Trace maps for Mackey algebras

Let $G$ be a finite group and $R$ be a commutative ring. The Mackey algebra $\mu_{R}(G)$ shares a lot of properties with the group algebra $RG$ however, there are some differences. For example, the group algebra is a symmetric algebra and this is not always the case for the Mackey algebra. In this paper we present a systematic approach to the question of the symmetricity of the Mackey algebra, by producing symmetric associative bilinear forms for the Mackey algebra. The category of Mackey functors is a closed symmetric monoidal category, so using the formalism of J.P. May for these categories, S. Bouc has defined the so-called Burnside trace. Using this Burnside trace we produce trace maps for Mackey algebras which generalize the usual trace map the group algebras. These trace maps factorise through Burnside algebras. We prove that the Mackey algebra $\mu_{R}(G)$ is a symmetric algebra if and only if the family of Burnside algebras $(RB(H))_{H\leqslant G}$ is a family of symmetric algebras with a compatibility condition. As a corollary, we recover the well known fact that over a field of characteristic zero, the Mackey algebra is always symmetric. Over the ring of integers the Mackey algebra of $G$ is symmetric if and only if the order of $G$ is square free. Finally, over a field of characteristic $p>0$ we show that the Mackey algebra is symmetric if and only if the Sylow $p$-subgroups of $G$ are of order $1$ or $p$.Comment: 21 pages. Second version: minor changes in the introduction and in the organisation of the proofs. The last part is generalized to commutative rings in which all prime except one are invertibl

Equivalences between blocks of p-local Mackey algebras

Let $G$ be a finite group and $(K,\mathcal{O},k)$ be a $p$-modular system. Let $R=\mathcal{O}$ or $k$. There is a bijection between the blocks of the group algebra and the blocks of the so-called $p$-local Mackey algebra $\mu_{R}^{1}(G)$. Let $b$ be a block of $RG$ with abelian defect group $D$. Let $b'$ be its Brauer correspondant in $N_{G}(D)$. It is conjectured by Brou\'e that the blocks $RGb$ and $RN_{G}(D)b'$ are derived equivalent. Here we look at equivalences between the corresponding blocks of $p$-local Mackey algebras. We prove that an analogue of the Brou\'e's conjecture is true for the $p$-local Mackey algebras in the following cases: for the principal blocks of $p$-nilpotent groups and for blocks with defect $1$. We also point out the probable importance of \emph{splendid} equivalences for the Mackey algebras.Comment: 24 pages. Second version. All the part about cohomological Mackey algebra has been remov

Equivalences between blocks of cohomological Mackey algebras

Let $G$ be a finite group and $(K,\mathcal{O},k)$ be a $p$-modular system "large enough". Let $R=\mathcal{O}$ or $k$. There is a bijection between the blocks of the group algebra $RG$ and the central primitive idempotents (the blocks) of the so-called cohomological Mackey algebra $co\mu_{R}(G)$. Here, we prove that a so-called permeable derived equivalence between two blocks of group algebras implies the existence of a derived equivalence between the corresponding blocks of cohomological Mackey algebras. In particular, in the context of Brou\'e's abelian defect group conjecture, if two blocks are splendidly derived equivalent, then the corresponding blocks of cohomological Mackey algebras are derived equivalent

On the wildness of cambrian lattices

In this note, we investigate the representation type of the cambrian lattices and some other related lattices. The result is expressed as a very simple trichotomy. When the rank of the underlined Coxeter group is at most 2, the lattices are of finite representation type. When the Coxeter group is a reducible group of type A 3 1 , the lattices are of tame representation type. In all the other cases they are of wild representation type

Fosfatgjødsling av Ringedalsvatnet – et oligotroft reguleringsmagasin

Det oligotrofe reguleringsmagasinet Ringedalsvatn (Odda kommune) ble i juni 2013 gjødslet med fosfat som et eksperimentelt tiltak for å stimulere primærproduksjonen og kvaliteten av ørret. Gjødslingen hevet den epilimniske konsentrasjonen av fosfat fra omlag 1 til 4 μg P/l. Utviklingen i plante- og zooplanktonets biomasse og sammensetning ble fulgt gjennom vekstsesongen. Ernæring, tilvekst, kondisjon, kjøttfarge og stabile isotoper (13C og 15N) hos fisken ble òg registrert. Fytoplanktonets biomasse (0-10 m) ble omlag fordoblet fra juli og til september, men maksimalt nivå var forholdsvis lavt (0,18 mm3/l). I littoralsonen ble det registrert en betydelig vekst av påvekstalger..

Røgden 2010. Vannkjemi og biologisk status

Etter at kalkingen opphørte i Røgden og i Løvbergsåas nedbørfelt i 2001, har det vært en nedgang i verdiene til forsurningsrelaterte variabler som pH, kalsium (Ca) og alkalitet fra 2003 til 2010. Dette har også skjedd i Varpåa etter at kalkingen av Søndre Baksjøen opphørte i 2009, men reduksjonen har vært mindre enn i Løvbergsåa. Beregninger av syrenøytraliserende kapasitet (ANC) viser en reduksjon på ca. 20 % fra 2003 til 2010. Likevel var ANC i Røgden, Løvbergsåa og Varpåa henholdsvis ca. 100 µekv/l, 100 µekv/l og 130 µekv/l i 2010. Dette er godt over 60 µekv/l som regnes som nedre grense for meget god syrenøytraliserende kapasitet, og langt fra grensa på 20 µekv/l hvor skader på fisk kan forventes. Røgden synes også å ha blitt litt mer produktiv fra 2003 til 2010 indikert ved økte konsentrasjoner av klorofyll a. Dette kan ha skapt et økt biologisk opptak av nitrat som ga utslag i lavere nitrat-konsentrasjoner i 2010 enn i 2003. Humuskonsentrasjonen (TOC) og tot N har vært uforandret. Middellengden av de viktigste vannloppeartene ble redusert fra 2003-2005 til 2007 for så å øke igjen fram mot 2010. Den meget lave andelen store arter i 2007, men en markert høyere andel i 2010 kan tyde på at predasjonspresset fra planktonspisende fisk var spesielt sterkt i 2007 og betydelig mindre i 2010

On Morita and derived equivalences for cohomological Mackey algebras

By results of the second author, a source algebra equivalence between two p-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories ofcohomological Mackey functors. The main result of this paper proves a partial converse: an equivalence (resp. Rickard equivalence) between the categories of cohomological Mackey functors of two blocks of finite groups induces a permeable Morita (resp. derived) equivalence between the two block algebras