Combinatorics of the nilpotent cone

$\global\def\Card{\operatorname{Card}}\global\def\GL{\mathrm{GL}}\global\def\im{\operatorname{im}}\gdef\KVar{\mathrm{KVar}}$

Let $n$ be an integer and $F$ be a field. Nilpotent matrices $N\in \mathrm M_n(F)$ are those matrices for which there exists an integer $p$ with $N^p=0$. Their characteristic polynomial is $\chi_N(T)=T^n$, and they satisfy $N^n=0$, which shows that the set $\mathscr N_n$ of nilpotent matrices is an algebraic variety. The equation $N^n=0$ is homogeneous of degree $n$, so that $\mathscr N_n$ 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, $p \times p$ matrices of the form \[ \begin{pmatrix} 0 & 0 & \dots & 0 & 0 \\ 1 & 0 & & & \vdots \\ 0 & 1 & \ddots & & 0 \\ \vdots & \ddots & \ddots & \ddots & 0 \\ 0 & & 0 & 1 & 0\end{pmatrix} \] whose minimal polynomial is $T^p$. The sum of the sizes of these blocks is $n$ and in this way, it is associated with any $n\times n$ nilpotent matrix a partition $\pi$ of~$n$. It is known that two nilpotent matrices are conjugate if and only if they are associated with the same partition. For any partition $\pi$ of~$n$, let us denote by $J_\pi$ the corresponding matrix whose sizes of blocks are arranged in increasing order, and $\mathscr N_\pi$ the set of nilpotent matrices that are associated with the partition $\pi$.

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 $F$ is a finite field.

Theorem (Fine, Herstein, 1958). — Let $F$ be a finite field with $q$ elements. The cardinality of $\mathscr N_n(F)$ is $q^{n^2-n}$. Equivalently, the probability that an $n\times n$ matrix with coefficients in $F$ be nilpotent is $q^{-n}$.

The initial proof of this results relies on the action of $\GL_n(F)$ on $\mathscr N_n(F)$: we recalled that the orbits correspond with the partitions of $n$, hence a decomposition \[ \Card(\mathscr N_n(F)) = \sum_{\pi} \Card(\mathscr N_\pi(F)). \] We know that $\mathscr N_\pi(F)$ is the orbit of the matrix $J_\pi$ under the action of $\GL_n(F)$. By the classic orbit-stabilizer formula, one thus has \[ \Card(\mathscr N_\pi(F)) = \frac{\Card(\GL_n(F))}{\Card(C_\pi(F))}, \] where $C_\pi(F)$ is the set of matrices $A\in\GL_n(F)$ such that $AJ_\pi=J_\pi A$. The precise description of $C_\pi(F)$ is delicate but their arguments go as follow.

They first replace the group $C_\pi(F)$ by the algebra $A_\pi(F)$ of all matrices $A\in\mathrm M_n(F)$ such that $AJ_\pi=J_\pi A$. For any integer, let $m_i$ be the multiplicity of an integer $i$ in the partition $\pi$, so that $n=\sum i m_i$. The block decomposition of $J_\pi$ corresponds with a decomposition of $F^n$ as a direct sum of invariant subspaces $V_i$, where $V_i$ has dimension $i m_i$. In fact, $V_1=\ker(J_\pi)$, $V_1\oplus V_2=\ker(J_\pi^2)$, etc. This shows that $A_\pi(F)$ is an algebra of block-triangular matrices. Moreover, the possible diagonal blocks can be shown to be isomorphic to $\mathrm M_{m_i}(F)$. In other words, we have a surjective morphism of algebras \[ A_\pi(F) \to \prod_i \mathrm M_{m_i}(F), \] whose kernel consists of nilpotent matrices. In particular, the proportion of invertible elements in $A_\pi(F)$ is equal to the proportion of invertible elements in the product $\prod_i \mathrm M_{m_i}(F)$.

Ultimately, Fine and Herstein obtain an explicit sum over the set of partitions of $n$ which they prove equals $q^{n^2-n}$, 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 $u\in \mathrm N_n(F)$: the least integer $p$ such that $\ker(u^p)=\ker(u^{p+1})$ coincides with the least integer $p$ such that $\im(u^p)=\im(u^{p+1})$, and one has $F^n=\ker(u^p)\oplus \im(u^p)$. The subspaces $N(u)=\ker(u^p)$ and $I(u)=\im(u^p)$ are invariant under $u$, and $u$ acts nilpotently on $\ker(u^p)$ and bijectively on $\im(u^p)$. In other words, we have associated with $u$ complementary subspaces $N(u)$ and $I(u)$, a nilpotent operator of $N(u)$ and an invertible operator on $I(u)$. This map is bijective.

