Roots of identity in finite groups of Lie type

Throughout the article it will be assumed that the finite fields under consideration are of odd characteristics.

The set of units in $\mathbb{Z}/n\mathbb{Z}$ will be denoted by $\mathbb{Z}/n\mathbb{Z}^{\times}$. By the notation $e(n)$, we will always denote the order of $q$ in the multiplicative group $\mathbb{Z}/n\mathbb{Z}^{\times}$. The Euler’s totient function will be denoted by $\phi(n)$. We denote the set of irreducible monic polynomials over $\mathbb{F}_{q}$ by $\Phi$. Let us also fix the notation for the set of all partitions of $n$ to be $\Lambda^{n}$. By the symbol $\left(\dfrac{u}{q}\right)_{i}$, we denote the quantity $\left(1-\dfrac{u}{q}\right)\left(1-\dfrac{u}{q^{2}}\right)\ldots\left(1-\dfrac{u}{q^{i}}\right)$. For a partition $\lambda\vdash n$, the number of occurrences of $i$ will be denoted as $m_{i}(\lambda)$. The transpose of a partition $\lambda$ will be denoted as $\lambda^{\prime}$. The number of partitions of a positive integer $n$ will be denoted as $\mathbf{P}(n)$ and those with lesser than or equal to equal to $m$ many (not necessarily distinct) parts will be denoted as $\mathbf{P}_{m}(n)$.

Given a $n$-dimensional vector space $V$ over $\mathbb{F}_{q}$, the set of all automorphisms of $V$ is denoted by $\mathrm{GL}(V)$. After fixing an appropriate basis, $\mathrm{GL}(V)$ can be identified with $\mathrm{GL}_{n}(q)$. The conjugacy classes of $\mathrm{GL}_{n}(q)$ is determined by functions $\lambda:\Phi\longrightarrow\Lambda$ by $f\mapsto\lambda_{f}$ such that $\sum\limits_{f}\deg(f)\cdot|\lambda_{f}|=n$. The combinatorial data attached via this correspondence, for an element $\alpha\in\mathrm{GL}_{n}(q)$ will be denoted by $\Delta(\alpha)$. The centralizer size is given by

This quantity sometimes can be shortened using a general expression, which we recall from [6, Theorem 2] below;

Consider the field $k=\mathbb{F}_{q^{2}}$, and the involution $\sigma\colon\mathbb{F}_{q^{2}}\longrightarrow\mathbb{F}_{q^{2}}$ defined as $\sigma(a):=\overline{a}=a^{q}$ of the field. We further get an automorphism of $\mathbb{F}_{q^{2}}[t]$, the polynomial ring over $\mathbb{F}_{q^{2}}$, by action on the coefficients of the polynomials. The image of $f\in\mathbb{F}_{q^{2}}[t]$ will be denoted as $\overline{f}$. Consider the Hermitian form given by the matrix

Then we identify the unitary group (corresponding to the above Hermitian form) with the set

Since all non-degenerate Hermitian forms on an $n$-dimensional vector space are unitarily congruent to the Hermitian form given by the above matrix, it is evident that all the unitary groups arising are conjugate with each other inside $\mathrm{GL}_{n}(q^{2})$ (see theorem $10.3$ of [12]). From the work of Wall, it is known that two elements $g_{1},g_{2}\in\mathrm{U}_{n}(q^{2})$ are conjugate to each other if and only if they are conjugate in $\mathrm{GL}_{n}(q^{2})$. These conjugacy classes are parameterized by special polynomials and partitions. We call a polynomial $f$ of degree $d$ to be $\sim$-symmetric if $\widetilde{f}=f$ where $\widetilde{f}(t)=\overline{f(0)}^{-1}t^{d}\bar{f}(t^{-1})$. A polynomial without a proper $\sim$-symmetric factor will be called a $\sim$-irrdeucible polynomial. Note that a $\sim$-irreducible polynomial can be reducible in the usual sense. Denote by $\widetilde{\Phi}$ to be the set of all $\sim$-irreducible polynomials. Let $\mathcal{P}_{n}$ denote the set of all partitions of numbers $\leq n$. Then a conjugacy class in $\mathrm{U}(n,\mathbb{F}_{q^{2}})$ is determined by a function $\lambda\colon\widetilde{\Phi}\longrightarrow\mathcal{P}_{n}$ which satisfies (a) $\lambda(f)=\lambda(\widetilde{f})$ and, (b) $\sum\limits_{\phi}|\lambda(\phi)|\deg\phi=n$. The centralizer six=ze can be found in [36] and we are not quoting it here.

