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

An arrangement of hyperplanes is a finite collection of hyperplanes in a real Euclidean space. To such a collection one associates the characteristic polynomial that encodes the combinatorics of intersections of the hyperplanes. Finding the…

We suggest new realizations of quantum groups corresponding to complex simple Lie algebras, and of affine quantum groups. These new realizations are labeled by Coxeter elements of the corresponding Weyl group and have the following key…

Let $G$ be a finite group admitting a coprime automorphism $\alpha$ of order $e$. Denote by $I_G(\alpha)$ the set of commutators $g^{-1}g^\alpha$, where $g\in G$, and by $[G,\alpha]$ the subgroup generated by $I_G(\alpha)$. We study the…

Let p be a prime number. Let G be a finite p-group and $\chi \in Irr(G)$. Denote by $\bar{\chi} \in Irr(G)$ the complex conjugate of $\chi$. Assume that $\chi(1)=p^n$. We show that the number of distinct irreducible constituents of the…

Let $k$ denote a field with nontrivial discrete valuation. We assume that $k$ is complete with perfect residue field. Let $G$ be the group of $k$-rational points of a reductive, linear algebraic group defined over $k$. Let $\gg$ denote the…

For any polynomial ideal $I$, let the minimal triangular set contained in the reduced Buchberger-Gr\"obner basis of $I$ with respect to the purely lexicographical term order be called the W-characteristic set of $I$. In this paper, we…

We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of $sl(n+1,\Bbb C)$. We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class…

For symplectic Lie algebras $\mathfrak{sp}(2n,\mathbb{C})$, denote by $\mathfrak{b}$ and $\mathfrak{n}$ its Borel subalgebra and maximal nilpotent subalgebra, respectively. We construct a relationship between the abelian ideals of…

We provide explicit combinatorial formulas for the Chow polynomial and for the augmented Chow polynomial of uniform matroids, thereby proving a conjecture by Ferroni. These formulas refine existing formulas by Hampe and by Eur, Huh, and…

Let $I$ be a perfect ideal of height two in $R=k[x_1, \ldots, x_d]$ and let $\varphi$ denote its Hilbert-Burch matrix. When $\varphi$ has linear entries, the algebraic structure of the Rees algebra $\mathcal{R}(I)$ is well-understood under…

Following the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in terms of n-th power residue symbols. This formalism leads to…

We begin with a brief introduction on N=1 gauge theories, focusing on the importance of the effective superpotential in light of the new techniques to compute it systematically. We then proceed to consider theories for which the Konishi…

Let $G_{\mathbb R}$ be the set of real points of a complex linear reductive group and $\hat G_\lambda$ its classes of irreducible admissible representations with infinitesimal integral regular character $\lambda$. In this case each cell of…

Let $\overline{\mathtt{X}}_\lambda$ be the closure of the $\mathtt{I}$-orbit $\mathtt{X}_\lambda$ in the affine Grassmanian $\mathtt{Gr}$ of a simple algebraic group $G$ of adjoint type, where $\mathtt{I}$ is the Iwahori group and $\lambda$…

Let G be an algebraic reductive group over a an algebraically closed field of positive characteristic. Choose a parabolic subgroup $P$ in $G$ and denote by $U$ its unipotent radical. Let $X$ be a $G$-variety. The purpose of this paper is to…

The present paper investigates properties of quasi-stable ideals and of Borel-fixed ideals in a polynomial ring $k[x_0,\dots,x_n]$, in order to design two algorithms: the first one takes as input $n$ and an admissible Hilbert polynomial…

In a finite group G, we consider nilpotent weights, and prove a pi-version of the Alperin Weight Conjecture for certain pi-separable groups. This widely generalizes an earlier result by I. M. Isaacs and the first author.

In his study of Kazhdan-Lusztig cells in affine type $A$, Shi has introduced an affine analog of Robinson-Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combinatorial realization of…

We compute $\mathrm{Ext}^{1}_B(\chi_1,\chi_2)$ between two characters $\chi_1,\chi_2$ of a Borel subgroup $B$ of a split reductive group $G$ over a finite field $\mathbb{F}_q,$ and make an application to the calculation of…

Let $V$ be a vertex algebra and $g$ be an automorphism of $V$ of order $T$. For any $n, m \in (1/T)\mathbb{N}$, we construct an $\tilde{A}_{g,n}(V)\!-\!\tilde{A}_{g,m}(V)$-bimodule $\tilde{A}_{g,n,m}(V)$, where $\tilde{A}_{g,n}(V)$ denotes…