Enumerating partial linear transformations in a similarity class

2020/05/13 by Arora, Akansha, Ram, Samrith
#05A05 #05A10 #15B33 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2005.06222

Abstract

Let V be a finite-dimensional vector space over the finite field \mathbb F_q and suppose W and \widetildeW are subspaces of V. Two linear transformations T:W→ V and \widetildeT:\widetildeW→ V are said to be similar if there exists a linear isomorphism S:V→ V with SW=\widetildeW such that S∘ T=\widetildeT∘ S . Given a linear map T defined on a subspace W of V, we give an explicit formula for the number of linear maps that are similar to T. Our results extend a theorem of Philip Hall that settles the case W=V where the above problem is equivalent to counting the number of square matrices over \mathbb F_q in a conjugacy class.

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