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The Hecke Algebra of a Frobenius P-Category
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We introduce a new avatar of a Frobenius P-category [Formula: see text] under the form of a suitable subring [Formula: see text] of the double Burnside ring of P — called the Hecke algebra of [Formula: see text] — where we are able to formulate: (i) the generalization to a Frobenius P-category of the Alperin Fusion Theorem, (ii) the “canonical decomposition” of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and (iii) the “basic P × P-sets” in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P × P-sets in [6]. We also explain the relationship with the usual Hecke algebra of a finite group.
Title: The Hecke Algebra of a Frobenius P-Category
Description:
We introduce a new avatar of a Frobenius P-category [Formula: see text] under the form of a suitable subring [Formula: see text] of the double Burnside ring of P — called the Hecke algebra of [Formula: see text] — where we are able to formulate: (i) the generalization to a Frobenius P-category of the Alperin Fusion Theorem, (ii) the “canonical decomposition” of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and (iii) the “basic P × P-sets” in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P × P-sets in [6].
We also explain the relationship with the usual Hecke algebra of a finite group.
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