For any integer $d$, let $\nu_d$ be the cardinality of nilpotent matrices in $\mathrm M_d(F)$, and $\gamma_d$ be the cardinality of invertible matrices in $\mathrm M_d(F)$. Let also $\mathscr D_d$ be the set of all pairs $(K,I)$, where $K$ and $I$ are complementary subspaces of dimensions $d$, $n-d$ of $F^n$. We thus obtain \[ n^2 = \sum_{(K,I)\in\mathscr D_d} \nu_d \cdot \gamma_{n-d}. \] We need to compute the cardinality of $\mathscr D_d$. In fact, given one pair $(K,I)\in\mathscr D_d$, all other are of the form $(g\cdot K,g\cdot I)$, for some $g\in\GL_n(F)$: the group $\GL_n(F)$ acts transitively on $\mathscr D_d$. The stabilizer of $(K,I)$ can be identified with $\GL_d(F)\times \GL_{n-d}(F)$. Consequently, \[ \Card(\mathscr D_d) = \frac{\Card(\GL_n(F))}{\Card(\GL_d(F)\Card(\GL_{n-d}(F))} = \frac{\gamma_n}{\gamma_d \gamma_{n-d}}. \] We thus obtain \[ q^{n^2} = \sum_{d=0}^n \frac{\gamma_n}{\gamma_d \gamma_{n-d}} \nu_d \gamma_{n-d} = \gamma_n \sum_{d=0}^n \frac{\nu_d}{\gamma_d}. \] By subtraction, we get \[ \frac{\nu_n}{\gamma_n} = \frac {q^{n^2}}{\gamma_n} - \frac{q^{(n-1)^2}}{\gamma_{n-1}},\] or \[ \nu_n = q^{n^2} - q^{(n-1)^2} \frac{\gamma_n}{\gamma_{n-1}}. \] It remains to compute $\gamma_n$: 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 \[ \gamma_n = (q^n-1) (q^n-q)\dots (q^n-q^{n-1}). \] Then, \[ \gamma_n = (q^n-1) q^{n-1} (q^{n-1}-1)\dots (q^{n-1}-q^{n-2}) = (q^n-1)q^{n-1} \gamma_{n-1}. \] We thus obtain \[ \nu_n = q^{n^2} - q^{(n-1)^2} (q^n-1) q^{n-1} = q^{n^2} - q^{(n-1)^2} q^{2n-1} + q^{(n-1)^2} q^{n-1} = q^{n^2-n}, \] as claimed.

The proof of Leinster (2021)

Leinster defines a bijection from $\mathscr N_n(F)\times F^n$ to $\mathrm M_n(F)$. The definition is however not very canonical, because he assumes given, for any subspace $V$ of $F^n$, a basis of $V$.

Take a pair $(u,x)$, where $u\in\mathscr N_n(F)$ and $x\in F^n$ and consider the subspace $V_x=\langle x,u(x),\dots\rangle$, the smallest $u$-invariant subspace of $F^n$ which contains $x$. To describe $u$, we observe that we know its restriction to $V_x$, and we need to describe it on the chosen complementary subspace $V'_x$.

To that aim, we have to give ourselves an endomorphism $u'_x$ of $V'_x$ and a linear map $\phi_x\colon V'_x\to V_x$. Since we want $u$ to be nilpotent, it is necessary and sufficient to take $u'_x$ nilpotent.

Instead of considering $\phi_x\colon V'_x\to V_x$, we can consider the map $y\mapsto y+\phi_x(y)$. Its image is a complement $W_x$ of $V_x$ in $F^n$, and any complement can be obtained in this way. The nilpotent endomorphism $u'_x$ of $V'_x$ transfers to a nilpotent endomorphism $w_x$ of $W_x$.

All in all, what the given pair $(u,x)$ furnishes is a subspace $V_x$ with a basis $(x_1=x,x_2=u(x),\dots)$, a complement $W_x$, and a nilpotent endomorphism $w_x$ of $W_x$. This is more or less what the Fitting decomposition of an endomorphism gives us! Recall that $V_x$ was assumed to have been given a basis $(e_1,\dots,e_p)$. There exists a unique automorphism of $V_x$ which maps $e_i$ to $u^{i-1}(x)$ for all $i$. In other words, we have a pair of complementary subspaces $(V_x,W_x)$, a linear automorphism of $V_x$, and a nilpotent automorphism of $W_x$. By the Fitting decomposition, these data furnish in a bijective way an endomorphism of $F^n$, 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 $F$, which hold in the Grothendieck ring of varieties $\KVar_F$. As an abelian group, it is generated by symbols $[X]$, for all algebraic varieties $X$ over $F$, with relations $[X]=[Y]+[X\setminus Y]$, whenever $Y$ is a closed subvariety of $X$. The ring structure is defined so that the formula $[X]\cdot[Y]=[X\times Y]$ for all algebraic varieties $X$ and $Y$ over $F$.

