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Register a new userA note on the action of $SL(m, mathbb{Z}_n)$ on the ring $mathbb{Z}^m_n$
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We know that $mathbb{Z}_n$ is a finite field for a prime number $n$. Let $m,n$ be arbitrary natural numbers and let $mathbb{Z}^m_n= mathbb{Z}_n timesmathbb{Z}_ntimes...timesmathbb{Z}_n$ be the Cartesian product of $m$ rings $mathbb{Z}_n$. In this note, we present the action of $SL(m, mathbb{Z}_n)={A in mathbb{Z}^{m,m}_{n} : det A equiv 1 (modsimn)}$, where $SL(m, mathbb{Z}_n)$ for $ngeq 2$ is a group under matrix multiplication modulo $n$, on the ring $mathbb{Z}^m_n$ as a right multiplication of a row vector of $mathbb{Z}^m_n$ by a matrix of $SL(m, mathbb{Z}_n)$ to determine the orbits of the ring $mathbb{Z}^m_n$. This work is an extension of [1]
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We consider an analogue of Wittens $SL(2,mathbb{Z})$ action on three-dimensional QFTs with $U(1)$ symmetry for $2k$-dimensional QFTs with $mathbb{Z}_2$ $(k-1)$-form symmetry. We show that the $SL(2,mathbb{Z})$ action only closes up to a multiplication by an invertible topological phase whose partition function is the Brown-Kervaire invariant of the spacetime manifold. We interpret it as part of the $SL(2,mathbb{Z})$ anomaly of the bulk $(2k+1)$-dimensional $mathbb{Z}_2$ gauge theory.
Let $Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $text{SL}_n(mathbb{Z})$. Borel-Serre proved that the cohomology of $Gamma_n(p)$ vanishes above degree $binom{n}{2}$. We study the cohomology in this top degree $binom{n}{2}$. Let $mathcal{T}_n(mathbb{Q})$ denote the Tits building of $text{SL}_n(mathbb{Q})$. Lee-Szczarba conjectured that $H^{binom{n}{2}}(Gamma_n(p))$ is isomorphic to $widetilde{H}_{n-2}(mathcal{T}_n(mathbb{Q})/Gamma_n(p))$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map $H^{binom{n}{2}}(Gamma_n(p)) rightarrow widetilde{H}_{n-2}(mathcal{T}_n(mathbb{Q})/Gamma_n(p))$ is always surjective, but is only injective for $p leq 5$. In particular, we completely calculate $H^{binom{n}{2}}(Gamma_n(5))$ and improve known lower bounds for the ranks of $H^{binom{n}{2}}(Gamma_n(p))$ for $p geq 5$.
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For a positive integer $g$, let $mathrm{Sp}_{2g}(R)$ denote the group of $2g times 2g$ symplectic matrices over a ring $R$. Assume $g ge 2$. For a prime number $ell$, we give a self-contained proof that any closed subgroup of $mathrm{Sp}_{2g}(mathbb{Z}_ell)$ which surjects onto $mathrm{Sp}_{2g}(mathbb{Z}/ellmathbb{Z})$ must in fact equal all of $mathrm{Sp}_{2g}(mathbb{Z}_ell)$. The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.
For every pair of distinct primes $p$, $q$ we prove that $mathbb{Z}_p^3 times mathbb{Z}_q$ is a CI-group with respect to binary relational structures.
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