Spectra of Symmetrized Shuffling Operators (Memoirs of the American Mathematical Society) by Victor Reiner (PDF)

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For a finite real reflection group W and a W -orbit O of flats in its reflection arrangement – or equivalently a conjugacy class of its parabolic subgroups – the authors introduce a statistic noninv O (w) on w in W that counts the number of “”O -noninversions”” of w . This generalises the classical (non-)inversion statistic for permutations w in the symmetric group S n. The authors then study the operator O of right-multiplication within the group algebra CW by the element that has noninv O (w) as its coefficient on w.

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