Let $n \geq 2$ be a positive integer, and $q$ be a prime power. We study the $number$ of additive decompositions of nonscalar matrices in the matrix ring $M_n(\mathbb{F}_q)$ as sums of elements from two prescribed conjugacy classes. Let $z \in M_n(\mathbb{F}_q)$ be nonscalar. We show that, except for the case $(n,q,\textrm{Tr}(z)) = (2,2,1)$, there exist conjugacy classes $X, Y \subset M_n(\mathbb{F}_q)$ such that the characteristic polynomial of $X$ is irreducible of degree $n$ and the characteristic polynomial of $Y$ is of the form $ (T- \lambda) h(T)$, where $\lambda \in \mathbb{F}_q$, $h$ is irreducible of degree $n-1$ and $h(\lambda) \neq 0$. These classes can be chosen so that $\textrm{Tr}(z) = \textrm{Tr}(X) + \textrm{Tr}(Y)$. For such $X$ and $Y$, let $$ N_{X,Y}(z) = \# \{ (x,y) \in X \times Y : x + y = z \}. $$ We prove the following estimate: $$ \left| N_{X,Y}(z) - q^{(n-1)^2} \right| \leq 42 q^{(n-1)^2 -1}. $$ Thus, for nonscalar matrices with matching trace, the number of such additive decompositions is $approximately$ the same, namely $q^{(n-1)^2}$, with an absolute error constant independent of $n$ and $q$.
For $n\geq 4$, let $T_n$ be the rank-3 matroid on $\mathbb{Z}/n\mathbb{Z}$ whose bases are the three-element non-zero-sum subsets. Let $X_1(n)^\circ$ denote the open subscheme of the modular curve $X_1(n)$ obtained by removing the cusps corresponding to reducible N\'eron polygons. For $n \geq 10$, we give a purely alge...
Given a field $\mathbb{F}$ and an integer $n \geq 2$, denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable. It was recently proved that $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ if and only if $\mathbb{F}$ is not quad...
For coprime $1<a<b$, let $M_n^{a,b}(\mathbb{F}_q)$ be the set of commuting pairs of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ with $X^a=Y^b$. Huang, Jiang, and Oblomkov assembled their orders as an Eulerian $q$-series $Z_{a,b}(q)$. They conjectured that it is an explicit product $P_{a,b}(q)$ involving Jacobi's...
Fix a prime $\ell$. Let $K = \mathbb{F}_q(t)$ be a global function field such that $\gcd(q,6) = 1$ and $q \equiv 1 \pmod \ell$. Let $E$ be a non-isotrivial elliptic curve over $K$. Given a fixed monic polynomial $Q$ over $\mathbb{F}_q$, and assuming some mild conditions on $E$, we show that the rank of $E$ does not cha...
Let $H_n(q)$ be the generic Iwahori--Hecke algebra associated to the symmetric group $\mathfrak{S}_n$ of degree $n$ and let $\mathscr{Y}_{n,i}$ be the sum of standard basis of $H_n(q)$ with Coxeter length $i$ for $1\leq i\leq \binom{n}{2}$. We show that $\mathscr{Y}_{n,1}$ and $\mathscr{Y}_{n,2}$ are noncommutative whe...
Let $K$ be an algebraically closed field of odd characteristic $p$, let $G= {\rm SL}_2(K)$, and let $V$ be the natural representation of $G$. Let $b_k$ denote the number of $G$-indecomposable factors of $V^{\otimes k}$, counted with multiplicity, and let $\delta_p=1-\log_{p^2}\!\bigl(\tfrac{p+1}{2}\bigr)$. Then there e...
Aranya Lahiri, Nai-Heng Sheu· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