By construction of this ring, equalities $[X]=[Y]$ in $\KVar_F$ imply that many invariants of $X$ and $Y$ coincide. In particular, when $F$ is a finite field, they will have the same number of points.

The question is thus to compute the class $[\mathscr N_n]$ of the variety $\mathscr N_n$, for any field $F$. The proofs that I described above can be more or less transferred to this context and imply the following theorem. We denote by $\mathbf L\in \KVar_F$ the class of the affine line $\mathbf A^1$.

Theorem. — One has an equality $[\mathscr N_n] \mathbf L^n = \mathbf L^{n^2}$ in the localization of the Grothendieck ring $\KVar_F$ by the element $(\mathbf L-1)\dots(\mathbf L^{n-1}-1)$.

The following question is then natural. (I have not thought about it at all.)

Question. — Does one have $[\mathscr N_n]=\mathbf L^{n^2-n}$ in $\KVar_F$?

Behaviour of conjugacy by reduction modulo integers

Let $A$ and $B\in\mathrm{M}_n(\mathbf Z)$ be two square matrices with integer coefficients. Assume that they are conjugate by $\mathrm{GL}_n(\mathbf Z)$, namely, that there exists a matrix $P\in\mathrm{GL}_n(\mathbf Z)$ such that $B=P^{-1}AP$. Then we can reduce this relation modulo every integer $d\geq 2$ and obtain a similar relation between the images of $A$ and $B$ in $\mathrm M_n(\mathbf Z/d\mathbf Z)$.

Almost the same holds if $A$ and $B$ are only conjugate by $\mathrm{GL}_n(\mathbf Q)$, except for a few exceptions: we just need to take care to reduce the relation modulo integers $d$ that are coprime to the denominators of the coefficients of $P$ or of $P^{-1}$.

I was quite surprised at first to learn that the converse assertion is false. There are matrices $A$ and $B$ in $\mathrm M_2(\mathbf Z)$ whose images modulo every integer $d\geq 2$ are conjugate, but which are not conjugate by a matrix in $\mathrm{GL}_2(\mathbf Z)$.

An example is given by Peter Stebe in his paper “Conjugacy separability of groups of integer matrices”, Proc. of the AMS, 32 (1), mars 1972, p. 1—7.
Namely, set
\[ A = \begin{pmatrix} 188 & 275 \\ 121 & 177 \end{pmatrix} = \begin{pmatrix} 11\cdot 17+1 & 25\cdot 11 \\ 11^2 & 11\cdot 16+1 \end{pmatrix} \]
and
\[ B = \begin{pmatrix} 188 & 11 \\ 3025 & 177 \end{pmatrix} =
\begin{pmatrix} 11\cdot 17+1 & 11 \\ 11^2\cdot 25 & 11\cdot 16+1 \end{pmatrix}. \]
These matrices $A$ and $B$ have integer coefficients, their determinant is $1$, hence they belong to $\mathrm{SL}_2(\mathbf Z)$. They also have the same trace, hence the same characteristic polynomial, which is $T^2-365T+1$. The discriminant of this polynomial is $3\cdot 11^2\cdot 367$. This implies that their complex eigenvalues are distinct, hence these matrices are diagonalizable over $\mathbf C$, and are conjugate over $\mathbf C$.

In the same way, we see that they remain conjugate modulo every prime number $p$
that does not divide the discriminant. Modulo $3$ and $11$, we check that both matrices become
conjugate to $\begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}$, while they become conjugate to $\begin{pmatrix}-1 & 1 \\ 0 & -1 \end{pmatrix}$ modulo $367$.

It is a bit more delicate to prove that if we reduce modulo any integer $d\geq 2$, then $A$ and $B$ become conjugate under $\mathrm{SL}_2(\mathbf Z/d\mathbf Z)$. Stebe's argument runs in two steps.
He first computes the set of matrices $V$ that conjugate $A$ to $B$, namely he solves the equation $VA=BV$. The answer is given by
\[ V=V(x,y) = \begin{pmatrix} x & y \\ 11 y & 25 x-y \end{pmatrix}. \]
Moreover, one has
\[ \det(V(x,y))=25x^2 - xy -11y^2. \]
Consequently, to prove that $A$ and $B$ are conjugate in $\mathrm{SL}_2(\mathbf Z/d\mathbf Z)$, it suffices to find $x,y\in\mathbf Z/d\mathbf Z$ such that $\det(V(x,y))=1 \pmod d$.
To prove that they are conjugate by $\mathrm{SL}_2(\mathbf Z)$, we need to find $x,y\in\mathbf Z$ such that $\det(V(x,y))=1$, and if we agree to be content with a conjugacy by $\mathrm{GL}_2(\mathbf Z)$, then solutions of $\det(V(x,y))=-1$ are also admissible.

