相关论文: Grobner-Shirshov Basis for affine Weyl Group $\wid…

We establish Gr\"obner--Shirshov bases theory for commutative dialgebras. We show that for any ideal $I$ of $Di[X]$, $I$ has a unique reduced Gr\"obner--Shirshov basis, where $Di[X]$ is the free commutative dialgebra generated by a set $X$,…

This paper is a detailed description of an algorithm based on a generalized Buchberger algorithm for constructing Groebner-type bases associated with polynomials of shift operators. The algorithm is used for calculating Feynman integrals…

We give a uniform, Lie-theoretic mirror symmetry construction for the Frobenius manifolds defined by Dubrovin-Zhang in arXiv:hep-th/9611200 on the orbit spaces of extended affine Weyl groups, including exceptional Dynkin types. The B-model…

Let v and w be nontrivial words in two free groups. We prove that, for all sufficiently large finite non-abelian simple groups G, there exist subsets C of v(G) and D of w(G) of size such that every element of G can be realized in at least…

Let $Di\langle X\rangle$ be the free dialgebra over a field generated by a set $X$. Let $S$ be a monic subset of $Di\langle X\rangle$. A Composition-Diamond lemma for dialgebras is firstly established by Bokut, Chen and Liu in 2010…

We present a complete classification for the epimorphisms from the affine Artin-Tits groups $A(\widetilde{A}_{n})$ to the corresponding Coxeter groups $W(\widetilde{A}_{n})$, for $n\geq 1$.

This note is an application of classification results for finite-dimensional Nichols algebras over groups. We apply these results to generalizations of Fomin--Kirillov algebras to complex reflection groups. First, we focus on the case of…

We define an action of the double coinvariant algebra $DR_n$ on the equivariant Borel-Moore homology of the affine flag variety $\widetilde{Fl}_n$ in type $A$, which has an explicit form in terms of the left and right action of the…

Several combinatorial actions of the affine Weyl group of type $\widetilde{C}_{n}$ on triangulations, trees, words and permutations are compared. Addressing a question of David Vogan, we show that, modulo a natural involution, these…

In this paper we describe an efficient involutive algorithm for constructing Groebner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain…

For the root system of type $B_l$ and $C_l$, we generalize the result of \cite{DZ1998} by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the…

We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants such as real roots. The crucial ingredient is a `reflection' in the class of such Nichols algebras. We conclude the classifications of…

Every affine Weyl group appears as the iterated monodromy group of a Chebyshev-like polynomial self-map of $\mathbb{C}^n$.

Let $\mathbf{G}$ be a connected split reductive group over a field of characteristic zero or sufficiently large characteristic, $\gamma_0\in(\operatorname{Lie}\mathbf{G})((t))$ be any topologically nilpotent regular semisimple element, and…

Let $X$ be a smooth complex projective algebraic variety. Let $\mathcal{G}$ be a $G$-banded gerbe with $G$ a finite abelian group. We prove an exact formula expressing genus $g$ orbifold Gromov-Witten invariants of $\mathcal{G}$ in terms of…

Developed by Buchberger for commutative polynomial rings, Groebner Bases are frequently applied to solve algorithmic problems, such as the congruence problem for ideals. Until now, these ideas have been transmitted to different in part…

The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide…

We present an approach to the computation of confluent systems of defining relations in associative conformal algebras based on the similar technique for modules over ordinary associative algebras.

We prove a descent result for affine/projective varieties defined over an algebraically closed field. The idea is to work with the reduced Groebner basis of the ideal where the variety vanishes and study it's behaviour under group action…

Let $U_q(A_N)$ be the Drinfeld-Jimbo quantum group of type $A_N$. In this paper, by using Groebner-Shirshov bases, we give a simple (but not short) proof of the Rosso-Yamane Theorem on PBW basis of $U_q(A_N)$.