Related papers: Characteristic and Ehrhart polynomials

For a not-necessarily commutative ring R we define an abelian group W(R;M) of Witt vectors with coefficients in an R-bimodule M. These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous…

We extend our combinatorial approach of decomposing the partition function of the Potts model on finite two-dimensional lattices of size L x N to the case of toroidal boundary conditions. The elementary quantities in this decomposition are…

We present an algorithm for growing the denominator $r$ polygons containing a fixed number of lattice points and enumerate such polygons containing few lattice points for small $r$. We describe the Ehrhart quasi-polynomial of a rational…

The lattice of flats $\mathcal L_M$ of a matroid $M$ is combinatorially well-behaved and, when $M$ is realizable, admits a geometric model in the form of a "Schubert variety of hyperplane arrangement". In contrast, the lattice of flats of a…

Let B be a reductive Lie subalgebra of a semi-simple Lie algebra of the same rank both over the complex numbers. To each finite dimensional irreducible representation $V_\lambda$ of F we assign a multiplet of irreducible representations of…

In this paper we are constructing integer lattice squares, cubes or hypercubes in $\mathbb R^d$ with $d\in \{2,3,4\}$. For squares and cubes we find a complete description of their Ehrhart polynomial. For hypercubes, we compute one of the…

Let $k$ be a finite field, and $L$ be a $q$-linearized polynomial defined over $k$ of $q$-degree $r$ ($L=\sum^r_{i=0}a_iZ^{q^i}$, with $a_i\in k$). This paper provides an algorithm to compute a characteristic polynomial of $L$ over a large…

In recent years, the notion of characteristic polynomial of representations of Lie algebras has been widely studied. This paper provides more properties of these characteristic polynomials. For simple Lie algebras, we characterize the…

In this paper we describe the polynomial identities of degree 4 for a certain subspace of the Weyl algebra A_1 over an infinite field of arbitrary characteristic.

In this paper, we obtain a general formula for the characteristic polynomial of a finitely dimensional representation of Lie algebra $\mathfrak{sl}(2, \C )$ and the form for these characteristic polynomials, and prove there is one to one…

We introduce a powerful connection between Ehrhart theory and additive number theory, and use it to produce infinitely many new classes of inequalities between the coefficients of the $h^*$-polynomial of a lattice polytope. This greatly…

Given an $\Bbb{F}$-represented matroid $(M,\rho)$ with the ground set $[m]$, the representation $\rho$ naturally defines a hyperplane arrangement $\mathcal{A}_\rho$. We will study its parallel translates $\mathcal{A}_{\rho,{g}}$ of…

We extend the asymptotic formula for counting integral matrices with a given irreducible characteristic polynomial by Eskin, Mozes and Shah in 1996 to the case of counting elements in a maximal order of certain central simple algebra with a…

We count matrices in the special linear group SL(n, Z) whose characteristic polynomials split completely over Q.

We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related…

Motivated by Stanley's generalization of the chromatic polynomial of a graph to the chromatic symmetric function, we introduce the characteristic polynomial of a representation of the symmetric group, or more generally, of a symmetric…

We show that the characteristic polynomial of a hyperplane arrangement can be recovered from the class in the Grothendieck group of varieties of the complement of the arrangement. This gives a quick proof of a theorem of Orlik and Solomon…

We present and study two families of polynomials with coefficients in the center of the universal enveloping algebra. These polynomials are analogues of a determinant and a characteristic polynomial of a certain non-commutative matrix,…

Chevalley's theorem states that for any simple finite dimensional Lie algebra G (1) the restriction homomorphism of the algebra of polynomials on G onto the Cartan subalgebra H induces an isomorphism between the algebra of G-invariant…

We show that for every subset $E$ of positive density in the set of integer square-matrices with zero traces, there exists an integer $k \geq 1$ such that the set of characteristic polynomials of matrices in $E-E$ contains the set of…