Using these information, we determine when a matrix $A$ is an $M$-th root of identity. This information is further used in subsequent sections to determine the desired generating functions, with the help of the concept of central join of two matrices, following [34], [33], [35].

Note that an element in $\mathrm{G}(q)$ is semisimple if and only if it has order coprime to $q$. In this subsection, we classify all the semisimple elements, which give $\mathrm{Id}$, when raised to the power $M$. In this subsection, we will assume that $(M,q)=1$. Let $f$ denote a product of distinct irreducible polynomials, and $A$ be the semisimple matrix with $f$ as its characteristic polynomial. Then if $A^{M}=\mathrm{Id}$, then $f$ must have irreducible factors coming from irreducible factors of $x^{M}-1\in\mathbb{F}_{q}[x]$. We recall some results and definitions from [21]. For a positive integer $n$, the splitting field of $x^{n}-1$ over a field $K$ is called the $n$-th cyclotomic field over $K$ and denoted by $K^{(n)}$. The roots of $x^{n}-1$ in $K^{(n)}$ are called the $n$-th roots of unity over $K$ and the set of all these roots is denoted by $E^{(n)}$.

A generator of the cyclic group is called primitive $n$-th root of unity over $K$. If $(n,p)=1$ and $\zeta$ is a primitive $n$-th root of unity over $K$, the polynomial

is called the $n$-th cyclotomic polynomial over $K$. When $(n,p)=1$, we have that $x^{n}-1=\prod_{d|n}Q_{d}(x)$, see [21, Theorem 2.45]. Note that if $A^{M}=1$, for each irreducible factor $g$ of $f$, there exists an $m(g)$ such that $f|Q_{m(g)}(x)$. Thus we need to know how $Q_{n}(x)$ factors over $\mathbb{F}_{q}[x]$. We recall the following result.

Since $(M,p)=1$, the polynomial $x^{M}-1$ is separable and hence for $d\neq d^{\prime}$ dividing $M$, the cyclotomic polynomials $Q_{d}(x)$ and $Q_{d^{\prime}}(x)$ are coprime. Hence we have the following lemma.

To deduce the result on the probability of a semisimple element to be an $M$-th root of identity, we need to know the order of the centralizer of each element. This has been discussed in Section 2.

To calculate the number of semisimple conjugacy classes in $\mathrm{Sp}_{2n}(q)$, which are $M$-th root of $\mathrm{Id}$, we need to know the factorization of $x^{M}-1$. Before proceeding further, for an integer $m$, let us define

The set $D_{m}$ plays a crucial role in counting the semisimple elements in finite symplectic and orthogonal groups, which gives $\mathrm{Id}$ on raising the power to $M$. We collect the following results from [37].

Let us now deduce the SCIM factors of $x^{M}-1$ in $\mathbb{F}_{q^{2}}$. According to [5, Lemma 2], if $f$ is a SCIM polynomial of odd degree $d$, then for any root $\zeta$ of $f$, it satisfies $\zeta^{q^{d}+1}=1$. This means that $\mathrm{ord}(f)|q^{d}+1$. For a positive integer $m$ define

It is evident that if $f$ is a SCIM polynomial of degree $d$, then $\mathrm{ord}(f)\in\widetilde{D}_{(d-1)/2}$. If $d\in D_{m}$, and $\beta$ is a primitive $d$-th root of unity, then the set $\{\beta,\beta^{q^{2}},\beta^{q^{4}},\cdots,\beta^{q^{2(d-1)}}\}$ is of cardinality $d$. Indeed, if $\beta^{q^{2i}}=\beta^{q^{2j}}$ for some $0\leq i<j\leq d-1$, then $d|q^{2(j-i)}-1$. Hence $2(j-i)$ is an even multiple of $d$, which forces $2(j-i)=0$, see [37, Prposition 1]. For $e\in\widetilde{D}_{m}$ and a primitive $e$-th root of unity, let us define