Let us first start with the equations modulo $d$. By the Chinese remainder theorem, we may assume that $d=p^m$ is a power of a prime number $p$. Now, if $p\neq 5$, we can take $y=0$ and $x$ such that $5x=1\pmod {p^m}$. If $p=5$, we take $x=0$ and we solve $y$ for $-11y^2=1\pmod {5^m}$, which is possible since $-11\equiv 4\pmod 5$ is a square, and it is easy, by induction (anyway, this is an instance of Hensel's lemma), to produce $y$ modulo $5^m$ such that $y\equiv 3\pmod 5$ and $-11y^2=1\pmod{5^m}$.

On the other hand, the equation $25x^2-xy-11y^2=\pm 1$ has no solutions in integers.
The case of $-1$ is easy by reduction modulo $3$: it becomes $x^2+2xy+y^2=2$, which has no solution since $x^2+2xy+y^2=(x+y)^2$ and $2$ is not a square modulo $3$.
The case of $+1$ is rather more difficult. Stebe treats it by reducing to the Pell equation $u^2=1101y^2+1$ and shows by analysing the minimal solution to this Pell equation that $y$ is divisible by $5$, which is incompatible with the initial equation.


From a more elaborate point of view, $V$ is a smooth scheme over $\mathbf Z$ that violates the integral Hasse principle. In fact, $V$ is a torsor under the centralizer of $A$, which is a torus, and the obstruction has been studied by Colliot-Thélène and Xu, precisely in this context. However, I did not make the calculations that could use their work to reprove Stebe's theorem.

Book review: Contemplative Practices in Higher Education

Contemplative Practices in Higher Education, by Daniel P. Barbezat and Mirabai Bush.
Jossey-Bass, 2014. 
[Center for Contemplative Mind in Society] [Library of Congress] [Amazon] [Fnac] [Barnes&Noble]

“Contemplative practices in higher education”? what the f...? Does this means that we should have to have our students meditate instead of practicing mathematics by doing more and more exercises? Again, what the f... ? And is it really appropriate, in our universities (which, in France, are mostly laïques et républicaines) to experiment such practices?

The subtitle of the book under review should perhaps reassure us: Powerful methods to transform teaching and learning. Indeed, as its authors explain to us in the very first lines of its preface, contemplative practices always has a well established place in the intellectual inquiry, a place which goes well beyond their vital role in all the major religions and spiritual traditions. The authors acknowledge many objectives to these practices, pointing out 4 of them whose importance can difficultly be denied:

• Development of attention and focus;
• Deeper understanding of the content of the course;
• Compassion, relation with self; deepening of the moral and spiritual component of education;
• Development of personality, and of creativity.

The largest part of this book develops twenty years of experiments of various contemplative practices in higher education, that were put forward to strengthen that quality of teaching, especially in the first grades of American college, and in almost all fields (law, economics, physics, chemistry, environmental sciences, music, literature, psychology).

Daniel Barbezat, a professor in economics at Amherst, explains for example how these methods allowed him to solve the following contradiction: how is it possible that his field (economics) pretends studying the mechanisms of decision that are supposed to lead people to well-being, without every considering the nature of well-being? He proposed to his class various alternatives, of the following kind:

1. The class is divided in ten groups of three people; the member of one group receive $1000 each, the other nothing
2. Everybody receives $200

He then asked everyone to choose between these two possibilities, and to guess with which proportion each possibility would be chosen. He returned to that exercise later following a meditation exercise about gratitude (think about things you are grateful for, then think about someone who is at the source of this gratitude). The results were not at all the same, therefore opening a way for thinking on the place of individual in society.

David Haskell, who teaches environmental sciences and biology, adapted the reading method of monks (as he says, lectio without too much divina) to have his class study problems of hunger and development. He asked his students to alternate between periods of quiet rest (say one minute) and the reading of one or two sentences of the text (each one reads by turns) and to brief commentaries by the students, etc. Other teachers propose the students to behold some text, or some graphic representation, and then to comment it. Examples are given of the probability distribution of the hydrogen bromide atom, according to its energy levels, or to two charts of industrial production (in absolute vs relative value). The authors claim that such exercises deepen the relation with self, with the studied document, and with other materials of the course.

