Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A ghost ring for the left-free double Burnside ring and an application
  to fusion systems   For a finite group $G$, we define a ghost ring and a mark homomorphism for
the double Burnside ring of left-free $(G,G)$-bisets. In analogy to the case of
the Burnside ring $B(G)$, the ghost ring has a much simpler ring structure, and
after tensoring with $\QQ$ one obtains an isomorphism of $\QQ$-algebras. As an
application of a key lemma, we obtain a very general formula for the Brauer
construction applied to a tensor product of two $p$-permutation bimodules $M$
and $N$ in terms of Brauer constructions of the bimodules $M$ and $N$. Over a
field of characteristic 0 we determine the simple modules of the left-free
double Burnside algebra and prove semisimplicity results for the bifree double
Burnside algebra. These results carry over to results about biset-functor
categories. Finally, we apply the ghost ring and mark homomorphism to fusion
systems on a finite $p$-group. We extend a remarkable bijection, due to
Ragnarsson and Stancu, between saturated fusion systems and certain idempotents
of the bifree double Burnside algebra over $\ZZ_{(p)}$, to a bijection between
all fusion systems and a larger set of idempotents in the bifree double
Burnside algebra over $\QQ.$

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The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci
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The categories of this paper are: cs