Related papers: Bruhat-Chevalley order on the rook monoid

We extend Grood's tableau construction of irreducible representations of the rook monoid and Steinberg's analogous result for the full transformation monoid. Our approach is characteristic-free and applies to any submonoid $\mathcal{M}(n)$…

Let $G$ be a finite group. A finite unordered sequence $S = g_1 \boldsymbol{\cdot} \ldots \boldsymbol{\cdot} g_{\ell}$ of terms from $G$, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their…

The combinatorially and the geometrically defined partial orders on the set of permutations coincide. We extend this result to $(0,1)$-matrices with fixed row and column sums. Namely, the Bruhat order induced by the geometry of a Cherkis…

Let W be a finite reflection group acting orthogonally on R^n, P be the Chevalley polynomial mapping determined by an integrity basis of the algebra of W-invariant polynomials, and h be the highest degree of the coordinate polynomials in…

We consider RoCK (or Rouquier) blocks of symmetric groups and Hecke algebras at roots of unity. We prove a conjecture of Turner asserting that a certain idempotent truncation of a RoCK block of weight $d$ of a symmetric group $\mathfrak…

In this paper, we extend Manin and Schechtman's higher Bruhat orders for the symmetric group to higher Bruhat orders for non-longest words $w$ in $S_n$. We prove that the higher Bruhat orders of non-longest words are ranked posets with…

We describe the structure of the monoid of natural-valued monotone functions on an arbitrary poset. For this monoid we provide a presentation, a characterization of prime elements, and a description of its convex hull. We also study the…

We define a subsemigroup $S_n$ of the rook monoid $R_n$ and investigate its properties. To do this, we represent the nonzero elements of $S_n$ (which are $n\times n$ matrices) via certain triplets of integers, and develop a closed-form…

In this paper we study those generic intervals in the Bruhat order of the symmetric group that are isomorphic to the principal order ideal of a permutation w, and consider when the minimum and maximum elements of those intervals are related…

In the current article we study structure of a Chevalley group $G(R)$ over a commutative ring $R$. We generalize and improve the following results: (1) standard, relative, and multi-relative commutator formulas; (2) nilpotent structure of…

In this paper we prove that every automorphism of a Chevalley group with the root system G_2 over a commutative ring R with 1/3, generated by all its invertible elements and the ideal 2R is a composition of ring and inner automorphisms.

In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank >1 over a commutative ring (with 1/2 for the systems A_2, F_4, B_l, C_l; with 1/2 and 1/3 for the system G_2) is…

The main purpose of this paper is to prove that the Bruhat-Chevalley ordering of the symmetric group when restricted to the fixed-point-free involutions forms an $EL$-shellable poset whose order complex triangulates a ball. Another purpose…

An inversion in a matrix of zeros and ones consists of two entries both of which equal $1$, and one of which is located to the top-right of the other. It is known that in the class $\mathcal{A}(R,S)$ of $(0,1)$--matrices with row sum vector…

We present some fundamental results on (possibly nonlinear) algebraic semigroups and monoids. These include a version of Chevalley's structure theorem for irreducible algebraic monoids, and the description of all algebraic semigroup…

We consider elements of finite order in the Riordan group $\cal R$ over a field of characteristic $0$. Viewing $\cal R$ as a semi-direct product of groups of formal power series, we solve, for all $n \geq 2$, two foundational questions…

Let $n>1$ and let $R$ be a commutative ring with identity $1\ne 0$ and $R[x_1,\ldots,x_n]^n$ the set of all $n$-tuples of polynomials of the form $(f_1,\ldots,f_n),$ where $f_1,\ldots,f_n\in R[x_1,\ldots,x_n]$. We call these $n$-tuples…

Based on the construction of simple Lie algebras via root category and following Chevalley's results, we construct Chevalley groups from the root category. Then we prove the Bruhat decomposition and the simplicity of the Chevalley groups,…

Let $W$ be a finite reflection group. For a given $w \in W$, the following assertion may or may not be satisfied: (*) The principal Bruhat order ideal of $w$ contains as many elements as there are regions in the inversion hyperplane…

In this article, we consider the monoid of all monotone and order-decreasing partial transformations denoted as $\mathcal{DORP}_{n}$ on an $n$ ordered chain $[n]=\{1, \ldots,n\}$, its two-sided ideal $I(n,p)= \{\rho \in \mathcal{DORP}_{n} :…