The signed random-to-top operator on tensor space (draft)
نویسنده
چکیده
The purpose of this note is to answer a question I asked in 2010 in [Grinbe10]. It concerns the kernel of a certain operator on the tensor algebra T (L) of a free module L over a commutative ring k (an operator that picks out a factor from a tensor and moves it to the front, and takes an alternating sum of the results ranging over all factors – an algebraic version of what probabilists call the “random-to-top shuffle”, albeit with signs). Originating in pure curiosity, this question has been tempting me with its apparent connections to the randomto-top and random-to-random shuffling operators as studied in [ReSaWe14] and [Schock02]. I have not (yet?) grown any wiser from these connections, but I was able to answer the question (with some help from a 1950 paper by Specht [Specht50]), and the answer seems (to me) to be interesting enough to warrant some publicity. We shall not use the notations of [Grinbe10] (indeed, our notations in the following will be incompatible with those in [Grinbe10]).
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