Burnside rings for Real 2-representation theory: The linear theory

Abstract

This paper is a fundamental study of the Real 2-representation theory of 2-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a 2-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real 2-representations as a Real variant of the Burnside ring of the fundamental group of the 2-group and study the Real categorical character theory. This paper unifies two previous lines of inquiry, the approach to 2-representation theory via Morita theory and Burnside rings, initiated by the first author and Wendland, and the Real 2-representation theory of 2-groups, as studied by the second author.

Matrix factorizations, Reality and Knörrer periodicity

“…We work in the framework of Real 2-representation theory, as developed by the second author [44,51], in which a 𝐶 2 -graded finite group Ĝ acts coherently on a ℂ-linear category  by functors 𝜌(𝜎) ∶  → , 𝜎 ∈ Ĝ, which are linear and covariant if 𝜋(𝜎) = 1 and, depending on the setting, antilinear and covariant or linear and contravariant if 𝜋(𝜎) = −1. In the context of matrix factorizations, a (possibly antilinear) action of Ĝ on 𝑅 for which the potential satisfies condition (1) or (3) defines a Real 2-representation on MF(𝑅, 𝑤); see Lemmas 3.2 and 5.6.…”

“…We work in the framework of Real 2‐representation theory, as developed by the second author [44, 51], in which a $$C_2$$‐graded finite group $$\hat{G}$$ acts coherently on a $$\mathbb {C}$$‐linear category $$\mathcal {C}$$ by functors $$\rho (\sigma ): \mathcal {C}\rightarrow \mathcal {C}$$, $$\sigma \in \hat{G}$$, which are linear and covariant if $$\pi (\sigma )=1$$ and, depending on the setting, antilinear and covariant or linear and contravariant if $$\pi (\sigma )=-1$$. In the context of matrix factorizations, a (possibly antilinear) action of $$\hat{G}$$ on $$R$$ for which the potential satisfies condition () or () defines a Real 2‐representation on $${\rm MF}(R,w)$$; see Lemmas 3.2 and 5.6.…”

Matrix factorizations, Reality and Knörrer periodicity

“…Strategy of proof. We work in the framework of Real 2-representation theory, as developed by the second author [You21], [RY21], in which a C 2 -graded finite group Ĝ acts coherently on a C-linear category C by functors ρ(σ) : C → C, σ ∈ Ĝ, which are linear and covariant if π(σ) = 1 and, depending on the setting, antilinear and covariant or linear and contravariant if π(σ) = −1. In particular, a (possibly antilinear) action of Ĝ on R for which the potential satisfies condition (1) or (2) defines a Real 2-representation on MF(R, w); see Lemmas 3.2 and 5.6.…”

Real representations of C2-graded groups: The antilinear theory

“…It is interesting that the Real 2-representation theory of C 2 -graded groups [25,22] is as developed as its 1-counterpart, the topic of the present paper. In 2representation theory, the step from split to general C 2 -graded groups goes back to Hahn [12], who extends hermitian Morita theory from algebras with involution to algebras with anti-structure.…”