In this paper we define an equivalence relation on the set of all xj in order to form a basis for a new descent algebra of Weyl groups of type A,. By means of this, we construct a new commutative and semi-simple desce...In this paper we define an equivalence relation on the set of all xj in order to form a basis for a new descent algebra of Weyl groups of type A,. By means of this, we construct a new commutative and semi-simple descent algebra for Weyl groups of type An generated by equivalence classes arising from this equivalence relation.展开更多
It's well known that a reflectin rα associated to every root α belongs to the Weyi group of a Lie algebra g(A) of finite type. When g(A) is a symmetrizable Kac-Moody algebra of indefinite type, one of can ...It's well known that a reflectin rα associated to every root α belongs to the Weyi group of a Lie algebra g(A) of finite type. When g(A) is a symmetrizable Kac-Moody algebra of indefinite type, one of can define a reflection rα for every imzginary root α satisfying (α, α) < 0. From [3] we know rα ∈-W or rα is an element of-W mutiplied by a diagram automorphism . How about the relationship between reflections associated to imaginary root and the Weyl group of a symmetrized Kac-Moody algebra (GKM algebra for short)? We shall discuss it for a special GKM algebra in present paper (see 3). In sections 1 and 2 we introduce some basic concepts and give the set of imaginary root of a class of rand 3 GKM algebras.展开更多
. In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type A1. We introduce a notion of root basis for these root systems.... In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type A1. We introduce a notion of root basis for these root systems, and using a unique expression of the elements of the Weyl group with respect to a set of generators for the Weyl group, we calculate the length function with respect to a very specific root basis.展开更多
For type B_(3), we show that Lusztig's conjecture on the structure of the based ring of the two-sided cell corresponding to the unipotent class in Sp_(6)(C) with three equal Jordan blocks needs modification.
Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra ac...Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module can be interpreted as a differential operator action on polynomials, and thus on the corresponding truncated formal power series. We prove that the space of truncated formal power series gives a differential-operator representation of the Weyl group W. We also introduce a system of partial differential equations to investigate singular vectors in the Verma module. It is shown that the solution space of the system in the space of truncated formal power series is the span of {w(1) | w ∈ W }. Those w(1) that are polynomials correspond to singular vectors in the Verma module. This elementary approach by partial differential equations also gives a new proof of the well-known BGG-Verma theorem.展开更多
Let(W,S)be the affine Weyl group of type B_(2),on which we consider the length function e from W to N and the Bruhat order≤.For y<w in W,letμ(y,w)be the coefficient of q^(1/2(e(w)-e(y)-1)) in Kazhdan-Lusztig poly...Let(W,S)be the affine Weyl group of type B_(2),on which we consider the length function e from W to N and the Bruhat order≤.For y<w in W,letμ(y,w)be the coefficient of q^(1/2(e(w)-e(y)-1)) in Kazhdan-Lusztig polynomial P_(y,w)∈Z[q].We determine someμ(y,w)for y∈c_(0) and w∈c_(2),where c0 is the lowest two-sided cell of B_(2) and c_(2) is the higher one.Furthermore,we get some consequences using left or right strings and some properties of leading coefficients.展开更多
Let V be a hyperbolic 5-dimensional indefinite space. W is the infinite Weyl group of an irreducible root system. The principal aim of this paper is to classify all crystallgraphic groups associated with W up to conju...Let V be a hyperbolic 5-dimensional indefinite space. W is the infinite Weyl group of an irreducible root system. The principal aim of this paper is to classify all crystallgraphic groups associated with W up to conjugation in the affine group A(V).展开更多
With the help of the local isomorphism between theHurwitz space M_(0;k−1,l−k−r+1,r−1) and the orbit spaceMk,k+r(Al),we will show the existence of a Frobenius manifold structure on the orbit space M_(k,k+r)(Al)\Σr of ...With the help of the local isomorphism between theHurwitz space M_(0;k−1,l−k−r+1,r−1) and the orbit spaceMk,k+r(Al),we will show the existence of a Frobenius manifold structure on the orbit space M_(k,k+r)(Al)\Σr of the extended affine Weyl groupW^((k,k+r))(A_(l))for 1≤k<k+r≤l.展开更多
文摘In this paper we define an equivalence relation on the set of all xj in order to form a basis for a new descent algebra of Weyl groups of type A,. By means of this, we construct a new commutative and semi-simple descent algebra for Weyl groups of type An generated by equivalence classes arising from this equivalence relation.
