Related papers: Hall polynomials for affine quivers

We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a…

We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order $n$, define a unit in the integral group ring for infinitely many positive integers $n$. We show that this happens if and only if…

We introduce a multivariate generalization of normalized Chebyshev polynomials of the second kind. We prove that these polynomials arise in the context of cluster characters associated to Dynkin quivers of type $\mathbb A$ and…

The direct and inverse spectral problems are solved for a wide subclass of the class of Schwarz matrices. A connection between the Schwarz matrices and the so-called generalized Hurwitz polynomials is found. The known results due to H. Wall…

We give a Morita equivalence theorem for so-called cyclotomic quotients of affine Hecke algebras of type B and D, in the spirit of a classical result of Dipper-Mathas in type A for Ariki-Koike algebras. As a consequence, the representation…

Let us consider a specialization of an untwisted quantum affine algebra of type $ADE$ at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases,…

For Dynkin quivers, we find the Laurent polynomials $\widetilde{X}_{a, c}^{b}(v)$ and use $\widetilde{X}_{a, c}^{b}(v)$ to construct the Hall algebra $\hc_v(\cc(\cp))$ over $\mz[v, v^{-1}]$, where $\widetilde{X}_{a, c}^{b}(|\mf_q|)$'s are…

In the paper we obtain an existence criterion for Hall subgroups of finite groups in terms of a composition series.

We prove a complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally…

We consider the finite set of isogeny classes of $g$-dimensional abelian varieties defined over the finite field $\mathbb{F}_q$ with endomorphism algebra being a field. We prove that the class within this set whose varieties have maximal…

An inifinite-dimensional representation of the double affine Hecke algebra of rank 1 and type $(C_1^{\vee},C_1)$ in which all generators are tridiagonal is presented. This representation naturally leads to two systems of polynomials that…

We give a criterion for a collection of polynomials to be a universal Gr\"{o}bner basis for an ideal in terms of the multidegree of the closure of the corresponding affine variety in $(\mathbb{P}^1)^N$. This criterion can be used to give…

K-theoretic Hall algebras (KHAs) of quivers with potential $(Q,W)$ are a generalization of preprojective KHAs of quivers, which are conjecturally positive parts of the Okounkov-Smironov quantum affine algebras. In particular, preprojective…

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…

A theorem by Hall asserts that the multiplication in torsion free nilpotent groups of finite Hirsch length can be facilitated by polynomials. In this note we exhibit explicit Hall polynomials for the torsion free nilpotent groups of Hirsch…

We define a family of quiver representation-valued invariants of oriented classical and virtual knots and links associated to a choice of finite quandle $X$, abelian group $A$, set of quandle 2-cocycles $C\subset H^2_Q(x;A)$, choice of…

An important invariant of a polynomial $f$ is its Jacobian algebra defined by its partial derivatives. Let $f$ be invariant with respect to the action of a finite group of diagonal symmetries $G$. We axiomatically define an orbifold…

General error locator polynomials are polynomials able to decode any correctable syndrome for a given linear code. Such polynomials are known to exist for all cyclic codes and for a large class of linear codes. We provide some decoding…

Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite…

We construct infinite classes of almost bent and almost perfect nonlinear polynomials, which are affinely inequivalent to any sum of a power function and an affine function.