arXiv subjects

Fumihito Oda

Publications and source records attributed to Fumihito Oda.

3 recordsLinked to original sources

arXiv2026

Deflation map and the sum of inverses of the element orders in finite groups

Let $G$ be a finite group. In this note, we construct an element ${\sigma}^G$ in the Burnside ring of $G$ over $\mathbb{Q}$, which gives a rational number $m(G)$ that is the sum of the inverses of the element orders in $G$, by using the deflation map ${\mathrm{Def}}^G_{G/N}$ induced by the $(G/N,G)$-biset $G/N$ for a normal subgroup $N$ of $G$. As a corollary to Theorem, we obtain another expression for the rank of the crossed Burnside ring $B^{\rm c}(G)$ of $G$ and the determinant of the Cartan matrix of $p$-local Mackey algebra of $G$ over a field $k$ of characteristic $p>0$ with big enough.

math.GR↗

arXiv2022

Crossed Burnside rings and cohomological Mackey $2$-motives

Balmer and Dell'Ambrogio introduced the pseudo-functor $P$ from the bicategory of $k$-linear Mackey $2$-motives to the bicategory of $k$-linear cohomological Mackey $2$-motives over a commutative ring $k$. They showed that $P$ maps the general Mackey $2$-motives to the cohomological Mackey $2$-motives by using the ring homomorphism from the crossed Burnside ring of a finite group $G$ over $k$ to the center $ ZkG$ of group algebra $kG$ ([BD21, Theorem 5.3]). We study the behavior of motivic decomposition of cohomological Mackey $2$-motives as images by $P$ of motivic decomposition of Mackey $2$-motives.

math.RT↗

arXiv2019

The lattice Burnside rings

We introduce the concept of lattice Burnside ring for a finite group G associated to a family of nonempty sublattices of a finite G-lattice assigned to subgroups of G. The slice Burnside ring introduced by S. Bouc is isomorphic to a lattice Burnside ring. Any lattice Burnside ring is isomorphic to an abstract Burnside ring. The ring structure of a lattice Burnside ring is explored on the basis of the fundamental theorem of abstract Burnside rings. We study the units and the primitive idempotents of a lattice Burnside ring. There are certain rings called partial lattice Burnside rings. Any partial lattice Burnside ring, which is isomorphic to an abstract Burnside ring, consists of elements of a lattice Burnside ring; it is not necessarily a subring. The section Burnside ring introduced by S. Bouc is isomorphic to a partial lattice Burnside ring.

math.GR↗