Let be an integer and be a field. Nilpotent matrices are those matrices for which there exists an integer with . Their characteristic polynomial is , and they satisfy , which shows that the set of nilpotent matrices is an algebraic variety. The equation is homogeneous of degree , so that is a cone.
The classification of nilpotent matrices is an intermediate step in the theory of Jordan decomposition: In an adequate basis, a nilpotent matrix can be written as a diagonal block matrix of “basic” nilpotent matrices, matrices of the form whose minimal polynomial is . The sum of the sizes of these blocks is and in this way, it is associated with any nilpotent matrix a partition of~. It is known that two nilpotent matrices are conjugate if and only if they are associated with the same partition. For any partition of~, let us denote by the corresponding matrix whose sizes of blocks are arranged in increasing order, and the set of nilpotent matrices that are associated with the partition .
The theorem of Fine and Herstein (1958)
Having to teach “agrégation” classes made me learn about a classic combinatorial result: counting the number of nilpotent matrices when is a finite field.
Theorem (Fine, Herstein, 1958). — Let be a finite field with elements. The cardinality of is . Equivalently, the probability that an matrix with coefficients in be nilpotent is .The initial proof of this results relies on the action of on : we recalled that the orbits correspond with the partitions of , hence a decomposition We know that is the orbit of the matrix under the action of . By the classic orbit-stabilizer formula, one thus has where is the set of matrices such that . The precise description of is delicate but their arguments go as follow.
They first replace the group by the algebra of all matrices such that . For any integer, let be the multiplicity of an integer in the partition , so that . The block decomposition of corresponds with a decomposition of as a direct sum of invariant subspaces , where has dimension . In fact, , , etc. This shows that is an algebra of block-triangular matrices. Moreover, the possible diagonal blocks can be shown to be isomorphic to . In other words, we have a surjective morphism of algebras whose kernel consists of nilpotent matrices. In particular, the proportion of invertible elements in is equal to the proportion of invertible elements in the product .
Ultimately, Fine and Herstein obtain an explicit sum over the set of partitions of which they prove equals , after an additional combinatorial argument.
Soon after, the theorem of Fine and Herstein was given easier proofs, starting from Gerstenhaber (1961) to Kaplansky (1990) and Leinster (2021).
A proof
The following proof is borrowed from Caldero and Peronnier (2022), Carnet de voyage en Algébrie. It can be seen as a simplification of the proofs of Gerstenhaber (1961) and Leinster (2021).
Let us start with the Fitting decomposition of an endomorphism : the least integer such that coincides with the least integer such that , and one has . The subspaces and are invariant under , and acts nilpotently on and bijectively on . In other words, we have associated with complementary subspaces and , a nilpotent operator of and an invertible operator on . This map is bijective.
For any integer , let be the cardinality of nilpotent matrices in , and be the cardinality of invertible matrices in . Let also be the set of all pairs , where and are complementary subspaces of dimensions , of . We thus obtain We need to compute the cardinality of . In fact, given one pair , all other are of the form , for some : the group acts transitively on . The stabilizer of can be identified with . Consequently, We thus obtain By subtraction, we get or It remains to compute : since an invertible matrix consists of a nonzero vector, a vector which does not belong to the line generated by the first one, etc., we have Then, We thus obtain as claimed.
The proof of Leinster (2021)
Leinster defines a bijection from to . The definition is however not very canonical, because he assumes given, for any subspace of , a basis of .
Take a pair , where and and consider the subspace , the smallest -invariant subspace of which contains . To describe , we observe that we know its restriction to , and we need to describe it on the chosen complementary subspace .
To that aim, we have to give ourselves an endomorphism of and a linear map . Since we want to be nilpotent, it is necessary and sufficient to take nilpotent.
Instead of considering , we can consider the map . Its image is a complement of in , and any complement can be obtained in this way. The nilpotent endomorphism of transfers to a nilpotent endomorphism of .
All in all, what the given pair furnishes is a subspace with a basis , a complement , and a nilpotent endomorphism of . This is more or less what the Fitting decomposition of an endomorphism gives us! Recall that was assumed to have been given a basis . There exists a unique automorphism of which maps to for all . In other words, we have a pair of complementary subspaces , a linear automorphism of , and a nilpotent automorphism of . By the Fitting decomposition, these data furnish in a bijective way an endomorphism of , and that concludes the proof.
A remark about motivic integration
The framework of motivic integration suggests to upgrade these combinatorial results into equalities valid for all field , which hold in the Grothendieck ring of varieties . As an abelian group, it is generated by symbols , for all algebraic varieties over , with relations , whenever is a closed subvariety of . The ring structure is defined so that the formula for all algebraic varieties and over .
By construction of this ring, equalities in imply that many invariants of and coincide. In particular, when is a finite field, they will have the same number of points.
The question is thus to compute the class of the variety , for any field . The proofs that I described above can be more or less transferred to this context and imply the following theorem. We denote by the class of the affine line .
Theorem. — One has an equality in the localization of the Grothendieck ring by the element .
The following question is then natural. (I have not thought about it at all.)
Question. — Does one have in ?