相关论文: Ad-nilpotent ideals and The Shi arrangement

Shi tableaux are special binary fillings of certain Young diagrams which arise in the study of Shi hyperplane arrangements related to classical root systems. For type $A$, the set $\mathcal T$ of Shi tableaux naturally coincides with the…

Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In particular, we find out a natural parametrization of maximal elements and dimension formulas…

For an irreducible polynomial $\chi(x)\in \mathcal{O}_k[x]$ of degree $n$, where $k$ is a number field and $\mathcal{O}_k$ its ring of integers, let $N(X, T)$ denote the number of $n \times n$ integral matrices whose characteristic…

We establish counting formulas and bijections for deformations of the braid arrangement. Precisely, we consider real hyperplane arrangements such that all the hyperplanes are of the form $x\_i-x\_j=s$ for some integer $s$. Classical…

This paper reveals some new analytical and geometrical properties of the generalized algebraic multiplicity, $\chi$, introduced in [7, 5] and further developed in [20, 23, 24]. In particular, it establishes a completely new connection…

We classify one-element extensions of a hyperplane arrangement by the induced adjoint arrangement. Based on the classification, several kinds of combinatorial invariants including Whitney polynomials, characteristic polynomials, Whitney…

We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for…

This is the second of two papers where we study polytopes arising from affine Coxeter arrangements. Our results include a formula for their volumes, and also compatible definitions of hypersimplices, descent numbers and major index for all…

We consider ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra. The goal of this paper is two-fold. First, we study the ad-nilpotent ideals lying inside the Heisenberg ideal. The Heisenberg ideal is the nilpotent radical of…

Let $G$ be a classical linear algebraic group over an algebraically closed field, and let $\mathfrak{n}$ denote the subset of nilpotent elements in its Lie algebra. In this paper we study a partial order on the $G$-orbits in $\mathfrak{n}$…

Given a finite group $G$ and an irreducible complex character $\chi$ of $G$, the codegree of $\chi$ is defined as the integer ${\rm cod}(\chi)=|G:\ker\chi|/\chi(1)$. It was conjectured by G. Qian in [13] that, for every element $g$ of $G$,…

We initiate and develop the theory of finite $W$-superalgebras $\mathcal{W}_\chi$ associated to the queer Lie superalgebra $\g=\q(N)$ and a nilpotent linear functional $\chi \in \ev\g^*$. We show that the definition of the $W$-superalgebra…

We prove that one of the conditions in M.V. Zaicev's formula for the PI-exponent and in its natural generalization for the Hopf PI-exponent, can be weakened. Using the modification of the formula, we prove that if a finite dimensional…

We construct an explicit Gr\"obner--Shirshov basis for free associative Rota--Baxter algebras of weight zero with nilpotent operator $R^n=0$, where $n\ge 2$. First, we define a monomial order on the standard linear basis $RS(X)$ of the free…

We present an effective method for computing parametric primary decomposition via comprehensive Gr\"obner systems. In general, it is very difficult to compute a parametric primary decomposition of a given ideal in the polynomial ring with…

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of GL_n(C) on the variety of x-nilpotent complex matrices. We obtain a criterion as to whether the action admits a finite number of orbits and specify a…

Let t be any integer and fix an odd prime ell. Let Phi(x) = T_ell^n(x)-t denote the n-fold composition of the Chebyshev polynomial of degree ell shifted by t. If this polynomial is irreducible, let K = bbq(theta), where theta is a root of…

Motivated by better understanding the bideterminant (=product of minors) basis on the polynomial ring in $n \times m$ variables, we develop theory \& algorithms for Gr\"obner bases in not only algebras with straightening law (ASLs or Hodge…

We study certain types of ideals in the standard Borel subalgebra of an untwisted affine Lie algebra. We classify these ideals in terms of the root combinatorics and give an explicit formula for the number of such ideals in type $A$. The…

For an irreducible character $\chi$ of a finite group $G$, let $\mathrm{cod}(\chi):=|G: \ker(\chi)|/\chi(1)$ denote the codegree of $\chi$, and let $\mathrm{cod}(G)$ be the set of irreducible character codegrees of $G$. In this note, we…