A Solomon descent theory for the wreath products đșâđ_{đ«}
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Abstract
We propose an analogue of Solomonâs descent theory for the case of a wreath product <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G wreath-product German upper S Subscript n"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo> â </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">S</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">G\wr \mathfrak S_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite abelian group. Our construction mixes a number of ingredients: Mantaci-Reutenauer algebras, Spechtâs theory for the representations of wreath products, Okadaâs extension to wreath products of the Robinson-Schensted correspondence, and Poirierâs quasisymmetric functions. We insist on the functorial aspect of our definitions and explain the relation of our results with previous work concerning the hyperoctaedral group.
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