We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation V of the affine Kac-Moody algebra g(E9). We describe an elementary algorithm for determining the decomposition of the submodule of V ⊗n whose irreducible direct summands have highest weights which are maximal with respect to the null-root. This decomposition is based on Littelmann’s pat...

A loop of order $n$ possesses at least $3n^2-3n+1$ associative triples. However, no $n>1$ that achieves this bound seems to be known. If the is involutory, then it $3n^2-2n$ Involutory loops with triples can obtained by prolongation certain maximally nonassociative quasigroups whenever $n-1$ a prime greater than or equal $13$ $n-1=p^{2k}$, $p$ an odd prime. For orders $n\le 9$ minimum number re...

A nonempty subset A of {1, 2, . . . , n} is relatively prime if gcd(A) = 1. Let f(n) and fk(n) denote, respectively, the number of relatively prime subsets and the number of relatively prime subsets of cardinality k of {1, 2, . . . , n}. Let Φ(n) and Φk(n) denote, respectively, the number of nonempty subsets and the number of subsets of cardinality k of {1, 2, . . . , n} such that gcd(A) is rel...

We give two applications of our prior work toward the Putman–Wieland conjecture. First, we deduce a strengthening result Marković–Tošić on virtual mapping class group actions homology covers. Second, let g ⩾ 2 $g\geqslant 2$ and Σ ′ , n → $\Sigma _{g^{\prime },n^{\prime }}\rightarrow \Sigma _{g, n}$ be finite H $H$ -cover topological surfaces. show action + 1 _{g,n+1}$ an -isotypic component ( ...

We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation V of the affine Kac-Moody algebra g(E9). We describe an elementary algorithm for determining the decomposition of the submodule of V ⊗n whose irreducible direct summands have highest weights which are maximal with respect to the null-root. This decomposition is based on Littelmann’s pat...

Let f (m, n) denote the number of relatively prime subsets of {m+ 1, m + 2,. .. , n}, and let Φ(m, n) denote the number of subsets A of {m + 1, m + 2,. .. , n} such that gcd(A) is relatively prime to n. Let f k (m, n) and Φ k (m, n) be the analogous counting functions restricted to sets of cardinality k. Simple explicit formulas and asymptotic estimates are obtained for these four functions. A ...

Given free modules $M\subseteq L$ of finite rank $f\geq 1$ over a principal ideal domain $R$, we give procedure to construct basis $L$ from $M$ assuming the invariant factors or elementary divisors $L/M$ are known. matrix $A\in M_{m,n}(R)$ $r$, its nullspace in $R^n$ is $R$-module $f=n-r$. We submodule $f$ naturally associated with $A$ and whose easily computable, determine quotient module then...

here we consider all finite non-abelian 2-generator $p$-groups ($p$ an odd prime) of nilpotency class two and study the probability of having $n^{th}$-roots of them. also we find integers $n$ for which, these groups are $n$-central.