1 is dual to the restricted enveloping algebra of the Lie algebroid of J
2-groups, 2-characters, and Burnside rings
No full textWe study 2-representations, i.e., actions of 2-groups on 2-vector spaces. Our main focus is character theory for 2-representations. To this end we employ the technique of extended Burnside rings. Our main theorem is that the Ganter–Kapranov 2-character is a particular mark homomorphism of the Burnside ring. As an application we give a new proof of Osorno's formula for the Ganter–Kapranov 2-character of a finite group
D-modules and projective stacks
No full textWe study twisted D-modules on weighted projective stacks. We determine for which values of the twist and the weight the global sections functor is an equivalence, thus, proving a version of Beilinson-Bernstein Localisation Theorem
Topological rigidity of -quotients
No full textWe investigate the homotopy type of a certain homogeneous space for a simple
complex algebraic group. We calculate some of its classical topological
invariants and introduce a new one. We also propose several conjectures about
its topological rigidity
