Related papers: Normal form in Hecke-Kiselman monoids associated w…
Let $\mathrm{HK}_{\Theta}$ denote the Hecke-Kiselman monoid associated to a finite simple oriented graph $\Theta$. We present a Boolean matrix monoid that is isomorphic to the endomorphism monoid $\mathrm{End}(\mathrm{HK}_{\Theta})$.
In this paper, we present algebraic tools to obtain normal forms of $\omega$-Hamiltonian vector fields under a semisymplectic action of a Lie group, by taking into account the symmetries and reversing symmetries of the vector field. The…
Version 2: Corrected Section 3.3: instead of lexicographical normal forms we now use a normal form due to V. Diekert and A. Muscholl. Consequent changes made and some misprints corrected. Using an analogue of Makanin-Razborov diagrams, we…
Definition of generalized normal form for a system of ODEs corresponding to an infinitesimal symplectic or contact transformation near a singular point, with an arbitrary polynomial unperturbed part, and a method of its finding are…
Combining the definition of 0-Hecke monoids with that of Kiselman semigroups, we define what we call Kiselman quotients of 0-Hecke monoids associated with simply laced Dynkin diagrams. We classify these monoids up to isomorphism, determine…
Normal form theory is developed deeply for planar smooth systems but has few results for piecewise-smooth systems because difficulties arise from continuity of the near-identity transformation, which is constructed piecewise. In this paper,…
We give unique analytic "normal forms" for germs of a holomorphic vector field of the complex plane in the neighborhood of an isolated singularity of saddle-node type having a convergent formal separatrix. We specifically address the…
We describe the second cohomology of a regular semisimple Hessenberg variety by generators and relations explicitly in terms of GKM theory. The cohomology of a regular semisimple Hessenberg variety becomes a module of a symmetric group…
Let $\Gamma$ be a torsion-free arithmetic group acting on its associated global symmetric space $X$. Assume that $X$ is of non-compact type and let $\Gamma$ act on the geodesic boundary $\partial X$ of $X$. Via general constructions in…
We obtain a parametric normal form for any non-degenerate perturbation of the generalized saddle-node case of Bogdanov--Takens singularity. Explicit formulas are derived and greatly simplified for an efficient implementation in any computer…
Let $K[HK_{\Theta}]$ denote the Hecke-Kiselman algebra of a finite oriented graph $\Theta$ over an algebraically closed field $K$. All irreducible representations, and the corresponding maximal ideals of $K[HK_{\Theta}]$, are characterized…
This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith…
Associated to any uniform finite layered graph Gamma there is a noncommutative graded quadratic algebra A(Gamma) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul.…
We discuss algebraic and combinatorial aspects of the Hamiltonian normal form theory. The main objective is to describe the normal form near a singular point purely in terms of the original Hamiltonian, avoiding the normalization procedure.…
To any element of a connected, simply connected, semisimple complex algebraic group G and a choice of an element of the corresponding Weyl group there is an associated Lusztig variety. When the element of G is regular semisimple, the…
We derive a normal form for a near-integrable, four-dimensional symplectic map with a fold or cusp singularity in its frequency mapping. The normal form is obtained for when the frequency is near a resonance and the mapping is approximately…
An element $\alpha \in \mathbb{F}_{q^n}$ is a normal element over $\mathbb{F}_q$ if the conjugates $\alpha^{q^i}$, $0 \leq i \leq n-1$, are linearly independent over $\mathbb{F}_q$. Hence a normal basis for $\mathbb{F}_{q^n}$ over…
We strengthen the standard bifurcation theorems for saddle-node, transcritical, pitchfork, and period-doubling bifurcations of maps. Our new formulation involves adding one or two extra terms to the standard truncated normal forms with…
We revisit Haiman's conjecture on the relations between characters of Kazdhan-Lusztig basis elements of the Hecke algebra over the symmetric group. The conjecture asserts that, for purposes of character evaluation, any Kazhdan-Lusztig basis…
Let H be a non semi-simple Ariki-Koike algebra. According to [18] and [14], there is a generalisation of Lusztig's a-function which induces a natural order (parametrised by a tuple m) on Specht modules. In some cases, Geck and Jacon have…