The probability that an operator is nilpotent

2019/12/29 by Leinster, Tom
#05B20 #15A99 #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1912.12562

Abstract

Choose a random linear operator on a vector space of finite cardinality N: then the probability that it is nilpotent is 1/N. This is a linear analogue of the fact that for a random self-map of a set of cardinality N, the probability that some iterate is constant is 1/N. The first result is due to Fine, Herstein and Hall, and the second is essentially Cayley's tree formula. We give a new proof of the result on nilpotents, analogous to Joyal's beautiful proof of Cayley's formula. It uses only general linear algebra and avoids calculation entirely.

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