Mathematics are absent from this book. In a blog post hosted by the American mathematical society, Luke Wolcott evokes this possibility, but acknowledges that he did not go further than personal meditation. In fact, I could not find other explicit examples in various sources, even none in the archives of the Center for contemplative mind in society that the authors of this book lead. However, it seems to me that some practical exercises organised by a teacher such as Adrien Guinemer in his middle/high school classes go in that direction (notably, the study of sections of cubes, cones, cylinder made from plasticine).

There are at least two methods that I find interesting and that could easily be implemented in our classes:

• Meditation exercises at the beginning of the class — first have everybody focus his attention on its breath during five minutes, and then report it on the subject of the class.
• Introspection techniques to fight failure anxiety — the student is asked to solve an exercise while writing on his sheet everything that comes to his mind, whatever relation it has with the exercise.

Moreover, isn't it our role to develop a profound sense of compassion to our students, especially those who prepare themselves to become teachers?

The first part of the book proposes a theoretical and practical background that is necessary to appreciate the variety of these methods, as well as some issues that need to be avoided. Three of them seem particularly crucial to me, all of them requiring from the teacher a quite deep personal involvement in these contemplative practices:

1. Assign to the contemplative exercises a clear pedagogical goal, whose impact can be evaluated;
2. Disjoint the practice of these exercises from the cultural and religious backgrounds in which they were first devised;
3. Be able of managing students who would not be at ease, or even would reject, such practices.

Anyway, the variety of possibilities that is described in this book is an invitation from its two authors that we embrace these millenary-old techniques to deeply transform our teaching. So, to the question that begins this book review, the author do much better than answering “Why not?” since they tell us “Follow us, try, and see!”.

So let us try, and see.

The evolution of higher education

After a few months of silence, a short blog post to indicate a few web links that I found interesting, rising concern about the evolution of higher education.

In February 2014, Counterpunch published a series of remarks by Noam Chomsky under the title On Academic Labor. (I found it first on Alternet, under the alternate title How America's Great University System Is Getting Destroyed.)

More recently (May 2014), the New York Times published an editorial, Fat-Cat Administrators at the Top 25, where they quote a report from the Institute for Policy Studies indicating that "student debt and low-wage faculty labor are rising faster at state universities with the highest-paid presidents."

In fact, I had been made aware of the problem by a few posts from the blog The Homeless Adjunct, notably this post from 2012 that clearly explains how the American university system was killed in five easy steps:

1. Defund the university system;
2. Deprofessionalize and impoverish the  professors;
3. Install a managerial/administrative class who take over governance of the university;
4. Move in corporate culture and corporate money;
5. Destroy the students.

This probably looks a radical point of view, and had looked a bit radical to me at that time. Except that it is really how it now happens in France where we are clearly somewhere in between steps 3 and 4. Of course, the fact that our university system is mainly public delays the process a little bit, but look:

1. Decisive progress towards defunding was made in 2009 by the Sarkozy-Pécresse LRU-law. While the acronym stands for Liberty and Responsability of Universities, this law has been infamously referred to Autonomy of Universities. The French public universities are now allocated a global budget by the State, which they are now supposed to manage as they wish, except that the allocated budget is insufficient, and that they have almost no control of whatever. Many universities are on the edge of defaulting. So what we have under the eyes is nothing but a defunding of the system disguised as a change of allocation model.
2. The number of permanent positions is sharply decreasing. Of course, the age-pyramid of the present professors is also a cause for this evolution, since almost all baby-boomers have now retire. But the decrease is not at all the same in all fields—for example, this year, there were many more open positions in applied mathematics than in pure mathematics. Probably, when it comes about cutting positions, the "applied"-color makes it nicer for university boards. Probably too, applied mathematicians have been better at explaining their rôle in society.
3. Meanwhile, the administration is getting fatter. To manage the global budget, it has been necessary to hire full-time "managers". And to be able to attract them, it seems that their pay has nothing to do with the usual range among French public servants. At the same time, a new law reorganizes the higher-education system by forcing universities (as well as our innumerous engineering schools) to regroup themselves. This will create enormous beasts that will look like the Lernean Hydra. For example, the Paris-Saclay University regroups 22 higher education schools, among which 2 universities and 10 "grandes écoles"; it will host around 50.000 students and more than 10.000 professors and researchers! No doubt that it will require a heavy bureaucracy to manage this high number of people. And since we're split in many institutions, it will be hard to have the voice of academic freedom be listened to.

  All of this is very depressing...