文摘It's well known that a reflectin rα associated to every root α belongs to the Weyi group of a Lie algebra g(A) of finite type. When g(A) is a symmetrizable Kac-Moody algebra of indefinite type, one of can define a reflection rα for every imzginary root α satisfying (α, α) < 0. From [3] we know rα ∈-W or rα is an element of-W mutiplied by a diagram automorphism . How about the relationship between reflections associated to imaginary root and the Weyl group of a symmetrized Kac-Moody algebra (GKM algebra for short)? We shall discuss it for a special GKM algebra in present paper (see 3). In sections 1 and 2 we introduce some basic concepts and give the set of imaginary root of a class of rand 3 GKM algebras.
文摘. In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type A1. We introduce a notion of root basis for these root systems, and using a unique expression of the elements of the Weyl group with respect to a set of generators for the Weyl group, we calculate the length function with respect to a very specific root basis.
基金supported by National Key R&D Program of China (Grant No. 2020YFA0712600)National Natural Science Foundation of China (Grant No. 11688101)the AMSS for hospitality and for financial supports。
文摘For type B_(3), we show that Lusztig's conjecture on the structure of the based ring of the two-sided cell corresponding to the unipotent class in Sp_(6)(C) with three equal Jordan blocks needs modification.
基金supported by National Natural Science Foundation of China(Grant No.11326059)
文摘Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module can be interpreted as a differential operator action on polynomials, and thus on the corresponding truncated formal power series. We prove that the space of truncated formal power series gives a differential-operator representation of the Weyl group W. We also introduce a system of partial differential equations to investigate singular vectors in the Verma module. It is shown that the solution space of the system in the space of truncated formal power series is the span of {w(1) | w ∈ W }. Those w(1) that are polynomials correspond to singular vectors in the Verma module. This elementary approach by partial differential equations also gives a new proof of the well-known BGG-Verma theorem.
基金Supported in part by the Natural Science Foundation of China(Grant no.11101233)Beijing Youth Top-notch Talent Support Program(Grant no.21351918007).
文摘Let(W,S)be the affine Weyl group of type B_(2),on which we consider the length function e from W to N and the Bruhat order≤.For y<w in W,letμ(y,w)be the coefficient of q^(1/2(e(w)-e(y)-1)) in Kazhdan-Lusztig polynomial P_(y,w)∈Z[q].We determine someμ(y,w)for y∈c_(0) and w∈c_(2),where c0 is the lowest two-sided cell of B_(2) and c_(2) is the higher one.Furthermore,we get some consequences using left or right strings and some properties of leading coefficients.
文摘Let V be a hyperbolic 5-dimensional indefinite space. W is the infinite Weyl group of an irreducible root system. The principal aim of this paper is to classify all crystallgraphic groups associated with W up to conjugation in the affine group A(V).
基金supported by NSFC(Nos.12071451,11871446,12061131014)Wu Wen-Tsun Key Laboratory of Mathematics,USTC,CAS.
文摘With the help of the local isomorphism between theHurwitz space M_(0;k−1,l−k−r+1,r−1) and the orbit spaceMk,k+r(Al),we will show the existence of a Frobenius manifold structure on the orbit space M_(k,k+r)(Al)\Σr of the extended affine Weyl groupW^((k,k+r))(A_(l))for 1≤k<k+r≤l.
