Related papers: On Bohr sets of integer valued traceless matrices

Let X be a smooth curve over a finite field of characteristic p, let E be a number field, and consider an E-compatible system of lisse sheaves on the curve X. For each place lambda of E not lying over p, the lambda-component of the system…

Let $\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq \lambda_5 \geq -\lambda_1$ be real numbers such that $\sum_{i=1}^5 \lambda_i =0$. In \cite{oren}, O. Spector prove that a necessary and sufficient condition for $\lambda_1,…

In this investigation of character tables of finite groups we study basic sets and associated representation theoretic data for complementary sets of conjugacy classes. For the symmetric groups we find unexpected properties of characters on…

Let $K$ be an algebraically closed field of characteristic $0$ and let $M_n(K)$, $n \ge 3$, be the matrix ring over $K$. We will show that the image of any multilinear polynomial in four variables evaluated on $M_n(K)$ contains all matrices…

Let $F_n$ be an $n$ by $n$ symmetric matrix whose entries are bounded by $n^{\gamma}$ for some $\gamma>0$. Consider a randomly perturbed matrix $M_n=F_n+X_n$, where $X_n$ is a random symmetric matrix whose upper diagonal entries $x_{ij}$…

We introduce a new family of symmetric polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_{\lambda}$ arising from exactly solvable lattice models associated with the quantised loop algebra $\mathcal{U}_{q}(\mathfrak{sl}_{2}[z^\pm])$. The…

The radical of the Brauer algebra B_f^x is known to be non-trivial when the parameter x is an integer subject to certain conditions (with respect to f). In these cases, we display a wide family of elements in the radical, which are…

Let P a locally finite partially ordered set, F a field, G a group, and I(P,F) the incidence algebra of P over F. We describe all the inequivalent elementary G-gradings on this algebra. If P is bounded, F is a infinite field of…

Given a set $\Gamma$ of low-degree k-dimensional varieties in $\mathbb{R}^n$, we prove that for any $D \ge 1$, there is a non-zero polynomial $P$ of degree at most $D$ so that each component of $\mathbb{R}^n \setminus Z(P)$ intersects…

Given any commutative ring $R$, a commutator of two $n\times n$ matrices over $R$ has trace $0$. In this paper, we study the converse: whether every $n \times n$ trace $0$ matrix is a commutator. We show that if $R$ is a B\'{e}zout domain…

Let ${\mathcal B}=\{b_i \}_{i=1}^\infty$ be a fixed sequence of pairwise distinct elements of a number field $k$. Given the integers $2\leq s \leq r$, assuming a quantitative version of Vojta's conjecture on the bounded degree algebraic…

Let $f$ bea noncommutativepolynomial of degree $m\ge 1$ over an algebraically closed field $F$ of characteristic $0$. If $n\ge m-1$ and $\alpha_1,\alpha_2,\alpha_3$ are nonzero elements from $F$ such that $\alpha_1+\alpha_2+\alpha_3=0$,…

In this two-part series of articles, we present a new proof comparing the trace formula for a general linear group with that of one of its inner forms. Our methodology relies on the trace formula for Lie algebras, incorporating the notion…

The purpose of this note is to extend in a simple and unified way the known results on interlacing of zeros of paraorthogonal polynomials on the unit circle. These polynomials can be regarded as the characteristic polynomials of any matrix…

Finite matroids are combinatorial structures that express the concept of linear independence. In 1964, G.-C. Rota conjectured that the coefficients of the "characteristic polynomial" of a matroid $M$, polynomial whose coefficients enumerate…

We consider integrable category $\mathcal{O}$ representations of Borcherds--Kac--Moody algebras whose Cartan matrix is finite dimensional, and determine the necessary and sufficient conditions for which the tensor product of irreducible…

The main aim of this work is to apply the study of the asymptotic behaviour of generalized eigenvalues between infinite Hermitian definite positive matrices in an important question regarding the location of zeros of Sobolev orthogonal…

A finitely generated module over the ring L=Z[t, t^{-1}] of integer Laurent polynomials that has no Z-torsion is determined by a pair of sub-lattices of L^d. Their indices are the absolute values of the leading and trailing coefficients of…

We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show…

We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating…