19 research outputs found
Trace maps for Mackey algebras
Let be a finite group and be a commutative ring. The Mackey algebra
shares a lot of properties with the group algebra 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 is a symmetric algebra if and only if the
family of Burnside algebras 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 is
symmetric if and only if the order of is square free. Finally, over a field
of characteristic we show that the Mackey algebra is symmetric if and
only if the Sylow -subgroups of are of order or .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 be a finite group and be a -modular system.
Let or . There is a bijection between the blocks of the
group algebra and the blocks of the so-called -local Mackey algebra
. Let be a block of with abelian defect group . Let
be its Brauer correspondant in . It is conjectured by Brou\'e
that the blocks and are derived equivalent. Here we look at
equivalences between the corresponding blocks of -local Mackey algebras. We
prove that an analogue of the Brou\'e's conjecture is true for the -local
Mackey algebras in the following cases: for the principal blocks of
-nilpotent groups and for blocks with defect . 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 be a finite group and be a -modular system
"large enough". Let or . There is a bijection between the
blocks of the group algebra and the central primitive idempotents (the
blocks) of the so-called cohomological Mackey algebra . 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
Intervalles exceptionnels et modernes du treillis de Tamari
International audienceIn this article we use the theory of interval-posets recently introduced by Châtel and Pons in order to describe some interesting families of intervals in the Tamari lattices. These families are defined as interval-posets avoiding specific configurations. At first, we consider what we call exceptional interval-posets and show that they correspond to the intervals which are obtained as images of noncrossing trees in the Dendriform operad. We also show that the exceptional intervals are exactly the intervals of the Tamari lattice induced by intervals in the poset of noncrossing partitions. In the second part we introduce the notion of modern and infinitely modern interval-posets. We show that the modern intervals are in bijection with the new intervals of the Tamari lattice in the sense of Chapoton. We deduce an intrinsic characterization of the new intervals in the Tamari lattice. Finally, we consider the family of what we call infinitely modern intervals and we we prove that there are as many infinitely modern interval-posets of size n as there are ternary trees with n inner vertices
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Invitation to quiver representation and Catalan combinatorics
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