arXiv subjects

Hatice Mutlu

Publications and source records attributed to Hatice Mutlu.

5 recordsLinked to original sources

arXiv2026

Plus Constructions and Canonical Induction for Green Categories

We develop canonical induction for Green categories associated with Mackey systems. We establish a change-of-rings framework in which the lower-plus and upper-plus constructions are realized by induction and coinduction of modules. The two plus adjunctions yield the mark morphism, and, when the relevant group orders are invertible in the coefficient ring, Möbius inversion gives an explicit inverse. Using the resulting linearization and tom Dieck maps, we construct a canonical induction map, give an explicit formula for it, and characterize when it is a right inverse of linearization. In the classical setting of Mackey functors, our construction and splitting criterion recover Boltje's results.

math.RT↗

arXiv2026

Mackeyfication of equivariant categories I

Colloquially speaking, `equivariant categories' refer to families of additive categories $\mathcal{A}(G)$ depending 2-functorially on a finite group $G$. We construct approximations of equivariant categories by Mackey 2-functors, both on the left and on the right. The idea is to enlarge $\mathcal{A}$ in a minimal way to make induction appear. These `mackeyfications' are inspired by Boltje's work with ordinary Mackey 1-functors. We also relate our left and right mackeyfications via a mark transformation. Finally we discuss examples.

math.RT↗

arXiv2023

Monomial structures, I

The goal of a series of papers is to define $G$-actions on various $A$-fibered structures, where $G$ is a finite group and $A$ is an abelian group. One prominent such example is the $A$-fibered Burnside ring. If $A=\mathbb{C}^\times$, it is also called the ring of monomial representations (introduced by Dress in \cite{Dress1971}) and is the natural home for the canonical induction formula (see \cite{Boltje1990}). In this first part of the series, motivated by constructions in \cite{BoucMutlu}, we introduce $A$-fibered structures on posets, on abstract simplicial complexes, and on $A$-bundles over topological spaces, together with natural notions of homotopy, and functors between these structures respecting homotopy. In a sequel we will continue with $G$-representations in these $A$-fibered structures and associate to them elements in the $A$-fibered Burnside ring.

math.RT↗

arXiv2019

Monomial $G$-posets and their Lefschetz invariants

Let $G$ be a finite group, and $C$ be an abelian group. We introduce the notions of $C$-monomial $G$-sets and $C$-monomial $G$-posets, and state some of their categorical properties. This gives in particular a new description of the $C$-monomial Burnside ring $B_C(G)$. We also introduce Lefschetz invariants of $C$-monomial $G$-posets, which are elements of $B_C(G)$. These invariants allow for a definition of a generalized tensor induction multiplicative map $\mathcal{T}_{U,λ}: B_C(G)\to B_C(H)$ associated to any $C$-monomial $(G,H)$-biset $(U,λ)$, which in turn gives a group homomorphism $B_C(G)^\times\to B_C(H)^\times$ between the unit groups of $C$-monomial Burnside rings.

math.GR↗

arXiv2018

A new canonical induction formula for $p$-permutation modules

Applying Robert Boltje's theory of canonical induction, we give a restriction-preserving formula expressing any $p$-permutation module as a $\mathbb{Z}[1/p]$-linear combination of modules induced and inflated from projective modules associated with subquotient groups. The underlying constructions include, for any given finite group, a ring with a $\mathbb{Z}$-basis indexed by conjugacy classes of triples $(U, K, E)$ where $U$ is a subgroup, $K$ is a $p'$-residue-free normal subgroup of $U$ and $E$ is an indecomposable projective module of the group algebra of $U/K$.

math.RT↗